Radiation Trapping in Atomic Vapours --- # Radiation Trapping in # Atomic Vapours ANDREAS F. MOLISCH AND BERNHARD P. OEHRY *Institute for Communications and Radio Frequency Engineering,* *Technical University, Vienna* CLARENDON PRESS · OXFORD --- *This book has been printed digitally and produced in a standard specification in order to ensure its continuing availability* **OXFORD** UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan South Korea Poland Portugal Singapore Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Andreas F. Molisch and Bernhard P. Oehry, 1998 The moral rights of the author have been asserted Database right Oxford University Press (maker) Reprinted 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover And you must impose this same condition on any acquirer ISBN 978-0-19-853866-0 --- *People who don't question the assumptions made going into a problem often end up solving the wrong problem.* Marcian T. Hoff, Jr. *It was probably the most remarkable book ever to come out of the great publishing corporations of Ursa Minor. . . . Though it has many omissions and contains much that is apocryphal, or at least wildly inaccurate, it scores (. . .) in two important respects: first, it is slightly cheaper, and secondly, it has the words* DON'T PANIC *inscribed in large friendly letters on its cover.* Douglas Adams, 'The Hitchhiker's Guide to the Galaxy' --- ## PREFACE Atoms can both absorb and emit electromagnetic radiation like visible light. In an assembly of many atoms, e.g., in a vapour or in a gas, radiation emitted by one atom may be reabsorbed by one of its neighbours, just to be reemitted a little later. The game can go on for quite some time—until the photon manages somehow to escape from this prison. At first glance, quite easy to understand. On the other hand, there must be some reason for this voluminous book full of sums, integrals and matrices. Our interest in this subject originally stemmed from investigations on atomic line filters we did for the European Space Agency, ESA. We believed that radiation trapping would have some influence on the performance of these filters and were looking for a simple means to take it into account. We started out the same way as probably anyone new to the field would have done when faced with the problem of trapping. We read the classical paper of Holstein, and thought that was all there was to it. We soon had to realize, however, that the assumptions made by Holstein were not fulfilled in our problem, and decided to give the whole matter a closer look. That ‘closer look’ obtained a momentum of its own, and trapping became the centre of our own research in the following years. As our overview of the subject increased, we found that trapping plays a role in a staggering variety of situations. Trapping appears to be important in most experimental setups and in most engineering applications where atomic vapours are involved. Apart from the laboratory-scale situations that we treat in this book, radiation trapping also has an eminent role in astrophysics. Surprisingly, there is a considerable number of books on trapping from an astrophysical point of view, but none that treats trapping under laboratory conditions. This book tries to fill the gap.[^1] There have been several preliminary titles of this book. The first idea was ‘Everything you always wanted to know about radiation trapping but were afraid to ask’. That title was mildly amusing (at least for Woody Allen fans) but unfortunately also misleading. First, it gives the idea that the book answers all questions about radiation trapping—this is certainly not the case. There are several aspects that we have intentionally omitted from our coverage (quantum-mechanical treatments and completely analytical solutions of special trapping problems, to name but two). On the other hand, our guess is that you did not ‘always want to know’ all the things about radiation trapping that you are going [^1]For the reader who also wants to learn about the astrophysical point of view, we recommend the books by J. T. Jefferies ‘Spectral Line Formation’, Blaisdell (1968) and C. J. Cannon ‘The transfer of spectral line radiation’, Cambridge (1985). The former is aimed mainly at physical insight. The latter gives a good overview of the mathematical (especially numerical) methods that have been developed for astrophysical trapping problems. A didactically very well written book, which includes both physical and mathematical aspects, is the textbook by D. Mihalas, ‘Stellar atmospheres’, Freeman (2nd ed. 1978). --- to find in this book. Our hope is that the book will first present the variety of the subject, raising curiosity and questions, which it should subsequently answer. The second title we considered was 'Introduction to radiation trapping in laboratory environments'. This would also be partly misleading. On one hand, the text is introductory in the sense that it requires no prior knowledge apart from the basic physical and mathematical knowledge every scientist has. On the other hand, the book tries to cover most of the existing primary literature, and leads the reader to a level where he can either do advanced research of his own, or use even the most advanced existing results for applications in his own research. Thus, as so often, 'introduction' would have been an understatement. Furthermore, while this book provides the *mathematical* tools to deal with all laboratory situations, the emphasis in the *physical* interpretation and in the examples is on chemical physics, and not on laboratory plasmas (even though also the latter are of great practical importance, e.g. in fusion and X-ray lasers). Our choice is motivated mainly by the fact that inclusion of plasmas would have required a complete description of plasma physics itself, increasing the size of the book to unmanageable proportions. We thus finally ended up with the rather dry and short title 'Radiation trapping in atomic vapours'. The two ambitions of this book are to provide the reader with an insight into the physical mechanisms of radiation trapping, and to give him a toolbox of mathematical methods to solve the radiation trapping problems he will encounter in practice. As such, the book is essentially a monograph intended for self-study. It could also be used for a graduate course in radiation trapping, but we do not think that such a course currently exists at many universities. Perhaps the existence of a comprehensive text will remedy this situation. In any case, we hope that we can convey some of the fascination that this subject has for us. ### Acknowledgements This book would not have been possible without the support of many people, to whom we want to express our thanks and appreciation. Prof. Dr. Gottfried Magerl, head of the laser spectroscopy group at the Institut für Nachrichtentechnik und Hochfrequenztechnik (INTHFT) at the Technical University Vienna (Vienna, Austria), initiated and guided the project during which we made our first acquaintance with radiation trapping. He spent countless hours discussing various aspects of trapping with us, as did two other members of the laser spectroscopy group, Dr. Walter Ehrlich-Schupita and Dr. Brigitte Sumetsberger. To all of them, we express our deep gratitude. We also want to thank our colleagues from other research institutions, with whom we had very fruitful collaborations about various aspects of radiation trapping: at the *University of Pisa* (Italy), Prof. Dr. Maria Allegrini, Prof. Dr. Ennio Arimondo, Dr. Francesco Fuso, and Dr. Andrea Fioretti; at the *University of St. Petersburg* (Russia), Prof. Dr. Nikolai Bezuglov; at the *Lehigh University* (Bethlehem, USA), Prof. Dr. --- PREFACE John Huennekens; at *Lawrence Livermore National Laboratories* (Livermore, USA), Dr. Greg Parker. No scientific work can be done without appropriate financial support. A great part of our work on trapping was done while we were working under contract to the European Space Agency (ESA). The ESA contract officers were Dr. Roland Meynard and Dr. Andrea Marini. Their support and patience proved very helpful. Later on, the support and encouragement by Prof. Dr. Ernst Bonek, head of the radio-frequency engineering department at the Technical University Vienna, was indispensable for one of us (A.F.M.). We thank the anonymous reviewer asked by the publisher to critically read and comment on the manuscript. Several researchers sacrificed their time to give detailed reviews of specific chapters: Prof. Dr. Nikolai Bezuglov (Chapters 4–8 and 10), Dr. Lawrence Auer (Chapters 8, 9 and 13), Prof. Dr. Alan Streater (Chapters 11, 13.2, and App. G), Prof. Dr. Martin Stift (Chapter 12), Prof. Dr. John Huennekens (Chapter 15), Prof. Dr. James Lawler (Chapter 18), and Prof. Dr. Karl Blum (App. F). To all of them goes our profound gratitude. Of course, the responsibility for any remaining errors rests with us. We thank our editors at Oxford University Press for their help and guidance in the realization of the manuscript. Dr. Donald Degenhardt was our editor in the initial stages of the book; the final steps were guided by Dr. Sönke Adlung. *Vienna* October 1997 A.F.M. B.P.O. --- ## LIST OF SYMBOLS ### Symbol Modifiers Scalar quantities are printed in *italics*. The following modifiers are used to signify the most common mathematical transformations or certain properties of a symbol. | | | | | | :--- | :--- | :--- | :--- | | $x^*$ | approximation to $x$ | $\delta x$ | small correction to $x$ | | $\bar{x}$ | average of $x$ | $\Delta x$ | a change in $x$, difference in $x$ | | $\hat{x}$ | normalized quantity $x$ | $\underline{x}$ | complex quantity $x$ | | $\tilde{x}$ | Fourier-transform of $x$ | $\mathbf{x}$ | vector or matrix quantity (bold, upright) | ### Enumerating Indices As far as possible, the symbols commonly used as enumerating indices for vector and matrix elements in the literature ($i, j, k, l, m, n$) are employed as listed below. Note that the indices $i$ and $j$ have double uses. | | | | | | :--- | :--- | :--- | :--- | | $i$ | frequency index or index for number of absorptions | $k$ | spatial index (absorption) | | $j$ | index for mode-number or index for discrete angles | $m$ | spatial index (emission) | | | | $l$ | index for atomic level | ### Denoting Indices Denoting sub- or super-indices are used to indicate that a symbol is used in a certain context, e.g. $I_{\text{inc}}$ denotes incident radiation intensity, $I_{\text{ext}}$ stands for external intensity, $I_{\text{tot}}$ for total intensity, and $I_{\text{coh}}$ for coherently emitted intensity. Most denoting indices have only local use and meaning, but the following ones are employed throughout the text. | | | | | | :--- | :--- | :--- | :--- | | n | naturally broadened | 1(l) | atomic level 1 (the lower level) | | D | Doppler broadened | 2(u) | atomic level 2 (the upper level) | | L | Lorentz broadened | 21 | transition from level 2 to level 1 | | stat | statistically broadened | ul | transition from upper to lower level | | abs | absorbed | em | emitted | | ARF | atomic rest frame | esc | escape | | coh | coherent | exc | excitation | | coll | collisional | inc | incident | | cyl | cylinder geometry | max | maximum | | diff | diffusion | tot | total | | e | electron | $\nu$ | spectral quantity | | eff | effective | | | --- ## LIST OF SYMBOLS ### Global Symbols Symbols with local use are introduced and explained at the appropriate places in the text. Other symbols occur frequently and hence have a global meaning. These symbols are only explained on first occurrence and are listed below. Note that some symbols had to be given more than one meaning. At any occurrence it should be clear from the context which meaning applies. | Symbol | Description | | :--- | :--- | | **— a —** | | | $a$ | The "Voigt parameter", giving the ratio of natural to Doppler broadening in Voigt lines | | $a_0$ | Bohr radius | | $a_i, a_j, ...$ | general coefficients for expansions or for quadrature weights | | $a_{\text{fs}}, a_{\text{hfs}}$ | fine-splitting and hyperfine-splitting constants | | $A_{21}$ | Einstein coefficient of spontaneous emission | | $A_{k,m}$ | matrix elements for PCA method | | $\mathbf{A_j}$ | Feautrier submatrix | | $\mathbf{A1}, \mathbf{A2}$ | auxiliary matrices | | **— b —** | | | $b_i$ | weights for angle quadrature in discrete ordinate solution | | $b_{j,i}$ | finite-differencing constants for Feautrier technique | | $b_i(\mathbf{r})$ | basis functions for the variational and for the PCA solution of the Holstein equation | | $\mathbf{b}^{\text{acc}}$ | vector for the computation of the acceleration vector | | $B_{12} (B_{lu})$ | Einstein coefficient of absorption | | $B_{21} (B_{ul})$ | Einstein coefficient of stimulated emission | | $\mathbf{B}$ | magnetic field | | $\mathbf{B_j}$ | Feautrier submatrix | | **— c —** | | | $c$ | speed of light | | $c_i, c_j, c_{i,j}, ...$ | generally used for expansion coefficients | | $c1, c2, c3, ...$ | generally used for auxiliary constants | | $\text{cdf}(.)$ | cumulative distribution function | | $C_{12}, C_{ul}, ...$ | rate of collisional transfer from one state to another | | $C$ | with denoting index: rate of some process, e.g., $C_{\text{exc}}$ excitation rate, $C_{\text{ela}}$ rate of elastic collisions, $C_{\text{vc}}$ rate of velocity-changing collisions | | $C_\nu$ | normalization constant for lineshapes | | $C_x$ | normalization constant for lineshapes with normalized frequency | | $\mathbf{C_j}$ | Feautrier submatrix | | **— d —** | | | $d$ | distance | | $d_1, d_2, ...$ | constants for fitting equations of modal shapes | | $D$ | diffusion constant | --- ## LIST OF SYMBOLS | Symbol | Description | | :--- | :--- | | $D_x, D_y, D_z$ | squared amplitudes of dipoles along $x$-, $y$-, or $z$-axis | | **— e —** | | | $e$ | Euler's number (2.71828...) | | $e_-$ | electron charge | | $\text{erfc}(.)$ | complementary error function | | $\mathbf{e}_{\mathbf{rs}}$ | unit vector from $\mathbf{r}$ to $\mathbf{s}$ | | $\mathbf{e}_{\mathbf{n}}, (\mathbf{e}'_{\mathbf{n}})$ | unit vector in direction of emitted (absorbed) photon | | $E$ | energy ($E_e$ electron energy, $E_l$ energy of level $l$, ...) | | $E(\mathbf{r}, t)$ | excitation function | | $\text{Ei}(.)$ | exponential integral | | $\mathbf{E}$ | electric field | | $\mathbf{E}_{\mathbf{j}}, (\mathbf{E}_{\mathbf{j}}^{\mathbf{R}})$ | Feautrier (Rybicky) submatrix for excitation | | **— f —** | | | $f_{12}$ | oscillator strength | | $f(.)$ | arbitrary function | | $f1, f2, f3, ...$ | generally used for auxiliary functions | | $f_H, f_K$ | variable Eddington factors | | $f_r, f_z, f_{zz},$
$f_{rr}, f_{rz}, f_{\varphi\varphi}$ | Eddington factors in $r, z, ...$ directions | | $F$ | radiation flux density | | $F_s$ | saturation flux density | | $\text{FC}(x)$ | fraction of coherently scattered photons | | $\text{FF}(a, x)$ | Faraday function | | $\text{FV}(x)$ | function for the computation of Voigt profiles | | **— g —** | | | $g, g_0$ | trapping factor for lowest-order mode | | $g_j$ | trapping factor for $j$th mode | | $g_l, (g_s, g_I)$ | normalized magnetic (spin, nuclear) momentum | | $g'_1, g'_2$ | statistical weights of levels 1 and 2 | | $G(\mathbf{r},\mathbf{r}')$ | Kernel function (Green's function) | | $\text{GE}(.)$ | generalized error function | | **— h —** | | | $h$ | Planck constant | | $\hbar$ | $h/2\pi$ | | $H(a, x)$ | Voigt function | | $\mathbf{H}^{\text{H}}(\mathbf{r},\mathbf{p})$ | Hamiltonian | | $\text{He}_k(x)$ | Hermite polynomial | | $\text{Hs}(.)$ | Heaviside step function | | $H_r, H_z, H_\varphi$ | first moment of intensity in $r$-, $z$-, and $\varphi$-direction | | **— i —** | | | $i$ | enumerating index; preferentially for frequency or for number of absorptions | --- ## LIST OF SYMBOLS | | | | :--- | :--- | | $i_e$ | electron current | | $I$ | radiation intensity | | $I_s$ | saturation intensity | | $\mathbf{I}$ | Stokes vector | | $I_i^{\text{B}}(.)$ | modified Bessel functions of the second kind, $i$th order | | **— j —** | | | $j$ | enumerating index; preferentially for mode-number or for discrete angles | | $j$ | the imaginary unit | | $J$ | angle-averaged intensity | | $J_i^{\text{B}}(.)$ | $i$th-order Bessel function | | $\mathbf{J}^{\text{R}}$ | Rybicki intensity matrix | | **— k —** | | | $k$ | enumerating index; preferentially as spatial index (absorption) | | $k(\nu)$ | absorption coefficient | | $k_0$ | centre-of-line absorption coefficient | | $\bar{k}$ | Milne/Samson absorption coefficient | | $k_{\text{B}}$ | Boltzmann constant | | $\mathbf{k}$ | lineshape matrix for polarization | | $K, K(z, x)$ | second moment of intensity | | $K_i^{\text{B}}(.)$ | modified Bessel function of order $i$ | | $Ki(z)$ | repeated integral of Bessel function | | **— l —** | | | $l$ | enumerating index; preferentially as index for atomic level | | $L$ | without denoting index: cell length, else some other length, e.g., $L_{\text{th}}$ thermalization length, $L_{\text{sob}}$ Sobolev length, $L_{\text{m}}$ mean free path length | | $L(\nu)$ | Lorentzian lineshape | | **— m —** | | | $m$ | enumerating index; preferentially as spatial index (emission) | | $m$ | with denoting index; mass, e.g., $m_e$ electron mass, $m_{\text{atom}}$ atomic mass | | $m_j^{\text{D}}, m_j^{\text{L}}$ | geometry-dependent high-opacity proportionality constants for the trapping factor | | $\text{mfp}$ | mean free path | | $M_{i,j}^{\text{acc}}$ | elements of matrix for computation of acceleration vector | | $\mathbf{M}$ | matrix for phase matrix method, total matrix in double-splitting method | | **— n —** | | | $n = n_2 = n_{\text{u}}$ | excited-stated density (when only two levels are involved and only linear effects occur) | | $n$ | with denoting index; density [$\text{m}^{-3}$] in some state, e.g. $n_{\text{m}}$ | --- ## LIST OF SYMBOLS | Symbol | Description | | :--- | :--- | | | metastable density, $n_{7s}$ 7s-state density, $n_e$ electron density, $n_{\text{max}}$ maximum excited-state density, etc. | | $n(\mathbf{r})$ | spatial profile of excited atoms | | $\mathbf{n}$ | normal vector on the boundary surface | | $N$ | total particle density; equal to ground-state density in the linear regime | | $N$ | with denoting index: natural number, e.g. $N_{\text{sim}}$ number of simulated photons in MC simulation, $N_{\text{x}}$ number of frequency points in Feautrier technique, $N_{\text{r}}$ number of spatial discretization points, etc. | | — **o** — | | | $\mathbf{O}(\tau, \tau')$ | evolution operator for polarized transfer | | — **p** — | | | $p$ | pressure | | $p_0$ | unit pressure | | $p_i$ | probability that photon escapes after the $i$th emission | | $\bar{p}$ | average number of reabsorptions | | $p(\mathbf{r}_k)$ | pulse function at position $\mathbf{r}_k$ | | pdf(.) | probability density function | | $\mathbf{p}$ | (dipole) momentum | | $P$ | with denoting index: power, e.g., $P_{\text{ex}}$ excitation power, $P_{\text{abs}}$ absorbed power | | $P_j(I)$ | probability that photon in the $j$th mode is still in the cell after $I$ emissions | | PF($x, x'$) | frequency propagator | | PS(.) | spatial propagator | | $P_i(\mu)$ | Legendre polynomials | | $P_i^j(\mu)$ | associated Legendre functions | | $\mathbf{P}$ | phase matrix | | — **q** — | | | $q$ | $q^2 = r^2 + r'^2 - 2rr' \cos(\varphi - \varphi')$ | | $q_k$ | quadrature weights for Feautrier technique | | $Q$ | quenching rate | | $Q$ | second component of Stokes vector | | — **r** — | | | $r$ | radial coordinate in cylindrical or spherical coordinate system, especially: | | $r$ | radial point of absorption in cylinder or sphere | | $r'$ | radial point of emission in cylinder or --- ## LIST OF SYMBOLS | | | | :--- | :--- | | $\mathbf{r}'$ | point of photon emission | | $R$ | (cell) radius | | $R(.)$ | redistribution function | | $\text{Ry}$ | Rydberg constant | | — **s** — | | | $s$ | piece of length (e.g. along an integration path) | | $\text{si}(.)$ | si-function, $\text{si}(x) = \sin(x)/x$ | | $\mathbf{s}$ | position vector where the photon path intersects the vessel boundary | | $S$ | source function | | $\mathbf{S}$ | vector of source function | | — **t** — | | | $t$ | time | | $T$ | absolute temperature | | $T^{\text{coll}}, T^{\text{D}}, \dots$ | transmission factor for collisional, Doppler, etc. broadening | | $T(\rho, x)$ | frequency dependent transmission factor | | $T(\rho)$ | frequency-averaged transmission factor | | $\mathbf{T}_i^{\text{R}}$ | Rybicki submatrix | | — **u** — | | | $u$ | auxiliary variable for general use | | $U$ | third component of Stokes vector | | $U_m(.)$ | Chebycheff polynomials of the second kind | | --- ## LIST OF SYMBOLS | Symbol | Description | | :--- | :--- | | **— y —** | | | $y$ | $y$-coordinate | | $y_0$ | first-order solution in CLT | | $\mathbf{y}$ | solution vector in linear problem | | $Y(t)$ | emergent radiation from the cell | | $Y_i$ | emergent intensity for photons escaping after $i$ emissions | | $Y(y)$ | function for excited-state density in $y$-direction | | **— z —** | | | $z$ | $z$-coordinate, especially; | | $z$ | point of absorption in slab | | $z'$ | point of emission in slab | | $Z$ | proton number | | $Z(z)$ | function for excited-state density in $z$-direction | | **— Greek Letters —** | | | $\alpha_j$ | modal expansion coefficient | | $\alpha(\mathbf{r}, x, t)$ | spectral relaxation rate | | $\beta$ | branching ratio | | $\gamma$ | lineshape factor, $\gamma = (\kappa - 1)/2\kappa$ | | $\gamma$ | decay time constant | | $\gamma(\mathbf{r}, t)$ | local de-excitation rate | | $\delta(.)$ | Dirac pulse | | $\delta_{i,j}$ | Kronecker delta symbol | | $\varepsilon$ | a small value | | $\varepsilon_0$ | vacuum permeability | | $\zeta$ | parameter for fitting equations of modal shapes | | $\bar{\eta}$ | spatially averaged escape factor | | $\eta(\mathbf{r})$ | space-dependent escape factor | | $\eta(x, \mathbf{r}_k, \mathbf{r}_i)$ | probability that photon emitted at $\mathbf{r}_k$ escapes through boundary at $\mathbf{r}_i$ | | $\vartheta$ | angle with $z$-axis | | $\kappa$ | lineshape factor; wings of a line are modelled as $x^{-\kappa}$ | | $\lambda_j$ | $j$th eigenvalue of the Holstein equation | | $\mu$ | cosine of angle, i.e. short for $\cos(\vartheta)$ | | $\mu_{\mathrm{B}}$ | Bohr magneton | | $\nu$ | radiation frequency | | $\nu'$ | absorption frequency | | $\nu_0$ | frequency at line centre | | $\xi$ | optical density | | $\pi$ | 3.14159 ... | | $\rho$ | distance between photon emission and absorption points | | $\rho_{j,k}$ | density matrix elements | | $\boldsymbol{\rho}$ | density matrix | --- ## LIST OF SYMBOLS | Symbol | Description | | :--- | :--- | | $\sigma$ | (absorption) cross-section | | $\sigma$ | with denoting index: cross-section in general, e.g., $\sigma_{\text{coll}}$ collisional cross-section, $\sigma_{\text{geom}}$ geometrical cross-section, $\sigma_{12}$ collision cross-section from state 1 to state 2, etc. | | $\zeta$ | parameter for modal shape with diffusion | | $\tau$ | natural lifetime of an atomic level | | $\varphi$ | angle in spherical coordinate system, also rotation angle | | $\varphi$ | especially: angle of absorption point in a spherical coordinate system | | $\varphi'$ | angle of emission point in a spherical coordinate system | | $\varphi^{\text{ph}}$ | phase shift | | $\chi^{\text{a}}, \chi^{\text{e}}, \chi^{\text{s}}$ | coefficient for absorption, for stimulated emission, and for spontaneous emission | | $\psi(.)$ | spatial eigenmode (of the Holstein equation) | | $\omega$ | natural frequency, $\omega = 2\pi \cdot \nu$ | | $\Gamma(.)$ | Gamma function | | $\Gamma(\mathbf{e}_{\text{n}}, \mathbf{e}_{\text{n}}')$ | probability that a photon is emitted in direction $\mathbf{e}_{\text{n}}$ after absorption from direction $\mathbf{e}_{\text{n}}'$ | | $\Delta$ | width of stripes in PCA method | | $\Delta\nu^{\text{n}}, \Delta\nu^{\text{D}}, ...$ | full width at half maximum (FWHM) of natural, Doppler, etc. broadening | | $\Theta$ | intensity sum in Feautrier technique | | $\mathbf{\Theta_j}$ | Feautrier vector | | $\Lambda$ | with denoting index: operator, e.g., $\Lambda^{\text{a}}$ angular derivative operator, $\Lambda^*$ approximate operator, $\Lambda^{\text{diff}}$ diffusion operator, $\Lambda_{\text{int}}$ integral operator, etc. | | $\Xi$ | reflection coefficient of cell walls | | $\Upsilon 1, \Upsilon 2, ...$ | auxiliary functions | | $\Phi(\nu)$ | normalized absorption lineshape | | $\Psi(\nu), \Psi(x)$ | spectral emission lineshape | | $\Omega$ | spatial angle | | $\mathbf{\Omega}$ | vector of spatial angle | | **— Other symbols —** | | | $\nabla$ | Nabla operator | | $\nabla_r$ | divergence with respect to $r$ | | $\partial$ | partial derivative | | $\mathcal{L}$ | Laplace transform operator | | $\mathbf{1}$ | identity matrix | | $\mathrm{d}\mathbf{r}, \mathrm{d}\mathbf{\Omega}$ | Strictly speaking, the differential element in integrals over volumes should read $\mathrm{d}V(\mathbf{r})$. To simplify notation, we use $\mathrm{d}\mathbf{r}$ instead. Similarly, we use $\mathrm{d}\mathbf{\Omega}$ for integration over the spatial angle. | --- # 1 # INTRODUCTION ## 1.1 The physical process of radiation trapping Since the beginning of this century we have a rough understanding about the structure of atoms. They consist of a nucleus and electrons orbiting around this nucleus. We know that electrons occupy only certain discrete energy states. When they decay from a high-energy state to a low-energy state, they emit electromagnetic radiation. Usually, the low-energy state will be the ground state. In that case, the emitted radiation is called resonance radiation.[^1] Since the resonance photons have an energy that corresponds to the energy of the transition, they can easily be absorbed by other ground-state atoms. A ground-state atom that absorbs the resonance photon is excited to the upper state for some time and will eventually release the photon again. This absorption/reemission process can be repeated many times, until the photon finally escapes from the vapour. This process is called 'radiation trapping' (also known as 'imprisonment of resonance radiation', 'radiative transfer of spectral lines', 'line transfer', or 'radiation diffusion'). The physical effects of radiation trapping are most easily seen in pulsed experiments. We start out with a certain initial distribution of excited-state atoms (created, e.g. by a short laser pulse), and observe the time behaviour of the atoms and the photons. When photons are not trapped at all, they will emerge from the cell with a time dependence proportional to $\exp(-A_{21} \cdot t)$, where $A_{21}$ is the Einstein coefficient of spontaneous emission of the higher-energy atomic state. The excited-state density will display the same simple temporal decay. If, however, each photon is absorbed and reemitted $g_0$ times, then the time behaviour of the emergent radiation and of the excited-state density will be $\exp(-A_{21} \cdot t / g_0)$. The average number of absorption/reemission processes, $g_0$, is known as the 'trapping factor'. Its inverse is called the 'escape probability'.[^2] The direction of the photon changes at every absorption/reemission process—the statistical distribution of the direction of emission of a spontaneously emitted photon is isotropic. This description of photons wandering through a vapour displays a striking analogy to particle diffusion. In particle diffusion, a particle travels undisturbed through free [^1]: There are two definitions of 'resonance radiation'. The strict one defines 'resonance radiation' as radiation emitted when an excited-state atom radiatively decays to the ground state. In more general usage, the lower state can be an arbitrary excited state of the atom (whose energy must of course be lower than the state before the emission). [^2]: In order to gain some basic insight into the nature of the problem, we will use many simplifying pictures in this introduction, without stating in detail under which assumptions these simplifications are justified. Some of the introductory explanations will appear grossly inexact to the experienced radiation trapping researcher. More accurate descriptions will be used in the subsequent chapters. 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- ## INTRODUCTION space until it suffers a collision with some other particle. The collision changes its direction and speed, and leads to a statistical wandering about of the particle. The main parameter describing its behaviour is the particle's 'mean free path' between collisions. For radiation diffusion, one is tempted to say that the mean free path of a photon is the inverse of the absorption coefficient. One might think at first glance that the analogy to particle diffusion is perfect, and that radiation trapping can be described by a diffusion equation. All the sophisticated mathematical tools developed for that purpose, see e.g. Morse and Feshbach (1953), could be applied to the problem of radiation trapping (this is how the name 'radiation diffusion' came into existence). If that were the case, you could close the book right now, and return to doing something meaningful. Unfortunately, the situation is not that simple. The first important difference between particle diffusion and radiation trapping is that in particle diffusion, the time between two 'scattering' (i.e. collision) processes is rather long, while the scattering process itself is almost instantaneous. In radiation trapping, the 'scattering' (i.e. absorption/reemission process) takes rather long (about $1/A_{21}$ seconds), while the time between the scattering processes is very short (the flight time of the photon). This difference is, however, not *that* serious, and could be remedied by addition of one term in the diffusion equation. The second, considerably more important difference lies in the definition of the mean free path. Above, we implicitly assumed that a resonance line was a sharp line, so that there is a certain chance of absorption when the photon's energy exactly matches the energy difference between the two atomic states, but no absorption otherwise. However, this simple picture is not even coarsely correct. There are several unavoidable mechanisms that broaden spectral lines, so that every line has a certain *shape*. This means that the probability of absorption is largest when the photon energy $h\nu$ (where $h$ is Planck's constant, and $\nu$ is the frequency) matches the 'nominal' energy difference between the energy states. The absorption probability is somewhat smaller at a distance $\Delta\nu$ from the line-centre, see Fig. 1.1. The dependence of the absorption coefficient on the frequency displacement $\Delta\nu$ is called the lineshape—it can have, e.g. a Gaussian form. The relative width of a line is very small (typically $\Delta\nu \approx 10^9 \text{ Hz}$ compared to $\nu_0 \approx 10^{14} \text{ Hz}$), but the fact that there is a lineshape at all has dramatic consequences for the radiation trapping. The mean free path, i.e. the inverse of the absorption coefficient, depends strongly on the frequency displacement, so the question arises: what is the 'average' mean free path? If the photons would exactly retain their frequencies at each absorption/reemission process, then we would even be able to define a meaningful 'average' mean free path. We would just have to take the initial frequency distribution of the photons $\text{pdf}(\Delta\nu)$, normalized so that $\int \text{pdf}(\Delta\nu)\mathrm{d}\Delta\nu = 1$, and compute the average mean free path resulting from the frequency dependent absorption coefficient $k(\Delta\nu)$, $$ \text{average mean free path} = \int \frac{1}{k(\Delta\nu)} \text{pdf}(\Delta\nu) \, \mathrm{d}\Delta\nu \qquad (1.1) $$ This expression would be meaningful as long as the initial excitation has finite bandwidth. However, let us consider a vapour cell where the vapour is very dense, so that --- THE PHYSICAL PROCESS OF RADIATION TRAPPING [FIGURE: FIG. 1.1. An unrealistic 'on-off' absorption lineshape, and a more realistic one.] there are many atom–atom collisions. Any collision will slightly disturb the energy of the upper-state atom and it will reemit the photon at some slightly different frequency. The statistics of the emission frequency is related to the lineshape, as we will see in Chapter 4. Thus, the photon changes frequency at each absorption/reemission process. This effect is known as 'frequency redistribution' (FR).[^3] FR makes it impossible to define an average mean free path—the average mean free path is infinitely large. Therefore, radiation trapping cannot be described by a diffusion equation (a differential equation), but must be described by a different type of equation, namely a Boltzmann-type integral equation (the so-called Holstein equation). Due to the dependence of the absorption coefficient on the frequency, photons in the 'line centre' (i.e. with very small $\Delta\nu$) behave differently from 'wing photons' (photons with large $\Delta\nu$). A photon in the line centre has a very small mean free path, and thus can cover only a small distance before being reabsorbed, while a photon in the wings can cover a large distance, and perhaps even escape from the vapour. Since a photon can change frequency at each absorption/reemission process, it also has a certain chance of getting from the line centre into the wings (and vice versa). The chance of getting into the wing is related to the emission statistics. We will see later that a photon has a much larger chance of being emitted in the line centre than in the wings. The usual trapping procedure will thus be that a photon spends a lot of time in the line centre, being absorbed and reemitted, while covering only a small distance each time. After some time (or rather, with a certain probability), it will get into the wing of the line, where it covers a large distance. The next reemission will probably be again in the line [^3]: FR does not only occur in dense vapours (due to collisions) but in principle in all vapours, due to the Doppler effect. When the emission frequency becomes completely independent of the absorption frequency, we speak of 'complete frequency redistribution' CFR. --- centre, and the whole process will be repeated until the photon finally escapes from the vapour cell. Since photons in the wings can escape from the vapour more easily than line-centre photons, radiation trapping also leads to a distortion of the lineshape. In the most extreme case, there are no line-centre photons in the emergent intensity at all, but only wing photons. The emergent radiation thus has a 'dip' at the centre frequency, and relative maxima in the wings. This effect is known as 'line self-reversal'. For steady-state problems, the 'visible' physical effect of radiation trapping is not a prolongation of the lifetime (we are, after all, in steady-state), but an increase in the excited-state density. This can be easily seen by setting up a rate equation. Without radiation trapping, the excitation rate $E_{\text{exc}}$ (e.g. from absorbing laser radiation) must equal the natural decay, so that $E_{\text{exc}} = A_{21}n_2$, where $n_2$ denotes the density of upper-state atoms (excited atoms per cubic metre). When radiation trapping is present, a photon does not leave the vapour after time $1/A_{21}$, but remains in the vapour $g_0$ times as long (hopping from atom to atom), so that the rate equation becomes $E_{\text{exc}} = n_2 \cdot A_{21}/g_0$. We can easily see that the excited-state density is increased by a factor $g_0$. In all the above explanations, we have imagined the excited-state distribution to be uniform throughout the cell. This is not the case even when the initial distribution (for pulsed experiments) or the excitation function (for steady-state experiments) have been uniform. Photons that are emitted close to the boundary of the vapour cell can escape more easily than photons that are created in the cell centre. The excited-state density will have its maximum in the cell centre, and will become smaller towards the cell walls. When dealing with trapping, we essentially want to answer two questions: a) how often is a resonance photon absorbed before it leaves the vapour, and b) what is the spatial distribution of the excited-state atoms. All other physical parameters can be derived from the answers to these two questions. The search for these answers will directly lead to the Holstein equation. ## 1.2 Historical overview Resonance radiation is one of our most important sources of information about atoms. As a matter of fact, resonance radiation had been studied long before one had an idea of what atoms look like. Most of our understanding of the atomic structure stems from experiments with resonance radiation. The first systematic analysis of resonance radiation was done by Kirchhoff, Fraunhofer, and Bunsen in the 19th century. Kirchhoff and Bunsen used prisms to spectrally analyse the light emitted by flames. When certain materials were added to a flame, bright lines occurred in the spectrum. Fraunhofer analysed the solar spectrum the same way and noted a wealth of dark lines. Soon, it became clear that the lines in the flames and in the solar spectrum had their origin in the same effect, and that each line could be attributed to a chemical element. This is actually the origin of the name helium. Its lines were first discovered in the solar spectrum, and it was believed for some time that this element occurred only on the sun (greek *helios*). --- One set of lines analysed very carefully by astrophysicists was due to atomic hydrogen. The frequencies of its line spectrum showed a special regularity. The line frequencies can be written as $$ \nu \propto \frac{1}{n^2} - \frac{1}{n'^2} $$ (1.2) where $n$ and $n'$ are integers. This inspired Bohr to his famous postulates, the very first theory of the atomic structure that agreed with the then available experimental results. In subsequent years, experiments were usually one step ahead of theory: new facts were gathered from the study of resonance radiation, and the theoretical physicists built their groundbreaking theories to explain these facts. In modern times, experiments with resonance radiation remain the acid test for theoretical computations of the atomic structure. Alkali atoms are the favourite atoms used in theory, since their structure is reasonably simple. When theory and experiment disagree, there are two possibilities: the theory is wrong or there is some fault in the experiment (not to speak of the third possibility, both theory and experiment being flawed). One of the nastiest problems in experiments with resonance radiation is the trapping or imprisonment of the radiation (Mitchell and Zemansky 1961). The very first analysis of trapping, done by Compton (1922), assumed that there is a strict analogy between particle and radiation diffusion. Basically, Compton treated trapping as a conventional diffusion problem. This had a strange effect: the excitation in the vapour could die out faster than with the natural lifetime, which is obviously unreasonable. Milne (1926) noted this discrepancy and derived a modified diffusion equation that remedied this situation. However, he neglected the influence of the spectral shape of the emission line on the trapping process. Later attempts by Samson (1932) and Kenty (1932) to refine this by introducing an 'equivalent' absorption coefficient of the atoms (averaged over the lineshape) somewhat improved the situation. Still, experiment and theory often differed by orders of magnitude at high atomic densities (Zemansky 1932), (Michael and Yeh 1970). It was only in the late 1940s that Biberman (1947) and Holstein (1947) independently did away with the concept of the diffusion equation and showed that only an integro-differential equation can describe the problem adequately. This integro-differential equation is the famous Holstein equation (also called the Holstein–Biberman or Biberman–Holstein equation). We will derive it in Chapter 4 and it will form the basis of our considerations for most of this book. With the then revolutionary concept of complete frequency redistribution, CFR, they showed that the representation of trapping by a diffusion equation is not possible. Good agreement with experiments was achieved (Corney 1977). Still, they used a lot of simplifying assumptions for deriving and solving the Holstein equation. The work of researchers since 1950 has mostly been dedicated to relaxing these simplifications. For a long time, research was restricted to those cases that could be solved or at least approximated analytically. This is the reason why so many papers use, e.g., the 'high-opacity approximation' (see, e.g., the papers of van Trigt (1969–75)). This restriction --- actually was quite beneficial: it led to the development of efficient mathematical methods and, in many cases, gave closed-form results that are easy to use and give an intuitive understanding of the relation between the operating parameters and the number of re-absorptions. In recent years, the availability of ever faster computers has changed the situation somewhat. It is now possible to lift restrictions that are a prerequisite for the analytical computations, and to check their influence quantitatively. Parallel to the work of atomic physicists, astrophysicists have also tried to find ways for computing trapping. Their motivation is clear: stellar atmospheres are, after all, giant spherical ‘atomic vapour cells’. Much stimulation has come from their work, most notably in the treatment of incomplete frequency redistribution that we will encounter in Chapter 11. It is strange that, on the other hand, no astrophysicist seems to have read Holstein’s landmark paper. It was only in the 1950s that a group of scientists at JILA (Joint Institute of Laboratory Astrophysics, one of the foremost centres in the research of radiative transfer) independently arrived at a formulation that is equivalent to the Holstein equation. This series of papers (Thomas 1957), (Jefferies and Thomas 1958, 1959), (Thomas 1960), (Jefferies and Thomas 1960), (Jefferies and White 1960), (Jefferies 1960), and (Thomas and Zirker 1961), created enormous interest in what was called non-LTE stellar atmospheres (i.e. atmospheres where the structure is determined at least partly by radiation trapping, see e.g. (Hummer 1968b)), and many of the research results are also applicable to laboratory situations. Cross-reading between the papers on trapping in atomic physics and in astrophysics is hampered by the fact that though the basic problem is the same, the ‘boundary conditions’ are completely different. In atomic physics, we usually deal with the temporal decay of the excited-state density and the considered cells are small. Astrophysics is mostly concerned with the steady-state (we would be in a real mess if solar radiation should suddenly start to decay) and with large geometries. In this book, we will thus review quite a lot of mathematical methods that originally stemmed from astrophysics, but not go into the results—that has to be left to astrophysical textbooks. There are essentially two main directions in the research of radiation trapping. The first direction is the exploration of the very physical processes that underlie the radiation trapping. Examples for this are investigations of the relation between the frequencies of the absorbed and reemitted photons, and the relation of their polarizations. In the 1960s and 1970s, these investigations were done, e.g. by Hummer, Stenflo, Rees, and many others, using semi-classical models of the atoms. However, for an exact analysis, they must be based on quantum-mechanical formulations. Pioneering quantum-mechanical work has been done, e.g., by Cooper, Burnett, and colleagues, and by Landi-degli’Innocenti. These aspects become especially important in experiments with very strong lasers, where a semi-classical description of atoms is troublesome. The second main direction of research is related to the fact that although the semi-classical formulation can be written down in a few lines, the actual solution of such a system of equations is tremendously difficult. Brute-force numerical solutions of complicated multi-dimensional problems exceed even the capabilities of today’s most ad- --- vanced computers. Thus many scientists work on new or improved approaches towards the solution of the Holstein equation (or of the transfer equation, when they are astrophysicists). Some of these solution methods are tailored to specific physical situations, some use analytical tricks to reduce the numerical load, and some make use of special numerical properties. This research is directed not only at the very advanced fields of partial frequency redistribution and non-linear radiation trapping, but also at the ‘classical’ problem described in Part II. In 1988, a review paper stated that the subject of linear radiation trapping with complete frequency redistribution could be considered as complete from both a theoretical and a practical point of view. To disprove this statement, we mention here that more than one third of the methods described in Chapters 5–10 were developed *after* 1988: the fitting equations, the geometrical quantization technique, the propagator function method, and the analytical solution of the multiple scattering representation. A third field where radiation trapping is of central importance is plasma physics–a kind of union of the properties of chemical physics and astrophysics. On the one hand, the temperature and pressure range, and the (non-radiative) physical processes are reminiscent of stellar atmospheres. On the other hand, the study of plasmas involves complicated boundary conditions, and we are often interested in time-decay phenomena. Most research in plasmas is somehow related to military uses like nuclear fusion (of course there is also the peaceful aspect of controlled fusion) or creation of X-ray lasers in recombination plasmas. Research in this area certainly has been going on for a long time, but only after the end of the cold war were many projects declassified, and their results reported in the open literature. It is ironic that while trapping theory has made huge advances and allows now quite accurate computations in many cases, most *appliers* of trapping theory still use approximate equations that were developed 40–70 years ago. In itself this situation is not all that bad—as a matter of fact, it shows the high quality of these approximations. The real problem is that these approximations are valid only under certain, well-defined conditions, but they are often applied even when these conditions are *not* valid. This book should present the tools required to deal with a trapping problem efficiently, which means with just the right amount of effort. When we use simple approximations unthinkingly, we will end up with serious errors in our results. When, in contrast, we always use high-accuracy numerical methods that are universally valid, we should seriously think about first buying several state-of-the-art supercomputers. ## 1.3 Applications of radiation trapping The main body of the book, Parts II and III, will present methods for the computation of radiation trapping. The situation considered will be always the same as in this introduction—a glass cell filled with an atomic vapour, and excited by some external radiation. When working through all the theory of these chapters, you might start to wonder what use the whole effort is. To provide you with some motivation, we now --- have a glance over some applications of radiation trapping. We will see that the applications are quite diverse, and that understanding trapping is vital in many areas. These applications will then be treated in a more thorough way in Part IV, where we will apply the theory developed in Parts II and III. Let us first consider some effects of trapping on **measurements of physical properties**. First, there is the increase in the apparent lifetime of excited-state atoms described in Sec. 1.1. The simplest method for determining an atomic lifetime is sending a short laser pulse of appropriate frequency into a cell filled with an atomic vapour and observing the emitted fluorescence radiation. In case the radiation is trapped, however, when we simply set the decay time of the excited atom equal to the decay time of the radiation, we overestimate the natural lifetime of the excited state by a factor $g_0$. To determine the actual lifetime, we must take the number of reabsorptions into account. This effect can also be used the other way round when we already know the 'true' lifetime of the excited atoms. Then the measurement of the apparent lifetime gives the opacity, and thus the particle density of the vapour. This can be a valuable tool especially at higher temperatures, where other methods, like the vapour pressure curve, become unreliable. Another example is the steady-state measurement of ionization cross-sections. Ionization cross-section measurements are usually done by exciting a vapour with a certain known rate, from which the excited-state density is computed, and simultaneously measuring the number of created ions. The ionization is related to the quotient of the creation rates of ions and of excited-state atoms. We have seen in Sec. 1.1 that the radiation trapping causes an increase in the steady-state excited density, and thus enters this ratio. Gross errors will occur when that effect is not considered in the computation of the ionization cross-section. Similar effects arise in the measurement of collision, quenching, and intermixing cross-sections. In the last five years, **laser cooling** has become a 'hot' topic. The cooling has pushed the achievable temperature limit to the range of nano-Kelvins, and a new state of matter (Bose–Einstein condensation) has been realized. Many interfering effects have been eliminated, and currently it seems that radiation trapping presents the ultimate limit for the density and temperature that can be achieved. Radiation trapping also determines the efficiency of **gas lasers**. Imagine the simplest laser scheme, a three level atom. Atoms are excited (e.g. by electron collisions) from the ground state to an upper state, whence they decay to a middle state. If the lifetime of the middle level is shorter than the lifetime of the upper level, we get population inversion between the upper and the middle levels, and the gas becomes amplifying. The density of middle-state atoms will be higher when the transition from the middle to the lower level is trapped. In other words, the effective decay rate from the middle level will be lower (i.e. the effective lifetime is higher by a factor $g_0$). This effect decreases the population inversion, and thus the amplification. The opacity of the gas need not even be very large—just a 5% increase in the middle-state population can make the difference between amplification and attenuation. In **atomic line filters**, resonance transitions are employed as very selective optical --- filters. As for the laser, we need at least a three-level atom. In the simplest filter scheme, photons from a signal source are absorbed by ground-state atoms, exciting them to an upper state. Some of these decay then to a middle state, emitting fluorescence photons of a different wavelength. This wavelength conversion is very selective—it only works for photons with a certain wavelength, namely those that can be resonantly absorbed by ground-state atoms. By combining a vapour cell with simple glass filters, this wavelength conversion can be used to block broadband radiation and to obtain a filter that is as narrow and stable as an atomic line. Without radiation trapping, the conversion efficiency would be determined by the branching ratio, i.e. by how many of the input photons are reemitted at the 'new' wavelength instead of being reemitted again at the input wavelength. When the input transition is radiation trapped, then input photons that are not converted in the initial absorption/emission process get another chance of being converted. This effect thus increases the conversion efficiency of the filter. A good knowledge of radiation trapping effects and an accurate mathematical model are essential to optimize these filters with regard to both efficiency and speed. The commercially most important application where radiation trapping plays a major role are **fluorescent lamps**, representing a multi-billion dollar market. In these devices, atoms are excited by collisions with electrons, and the resulting resonance radiation is either directly used for lighting purposes, or is indirectly used after causing a phosphor layer on the bulb to fluoresce. The longer the resonance radiation stays in the vapour because of the trapping, the higher is the chance that it is quenched and destroyed to useless heat (e.g. when the excited atom collides with the wall of the bulb). Increasing the efficiency has high economic potential, and it is no accident that the Holstein equation originated in a research lab for fluorescent lamps. Apart from these areas that we will discuss in detail later on, there are many more applications—quite generally, we can state that > Radiation trapping has to be considered > whenever radiation interacts with atomic vapours. Of course there are some special cases where trapping can be neglected, but we have to *prove* that trapping does not play a role, not the other way round. We always have to remember that trapping computations are the art of the possible. For more than 20 years, we have known a very general formulation (Bulyshev *et al.* 1975) (a semi-classical formulation). This formulation can be written down in two lines; nonetheless it is hopeless to solve it for a two- or three-dimensional geometry. For results, we have to make simplifications. The science and art is to find those simplifications that are valid under the considered circumstances. The degree of simplification is always a compromise between accuracy and computational effort. The faster computers get, the more this balance will shift towards high-accuracy computations. --- ## 1.4 Outline of the book The book is divided into four parts: **Part I, Background**, gives information that serves as a basis towards a thorough understanding of radiation trapping. After the brief introduction you have just read, Chapter 2 discusses the lineshapes of atomic transitions. It is of course impossible to thoroughly treat this subject in a few pages; it deserves (and has been given) several books. However, it might be a useful refresher and is also used to establish the notation of the book. We have included some unconventional topics, like the efficient computation of Voigt profiles, that in our experience are helpful for practical computations. The depth of Chapter 3, which deals with collisional processes and diffusion, is similar. We give a brief refresher because these effects are essential in setting up the rate equations and are also important for the frequency redistribution effect mentioned in Sec. 1.1. Furthermore, it is useful to have a description of the diffusion process in order to find the similarities and differences to the radiation trapping. **Part II, The classical radiation trapping problem**, deals with a model problem that uses many simplifying assumptions (like complete frequency redistribution, no particle diffusion, simple geometrical shapes of the vapour cell, etc.). This model problem has been given the most attention in the literature for two reasons. First it is reasonably simple, which allows one to gain better insights into the physical processes, and also leads to an easier mathematical treatment. Secondly, these simplifying assumptions are fulfilled (at least approximately) in a lot of practical situations. Chapter 4 describes various ways to formulate the classical problem mathematically: three exact methods (the Holstein equation, the multiple-scattering representation, and the equation of radiative transfer combined with the rate equation) and two approximate ones (the Milne equation and the escape factor method). Chapters 5 and 6 then describe mathematical methods to solve the Holstein equation and the multiple-scattering representation, respectively. We will see that even the comparatively simple 'classical' problem requires considerable mathematical effort. A number of methods, ranging from semi-analytical to purely numerical ones, has been developed, and each has its advantages and drawbacks. After the minefield of mathematics and equations in these two chapters, Chapter 7 then gives a unified representation of the results (by means of fitting equations for the solution of the Holstein equation) and a physical interpretation. During a first reading of the book, you might prefer to glance only briefly through Chapters 5 and 6. However, we strongly recommend you to read them thoroughly at a later time. Only with a good grasp of the methods developed there it will be possible for you to treat realistic trapping problems and to do research of your own. Chapter 8 describes the Milne theory, an approximate method to treat trapping. It does not contain a correct description of the frequency redistribution, and also approximates the angular distribution of the radiation in a rather crude way. Nonetheless, the Milne approach gives good results at low opacities, and is still in use today because it is so simple. Chapter 9 finally gives mathematical methods for the solution of the transfer-plus-rate-equation formulation. These methods are mainly suitable for steady- --- state problems. They originated in astrophysics and are rarely used for solving the classical trapping problems of chemical physics. However, the same methods can be used to great advantage in generalized trapping problems. For this chapter, the comments made for Chapters 5 and 6 are also valid—it might be the best strategy to first get an impression of the whole book, and then to return to the chapter for a more detailed study. **Part III, Generalized trapping problems**, treats radiation trapping for those cases where the simplifications of the classical problem are no longer fulfilled. Chapter 10 deals with generalizations that require only more sophisticated mathematics, but no new physical concepts. These generalizations include multi-level problems, particle diffusion, reflecting cell walls, inhomogeneous distributions of absorbers, and more complicated vapour cell geometries. Chapter 11 does away with the assumption of complete frequency redistribution, i.e. we take into account the fact that the frequency of the absorbed photon might be somehow correlated to the frequency of the emitted photon. In Sec. 11.1, we describe the physical mechanisms that can lead to such a correlation and set up the generalized Holstein equation that describes trapping under such conditions. Sections 11.2–11.4 then describe methods to solve this generalized Holstein equation. Sections 11.5 and 11.6 formulate and solve the problem of partial frequency redistribution in terms of the multiple-scattering representation (Monte Carlo simulations) and of the transfer equation. After a brief description of approximate methods, we deal with the case that the vapour suffers large-scale movements. This case is of little interest in a vapour cell but becomes important in an expanding plasma. Chapter 12 shows how to include polarization effects into the computations. Chapter 13 deals with non-linear problems, which are the most difficult to solve. First, we discuss under what conditions non-linearities have to be included in our computations, and give a brief discussion of the physical effects of the interaction between atoms and high-intensity radiation. We then deal with mathematical methods for treating non-linear problems. Essentially three techniques, all originally stemming from astrophysics, have been developed: complete linearization, operator perturbation, and simple iteration. To complicate matters, the steady-state and the transient solutions are no longer related, as they were for the linear case. They have to be treated separately. Non-linearities also lead to some new physical effects, which are also discussed. Chapter 14 shows how various approaches can be combined to yield efficient solutions for specific problems. **Part IV, Applications**, describes various applications of the trapping theory to practically important situations. We describe in what situations what method can be used, and also describe some techniques that are tailored to some specific experiment (and thus have not been described in detail in Parts II and III, which deal with the general theory). Chapter 15 describes measurements in chemical physics. This includes such diverse applications as the measurement of collision cross-sections and of atomic lifetimes, and laser cooling. We also discuss the existing literature and point out possible problems. Chapter 16 deals with trapping in gas lasers. We describe a computation --- method that is suitable for weakly trapped transitions and includes the effects of partial frequency redistribution. We also outline a method suitable for lasers with a strongly trapped pump transition. Chapter 17 describes radiation trapping in atomic line filters (ALFs), a type of extremely-narrow-band optical filters. These are ‘textbook’ examples for trapping. The trapping theory developed in Parts II and III can be applied without modifications, and the effects of trapping can change the filter characteristics by orders of magnitude. Chapter 18 finally describes discharge lamps, the economically most important application involving trapping theory. After a brief introduction to discharge lamps in general, we look into the effects of trapping with regard to lamp efficiency, and how efficiency can be improved. We also give a short description of trapping in plasmas, plus an overview of the relevant literature. The **Appendices** contain material that was deemed to be important, but would disturb the flow of the book. Appendix A gives a brief refresher on the basic atomic structure. This is mainly intended for engineers that have not dealt with the subject for some time; most physicists can be expected to be well versed in it. Appendix B derives some auxiliary quantities for a numerical solution of the Holstein equation. Appendix C presents an overview of publicly available computer programs for the solution of trapping problems. Appendix D lists the numerical factors for the fitting equations for the solutions of the ‘classical’ Holstein equation, and Appendix E gives auxiliary quantities for the numerical solution of the simplified transfer equation. Appendix F gives a very brief overview of density matrix computations, and Appendix G describes the interaction of atoms with very intense radiation, like Rabi oscillations, power broadening, etc. The References finally contain the citations for all the papers that we found to be relevant in connection to radiation trapping. The various parts of the book are connected so that it is recommended to read them in the proper sequence. As mentioned above, some very mathematical parts can be glanced through during a first reading, but for the serious researcher there is no way around studying them in detail. Only by understanding the mathematics can the physics be described in a meaningful way. Of course, the mathematics is only useful when the equations are interpreted physically. --- # 2 # ATOMIC LINESHAPES It is already evident from the basic physical picture developed in Chapter 1 that mainly two parameters determine radiation trapping: the strength of the absorption and the spectral shape of the absorption. The strength of the absorption is characterized by the opacity of the atomic vapour, which is proportional to the absorption cross-section of the atoms and to the density of atoms. Hence, before we enter the mathematics of the trapping problem, we first have to determine the absorption cross-section and lineshape. We will not go into details (these are beyond the scope of this book), but only review the facts needed to approach our problem. In addition, there is another motivation for this chapter. It is not only nice but absolutely necessary to have all required formulas compiled in one consistent notation. One has to take really great care not to become completely lost by widely differing notation in the literature—the cgs-system and esu-units versus the SI-system, frequency versus angular frequency, atomic constants referenced to photon density, to energy density or to light intensity, the ubiquitous factor of two for polarization, and the like. ## 2.1 The Einstein theory of radiation Atoms can undergo a transition from one energy state to another by emitting or absorbing a photon, i.e. a quantum of electromagnetic radiation. For these optical transitions we distinguish between three processes, which are characterized by the three **Einstein coefficients** (Mitchell and Zemansky 1961). * Spontaneous emission. An atom passes from the higher level 2 to level 1 with the emission of a photon of frequency $\nu_{21}$ into a random direction. The energy difference between the two levels is carried away by the photon energy $h\nu_{21}$. This emission happens spontaneously, with a 'probability per second' of $A_{21}$—that [FIGURE: Diagram showing energy levels 1 and 2 with transition arrows labeled A_21, B_12 I_\nu, and B_21 I_\nu] FIG. 2.1. The three optical transition processes. 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0YGPSE6OJWFSJU Z1SFTT•0YGPSE6OJWFSJU Z 1SFTT%0*PTP --- means, on average, an atom in level 2 decays after a time $1/A_{21}$ to the lower state, level 1. This Einstein A-coefficient is a characteristic property of every atomic transition. It has the dimension of 1 over time, $[A_{21}] = 1/s$. The value of the A-coefficient can either be measured, or it can for some transitions be computed from quantum-mechanical models of the atoms. Such measurements and computations are a science of their own, and a great many results have been published in the literature. The transitions are not infinitely sharp but are characterized by a certain lineshape $k(\nu)$. However, the spectral width of the line is **much** smaller than the absolute transition frequency, $\nu_{21}$. In Sec. 2.2 we will discuss in detail the processes that determine the lineshape. When an ensemble of undisturbed atoms is in an excited level 2 at time $t = 0$, the excitation decays according to an exponential time dependence. The number of excited atoms decays with $\exp(-t/\tau)$, where $\tau$ is the natural lifetime of the excited level. Hence, when a single atom is excited to a higher state 2 by a short pulse of light at time $t = 0$, the probability that it is still in the higher state after time $t$ is $\exp(-t/\tau)$. The **natural lifetime** of the upper state, $\tau$, is given by summing over all decay probabilities to lower states, $$ \tau = 1 \Big/ \sum_i A_{2i} $$ (2.1) The summation is done over all transitions linking state 2 to lower levels. * **Absorption**. An atom in state 1 absorbs a photon of frequency $\nu_{12} = \nu_{21}$ and passes into state 2. The 'probability per second' for such a process is $B_{12} \cdot I_\nu$, where $B_{12}$ is the Einstein B-coefficient of absorption. $I_\nu$ is the spectral intensity of the radiation that the atom experiences. It is assumed that the angular distribution of the radiation is isotropic and that the spectrum of the radiation is white in the vicinity of the transition frequency $\nu_{12}$. The spectral intensity $I_\nu$ gives the intensity per unit frequency of the white excitation radiation. $I_\nu$ has units of W m$^{-2}$ Hz$^{-1}$ sr$^{-1}$. Note that there are, depending on the scientific discipline, at least three quantities that are alternatively denoted as 'intensity', with units of W sr$^{-1}$, W m$^{-2}$, and W m$^{-2}$ sr$^{-1}$. Although not correct from a photometric point of view, we adopt the radiation-trapping usage of intensity, namely power per unit area and per unit spatial angle; specifically, we define the intensity $I$ so that $I(\mathbf{r}, \mathbf{\Omega}, \nu, t) \cdot \delta A$ is the energy flux of photons of frequency $\nu$ and direction $\mathbf{\Omega}$ traversing the area $\delta A$, where $\delta A$ is perpendicular to $\mathbf{\Omega}$. Quantities with subindex $\nu$ denote spectral quantities, i.e., they are referred to 1 Hz of bandwidth (except for constant $C_\nu$ below). When the spectral light intensity around $\nu_{12}$ depends on frequency, then the transition rate depends on the actual lineshape and is given by a weighted integration of the spectral intensity over the spectral line, --- $$ B_{12}C_{\nu} \int_{-\infty}^{\infty} k(\nu)I_{\nu}\mathrm{d}\nu \quad \text{with} \quad C_{\nu} = 1 \bigg/ \int_{-\infty}^{\infty} k(\nu)\mathrm{d}\nu $$ (2.2) In case that the excitation source is a narrowband laser, the transition rate is $$ \frac{1}{4\pi} B_{12} C_{\nu} F_{\text{laser}} k(\nu_{\text{laser}}) $$ (2.3) $F_{\text{laser}}$ denotes the laser flux density, in units of $\mathrm{W\,m}^{-2}$. The factor $1/4\pi$ stems from the laser radiation not being isotropic. For an approximately unidirectional beam which fills the spatial angle $\mathrm{d}\Omega$, the transition rate is $\mathrm{d}\Omega/4\pi$ of the transition rate for isotropic radiation. * **Stimulated emission.** An atom in state 2 is exposed to radiation of spectral intensity $I_{\nu}$ at the frequency $\nu_{21}$, and itself emits a photon of frequency $\nu_{21}$ in the same direction as the stimulating photon. Stimulated emission is coherent, the stimulation photon and the emitted photon are indistinguishable—they have the same frequency, direction, polarization and phase. The 'probability per second' for stimulated emission is $B_{21} I_{\nu}$, where $B_{21}$ is the Einstein B-coefficient of emission. This again applies to a white spectrum of $I_{\nu}$ around $\nu_{21}$. For a non-white stimulating spectrum, the formulas above apply accordingly. Using thermodynamic arguments, Einstein showed that the two B-coefficients are connected by $$ \frac{B_{21}}{B_{12}} = \frac{g'_1}{g'_2}, $$ (2.4) where $g'_1$ and $g'_2$ are the statistical weights of levels 1 and 2. The statistical weights $g'$ are the quantum-mechanical degeneracies of the levels—there are $g'$ quantum-mechanically different and distinguishable states hidden under a level with a certain energy. How many states one summarizes into one 'level' depends on the energy resolution one makes. This does not influence our results since only ratios of degeneracies are physically meaningful. For the relation of the B-coefficients to the A-coefficient, Einstein found[^4] $$ \frac{A_{21}}{B_{12}} = \frac{2h\nu^3}{c^2} \frac{g'_1}{g'_2}. $$ (2.5) The Einstein B-coefficients here are defined in terms of spectral radiation intensity, $[\mathrm{W/m}^2\text{ Hz sr}]$, while the coefficients used in Einstein's original paper were defined in [^4]: These simple Einstein relations are valid for level 1 being the ground state. When level 1 is some excited level, the following relation holds: $$ \frac{g'_2 A_{21} + g'_1 \sum_{i<1} A_{1i}}{g'_2 B_{21}} = \frac{g'_2 A_{21} + g'_1 \sum_{i<1} A_{1i}}{g'_1 B_{12}} = \frac{2h\nu^3}{c^2} $$ --- terms of spectral radiation density, [J/m$^3$ Hz]. For isotropic radiation these B-coefficients are related by $$ B_{[\text{density}]} = \frac{c}{4\pi} B_{[\text{intensity}]}, $$ (2.6) The probability of an emission or absorption process is also often described by the **oscillator-strength** or $f$-value, which is related to the Einstein A-coefficient by $$ f_{12} = A_{21} \frac{m_e \epsilon_0 c^3}{2\pi e^2 \nu^2} \frac{g_2'}{g_1'}, $$ (2.7) where $m_e$ is the electron mass, $\epsilon_0$ is the vacuum dielectric constant, and $e_-$ is the electron charge. The $f$-value is defined via the absorption properties in the classical view of an electron oscillating in the atom. The A-coefficient is defined via the emission, and is quantum-mechanically meaningful. In the following, we will consistently use the A-coefficients; Equation (2.7) can be used to convert $f$-values found in the literature into A-coefficients. For collimated radiation sent through a homogeneous absorbing medium, the intensity of the radiation still present after a length $z$ is given by **Beer's law**, $$ I_\nu(z, \nu) = I_\nu(0, \nu) \exp(-k(\nu) \cdot z) $$ (2.8) where $k(\nu)$ is the spectral lineshape. The absorption coefficient at the line centre is given by $k(\nu_0)$, which we usually abbreviate to $k_0$. The integral over the absorption coefficient is related to the A-coefficient via the **Ladenburg relation** $$ \int k(\nu)\mathrm{d}\nu = \frac{c^2}{8\pi \nu_0^2} \frac{g_2'}{g_1'} n_1 A_{21} \left( 1 - \frac{g_1'}{g_2'} \frac{n_2}{n_1} \right). $$ (2.9) $n_2$ and $n_1$ are the densities of the upper- and lower-state atoms, respectively. If $n_2 << n_1$, i.e. in the absence of stimulated emission, this can be simplified to $$ \int k(\nu)d\nu = \frac{c^2}{8\pi \nu_0^2} \frac{g_2'}{g_1'} n_1 A_{21} = n_1 \frac{h\nu_0}{4\pi} B_{12}. $$ (2.10) The Ladenburg relation expresses that the integral over the absorption coefficient depends only on the particle density and on the atomic constant $A_{21}$, but not on the shape of the line. Different line-broadening mechanisms just 'smear' the absorptive power of the atoms over a spectral region, but do not change it. Finally, we introduce the concept of the **absorption cross-section** $\sigma$. We imagine that each atom is a sphere with an effective radius $r_{\text{eff}}$ and a cross section $\sigma = r_{\text{eff}}^2 \pi$. If a photon hits this sphere, it is absorbed by the atom. The absorption cross-section is frequency dependent, and is related to the absorption coefficient by $$ \sigma(\nu) \cdot n_1 = k(\nu). $$ (2.11) Usually, only the absorption cross-section $\sigma$ at line centre is given. --- It is very common in atomic physics to describe interaction probabilities by 'effective' cross-sections. These cross-sections can be totally unrelated to any geometrical dimensions of the atom. Geometrical cross-sections are on the order of $10^{-14}$ to $10^{-16}$ cm$^2$, while absorption cross-sections can be as high as $10^{-10}$ cm$^2$. The absorption cross-section is related to the Einstein A-coefficient by $$ \sigma(\nu) = \frac{g_2'}{g_1'} \frac{c^2}{8\pi\nu_0^2} A_{21} C_\nu k(\nu) $$ (2.12) When the radiation intensity is very low, then the rate of stimulated emission is much smaller than the rate of spontaneous emission, so that we can neglect stimulated emission. This is extremely convenient for the computations. If the spectrum of the isotropic radiation is white, we require that $A_{21} \ll B_{21} I_\nu$, or that $$ I_\nu \ll \frac{A_{21}}{B_{21}} = \frac{2h\nu^3}{c^2}. $$ (2.13) When the spectral intensity depends on frequency, we require that $$ C_\nu \int I_\nu(\nu)k(\nu)d\nu \ll \frac{2h\nu^3}{c^2}. $$ (2.14) The case most often encountered in practice is a narrowband laser source tuned to the centre of the line, $\nu_0$. For stimulated emission to be negligible, the laser flux density must fulfil $$ F_{\mathrm{laser}} \ll F_{\mathrm{s}} = 4\pi I_{\mathrm{s}} = 8\pi \frac{h\nu^3}{c^2} \frac{1}{C_\nu k(\nu_0)} $$ (2.15) The radiation intensity has to be much less than the saturation intensity $I_{\mathrm{s}}$ if we want to neglect stimulated emission (see also Sec. 13.2). The saturation intensity $I_{\mathrm{s}}$ is defined as the intensity where the population in level 2 reaches half the value of the fully saturated population. When we have infinite intensity, then spontaneous emission is negligible, and only absorption and stimulated emission are important. In that case, the populations in the levels 1 and 2 are in the ratio of their statistical weights, $g_1'/g_2'$. Looking at Eq. (2.15) we see that the saturation intensity depends only on the frequency and on the lineshape, but not on the strength of the transition. This is an extremely important fact, whose consequences are often not recognized in the literature. Equation (2.15) tells that it takes the same laser power to saturate a strong, dipole-allowed resonance line as it takes to saturate a weak, 'forbidden' line to a metastable state. For all transitions in the visible spectral range, the saturation flux density is on the order of 10 W/cm$^2$. Only for infrared transitions does the saturation intensity (and the saturation flux density) become markedly smaller. --- ## 2.2 Broadening mechanisms The spectral lineshape is determined by several broadening mechanisms. The most important mechanisms are natural broadening, Doppler broadening, and pressure broadening. The relative importance of these mechanisms depends on the experimental situation. ### 2.2.1 Natural broadening Natural broadening is caused by the finite lifetime of the atomic levels. The Heisenberg uncertainty principle states that energy and time can be determined simultaneously only with an uncertainty of $$ \Delta E \cdot \Delta t = \hbar. $$ (2.16) In our case, this relation applies to the energy and lifetime of an atomic level. An atomic level has an energy uncertainty due to its natural lifetime, $\Delta t = \tau$. Photons are emitted or absorbed in a range of energies, $\Delta E$, or in a frequency range $\Delta \nu^n = \Delta E / h$. The shorter the lifetime, the larger the broadening. The ground state has practically eternal life, hence its energy uncertainty is $\Delta E = 0$. The frequency uncertainty of a transition terminating on the ground state is entirely due to the finite upper-state lifetime, so that $\Delta \nu^n = 1 / (2 \pi \tau_2)$. When the lower state also has a finite lifetime, then $\Delta E = \Delta E_1 + \Delta E_2$, so that $$ \Delta \nu^n = \frac{1}{2\pi} \left( \frac{1}{\tau_1} + \frac{1}{\tau_2} \right). $$ (2.17) The probability that a photon with frequency $\nu$ is absorbed can be shown to be proportional to $$ k(\nu) = \frac{k_0}{1 + \left[ \frac{2(\nu - \nu_0)}{\Delta \nu^n} \right]^2}. $$ (2.18) The lineshape $k(\nu)$ of Eq. (2.18) is called a Lorentzian shape. The parameter $\Delta \nu^n$ is the full-width-at-half-maximum, FWHM, of the lineshape, see Fig. 2.2. For Lorentzian lineshapes, we define a normalized frequency $x$, $$ x = \frac{2(\nu - \nu_0)}{\Delta \nu^n} $$ so that the Lorentzian can be written as $$ k(x) = \frac{k_0}{1 + x^2} $$ ### 2.2.2 Doppler broadening When a monochromatic source of light is moving towards or away from an observer, the light appears to be shifted in frequency due to the Doppler effect. When the light --- BROADENING MECHANISMS [FIGURE: FIG. 2.2. Definition of the FWHM.] source moves with a velocity component $v_{\text{ls}}$ along the line of sight, the frequency seen by the observer is shifted by $\Delta \nu = \nu_0 \cdot v_{\text{ls}}/c$. Here, $\nu_0$ is the frequency emitted in the rest frame of the light source. Gas atoms in a vapour cell are in thermal motion and both emission and absorption frequencies are changed by the Doppler effect. In thermal equilibrium, the velocity distribution of the atoms is Maxwellian, which results in an absorption coefficient with the shape of a Gaussian profile, $$ k(\nu) = k_0 \cdot \exp \left[ - \left( \frac{2(\nu - \nu_0)}{\Delta \nu^{\text{D}}} \sqrt{\ln(2)} \right)^2 \right] \qquad (2.19) $$ The Doppler width $\Delta \nu^{\text{D}}$ is the FWHM of a purely Doppler broadened absorption or emission coefficient, $$ \Delta \nu^D = \frac{2}{c} \sqrt{\frac{2 \ln(2) k_B T}{m_{\text{atom}}}} \nu_0 \qquad (2.20) $$ with $T$ the absolute temperature, $k_B$ the Boltzmann constant, and $m_{\text{atom}}$ the atomic mass. We can draw some important conclusions from Eq. (2.20). - The Doppler width $\Delta \nu^{\text{D}}$ is proportional to the square root of the temperature, which is a rather weak dependence as compared to, e.g., the strong dependence of the vapour pressure on temperature. Hence, in most experiments the change of the Doppler width with temperature can be neglected. - Heavier atoms give smaller Doppler widths. - The Doppler width is proportional to the absolute frequency, hence, UV lines display more Doppler broadening than red or IR lines. --- For Doppler (and Voigt) lineshapes we use the normalized frequency $$x = \frac{2(\nu - \nu_0)}{\Delta\nu^\mathrm{D}}\sqrt{\ln(2)}.$$ (2.21) Hence, in terms of normalized frequency $x$, the Doppler profile is the simple Gaussian profile $$k(x) = k_0 \exp(-x^2).$$ (2.22) ### 2.2.3 *Pressure broadening* Atomic lines are also broadened by the presence of other atoms. The energy levels are disturbed by atoms colliding or moving by. Two groups of theories have evolved to describe these processes, the Lorentz (collisional) theory and the statistical approach (Berman and Lamb 1969). The Lorentz theory is the simpler approach; it has an obvious physical interpretation, and it is valid for most practically occurring situations. At very high pressures, however, the basic assumptions of the Lorentz theory break down and the statistical theory becomes a better description. In the **Lorentz theory**, the lineshape due to collisional broadening has the form $$k(\nu) = \frac{k_0}{1 + \left[ \frac{2(\nu - \nu_0)}{\Delta\nu^\mathrm{coll}} \right]^2},$$ (2.23) which is the same functional dependence as for the naturally broadened line, Eq. (2.18), only the FWHM, $\Delta\nu^\mathrm{coll}$ differs. The linewidth $\Delta\nu^\mathrm{coll}$ is proportional to the number of 'disturbing' atoms. The proportionality constant between the linewidth and the atomic density has to be determined experimentally. For many transitions, published data are available, especially for broadening by noble gases. In the Lorentz theory, the lineshape does not dependent on the strength or on the distance dependence of the forces interacting between the atoms, only the linewidth $\Delta\nu^\mathrm{coll}$ changes. The Lorentz theory does not account for the experimentally observed shift of the centre frequency, $\delta_\nu$, in the presence of other atoms. While it would be easy to incorporate the shift in the formula for the lineshape by simply replacing $\nu$ with $\nu - \delta_\nu$, this is rarely necessary in practice. For the radiation trapping process, such a shift is not relevant. The physical picture of broadening in the Lorentz theory is the following. It is assumed that the coherent wave train emitted by an atom is interrupted by a collision with another atom. The time between collisions is assumed to be much shorter than the natural lifetime. The Fourier transform of such a phase-disturbed wave train is a $\sin(\nu)/\nu$ function in the frequency domain. Averaging over the statistically distributed times between collisions results in the Lorentzian lineshape. The FWHM is thus determined by the particle velocities, by the effective cross-sections for the broadening --- collisions, and by the density of the disturbing atoms. Theoretical results for the radius of the cross-section, the so-called 'Weisskopf radius', are given in the literature (e.g. Sobelman (1972), Holstein (1951)) but are rather inaccurate. Measurements of the pressure-dependent FWHM are usually the only way to establish reliable results. The Lorentz theory is physically reasonable for collisions that influence line-broadening close to the line centre, while the statistical theory is valid for 'close encounters' that influence the lineshape far from the centre frequency. The transition frequency between these two regions, $\nu_t$, occurs at (Sobelman 1972, p. 394) $$ \nu_t - \nu_0 = \frac{\text{v}}{2\pi \rho_{\text{wk}}}, $$ (2.24) where $\text{v}$ is the relative velocity between the atoms and $\rho_{\text{Wk}}$ is the Weisskopf radius. As the Weisskopf radius is typically on the order of 5–10Å, the transition frequency lies at more than 100 GHz. For transitions in the visible spectral region this means that lineshapes are approximately Lorentzian up to 0.1 nm from the line centre. The transition frequency between the domains of the Lorentzian and of the statistical theories can also be written as (Sobelman 1972) $$ \nu_t - \nu_0 = \sqrt{\frac{N \text{v}^3}{2\pi \Delta\nu^{\text{coll}}}} $$ (2.25) where $N$ is the density of disturbing atoms. For the cesium resonance line (900 nm), for example, $\Delta\nu^{\text{coll}}/N = 1.5 \cdot 10^{-10} \text{ Hz/cm}^{-3}$, so that the transition frequency lies at 150 GHz (0.3 nm). There are some experimental confirmations of these considerations. Colbert and Huennekens (1991), e.g., measured radiation trapping in potassium and found that the experimental results agree with values computed under the assumption of Lorentzian broadening up to several hundred GHz. An additional condition for the validity of the Lorentz theory is that $2\pi \rho_{\text{Wk}}^3 N << 1$, which is valid for pressures below several atmospheres. The **statistical theory** is valid only very far in the wings of a line, see above. In this theory, the lineshape depends on the type of interaction force with the 'disturbing' atoms. The physical model of the statistical theory is to neglect the motion of the atoms and thus have stationary, statistically distributed atoms. These atoms create a potential that 'disturbs' the energy levels of excited atoms, which leads to a broadened spectral line. *Resonance interaction.* This interaction applies to self-broadening, i.e. to broadening by atoms of the same kind. A resonance interaction occurs when an atom is in an excited state with a strong transition to the ground state, and the disturbing atoms are in the ground state. The strength of a resonance interaction goes with the distance between collision partners as $1/r^3$, and is proportional to the strength of the resonant transition. *Van der Waals (or dispersion) interactions.* These interactions can occur between atoms in any state, and the atoms need not be of the same kind. Van der Waals --- interactions are most important for foreign-gas broadening, especially due to noble gases. Van der Waals forces arise from the interaction of permanent or instantaneous dipoles with the dipole induced in another atom. This interaction is much weaker than the resonance interaction. On the other hand, noble gas densities in an experiment can be quite high, since noble gases are often employed as buffer gases. (In a typical experiment, $10^{18} \text{ cm}^{-3}$ noble gas density as compared to $10^{13} \text{ cm}^{-3}$ alkali density.) Thus, van der Waals interactions can play an important role. The dispersion interaction goes like $1/r^6$ with the distance between collision partners. *Quadratic Stark effect.* This effect is due to interactions with electrons and ions. Charged atoms seldom occur in the usual experiments of chemical physics, but interactions involving charged species are naturally very important for plasmas and electric discharge lamps. The data on Stark broadening are far less comprehensive than those for self-broadening and noble-gas broadening. In some plasmas (e.g. in hydrogen plasmas), in addition, different forms of broadening are present, e.g., Holtsmark broadening. For the quadratic Stark effect, the interaction is proportional to $1/r^4$. *Repulsive forces:* Repulsive forces are due to the penetration of the electron shells of perturber and excited atom. They are usually very short-range ($1/r^{12}$) forces and have to be considered only when all other forces are very small. Repulsive forces can be neglected as broadening mechanisms in practically all experimental situations. Generally, the interaction is of the form $1/r^m$. It can be shown that in this case $$ k(\nu) \propto (\nu - \nu_0)^{-(m+3)/m} $$ (2.26) so that for van der Waals forces, $k(\nu) \propto (\nu - \nu_0)^{-3/2}$, while for the resonance interaction $k(\nu) \propto (\nu - \nu_0)^{-2}$. ### 2.2.4 *Voigt lineshapes* When natural broadening and Doppler broadening are simultaneously present, the lineshape is given by a convolution of a Gaussian and a Lorentzian lineshape. Mathematically, this can be written as (Mitchell and Zemansky 1961, p.101) $$ k(\nu) = k_0^{\mathrm{D}} \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{\exp(-u^2)}{a^2 + \left[ \frac{2(\nu - \nu_0)}{\Delta \nu^{\mathrm{D}}} \sqrt{\ln(2)} - u \right]^2} \mathrm{d}u, $$ (2.27) where $k_0^{\mathrm{D}}$ is the centre-of-line absorption coefficient when only Doppler broadening is present. The parameter $a$ determines the relative importance of natural and Doppler broadening. It is called the Voigt parameter, $$ a = \sqrt{\ln(2)} \frac{\Delta \nu^{\mathrm{n}}}{\Delta \nu^{\mathrm{D}}}. $$ (2.28) --- The Voigt integral, $$H(a, x) = \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{\exp(-u^2)}{a^2 + (x - u)^2} \mathrm{d}u (2.29)$$ gives the Voigt lineshape in terms of the (Doppler-) normalized frequency $x$. When there is (Lorentzian) pressure broadening present in addition to natural broadening, Eqs. (2.27)–(2.29) remain valid. In the formulas, $\Delta\nu^{\mathrm{n}}$ is simply replaced by $(\Delta\nu^{\mathrm{n}} + \Delta\nu^{\mathrm{coll}})$. Pure Doppler broadening, or pure collisional broadening, are quite rare in practical situations. Pure natural broadening occurs only for very cold atoms: a few mK are sufficient for Doppler broadening to be of the same order of magnitude as natural broadening (see also Chapters 11 and 15). It is thus of great practical value to have fast and accurate computation methods for the Voigt lineshape $H(a, x)$. The most important computation methods are summarized in Insert 2.1 below. The line-centre absorption coefficient of a Voigt lineshape is given by $k_0^{\mathrm{D}} \cdot H(a, 0)$, where $k_0^{\mathrm{D}}$ is the absorption coefficient for the pure Doppler line, with $$H(a, 0) = \exp(a^2) \cdot \mathrm{erfc}(a). (2.30)$$ This follows immediately from the Ladenburg relation, Eq. (2.9). When the Voigt parameter $a$ is very small, $H(a, 0)$ is close to unity, so that the centre-of-line absorption is not influenced very much by the presence of a small Lorentzian contribution. The line wings, however, are changed appreciably even for a small Voigt parameter. This is due to the fact that a Gaussian shape, $\exp(-x^2)$, goes to zero much quicker than the Lorentzian, $1/(1 + x^2)$. A Doppler shape has negligible values for $x > 3$, while a Lorentzian has non-negligible values even for $x > 10$. Thus, the wings of a Voigt shape are always Lorentzian, see Figs 2.3 and 2.4. The shape of the wings is of fundamental importance for radiation trapping, since photons can most easily escape via the wings, especially in high opacity vapours. To fully appreciate this, we have to remember the difference between the **absorption coefficient** of a line, $k(x)$, and the **spectral absorption**, through a length $L$ of the absorbing medium, $$\text{spectral absorption} = 1 - \exp[-k(x)L] (2.31)$$ At intermediate or high opacities, the absorption at the line centre is very close to unity, regardless of the lineshape. In the wings of the line, however, the absorption strongly depends on the lineshape. Figure 2.4 shows absorption versus frequency for three values of the Voigt parameter, and for three opacities per Voigt parameter. For a Doppler lineshape ($a = 0$), the absorption does not change very much with increasing opacity—a Doppler absorption coefficient results in an almost rectangular spectral absorption at high opacities. In the case of a Voigt lineshape, however, the wings of the line become more and more important as the opacity increases. --- [FIGURE: FIG. 2.3. Voigt profile for three values of the Voigt parameter a. The frequency is expressed in Doppler widths from the line centre.] We thus have to consider natural and collisional broadening even if their FWHM is much smaller than the Doppler FWHM, i.e. when their contribution to the absorption coefficient is small. We can only neglect the Lorentzian contributions when their influence is still negligible at a frequency where the vapour in an experimental vessel becomes transparent. This frequency $x_{\text{esc}}$ is given by the condition $k(x_{\text{esc}})L = 1$. For a Voigt parameter $a > 0.1$, the shape of the line centre is influenced by the Lorentzian contribution. At $a > 3$, the line is purely Lorentzian for all practical purposes. *** ### Insert 2.1 Computation methods for the Voigt profile **1. Direct numerical integration.** The most straightforward way to evaluate the Voigt profile is by numerical integration of the defining equation, Eq. (2.29). This equation can also be written in a form (Mitchell and Zemansky 1961, p. 320) more suitable for numerical integration (despite the oscillating factor $\cos(x \cdot u)$): $$H(a, x) = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \exp\left(-au - \frac{u^2}{4}\right) \cos(x \cdot u) du \qquad (2.32)$$ --- BROADENING MECHANISMS [FIGURE: Spectral absorption as a function of frequency for different Voigt parameters a and different opacities k_0L] FIG. 2.4. Spectral absorption as a function of frequency for different Voigt parameters $a$ and different opacities $k_0L$. Frequency is measured in Doppler widths from the line centre. --- ### 2. A very crude approximation. As a very rough first approximation, we can use the Doppler lineshape at the line centre and the function $$ \frac{a}{\sqrt{\pi}} \frac{1}{x^2} $$ (2.33) for the line wings. The transition from line centre to wings is made at the frequency where the Lorentzian contribution becomes larger than the Doppler contribution. ### 3. Approximations for small Voigt parameters. For small values of the Voigt parameter $a$, we have a predominantly Doppler-broadened line, only in the line wings the Lorentzian contribution becomes manifest. According to Mitchell and Zemansky (1961, p. 321), the Voigt lineshape can in this case be approximated by $$ H(a, x) = \exp(-x^2) - \frac{2a}{\sqrt{\pi}} [1 - 2xFV(x)], $$ where $$ FV(x) = \exp(-x^2) \int_0^x \exp(u^2)du $$ (2.34) The function $FV(x)$ is tabulated by Miller and Gordon (1931). For small values of the normalized frequency $x$, the term in square brackets approximates to $$ 1 - 2xFV(x) = 1 - 2x^2 + \frac{(2x^2)^2}{1 \cdot 3} - \frac{(2x^2)^3}{1 \cdot 3 \cdot 5} + \dots $$ (2.35) For large values of $x$, $$ 1 - 2xFV(x) = -\left[ \frac{1}{2x^2} + \frac{1 \cdot 3}{(2x^2)^2} + \frac{1 \cdot 3 \cdot 5}{(2x^2)^3} + \dots \right]. $$ (2.36) ### 4. Approximation for 3% accuracy. Anderson *et al.* (1985) have given a simple analytical approximation that is accurate to within about 3% for all values of $a$ and $x$: $$ \begin{aligned} c1(a) &= \frac{a + a^2}{2 + a} + 0.83255 \\ c2(a) &= c1(a) \left( 2.12839 \cdot \exp\left(-\frac{a}{1.23}\right) + \pi \left[ 1 - \exp\left(-\frac{a}{1.23}\right) \right] \right) \\ u &= \frac{x}{c1} \\ f1(u) &= \exp(-0.69315u^2) \\ f2(u) &= \frac{1}{1 + u^2} \\ f3(u, a) &= f1(u) \exp\left(-\frac{a}{1.23}\right) + f2(u) \left( 1 - \exp\left(-\frac{a}{1.23}\right) \right) \\ H(a, x) &= \frac{c2(0) f3(u, a)}{c2(a) f3(0, 0)} \end{aligned} $$ (2.37) --- **5. An approximation for $10^{-4}$ accuracy.** When more accurate computations are required, the polynomial approximations of Humlicek (1982) give $10^{-4}$ accuracy: $$ \begin{aligned} t &= a - j \cdot x \\ u &= t^2 \\ f1 &= t \cdot \frac{0.5641896}{0.5 + u} \\ f2 &= t \cdot \frac{1.410474 + u \cdot 0.5641896}{0.75 + u \cdot (3 + u)} \\ f3 &= \frac{16.4955 + t \cdot (20.20933 + t \cdot (11.96482 + t \cdot (3.778987 + t \cdot 0.5642236)))}{16.4955 + t \cdot (38.82363 + t \cdot (39.27121 + t \cdot (21.69274 + t \cdot (6.699398 + t))))} \\ f4 &= \exp(u) - t \cdot \frac{36183.31 - u \cdot (3321.9905 - u \ --- ## 2.3 Strength rules for fine and hyperfine splitting Atomic lines are often split into several components due to hyperfine- or isotope-splitting (see Appendix A). Quite often we can only find the Einstein A-values for the unsplit line in the literature. Fortunately, there exist strength rules that allow the approximate computation of the A-values of the components. These rules apply the better the closer the components are, i.e. they are reasonably accurate for most fine structures and very accurate for the hyperfine components. The strength rules were postulated by Dorgelo (1925), and Ornstein and Burger (1926), see also Kuhn (1962). **'For the transition between two term multiplets, the two sums of the intensities from any one level and from any other level are in the ratio of the statistical weights of these levels.'** As an example for the application of this rule, we examine the 3D–3P transition in sodium (Na, Fig.2.5). Both the upper and the lower level are split into two fine-structure components. There are four possible transitions between these levels, but we know from the selection rule $\Delta J = 0, \pm 1$ that the $3\text{D}_{5/2}$–$3\text{P}_{1/2}$ transition is disallowed. From the strength rules, we get for the relative strengths, $\Gamma$, of the three remaining lines, $$ \frac{\Gamma_\mathrm{b}}{\Gamma_\mathrm{a} + \Gamma_\mathrm{c}} = \frac{6}{4} \quad \text{and} \quad \frac{\Gamma_\mathrm{c}}{\Gamma_\mathrm{a} + \Gamma_\mathrm{b}} = \frac{2}{4}. \qquad (2.39) $$ From this relation, or from the explicit formulas of Eq. (2.40), it follows that $\Gamma_\mathrm{a} : \Gamma_\mathrm{b} : \Gamma_\mathrm{c} = 1 : 9 : 5$. [FIGURE: FIG. 2.5. The fine-structure components of the sodium 3D–3P transition. The $g'$-values are the statistical weights (degeneracies) of the levels.] --- --- ### Insert 2.2 Strength rules For any terms with L-S coupling, the following more general strength rules were derived by Kronig (1925), Sommerfeld and Hönl (1925), Russel (1925), and Dirac (1926), for $L - 1 \rightarrow L$ $$ \Gamma_{J-1 \rightarrow J} = c1 \frac{1}{J} (L+J+S+1)(L+J+S)(L+J-S)(L+J-S-1) $$ $$ \Gamma_{J \rightarrow J} = -c1 \frac{2J+1}{J(J+1)} (L+J+S+1)(L+J-S)(L-J+S)(L-J-S-1) $$ $$ \Gamma_{J+1 \rightarrow J} = c1 \frac{1}{J+1} (L-J+S)(L-J+S-1)(L-J-S-1)(L-J-S-2) $$ (2.40) and for $L \rightarrow L$ $$ \Gamma_{J-1 \rightarrow J} = -c2 \frac{1}{J} (L+J+S+1)(L+J-S)(L-J+S+1)(L-J-S) $$ $$ \Gamma_{J \rightarrow J} = c2 \frac{2J+1}{J(J+1)} [L(L+1) + J(J+1) - S(S+1)]^2 $$ $$ \Gamma_{J+1 \rightarrow J} = -c2 \frac{1}{J+1} (L+J+S+2)(L+J-S+1)(L-J+S)(L-J-S-1). $$ For the computation of the relative strength of hyperfine components, instead of the fine structure given above, the substitutions $L \rightarrow J$, $S \rightarrow I$, and $J \rightarrow F$ must be made. For $j - j$ coupling, $L \rightarrow j_1$, $S \rightarrow j_2$, and $J \rightarrow J$. Approximate strength values are tabulated by White and Elliason (1933). --- The strength rules apply to the line strength $\Gamma$. Line strength is a quantity that is proportional to the absorption coefficient $k$, which itself is proportional to the intensity emitted when the atoms are statistically excited, i.e. when the population in every sublevel is proportional to its statistical weight. To arrive at ratios for the $A$-coefficients, the strength ratios have to be divided by the statistical weights of the upper level: $$ A_{21} \propto \frac{\Gamma}{g_2'}. $$ (2.41) For ratios of absorption cross-sections and for $f$-values, the strength ratios are to be divided by the statistical weights of the lower level, $$ \sigma_{12} \propto f_{12} \propto \frac{g_2'}{g_1'} A_{21} \propto \frac{\Gamma}{g_1'} $$ (2.42) --- Thus we get for the 3D–3P transition in Na, see Fig. 2.5, $$ \begin{aligned} k_a : k_b : k_c &= \Gamma_{\mathrm{a}} : \Gamma_{\mathrm{b}} : \Gamma_{\mathrm{c}} = 1 : 9 : 5 \\ A_{\mathrm{a}} : A_{\mathrm{b}} : A_{\mathrm{c}} &= \frac{1}{4} : \frac{9}{6} : \frac{5}{4} = 1 : 6 : 5 \\ \sigma_{\mathrm{a}} : \sigma_{\mathrm{b}} : \sigma_{\mathrm{c}} &= \frac{1}{4} : \frac{9}{4} : \frac{5}{2} = 1 : 9 : 10. \end{aligned} \quad (2.43) $$ where $k_a$ stands for $\int k(\nu)\mathrm{d}\nu$ in transition $a$. We see that $A_{\mathrm{b}} = A_{\mathrm{a}} + A_{\mathrm{c}}$—as expected, the two D-sublevels have the same lifetime. Furthermore, $\sigma_{\mathrm{a}} + \sigma_{\mathrm{b}} = \sigma_{\mathrm{c}}$, which means, when the fine structure is not resolved, the two P sublevels have the same --- # 3 # COLLISIONS, QUENCHING, AND PARTICLE DIFFUSION In this chapter, we deal with non-radiative processes that can create or destroy excited atoms.[^5] The theory of collisional excitation and de-excitation is extremely complicated. A thorough review of the work up to 1970 can be found in the five-volume book by Massey and Burhop (1968–1975). The work done after that time would probably fill another ten volumes. We will take here a mostly phenomenological point of view. The processes are described by effective cross-sections, and no investigation of the actual physical processes is done. The concept of the effective cross-section can be interpreted as follows. We imagine that each atom is a hard sphere with a radius $r_0$ and thus a cross-section $\sigma_{\text{coll}} = r_0^2 \cdot \pi$. The effective cross-section is an approximate concept. It says that a collision takes place when the two spheres collide or touch. Otherwise, nothing happens. In reality, the range of interaction between the two atoms is more 'smeared'. ## 3.1 Collisional cross-sections The collision cross-section depends on the velocity of the colliding particles. If we consider the collision of an electron with an atom, then we have the basic process $$ n_l + e_- \rightarrow n_u + e_- $$ (3.1) When $n_l$ signifies a low-energy state and $n_u$ a high-energy state, then the atom gains energy in the collision, and this energy must come from the electron. It is thus obvious that only electrons with a sufficient energy (i.e. velocity) before the collision can cause such a process—there is a threshold below which the collision cross-section is zero. Similar considerations are valid for collisions between two atoms. The parameter that allows a qualitative estimate of the velocity dependence of the cross-section is the 'Massey parameter'. It is the ratio of two characteristic times, namely the collision time (the time that the disturbing atom requires to traverse the force field of the disturbed atom) and the time that is required for energy transfer between the two atoms. The collision time obviously depends on the relative velocity of the two atoms. When the Massey parameter is very large, then the disturbed atom can adjust its energy states almost adiabatically to the disturbing potential, so that after the disturbing atom has [^5]: This is a somewhat sloppy use of language. Of course the atoms are neither created nor destroyed, but they are excited to a higher state or lose their excitation non-radiatively. However, this often simplifies the notation, and the meaning is unambiguous. 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- COLLISIONS, QUENCHING, AND PARTICLE DIFFUSION left, no effective change has taken place. In that case, the collision cross-section will be very small. In the other extreme case of a very small Massey parameter, the atoms have not enough time to exchange energy, and the disturbing atom will 'whiz by' without noticeable effect. Again, the collision cross-section will be small. Only if the Massey parameter is on the order of unity we will have a large collision cross-section. It is, however, very difficult to measure the collision cross-section as a function of the particle velocity. A velocity-averaged collision cross-section is much simpler to determine. It is defined by $$ \sigma_{\text{coll}} = \int \text{pdf}(v)\sigma_{\text{coll}}(v)\text{d}v $$ (3.2) where pdf(v) is the probability density function of the velocity distribution. Usually, it is assumed to be Maxwellian. This assumption can break down, however, if the excitation is due to a narrow-band laser. Because of the Doppler effect, a narrow-band laser will tend to excite only atoms with a certain velocity, i.e. with a certain Doppler-shifted absorption frequency. We usually assume that the cross-section is independent of the velocity. This is not strictly valid, but often gives a good approximation. The total rate for the collisional process can then be written as $$ \text{collision rate} = \sigma_{\text{coll}}\bar{v}_{\text{rel}}N $$ (3.3) where $N$ is the density of the collision partners. Under the assumption of a Maxwellian velocity distribution, the average relative velocity $\bar{v}_{\text{rel}}$ of collision partners with masses $m_1$ and $m_2$ is $$ \bar{v}_{\text{rel}} = \sqrt{\frac{8k_{\text{B}}T}{\pi}} \sqrt{\frac{1}{m_1} + \frac{1}{m_2}} $$ (3.4) Measurements of cross-sections are done just the other way round. The rate at which the process happens is measured. From this, from the average particle velocity, and from the particle density (temperature) the cross-section is then computed. Using the rates computed from cross-sections found in the literature is often the only way to incorporate collisional processes in our computations. The big problem is that published cross-sections can sometimes differ by an order of magnitude or more. In that case, it is just a matter of taste which value is chosen. In the following, we will discuss some basic processes that can occur, and give the orders of magnitude that the associated cross-sections typically have. With these, one can make at least a first estimate of whether the process might play a role, or whether it can be neglected. In the former case, it is necessary to obtain more accurate values for the process of interest from critically evaluating published data in the literature. --- ## 3.2 Collisions between atoms of the same kind - The simplest case is the 'elastic' collision, where the energy state of both collision partners remains unchanged. The only effect is that the velocity vectors will change. These collisions lead to a diffusion of the atoms. - Another simple situation arises when one excited atom and one ground-state atom of the same kind collide. In that case, the excitation energy can be transferred from one atom to the other. Since no energy is converted to kinetic energy, the cross-section for such a process is very large—usually on the order of $10^{-10}$--$10^{-12}$ cm$^2$. The effect of these collisions is that the excitation diffuses through the vapour. The mean free path for the diffusion is given by the distance travelled between collisions. The mathematical treatment of the diffusion is described in Sec. 3.5. - The second possibility for a collision between an excited and a ground-state atom is self-quenching, leaving two ground-state atoms. The excitation energy is converted to kinetic energy of the two collision partners. Cross-sections for these collisions are typically on the order of $10^{-14}$--$10^{-18}$ cm$^2$. - A third result of a collision between an excited and a ground-state atom can be one ground-state atom plus one atom in a different excited state. The probability for this depends mainly on $\Delta E$, the energy difference between the former and the latter excited states. When $\Delta E$ is positive, i.e. the later state has lower energy, the excess energy is converted to kinetic energy of the collision partners, and vice versa. Generally, cross-sections for these processes are the larger the smaller $\Delta E$ is. One special case is the fine structure intermixing cross-sections, where $\Delta E$ is sometimes on the order of several $k_B T$ (see Sec. 3.7). Here, collision cross-sections are on the order of $10^{-14}$--$10^{-16}$ cm$^2$. - Finally, when two excited atoms of the same kind collide, then the result can be one highly excited atom plus one ground-state atom (energy-pooling collision). In some cases, the pooled energy is larger than the ionization energy, so that the results of such a collision are an ion, an electron, and a ground-state atom. Cross-sections for these collisions are on the order of $10^{-16}$--$10^{-18}$ cm$^2$. ## 3.3 Collisions between atoms of different kinds - When an excited atom collides with a foreign gas atom, this is usually an elastic collision, i.e. the collision partners change their directions, and perhaps also their kinetic energies, but the excitation stays the same. Since the energy levels of different elements do not match, there is a low chance of energy transfer. The diffusion constant $D$ in this case is usually given as $$ D = D_0 \frac{p_0}{p} \qquad (3.5) $$ where $D_0$ is the diffusion constant at the foreign gas pressure of $p_0 = 1$ bar and $p$ is the actual foreign gas pressure. The cross-sections for these collisions are much --- smaller than the cross-section for energy transfer between atoms of the same kind. Especially for noble gases, the cross-sections are on the order of the geometric cross-section of the atom (see Appendix A). * Collisions with foreign gas atoms can also lead to quenching of the excitation. Quenching is more probable when the foreign gas has an energy level close to the excited-state energy of the excited atom. Noble gases have very high lying energy levels, hence they have extremely small quenching cross-sections (typically on the order of $10^{-20}$–$10^{-24}$ cm$^2$). Molecules have much larger quenching cross-sections ($10^{-14}$–$10^{-18}$ cm$^2$), because they have a large number of (ro-vibrational) energy states that often match the excited-state energy of the collision partner. * In some cases, there is by chance a match of energy levels between two elements, and excitation can be efficiently transferred from one excited atom to an atom of a different kind. One example is a mixture of cesium and thallium. When Cs atoms are excited to their first resonant state (e.g. by light from a Cesium spectral lamp), and these collide with ground-state Tl atoms, there is a high probability that the excitation is transferred to the Tl atoms. One important rule for collisional processes is the law of detailed balance. If an atom can get from state $\{1\}$ to state $\{2\}$ by a collision process, then the process $\{2\} \rightarrow \{1\}$ is also possible. The transfer cross-section can be computed from the following consideration. Let us assume that only energy states $\{1\}$ and $\{2\}$ can interact. Then the rate equation $$n_1 v \sigma_{12} = n_2 v \sigma_{21}$$ (3.6) must be fulfilled. In thermodynamic equilibrium, the atomic densities obey the Boltzmann distribution, $$\frac{n_2}{n_1} = \frac{g_2'}{g_1'} \exp \left( - \frac{\Delta E}{k_B T} \right)$$ (3.7) so that $$\frac{\sigma_{12}}{\sigma_{21}} = \frac{n_2}{n_1} = \frac{g_2'}{g_1'} \exp \left( - \frac{\Delta E}{k_B T} \right)$$ (3.8) Since the cross-section must be independent of whether there is thermodynamic equilibrium or not, Eq. (3.8) must be universally valid. However, note that the derivation was based on the approximation that the cross-sections $\sigma$ are velocity independent—we thus cannot expect Eq. (3.8) to be unconditionally valid. Quite generally, cross-sections tend to deviate from Eq. (3.8) for large energy differences $\Delta E$. We see that processes that convert excitation energy into kinetic energy are more probable than those that convert kinetic to excitation energy. ## 3.4 Collisions with electrons; ionization and recombination Collisions involving charged particles are important for discharge lamps and plasmas. Scattering of an electron by a singly charged ion, or electron-electron scattering approximately has the cross-section --- $$ \sigma \approx 2.9 \cdot 10^{-13} \frac{1}{T_e^2} $$ (3.9) where $\sigma$ is in cm$^2$, and $T_e$ is the electron temperature in electron volts (note the implicit assumption of a Maxwellian velocity distribution). The cross-section for excitation of atoms by collisions with electrons is, as discussed before, strongly dependent on the velocity, i.e. on the electron energy. The average value of the cross-section is typically $10^{-15}$–$10^{-17}$ cm$^2$, just to get a feeling for the order of magnitude. For optically allowed transitions it is slightly larger than for optically forbidden transitions. A good estimate for optically allowed transitions can be obtained from the Bethe–Born approximation $$ \sigma(E_e) = 4\pi a_0^2 \left( \frac{\text{Ry}}{\Delta E_{12}} \right)^2 f_{12} \left\{ \frac{u}{(u+1)^2} \ln[1.25(u+1)] + \frac{c1}{u+1} \right\}, \quad u = \frac{E_e - \Delta E_{12}}{\Delta E_{12}} $$ (3.10) where $E_e$ is the electron energy, $\Delta E_{12}$ the energy difference between the two states, $a_0$ the Bohr radius, $f_{12}$ the oscillator strength, and Ry the Rydberg constant. The parameter $c1$ is 0.3 for transitions from the ground state to the first excited state, 0.2 from the ground state to higher excited states, and 0.1 elsewhere. When the upper state of the atom is the continuum, then we consider the ionization process $\text{n} + \text{e}_- \rightarrow \text{n}^+ + \text{e}_- + \text{e}_-$. An estimate for the cross-section can be obtained in the same way as for excitation to some discrete state, i.e. using the Bethe–Born approximation; we just have to replace $f_{12}$ by the oscillator strength for the bound–free transition. ## 3.5 Particle diffusion Excitation may not only wander about in a vapour by movement of photons, but also by particle movement. Since photons are that much faster than atoms at thermal velocity, particle movement mainly plays a role when the excitation stays with an atom for a long time, or in practical terms, when the considered excited level is a metastable one. In a thin vapour and in laboratory-scale cell sizes, atoms can move in free flight, i.e. they follow straight paths encountering essentially no collisions before they reach the cell walls. Upon a collision with a cell wall, an atom loses—at least in nearly all cases—its excitation, converting it to thermal energy of the wall. The time between free-flight wall collisions in a typical experiment will be at least on the order of tens of microseconds, i.e. quite long as compared to the tens of nanoseconds lifetimes of the majority of atomic levels, and also quite long as compared to the flight time of a photon through the cell. However, for experiments that involve long-lived metastable atomic states, particle movement may play a significant role. In experiments involving such atomic states, atoms typically are not left in free flight, but a buffer gas is added in order to hinder the movement of the atoms, i.e. to keep the atoms off the walls. Most often a noble gas is used for that purpose since the noble gases have rather high lying energy levels which in collisions do not easily take over the excitation of other atomic species. Hence, the --- buffer gas atoms appear as hard balls and collisions with other atoms are mainly elastic. Upon collision merely a change in direction takes place, but with only seldom an energy exchange, which would mean quenching the excited atom. Addition of an inert buffer gas has changed the free flight of the atoms into a random walk process that can be described by a diffusion equation. For excited atoms, the mean time before a de-exciting bump into a wall has dramatically risen. Essentially, there are two processes that occur in this particle diffusion. On one hand, there are velocity changing collisions with buffer gas atoms, i.e. with atoms of a different kind. The excited atom behaves like a pinball, changing direction every time it collides with a buffer gas atom. The cross-sections for these collisions are typically of the order of $10^{-15}\text{ cm}^2$. The other possibility is resonance-exchange collisions between atoms of the same kind. In that type of collision, the excitation 'hops' from one atom to another atom of the same kind that crosses its flight path. Neither atom changes its velocity, but since the excitation changes to a different atom, the net effect is that of a velocity changing collision. Cross-sections for these types of collision are typically of the order of $10^{-12}\text{ cm}^2$. Usually, we have the boundary condition $n = 0$ at the cell wall, which means approximately that all excited atoms that hit the cell wall are quenched (see also Sec. 10.4 for a discussion of this boundary condition). There have been attempts to make cell walls reflecting for excited atoms by covering the walls with paraffin. However, this introduces impurities into the cell, which leads to foreign gas quenching. An excited state density $n = 0$ at the walls is thus usually a good assumption. Mathematically, the random walk is described by the diffusion equation $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = D\nabla^2n(\mathbf{r}, t) + E(\mathbf{r}, t), $$ (3.11) where $n(\mathbf{r}, t)$ is the density of excited atoms, and $E(\mathbf{r}, t)$ is the excitation term. The general solution to the diffusion equation has the form $$ n(\mathbf{r}, t) = \sum_j \alpha_j \psi_j(\mathbf{r}) \cdot \exp(-t/\tau_j) $$ (3.12) This equation is easily interpreted. The distribution of excited atoms in the cell can be thought of as being built up by a sum of ensembles of atoms that belong to different spatial diffusion modes $\psi_j$, which decay with different time constants $\tau_j$. A certain initial distribution at time zero can be broken into a sum of modes with amplitudes or weights $\alpha_j$: $$ n(\mathbf{r}, 0) = \sum_j \alpha_j \psi_j(\mathbf{r}) $$ (3.13) This expansion into modes is unique since the modes $\psi_j$ are orthogonal to each other. For more information about modal solutions, like how to find the expansion coefficients, $\alpha_j$, or how to picture a temporal development of an initial distribution, a thorough discussion can be found in Chapter 4, (from Eq. (4.12) onwards—even when you are familiar with the concept, don't miss Fig. 4.2). --- [FIGURE: FIG. 3.1. The finite cylinder geometry.] In the rest of this section we list for your convenience the specific solution for the practically important finite cylinder geometry, from which you get with one stroke of your pen also the ready-made solutions for the one-dimensional slab and infinite cylinder geometries. Further, we take a look at steady-state excitation, i.e. at the question what happens when the excitation being lost to the walls by diffusion is continuously replenished by an excitation source. The finite cylinder geometry can be visualized as a two-dimensional cylindrical cell, extending from $0$ to $R$ in the $r$-direction and from $-L/2$ to $L/2$ in the $z$-direction, see Fig. 3.1. For the boundary condition $n = 0$ at the cell walls, the solution of the diffusion equation for the density of the metastables in this geometry is $$ n(r, z, t) = \sum_{j_r=0}^{\infty} \sum_{j_z=0}^{\infty} \left[ \alpha_{j_r, j_z}^a J_0^B \left( \frac{b_{j_r}}{R} r \right) \cos \left( \frac{z}{L} \pi (2j_z + 1) \right) \exp(-t/\tau_{j_r, j_z}^a) + \right. \\ \left. + \alpha_{j_r, j_z}^b J_0^B \left( \frac{b_{j_r}}{R} r \right) \sin \left( \frac{z}{L} \pi (2j_z + 2) \right) \exp(-t/\tau_{j_r, j_z}^b) \right], $$ (3.14) where the factors $\alpha_{j_r, j_z}$ are the expansion coefficients of the initial distribution into the even and odd eigenmodes, $J_0^B(\dots) \cdot \cos(\dots)$ and $J_0^B(\dots) \cdot \sin(\dots)$, respectively. The parameters $b_{j_r}$ are the $j_r$th roots of the zero-order Bessel function of the first kind, $J_0^B$. The associated decay times are $$ \frac{1}{\tau_{j_r, j_z}^a} = D \left[ \left( \frac{b_{j_r}}{R} \right)^2 + \left( \frac{2j_z + 1}{L} \pi \right)^2 \right], \quad \text{and} \quad \frac{1}{\tau_{j_r, j_z}^b} = D \left[ \left( \frac{b_{j_r}}{R} \right)^2 + \left( \frac{2j_z + 2}{L} \pi \right)^2 \right]. $$ (3.15) For many computations, it is sufficient to approximate the wall-quenching by writing $$ n(r, z, t) \approx n(r, z, 0) \exp(-t/\tau_{00}), $$ (3.16) --- which means, only the lowest-order diffusion mode is considered. In that case, we write for the wall-quenched lifetime $\tau_{\text{wall}}$ $$ \frac{1}{\tau_{\text{wall}}} = \frac{1}{\tau_{00}} = D \left[ \left( \frac{2.405}{R} \right)^2 + \left( \frac{\pi}{L} \right)^2 \right]. $$ (3.17) We now proceed to show why Eq. (3.16) is often a good approximation. During a pumping phase, i.e. while the vapour is being excited, an excitation term $E(t)$ is present. If the pump phase is long enough, a steady state is achieved, so that any time dependence $\partial/\partial t$ vanishes and Eq. (3.11) becomes $$ D \nabla^2 n(r, z) = -E(r, z). $$ (3.18) As the modal expansion Eq. (3.14) forms a complete set of orthogonal modes (under the given boundary conditions), both $E$ and $n$ can be expanded as $$ \begin{aligned} E(r, z) &= \sum_{j_r=0}^{\infty} \sum_{j_z=0}^{\infty} \left --- HYPERFINE-STRUCTURE INTERMIXING [FIGURE: FIG. 3.2. Steady-state distribution of excited atoms for a uniform excitation in a finite cylinder with wall quenching. The steady-state distribution bears a strong resemblance to the ground mode alone.] way: expand the excitation into the eigenmodes of the diffusion equation, Eq. (3.18), compute the expansion coefficients of the steady state distribution $n(r, z)$ by multiplying each expansion coefficient with its associated lifetime, and sum over all modes with these new expansion coefficients. Since $\tau_{00}$ is the largest time constant, the lowest-order mode will have a larger 'weight' in the series Eq. (3.19), so that the steady-state distribution $n(r, z)$ will usually bear a strong resemblance to the lowest-order mode. Figure 3.2 shows the steady-state distribution of diffusing metastable atoms for a uniform excitation across the vapour cell, $E(r, z) = \text{const}.$ ## 3.6 Hyperfine-structure intermixing Hyperfine levels are distinct energy levels. When the hyperfine splitting, hfs, of the considered line is large (several GHz) and we have a narrow-bandwidth laser, we can selectively excite a single hyperfine sublevel. The hyperfine sublevels cannot radiatively interact among themselves. However, they are energetically quite close together ($\Delta E \ll k_B T$), so that they can interact by energy-changing collisions. If there are many such collisions, the atomic level will be in thermodynamic equilibrium, and the number of atoms in each sublevel is distributed according to the degeneracies of the sublevels. If there are no collisions, the atoms will remain in the substates into which they got by absorption or emission processes. We thus have to know the collisional intermixing cross-sections for the hyperfine levels. Reliable measurements have been made only for a few states of some selected atoms. Franz and Franz (1966) developed a theory of hfs intermixing that allows us to make rough estimates of the hfs-intermixing cross-sections. The motivation for the work of Franz was hyperfine pumping—a spectroscopic technique that uses polarized --- light to excite atoms in a magnetic field into only one Zeeman sublevel. We are usually interested in situations where there are no magnetic fields, so that the Zeeman levels are indistinguishable and only the hfs levels have different energies. As the more general theory is available (involving magnetic fields), we will report its most important considerations here and then specialize to our case (no magnetic field). We will only consider collisions with buffer gas atoms. It is well known that spin-exchange collisions between atoms of the same kind lead to the thermalization of the distribution. The cross-sections for these collision are in the order of the geometrical cross-section, but due to the low particle densities used in most experiments, they are rarely important. The theory of Franz is valid explicitly for alkali atoms. One might conclude that the qualitative results also hold for the doublet part of the term schemes of trivalent elements; however, this conclusion should be used with care. Consider the system of alkali and noble gas atoms in a vessel. A magnetic field is applied which is strong enough to make the Zeeman sublevels of the alkali atoms energetically distinct, but weak enough that $F$ and $m_F$ remain good quantum numbers. With each collision, the direction of $\mathbf{L}$ is randomized. (See Appendix A for the designation of quantum numbers and momentum vectors.) This last statement can be justified by the following line of reasoning. The directions of $\mathbf{S}$ and $\mathbf{I}$ cannot be randomized by collisions with noble gas atoms, as this would require magnetic interaction. This is confirmed by the measurements of relaxation rates of the ground-state hyperfine splitting (about $100 \text{ s}^{-1}$). According to the theory of Franz, at least *one* vector should be randomized (in direction), so it has to be $\mathbf{L}$. The question is now, which of the sum vectors ($\mathbf{J}$ or $\mathbf{F}$) is randomized by the randomization of $\mathbf{L}$. Consider the characteristic times of rubidium. The hyperfine period is about $10^{-10} \text{ s}$ (hfs splitting is a maximum of $10 \text{ GHz}$). The fine structure period is around $10^{-13} \text{ s}$, and the duration of a collision is on the order of $10^{-12} \text{ s}$. As the hfs period is comparatively large, it follows that $\mathbf{I}$ stands still during a collision—the hfs cannot follow the rapid movements. The fine structure, however, *can* follow, so that a randomization of $\mathbf{L}$ also leads to a randomization of $\mathbf{J}$, but not of $\mathbf{F}$. In sodium, where the fine structure splitting is very small, the fine structure period might become larger than the collision time. In this case, $\mathbf{L}$ is randomized (note that by randomizing a vector, we mean a randomization of the direction with respect to the direction of the magnetic field). From these considerations it is obvious that *all* s-states have practically no intermixing; there is no $\mathbf{L}$ that could be randomized, and $\mathbf{S}$ and $\mathbf{I}$ would require magnetic interaction. In conclusion, as a rule of thumb, we can state that at noble gas pressures above 5–10 Torr, p-levels are completely intermixed, whereas s-levels are not intermixed at all. ## 3.7 Fine-structure intermixing Collisions with noble gases can also transfer atoms from one fine-structure level to another. For the same reasons that applied to hfs-intermixing, the knowledge of the --- # 4 ## FORMULATION OF THE CLASSICAL PROBLEM There are three principal mathematical approaches to describing trapping: the Holstein equation, the method of combining the equation of radiative transfer with the rate equation for the excited state atoms, and the multiple-scattering representation. Ultimately, these three descriptions are equivalent—as they should be, since they describe the same physical process. Their different formulations stem from different views of the trapping problem. Each method has its specific advantages and disadvantages for the evaluation, hence it is very helpful to fully understand all three formulations. We will also examine two approximate formulations, the Milne equation and the escape factor method. These two descriptions are quite convenient since they are mathematically very simple, but both formulations are based on wrong physical assumptions and are thus **not** equivalent to the Holstein equation. However, for a limited range of parameters, deviations from reality are not too strong. The **Holstein equation** is a rate equation for the excited-state atoms. Excited-state atoms are lost by natural decay, and are gained when ground-state atoms absorb resonance photons that were emitted elsewhere in the vapour. To account for this reabsorption, we have to calculate the photon flux at the absorber site by considering all points of emission, i.e. by a spatial integration over the geometry. The Holstein equation is thus an integral equation. For a complete formulation we have to find the Kernel function, giving the probability that a photon emitted at a point $\mathbf{r}'$ is reabsorbed at point $\mathbf{r}$. This can be done either by inspection—if the geometry is simple enough—or by a formal procedure which we will discuss in Sec. 4.2. The fact that the Holstein equation is an integral equation also suggests several solution procedures (which have been devised for integral equations in general); these will be discussed in Chapter 5. Essentially, they are based on finding eigenmodes and trapping factors for the Holstein equation (similar to the modes of the diffusion equation). Actual problems are then treated by a kind of Fourier analysis. The **multiple-scattering representation** differs from the Holstein equation insofar as it computes the values $p_i$, which are the probabilities for photons to escape after $i$ absorption/reemission processes—no modes occur in this description. The description is thus more physically intuitive, but sometimes mathematically more cumbersome. The usual way to get the $p_i$ is a Monte Carlo simulation, but also semi-analytical methods are possible (all these approaches will be discussed in Chapter 6). Anyway, we obtain the same information as from the Holstein equation, but in a slightly different form. In Sec. 4.3, we will show that it is possible to transform the results obtained from the multiple-scattering representation to the results of the Holstein equation, and vice versa. 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- In contrast to the Holstein equation, the **equation of radiative transfer** is a rate equation for the photons. The radiation depends not only on the position, but also on the considered direction. Photons are lost through absorption by ground-state atoms, and gained when an excited-state atom decays to the ground state and emits a photon in the considered direction. For a full description of the physics, the equation of radiative transfer has to be combined with the rate equation for the excited-state atoms, which balances the population loss due to spontaneous emission with the population gain due to absorption of radiation. In contrast to the Holstein equation, we do not explicitly consider the emission at other points in the vapour—this information is already contained in the radiation intensity—so that there is no need for a Kernel function. We can show by formal integration and substitution that the method of combining transfer equation and rate equation is completely equivalent to the Holstein equation description, although the mathematical formulation is quite different (Sec. 4.4). The transfer equation is most often used for steady-state problems; various mathematical methods for this range of problems have been developed (Chapter 9). As mentioned above, we will also deal with two approximate formulations, namely the Milne equation and the escape factor method. These methods are approximate in the sense that the mathematical descriptions of the basic physical problem are already approximate. With the three methods above, only the numerical solutions of the mathematical descriptions are approximate. The **Milne equation** is a diffusion-type equation for the density of the excited atoms. It is historically the oldest formulation and we will also introduce this method first, in the next section, since the defects of the Milne approach justify the need for the exact formulations. The Milne formulation neglects the influence of the lineshape on radiation trapping, although the lineshape is of vital importance for the trapping process. The Milne equation is thus based on wrong physical assumptions. Nevertheless, it gives reasonably accurate results at low opacities, and is thus still in use today. Its solution is described in Chapter 8. The **escape factor**, described at the end of this chapter in Sec. 4.5, just gives the probability that a photon emitted at some point in the vapour cell escapes from the cell. It says nothing about the distribution of the excited-state atoms, and thus gives less information than the Holstein equation. We will show, however, how this method can be used for ‘quick-and-dirty’ estimates of trapping effects. ## 4.1 The Milne equation At first glance, the physical process of radiation trapping bears a strong resemblance to particle diffusion. A photon is emitted, flies in a straight line for a certain length, is absorbed, and is then reemitted in a direction that is statistically independent of the previous direction. It is thus very tempting to derive a diffusion-type equation for the distribution of the excited atoms—the Milne equation. --- [FIGURE: FIG. 4.1. Approximation of a lineshape by an 'equivalent' box-shaped line.] Let us assume for the moment that we can replace the lineshape by an 'equivalent' box-shaped line, see Fig. 4.1. In this simplified problem, the photon mean free path is $1/\bar{k}$, and the equation for the excited-state atoms reads (Milne 1926) $$ \nabla^2 \left[ n(\mathbf{r}, t) + \tau \frac{\partial n(\mathbf{r}, t)}{\partial t} \right] = 4\bar{k}^2\tau \frac{\partial n(\mathbf{r}, t)}{\partial t}, $$ (4.1) where $n$ is the excited-state density, $\mathbf{r}$ is the position in the cell, and $t$ is the time. This equation is quite similar to the diffusion equation; the additional term in the square brackets, $\tau \cdot \partial n(\mathbf{r}, t)/\partial t$, accounts for the time that the 'scattering' (i.e. absorption-reemission) process takes. In particle diffusion, the scattering process is instantaneous; in radiation trapping, it lasts in the mean one natural lifetime of the atoms. In practice, no box-shaped lines occur. For Doppler, Lorentz, Voigt, or other lineshapes, the mean free path strongly depends on the frequency of the photon. Furthermore, the photon frequency changes with each absorption-reemission process, so that the mean free path changes, too. We will show later that a frequency-averaged mean free path does not makes sense either (it is infinite). So since we have no mean free path, we cannot have a diffusion equation. The Milne theory is thus based on wrong physical assumptions. At low opacities, the error of replacing, e.g., a Doppler profile by a boxed-shaped line, is quite small, so that the solution of the Milne equation will be quite close to the true solution. Because of its mathematical simplicity, the Milne equation is still in use for the low-opacity regime. We will describe it in some detail in Chapter 8. One should always keep in mind, however, that it is a mathematical tool, which gives, more or less by chance, approximately correct results in a limited range of parameters, but it is not a correct formulation of the physical process of trapping. ## 4.2 The Holstein equation ### 4.2.1 Derivation of the Holstein equation In the 1920s and 1930s, many experiments with trapping in high-opacity vapours were performed, and it became clear that Milne's theory of trapping was not a good description of trapping at these high opacities. Several authors tried to save the theory by --- inventing more and more esoteric definitions of the equivalent opacity, but these were usually just appropriate for one special case, and broke down for other experimental situations. It was only in the late 1940's that Holstein (1947) did away with the concept that trapping had to be a diffusion equation, and started afresh to derive a correct, completely new description of trapping. This description is an integro-differential equation commonly called the Holstein equation. At the same time, Biberman (1947) derived this equation independently. Hence, the Holstein equation is also called the Holstein–Biberman or the Biberman–Holstein equation (mostly in the Russian literature). In this subsection, we derive the Holstein equation; the simplifying assumptions that are implicit in this derivation will be discussed in detail in Sec. 4.2.3. The formulation of the Holstein equation in various important geometries will be given in Sec. 4.2.2. For the derivation of the Holstein equation, we first consider the probability that a photon with a certain (normalized) frequency $x$ traverses a distance $\rho$. According to Beer's law, this transmission probability is $$ T(\rho, x) = \exp(-k(x)\rho). \qquad (4.2) $$ The next step is to determine the probability that an excited atom emits a photon of frequency $x$. In other words, what is the emission spectrum $\Psi(x)$ of excited-state atoms? If an excited atom experiences several collisions between absorption and reemission of the photon, the reemitted frequency will be completely independent of the frequency of the absorbed photon (complete frequency redistribution, CFR). The emission spectrum will thus be proportional to the absorption coefficient $$ \Psi(x) = C_x k(x) \quad \text{where} \quad C_x = \frac{1}{\int_{-\infty}^{\infty} k(x)\mathrm{d}x}. \qquad (4.3) $$ Since $\Psi(x)$ is a probability, it must be normalized with the factor $C_x$ so that $\int \Psi(x)\mathrm{d}x = 1$. We will see later that Eq. (4.3) is not only valid when the excited atoms suffer collisions (i.e. for collisionally broadened lines) but also for Doppler lines. We now average the transmission probability $T(\rho, x)$ over the emission spectrum of the atoms. We get the frequency-averaged probability of traversing a distance $\rho$, $$ T(\rho) = C_x \int k(x) \exp(-k(x)\rho)\mathrm{d}x. \qquad (4.4) $$ We call this quantity the transmission factor $T(\rho)$. In contrast to the frequency-averaged mean free path of the Milne method, the (frequency-averaged) transmission factor does exist. Since it is a probability, it has a value between 0 and 1.[^6] We now can compute the probability that a photon is reabsorbed at a distance $\rho$ from the point of emission. The probability that a photon can cover a distance $\rho$ without [^6]: In chemical physics, the 'equivalent width' of a line is defined as $W(\rho) = \int [1 - \exp(-k(x)\rho)]\mathrm{d}x$. It is related to $T$ by $T(\rho) = C_x \mathrm{d}W(\rho)/\mathrm{d}\rho$ (Armstrong 1983). --- being absorbed is $T(\rho)$; the probability that it covers a distance $\rho + \mathrm{d}\rho$ is $T(\rho + \mathrm{d}\rho)$. The probability that the photon is absorbed between $\rho$ and $\rho + \mathrm{d}\rho$ is thus $$ T(\rho) - T(\rho + \mathrm{d}\rho) = - \frac{T(\rho + \mathrm{d}\rho) - T(\rho)}{\mathrm{d}\rho}\mathrm{d}\rho = - \frac{\partial T(\rho)}{\partial \rho}\mathrm{d}\rho $$ (4.5) When we use the symbol $\dot{T}$ for the negative derivative of $T$, $$ \dot{T}(\rho) = - \frac{\partial T(\rho)}{\partial \rho}, $$ (4.6) the probability of reabsorption can be written as $\dot{T}\mathrm{d}\rho$. Inserting $T(\rho)$ from (4.4), we get for the probability of reabsorption $$ \dot{T}(\rho) \cdot \mathrm{d}\rho = \int [\mathrm{d}x C_x k(x)] \exp[-k(x)\rho] [k(x)\mathrm{d}\rho] \, . $$ (4.7) Each term in (4.7) has a physical meaning. The first square bracket is the probability that a photon is emitted with a frequency between $x$ and $x + \mathrm{d}x$. The exponential term is the probability that the photon of frequency $x$ traverses a distance $\rho$. The third square bracket term is the absorption probability of a photon of frequency $x$ in a volume element of width $\mathrm{d}\rho$—the differential form of Beer's law. The total probability of reabsorption is given by integrating the three contributing (frequency-dependent) probability terms over frequency. Equation (4.7) gives the probability that a photon is absorbed at a distance $\rho$ from the point of emission, i.e. in a spherical shell between $\rho$ and $\rho + \mathrm{d}\rho$. Hence, the probability that a photon is absorbed in a volume element defined by the spherical angle $\mathrm{d}\Omega$ and the spheres with distance $\rho$ and $\rho + \mathrm{d}\rho$ from the point of emission is $$ \frac{\mathrm{d}\Omega}{4\pi} \dot{T}(\rho)\mathrm{d}\rho $$ (4.8) The volume of this element is $\rho^2\mathrm{d}\rho\mathrm{d}\Omega$. For the probability of reabsorption in a unit volume element, we have to divide Eq. (4.8) by the volume of the element, and arrive at $$ \frac{1}{4\pi\rho^2} \dot{T}(\rho) $$ (4.9) The probability that a photon emitted at point $\mathbf{r}'$ is reabsorbed at $\mathbf{r}$ is thus given by $$ G(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi\rho^2} \dot{T}(\rho) = - \frac{1}{4\pi\rho^2} \frac{\partial T(\rho)}{\partial \rho}. $$ (4.10) We call this probability density the Kernel function $G(\mathbf{r}, \mathbf{r}')$. It depends only on the distance $\rho = |\mathbf{r} - \mathbf{r}'|$ between the two points and not on the absolute positions in the cell containing the vapour. Note that $G(\mathbf{r}, \mathbf{r}') = G(\mathbf{r}', \mathbf{r})$. --- After this tedious derivation of the Kernel function, the derivation of the actual Holstein equation is fairly simple. As mentioned in the introduction, the Holstein equation is a rate equation for the excited-state atoms. At a certain point $\mathbf{r}$ in the vapour cell, the density of excited atoms is decreased by natural decay. It is increased by the absorption of photons emitted by atoms that decay somewhere else in the vapour cell. This increase at $\mathbf{r}$ is obviously the rate of emission at $\mathbf{r}'$, times the probability that a photon emitted at $\mathbf{r}'$ is reabsorbed at $\mathbf{r}$, integrated over the whole cell. The emission rate is $n(\mathbf{r}', t)/\tau$, the reabsorption probability is, according to its definition, $G(\mathbf{r}, \mathbf{r}')$. The rate equation for the density of excited-state atoms $n(\mathbf{r})$ can then be written down as $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau} n(\mathbf{r}, t) + \frac{1}{\tau} \int_V n(\mathbf{r}', t) G(\mathbf{r}, \mathbf{r}') d\mathbf{r}', $$ (4.11) where the integral is taken over the volume of the cell. Equation (4.11) is the Holstein equation. In contrast to the (incorrect) Milne equation, it is an integral equation, not a differential equation. > $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau} n(\mathbf{r}, t) + \frac{1}{\tau} \int_V n(\mathbf{r}', t) G(\mathbf{r}, \mathbf{r}') d\mathbf{r}', $$ > > The Holstein equation Equations of the form (4.11) are called Fredholm integral equations of the second kind and have been widely studied in mathematics; they are *linear* equations for $n(\mathbf{r}, t)$. These types of equations have some nice properties that make life much easier. The most important of these properties is that the general solutions are of the form $$ n(\mathbf{r}, t) = \sum_j \alpha_j \psi_j(\mathbf{r}) \exp\left[-t/(g_j \tau)\right]. $$ (4.12) Here, the $\psi_j$ are the (normalized) eigenmodes, $(1 - 1/g_j) = \lambda_j$ the corresponding eigenvalues and $\alpha_j$ the expansion coefficients of the initial distribution into the eigenmodes. For the interpretation of Eq. (4.12), let us assume for the moment that we know the eigenmodes and the eigenvalues—we will devote Chapter 5 to describing the methods to compute these results. At the time $t = 0$ we have a certain initial distribution of excited atoms $n(\mathbf{r}, 0)$, produced by some excitation mechanism, e.g. by light from a spectral lamp or from a laser. This initial distribution can be expanded into the eigenmodes $$ n(\mathbf{r}, 0) = \sum_j \alpha_j \psi_j(\mathbf{r}). $$ (4.13) This expansion of the initial distribution into eigenmodes is analogous to the well-known expansion of a function into a Fourier series, we just use different basis --- [FIGURE: FIG. 4.2. Temporal evolution of the eigenmodes and of the total distribution of excited atoms. The geometry is an infinite slab, see Fig. 4.3. (The traces are calculated for an opacity $k_0L = 4$ for a Lorentzian lineshape.)] functions—the eigenmodes of the Holstein equation instead of the sine or cosine functions of the Fourier series. We note in passing that the eigenmodes are orthonormal, so that the expansion is unique and the expansion coefficients can be computed from $$ \alpha_j = \int_V n(\mathbf{r}, 0)\psi_j(\mathbf{r})\mathrm{d}\mathbf{r} $$ (4.14) The expansion coefficients $\alpha_j$ can also be thought of as the 'amplitudes' of the eigenmodes at time 0. According to Eq. (4.12), the amplitude of each eigenmode decays with its own time constant $g_j\tau$. The shape of each eigenmode stays the same for all times. The shape of the **total** distribution of excited atoms, $n(\mathbf{r}, t)$, however, usually changes with time. At any time, the total distribution of excited atoms is the sum of the eigenmodes weighted with their amplitudes at that time, $\alpha_j \exp(-t/g_j\tau)$. Since the ratios of the amplitudes change with time, the shape of the total distribution must also change. Figure 4.2 shows a typical example of such a temporal evolution of the eigenmodes and the resulting total distribution. The shape of $n(\mathbf{r}, t)$ only stays the same when the initial distribution consists of just one eigenmode. When the initial distribution is identical to the lowest-order eigenmode, then Eq. (4.12) becomes --- $$ n(\mathbf{r}, t) = \alpha_0 \psi_0(\mathbf{r}) \exp(-t/g_0\tau) $$ (4.15) The lowest-order eigenmode has positive values for all $\mathbf{r}$. All higher-order eigenmodes can also take negative values, see Fig. 4.2. Since there are no negative atomic densities, higher-order modes cannot exist by themselves. In the initial distribution, the $\alpha_j$ must have such values that $n(\mathbf{r}, 0) = \sum \alpha_j \psi_j(\mathbf{r})$ is positive everywhere—we must have a physically reasonable initial distribution. The lowest-order mode decays slower than the higher-order modes, so that the 'negative' contributions to the particle density die out faster than the 'positive'. Thus, if the initial distribution does not have negative values of $n$, then negative $n$ are impossible for all times. This situation is somewhat similar to the modes of the solution of the diffusion equation, but is in strong contrast to the modes of a laser. In a laser, each mode can exist by itself, and we can even select a certain mode by imposing certain boundary conditions, e.g., by placing an appropriately shaped aperture in the cavity. Laser modes are solutions to equations for field strength, and field strength can take both positive and negative values. In trapping, however, we deal with particle densities, where only positive values make sense. The modes of the spatial distribution of excited atoms have no physical meaning, they are mathematical entities. In order to find the physical meaning of the *eigenvalues*, let us consider a photon that is created when an excited atom belonging to the $j$th eigenmode of the initial distribution decays—we say that such a photon belongs to the $j$th eigenmode. According to Eq. (4.12), the excitation in the $j$th mode decays with a time dependence of $\exp(-t/g_j\tau)$. The expectation value with respect to $t$ of this exponential function is $g_j \cdot \tau$. Hence, a photon belonging to the $j$th mode spends an average time $g_j \cdot \tau$ in the vapour before it escapes. Since $\tau$ is the mean atomic lifetime, $g_j$ is the mean number of absorption (or emission) processes a photon in the $j$th mode suffers. The $g_j$ are thus also known as trapping factors; they are always larger than (or equal to) one. Note that some authors denote as $g$ what in our notation is $1/g$. Henceforth we will speak of 'the' eigenmode and 'the' trapping factor $g$ when we imply that only the lowest-order mode exists. As mentioned above, higher-order modes cannot exist by themselves. ### 4.2.2 Idealized geometries For actual computations, we have to specify the geometry of the problem. The term 'geometry' is here not only used to describe the shape of the experimental vessel, but also includes the spatial distribution of the excitation. Usually, one makes some, hopefully small, idealizations to reduce the complexity of the problem. There are mainly three geometries in use to model experimental situations. Not surprisingly, these three most often used geometries are the simplest possible—all three are one-dimensional, i.e. only a variation with one spatial coordinate is allowed. (i) **The infinite slab.** The slab is an excellent model for many experimental situations. Due to its relative simplicity, it is furthermore important for investigating basic effects of trap- --- THE HOLSTEIN EQUATION [FIGURE: FIG. 4.3. The geometry of the infinite slab. A cylinder with reflecting walls is equivalent to the infinite slab.] ping. An idealized, plane-parallel slab extends infinitely in the $x$ and $y$ directions, and from $-L/2$ to $L/2$ in the $z$-direction, see Fig. 4.3. It is a one-dimensional geometry, i.e., it is assumed that there is only variation in the $z$-direction. Mathematically, the slab model requires that $\partial/\partial x = \partial/\partial y = 0$. Physically, this means not only that the vessel has to resemble a slab, but also that the initial excitation has to be uniform in the $x$ and $y$ directions. Thin, large slabs are a practical realization. More importantly, cylindrical vessels with arbitrarily shaped cross-sections and perfectly reflecting side walls are equivalent to a slab. This can be shown in the following way: A photon in such a mirrored cell cannot escape through the side walls, but only through the top or bottom. For the trapping process, it is sufficient to monitor the $z$-coordinate of the photon. The $z$-component of the photon velocity remains unchanged by a reflection at a mirror so that the $z$-coordinate of the point of reabsorption is the same as in the slab case, see Fig. 4.3. The mirror just folds the photon path. For the derivation of the Kernel function in a slab, we note that the distance $\rho$ between the points of absorption and reemission is $\rho = |z - z'|/\mu$ where $\mu = |\cos(\vartheta)|$ and $\vartheta$ is the angle between the $z$-axis and the direction of the photon. Furthermore, $\mathrm{d}\mathbf{r} = \rho^2 \sin(\vartheta)\mathrm{d}\vartheta\mathrm{d}\varphi\mathrm{d}z/\mu$, where $\varphi$ is the angle between the $x$-axis and the photon direction. We insert these relations into Eq. (4.9) and (4.10), and integrate over $\varphi$ to get an equation for the Kernel function (Biberman 1947). As expected from the argumentation above, this Kernel function has the form $G(|z - z'|)$, i.e. it depends only on the difference in the $z$-coordinates of the points of absorption and reemission. The explicit equation is given in Insert 4.1. Series representations that might be useful for computer evaluation are given by Hummer (1981, 1982). We also note that in slab geometries, it is often more --- convenient to think in ‘optical depth’ coordinates, and not in physical (spatial) coordinates. This is mainly an advantage when the distribution of absorbers is inhomogeneous, and will be discussed in Chapter 10 in more detail. **(ii) The infinite cylinder** A cylinder is obviously the most important geometry in situations where a laser is used to excite the atoms. The pencil beam coming out of the laser usually dictates a cylindrically symmetric experiment. In most cases, the experimental vessel will be long and thin, and we shine in the laser along the axis of the vessel. It is then obvious to make the idealization that the cylinder is infinitely long—in practice it is most often sufficient that the length of the cylinder is larger than its diameter. The accuracy of such a simplification has been numerically justified; Avery *et al.* (1969) have shown that in steady-state problems, the radial distribution of excited atoms in a cylinder with a height-to-diameter ratio exceeding 1 is practically identical to the radial distribution in a cylinder of infinite height. Furthermore, a cylinder of finite height with reflecting top and bottom is equivalent to a cylinder of infinite height. This can be proven the same way as above for the equivalence of slab and cylinder with reflecting side walls. For the infinite cylinder model to be valid, we not only have to make these geometrical considerations, but also to demand that the initial excitation is uniform along the axis of the cylinder. In other words, when we have a quite long cell, send in a laser beam along the cell axis, and the laser radiation is absorbed appreciably, we have a **finite** cylinder. The infinite cylinder model is only valid if, e.g., the laser is detuned from the absorption line centre of the atoms in the cell so that the laser is not attenuated appreciably on its way through the vapour. Mathematically, the ideal (infinite) cylinder geometry extends infinitely in the $z$-direction, from $0$ to $R$ in the $r$-direction and is independent of the azimuthal angle $\varphi$. The initial excitation fulfills the conditions $\partial/\partial z = 0$ and $\partial/\partial \varphi = 0$. For the derivation of the Kernel function, we note that in a cylinder, $\rho$ becomes $(z'^2 + r^2 + r'^2 - 2rr' \cos \varphi)^{1/2}$ when we set $z = 0$. Inserting into Eq. (4.10), we get the Kernel function shown in Insert 4.1 (Golubovskii and Lyagushchenko 1976). Note that in the cylinder and sphere geometries one has to pay attention not to confuse the three-component spatial vector $\mathbf{r}$ with the spatial coordinate $r$, which is one component of $\mathbf{r}$. **(iii) The sphere** In plasma research and in astrophysics, the most important problems are spherically symmetric. Most fusion plasma geometries, and of course all stars, are spherical. On the other hand, spheres are of little interest in chemical physics. The sphere extends from $0$ to $R$ in the $r$-direction. Again, the geometry is one-dimensional, $\partial/\partial \vartheta = \partial/\partial \varphi = 0$, where $\vartheta$ and $\varphi$ are the coordinates in a spherical coordinate system. For the derivation of the Kernel function, we assume that the point of emission lies at $(r', 0, \vartheta')$, and the point of absorption at $(r, 0, 0)$. The --- THE HOLSTEIN EQUATION distance between the two points is $\rho^2 = r^2 + r'^2 - 2rr' \cos(\vartheta')$. Inserting into Eq. (4.10), we get the Kernel function given in Insert 4.1 (Cuperman *et al.* 1963). *** **Insert 4.1 Kernel functions in simple geometries** In most trapping computations, we deal with one of the three one-dimensional geometries: the infinite slab, the infinite cylinder, and the sphere. **Note** that the geometries 'slab', 'cylinder' and 'sphere' do not just describe the shape of the experimental vessel, but characterize the symmetry of the problem. For the **slab**, the Kernel function (Biberman 1947) is $$ G(|z - z'|) = \frac{C_x}{2} \int_1^\infty \int_{-\infty}^\infty \frac{k^2(x)}{u} \exp[-k(x)|z - z'|u] \mathrm{d}x \mathrm{d}u, $$ (4.16) where $u = 1/|\cos(\vartheta)|$ and $\vartheta$ is the angle between the $z$-axis and the direction of the photon. For the **cylinder**, the Kernel function (Golubovskii and Lyagushchenko 1976) is $$ G(r, r') = \frac{C_x}{4\pi} \int_0^{2\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{k^2(x)}{z'^2 + r^2 + r'^2 - 2rr' \cos(\varphi)} \exp\left[-k(x)\sqrt{z'^2 + r^2 + r'^2 - 2rr' \cos(\varphi)}\right] \mathrm{d}x \mathrm{d}z' \mathrm{d}\varphi. $$ (4.17) For the **sphere**, the Kernel function (Cuperman *et al.* 1963) is $$ G(r, r') = \frac{C_x}{2} \int_{-\infty}^\infty \int_0^\pi \frac{k^2(x)}{r^2 + r'^2 - 2rr' \cos(\vartheta)} \exp\left[-k(x)\sqrt{r^2 + r'^2 - 2rr' \cos(\vartheta)}\right] \sin(\vartheta) \mathrm{d}\vartheta \mathrm{d}x. $$ (4.18) *** ### 4.2.3 *Simplifying assumptions in the derivation of the Holstein equation* We have already discussed these assumptions briefly in the introduction to Part II. Now, we give a more detailed description, and show where they are relevant in the derivation of the Holstein equation. **Assumption (i)** *Only two levels in the atomic structure are relevant for the trapping process.* This assumption is necessary so that we can keep our simple physical picture that a photon is absorbed and reemitted on a single transition. If more levels were involved, the wavelength of the photon could change appreciably in an absorption–reemission process. --- [FIGURE: FIG. 4.4. Energy level diagram of Na.] The two levels involved are usually the ground state and a resonant state, e.g., the 3s and the 3p states in sodium (Na, see Fig. 4.4). All the higher-lying levels cannot interact with the radiation, and the 3p atoms can decay to no other state than the 3s ground state. Such a situation occurs quite often in situations of interest for spectroscopy; for a long time, experiments with the resonance lines of alkali atoms (a typical two-level situation) accounted for the majority of spectroscopic experiments. **Assumption (ii)** *The spatial distribution of lower-state atoms is uniform.* We need this assumption for the definition of the transmission factor $T(\rho, x)$, where we state that the opacity for a photon of frequency $x$ along a distance $\rho$ is $k(x)\rho$. If the density of the absorbers were not uniform, the opacity would be the integral of the absorption coefficient along the photon path, $\int k(\mathbf{r}, x)d\rho$, leading to enormous complications in the derivation of the Kernel function. When the lower state is the ground state, this simply means that the vapour density is uniform throughout the cell. This is always fulfilled in the situations of chemical physics. There we usually have some vapour cell that has a more or less uniform temperature. In plasma physics and in astrophysics, on the other hand, this condition is frequently violated—for example when we produce a plasma by evaporating some target with a high-intensity laser beam. The situation is somewhat different when the absorbing, lower state is itself an excited state, e.g., a metastable level. Metastables can move through the cell and are quenched when they collide with a cell wall. The metastable density will thus be very low near the cell walls and higher near the cell centre (see also Chapter 3). In such a case, the distribution of the absorbing lower-state atoms is markedly non-uniform, and the Holstein equation is not valid. Similarly, the lineshape (i.e. the broadening mechanism) must be uniform throughout the vapour. This is fulfilled well in vapour cells, but can be violated in exploding plasmas, see Sec. 11.8. --- ### Assumption (iii) *The density of ground-state atoms is much larger than the density of upper-state atoms.* When this is not true, the density of the absorbers depends on the density of the excited-state atoms, and the whole problem becomes non-linear. When, on the other hand, the ground state density greatly exceeds the excited state density, the ground state density is not noticeably increased by decaying excited atoms. This condition is equivalent to the *absence of stimulated emission*. As discussed in Chapter 2, all transitions in the visible have saturation flux densities on the order of $10 \text{ W/cm}^2$. This value is true regardless of whether we have very weak or very strong transitions. However, in a vapour with trapping, the saturation intensity is decreased roughly by the average number of reabsorptions in the vapour. For high-opacity vapours, this means that the saturation flux density can decrease to a few $\text{mW/cm}^2$. Physically, this is easily understood. The saturation intensity is reached when the spontaneous emission rate becomes equal to the stimulated emission rate. Trapping does not influence the stimulated emission rate but decreases the 'effective' spontaneous emission rate—the 'effective' upper state lifetime is increased. Thus, the saturation threshold is decreased. For checking the validity of the assumption, we need the average number of reabsorptions, which we can get only from a solution of the Holstein equation. We thus have to make an *a posteriori* consistency check of our computations. We know the average number of reabsorptions after we have computed the solution of the Holstein equation. We then check whether the excitation intensity is much lower than the (untrapped) saturation intensity divided by the average number of reabsorptions. Only if this is the case do we have computed meaningful results. In many cases, however, one can make a rough order-of-magnitude guess in order to check the validity of the assumptions before going through all the computations. If two excited atoms collide, and both are quenched (i.e., the excitation energy of both is converted to kinetic energy), then we also have a non-linear process. The decay rate due to this process is proportional to $n^2(\mathbf{r}, t)$. However, these processes are usually negligible. The quenching cross-section by excited-state atoms is usually smaller than the quenching cross-section by ground-state atoms for energetic reasons. Furthermore, we require the density of excited-state atoms to be much smaller than the density of ground-state atoms anyway, so that quenching by excited atoms can be neglected against quenching by ground-state atoms. Similarly, other collisional processes where two excited-state atoms are involved in a reaction that ends up with both atoms no longer in the exited state (like energy pooling or Penning ionization) must be negligible. ### Assumption (iv) *All photons that arrive at the cell walls leave the vapour.* In the derivation of the Holstein equation, we have implicitly assumed that a photon can get from the point of emission $\mathbf{r}'$ to the point of absorption $\mathbf{r}$ only on a straight flight path. It may not take a detour via a mirror. Thus, we demand that the surface of the vapour cell is non-reflecting, i.e. either completely absorbing or completely transparent. --- This is no contradiction to the statement that a cylindrical vapour cell with completely mirrored side walls is optically equivalent to a plane-parallel slab. In this case, the mirrors just fold the photon flight path in a direction where no changes occur due to the one-dimensional geometry—we still require that the top and bottom of the cell are transparent. ### Assumption (v) *The flight time of the photons is negligible compared with the natural lifetime of the excited atoms.* When writing the Holstein equation, Eq. (4.11), as a rate equation, we assumed that the transfer of excitation from point $\mathbf{r}'$ to point $\mathbf{r}$ happens instantaneously. Fortunately, this can always be assumed in the situations of chemical physics. The shortest natural lifetimes are on the order of a few nanoseconds, and typical lifetimes are on the order of a few ten to a few hundred nanoseconds. The important first resonance states of alkali atoms all have about 30 ns lifetime. If the typical dimension, $L$, of a vapour cell is 30 cm, it takes a photon only one nanosecond to fly through the cell end to end—much less than an atomic lifetime. Mathematically, this condition can be formulated as $L/c \ll \tau$. In astrophysics, the 'vapour cells', i.e. the stellar atmospheres, are much larger so that there the flight time can play an important role (Kunasz 1983). Also in plasmas, kinetic processes may be so fast that properties may change during flight time so that the flight time must be taken into account (Bond 1983a). ### Assumption (vi) *The positions of the atoms do not change between absorption and reemission.* The Holstein equation implies that a transfer of excitation from point $\mathbf{r}'$ to point $\mathbf{r}$ can happen only by radiative transfer. Since the excited atoms migrate through the vapour cell by thermal movement, this is also a transfer of excitation. We can certainly neglect this movement if the distance the atoms cover during one natural lifetime is much smaller than both the cell size and the typical length between absorption and reemission, $1/k_0$. During a natural lifetime of 30 ns an atom moves about 30 $\mu$m in free flight under typical laboratory conditions; it moves much less if it suffers direction-changing collisions. This is certainly smaller than any practical cell size. Atoms with very large excited state lifetimes, i.e. metastable atoms, might cover large distances during one lifetime, but their oscillator strength is so small that usually no trapping will occur anyway. Particle movement is especially significant for the treatment of wall quenching. Chung (1987) gave a very simple estimate for the importance of wall quenching. We first compute the distance a photon can cover during one 'effective' lifetime, which is the upper-state lifetime times the average number of reabsorptions. We then state that one-sixth of the atoms that are in a layer of this distance from the wall will be quenched, since the photons can move in six directions $(\pm x, \pm y, \pm z)$, only one of which will lead to quenching. When the number of wall-quenched atoms is much smaller than the total number of excited-state atoms in the vessel, then we can neglect wall quenching. More quantitative criteria on when to neglect particle diffusion will appear in Sec. 10.4. --- **Assumption (vii)** *There is complete frequency redistribution (CFR) in the laboratory rest frame.* In the derivation of the Holstein equation, we assumed that the emission spectrum of an atom is equal to its absorption spectrum. This implies that the emitted frequency is completely independent of the frequency of the absorbed photon. This is a very important point in the Holstein theory, and it is this assumption that distinguishes the Holstein theory from the (wrong) Milne theory. Still, the CFR assumption is only valid when the lineshape is caused by certain broadening mechanisms. It is certainly valid when the line is collisionally broadened, and it is also a good assumption when the line is Doppler-broadened. However, CFR is not valid when the line is dominantly naturally broadened. When we have only natural broadening, then, due to energy conservation, the absorbed and reemitted frequencies are the same (see also Sec. 11.1). We can neglect natural broadening when a) collisional broadening is larger than natural broadening. This means that the atom suffers at least one collision during one natural lifetime. b) Doppler broadening is larger than natural broadening. This requires that the contribution of the Doppler broadening at the frequency $x_{\text{esc}}$ where a photon can escape easily, i.e. where $k(x_{\text{esc}})L = 1$, must be larger than the contribution of the natural broadening. When either of these two conditions is fulfilled, CFR is valid. A more detailed discussion and formulas that define the conditions a) and b) will be given in Chapter 11. If we have a hyperfine-split line, then CFR also means that the photon does not ‘remember’ in which hyperfine transition is was absorbed. This implies that the hfs-components are completely intermixed as was discussed in Chapter 3. **Assumption (viii)** *The reemission of the photons is isotropic.* We used this assumption in writing down Eq. (4.8). It implies that the direction of emission is independent of the direction of the absorbed photon. Since we already assumed that any frequency coherence is destroyed between absorption and reemission, it is clear that also any polarization coherence is destroyed. **Assumption (ix)** *No external radiation is incident for $t > 0$.* The Holstein equation, (4.11), describes the decay of an initial distribution of excited atoms. This is also the case of interest in most problems in chemical physics. In many other experiments, one is interested in the steady-state distribution that arises in a vessel under continuous excitation, e.g. with a laser. In that case, the Holstein equation becomes $$ E(\mathbf{r}) = \frac{1}{\tau}n(\mathbf{r}) - \frac{1}{\tau} \int_V n(\mathbf{r}')G(\mathbf{r}, \mathbf{r}')\text{d}\mathbf{r}', \qquad (4.19) $$ where $E(\mathbf{r})$ is the number of atoms transferred per second into the excited state by external excitation (light incident from **outside** the cell). We will see in Chapter 7 how --- we can easily compute the solutions to Eq. (4.19) once we know the solutions to the time-dependent Holstein equation, Eq. (4.11). When the excitation term is time-dependent, one can calculate the convolution between the solution of the time-dependent Holstein equation and the excitation term. Using standard Laplace- transform techniques, this is a trivial problem. This simple approach is possible because the whole problem is linear under our assumptions. Apart from these assumptions, we also require that the absorption and emission of photons can be described by the equation of radiative transfer. Strictly speaking, our semi-classical picture of photons 'hopping' from atom to atom is only an approximation that can become problematic when e.g. the 'length' of the photon (which is determined from quantum-mechanical considerations) becomes larger than the distance between the atoms. The validity of the semi-classical picture is equivalent to the validity of the equation of radiative transfer. Some conditions for this are given in Appendix G. For our purposes, we always assume that these conditions are fulfilled. ## 4.3 The multiple-scattering representation The output of the Holstein equation are eigenmodes and eigenvalues. These can be combined to yield physically reasonable solutions for arbitrary initial distributions of excited-state atoms. However, the eigenmodes by themselves are not physically reasonable. The multiple-scattering (MS) representation is closer to physical reality. In the MS approach one computes the probabilities that a photon escapes after exactly $i$ absorption–reemission processes, henceforth just called the $p_i$. The probabilities $p_i$ may, e.g., be obtained from a Monte Carlo simulation, see Chapter 6. A typical result of such a computation is shown in Fig. 4.5. For the time being we consider the escape probabilities as given and look how this representation of the classical problem connects to the Holstein formulation. From the $p_i$ we can compute the average number of reabsorptions. Similarly, we can compute the density of excited-state atoms that are created by photons that have suffered $i$ absorptions/reemissions. All these quantities have a real physical interpretation. In many cases, we are not interested in the $p_i$, but in the temporal evolution of the number of excited atoms or of the intensity of the light that escapes from the vapour. Let us first compute the emergent intensity. For photons that escape directly after the first emission, the intensity decays with time according to an exponential law, $$Y_1(t) = \frac{1}{\tau} \exp(-t/\tau).$$ (4.20) The intensity is normalized such that $$\int_0^\infty Y(t)\mathrm{d}t = 1.$$ (4.21) For a photon that undergoes one reabsorption (i.e. two emissions), the probability that the first emission happens at $t_1$ is $Y_1(t_1)$. Since we neglect the flight time of the --- [FIGURE: Bar chart showing Probability of being reabsorbed i times vs Number of reabsorptions, i] FIG. 4.5. Result of a multiple-scattering computation: probabilities $p_i$ that a photon escapes after exactly $i$ absorption–reemission processes. (The geometry is a slab and the lineshape is Doppler, with opacity $k_0L = 5$.) photon, $t_1$ is also the time of reabsorption. The probability that the reemission happens at time $t$ *after* the reabsorption process is also $Y_1(t)$, since the emission processes are statistically independent. The probability that the second emission process happens at time $t_2$ is thus $$ Y_2(t_2) = \int_0^{t_2} Y_1(t_2 - t_1)Y_1(t_1)\mathrm{d}t_1, $$ (4.22) which is a convolution of the two time functions. The temporal intensity of the photons that escape after $i$ emission processes is thus the convolution of $i$ exponential time functions. In the Laplace transform domain, the convolutions become multiplications (Wozencraft and Jacobs 1965), $$ \mathcal{L}[Y_i(t)] = \{\mathcal{L}[Y_1(t)]\}^i, $$ (4.23) where $\mathcal{L}$ denotes the Laplace transform. $Y_i(t)$ is the time behaviour of the photons that escape after exactly $i$ absorption–reemission processes. After an inverse Laplace transform, we find that $$ Y_i(t) = \frac{1}{\tau} \left( \frac{t}{\tau} \right)^{i-1} \frac{\exp(-t/\tau)}{(i-1)!}. $$ (4.24) Hence, the total output signal is --- FORMULATION OF THE CLASSICAL PROBLEM [FIGURE: FIG. 4.6. The contributions of the photons that escape after i absorption–reemission processes, Y_i(t), sum up to give the total emergent intensity, Y(t).] $$ Y(t) = \sum_i p_i Y_i(t) = \frac{1}{\tau} \sum_i p_i \left( \frac{t}{\tau} \right)^{i-1} \frac{\exp(-t/\tau)}{(i - 1)!}, \qquad (4.25) $$ a result that contains only information from the multiple-scattering representation (Wiorkowski and Hartmann 1985, Braun *et al.* 1985). Figure 4.6 shows how the $Y_i(t)$ add up to the total $Y(t)$. To compute the temporal behaviour of the total number of excited atoms in the vapour cell, $$ n^\Sigma(t) = \int_V n(\mathbf{r}, t)\mathrm{d}\mathbf{r}, \qquad (4.26) $$ we need a relation between $n^\Sigma(t)$ and $Y(t)$. Every photon that leaves the vapour cell means that the number of excited atoms is decreased by one. This fact gives us the following proportionality between the emergent light intensity and the decaying number of excited atoms, $$ -\frac{\mathrm{d}n^\Sigma(t)}{\mathrm{d}t} \propto Y(t). \qquad (4.27) $$ After integration (with the 'boundary condition' $n^\Sigma(\infty) = 0$ ) we arrive at $$ n^\Sigma(t) = n^\Sigma(0) \left[ 1 - \int_0^t Y(u)\mathrm{d}u \middle/ \int_0^\infty Y(u)\mathrm{d}u \right]. \qquad (4.28) $$ --- The Holstein equation and the MS computation describe the same physical process; they are just different approaches to the same problem. Yet their solutions are given in different terms—modes of excited atoms and decay constants for each mode in the Holstein approach, and the escape probabilities $p_i$ in the MS approach. We would like to find an interrelation between the solutions of the two formulations. This interrelation can be found by computing the total emergent intensity, $Y(t)$, with both approaches. Equation (4.25) already gives the result for the MS approach. For the Holstein equation, we represent $n(\mathbf{r}, t)$ by a modal expansion, calculate $Y(t)$ with the help of Eq. (4.27), and normalize so that $\int Y(t)\mathrm{d}t = 1$. We get $$ Y(t) = \frac{1}{\sum_j \hat{\alpha}_j g_j \tau} \sum_j \hat{\alpha}_j \exp\left(-\frac{t}{g_j \tau}\right), \quad \text{with} \quad \hat{\alpha}_j = \frac{\alpha_j}{g_j} \int_V \psi_j(\mathbf{r})\mathrm{d}\mathbf{r}. $$ (4.29) Now we can set this result equal to the result for $Y(t)$ from the MS computation, Eq. (4.25), $$ \sum_i p_i \left(\frac{t}{\tau}\right)^{i-1} \frac{\exp\left(-\frac{t}{\tau}\right)}{(i-1)!} = \frac{1}{\sum_j \hat{\alpha}_j g_j} \sum_j \hat{\alpha}_j \exp\left(-\frac{t}{g_j \tau}\right). $$ (4.30) We now multiply both sides by $\exp(t/\tau)$ and expand the exponential terms on the right-hand side into a Maclaurin series. By comparing equal powers of $t$, we arrive at $$ p_i = \frac{1}{\sum_j \hat{\alpha}_j g_j} \sum_j \hat{\alpha}_j \left(1 - \frac{1}{g_j}\right)^{i-1}. $$ (4.31) The left-hand side is the MS output, and on the right-hand side are the solutions of the Holstein equation. Equation (4.31) thus gives the sought connection between the MS and Holstein computations (Molisch 1990), (Falecki *et al.* 1991), (Molisch *et al.* 1992a). This connection is extremely important, because it permits a comparison between results achieved with two completely different methods. This result also forms the basis for possible efficient combinations between MS computation and solutions of the Holstein equation (see also Chapter 14). ## 4.4 The equation of radiative transfer Another possibility to describe radiation trapping is the equation of radiative transfer coupled with the rate equation of the excited-state atoms. This approach is especially common in the astrophysical literature (see, e.g., Cannon (1985) and references therein). Theoretically, it is completely equivalent to the Holstein equation—we will see below how the transfer and rate equations can be combined to give the Holstein equation. However, the numerical methods of evaluation are quite different, so that both methods have --- their specific advantages and disadvantages. Without going into details, it seems that for the time-decay problems without excitation terms common in atomic and chemical physics, the Holstein equation is more suitable. For steady-state problems, the transfer-plus rate-equation approach is more suitable than the Holstein equation, since it is more easily generalized to non-linear, multi-level, and inhomogeneous problems. The approach treated in this section is thus often used for trapping in plasmas and in stellar atmospheres. The equation of radiative transfer is the rate equation for photons. We consider the intensity $I(\mathbf{r}, \mathbf{\Omega}, \nu, t)$ of light with frequency $\nu$ travelling in a certain direction $\mathbf{\Omega}$, at a certain place $\mathbf{r}$ at time $t$. The intensity $I$ is reduced by absorption through ground-state atoms, while it is increased by spontaneous and stimulated emission. The equation of radiative transfer is then (Cannon 1985, p. 7) $$ (\mathbf{\Omega} \cdot \nabla)I(\mathbf{r}, \mathbf{\Omega}, \nu, t) = -k(\nu) \left[ I(\mathbf{r}, \mathbf{\Omega}, \nu, t) - S(\mathbf{r}, t) \right], $$ (4.32) where $S(\mathbf{r}, t)$ is the 'source function' of astrophysics. The term $k(\nu)S(\mathbf{r}, t)$ is the number of photons of frequency $\nu$ emitted at the point $\mathbf{r}$ in the direction $\mathbf{\Omega}$. Under the assumptions of Sec. 4.2.3 (complete redistribution, isotropic emission, no stimulated emission), the source function becomes $$ S(\mathbf{r}, t) = \frac{n(\mathbf{r}, t)A_{21}}{N B_{12}}. $$ (4.33) The source function is thus proportional to the density of the excited-state atoms, $n(\mathbf{r}, t)$. In the case of a slab, Eqs. (4.32) and (4.33) give $$ \cos(\vartheta) \frac{\partial I(z, \vartheta, \nu, t)}{\partial z} = -k(\nu) \left[ I(z, \vartheta, \nu, t) - \frac{A_{21}}{B_{12}} \frac{n(z, t)}{N} \right]. $$ (4.34) For the rate equation of the excited atoms, we describe the processes that can increase or decrease $n(z, t)$. Atoms are excited by absorption of photons, and de-excited by spontaneous emission (stimulated emission is negligible with our assumptions). Under complete frequency redistribution, these processes are described by (Cannon 1985) $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -A_{21}n(\mathbf{r}, t) + \frac{N B_{12}}{4\pi} \iint I(\mathbf{r}, \mathbf{\Omega}, \nu, t)C_\nu k(\nu)\mathrm{d}\nu\mathrm{d}\mathbf{\Omega}, $$ (4.35) where the normalization constant is $C_\nu = 1 / \int k(\nu)\mathrm{d}\nu$. For the slab geometry, this rate equation can be written as $$ \frac{\partial n(z, t)}{\partial t} = -A_{21}n(z, t) + \frac{N B_{12}}{2} C_\nu \int_{-\infty}^{\infty} \int_{0}^{\pi} I(z, \vartheta, \nu, t)k(\nu) \sin \vartheta \mathrm{d}\vartheta \mathrm{d}\nu. $$ (4.36) Equations (4.34) and (4.36) combined describe trapping in a slab under the same assumptions as the Holstein equation. We can see that this formulation is truly equivalent --- to the Holstein equation by formal integration of Eq. (4.34), the equation of radiative transfer for the slab geometry, $$ I(z, \vartheta, \nu, t) = \int k(\nu) \exp \left[ - \frac{1}{\cos(\vartheta)} k(\nu) |z - z'| \right] \frac{1}{\cos(\vartheta)} \frac{A_{21}}{B_{12}} \frac{n(z', t)}{N} dz'. $$ (4.37) When we insert Eq. (4.37) into Eq. (4.36), we get the Holstein equation for the slab case—that is, Eq. (4.11) with the Kernel function Eq. (4.16). This equivalence is also more general in that it is valid for all geometries—this can be shown by appropriate tricks of vector algebra (Cayless 1986). ## 4.5 The escape factor The Holstein equation completely describes the density of the excited-state atoms. Sometimes, however, there are cases where we only want to compute the probability that a photon emitted at a point $\mathbf{r}$ escapes from the vapour cell without being absorbed. This probability is called the escape factor $\eta(\mathbf{r})$. It provides less information than the solution of the Holstein equation, but can be computed more easily. The escape factor is thus often used for a 'quick and dirty' account of trapping effects.[^7] One possible approach is to assume a certain distribution of excited-state atoms and to average the escape factor over this distribution. If we have some 'magic' process that fixes the excited-state distribution, $n(\mathbf{r})$, at all points to certain known values—**regardless of the reabsorption rate**—then we can use this $n(\mathbf{r})$ to compute the average escape factor $\bar{\eta}$. When the spatial redistribution of excitation due to the trapping is small compared to the creation or destruction of excitation by other mechanisms, then we can predict the actual distribution of $n(\mathbf{r})$ quite well (i.e., we have a 'magic' process), and the above approach is justified. This is the case in some fluorescent lamp applications, in many plasmas (Biberman *et al.* 1987), and also in astrophysics (where the excited-state density is mainly determined by thermal excitation). However, in a cell where natural decay and reabsorption of photons are the dominant mechanisms for the destruction and creation of excited-state atoms, the excited state distribution is determined by trapping, which in turn is influenced by the escape factor $\eta$, i.e. by the parameter we want to compute. The computation of the average number of reabsorptions with the escape factor technique might even give reasonably accurate results, since this parameter is not very sensitive to the actual distribution of the excited-state atoms (see also Sec. 6.1). However, we forfeit all knowledge about the spatial distribution of the excited-state atoms, and also information about the time behaviour of the emergent radiation will be lost. A good description is **only** possible by the Holstein equation or its equivalents, which give the spatial distribution of the excited-state atoms simultaneously with the trapping factors. [^7]: There are some authors who use a different definition of the escape factor, based upon the difference between absorbed and reemitted radiation at any point in the vapour, see e.g. Cho and Eddy (1989). With this definition, also negative escape factors are possible. In order to distinguish, we will call this definition the 'escape function'; it will be used in Sec. 11.2. --- In plasma research, the escape factor technique is often used in conjunction with the ‘effective-lifetime’ approximation to compute the excited-state distribution. In this approach, the natural lifetimes of excited atoms are multiplied with the escape factor $\eta(\mathbf{r})$ at all points of the vapour to give the ‘effective lifetimes’. This leads to a drastic simplification. The trapping process is basically a **non-local** process. The excited-state distribution at some point in the vapour is influenced by the excited-state distributions at all other points. The ‘effective-lifetime’ approximation reduces the whole process to a **local** process, where the excited-state distribution at a certain point becomes independent of the other points. Obviously, such a simplification can also lead to considerable errors (Abramov and Kogan 1967). We have exact agreement between the escape factor and the Holstein solution only in the limit that the excited-state distribution is spatially uniform (Irons 1991). Another useful application of the escape factor is to obtain upper and lower bounds for the trapping factor. From physical reasoning, we know that the lowest-order eigenfunction (for all opacities and lineshapes) is always positive, has a maximum in the middle of the cell and decays monotonically toward the cell boundary. Thus, the two most extreme shapes we can imagine for the lowest-order mode are the uniform distribution and the delta function at the centre of the cell. Hence, the trapping factors associated with $n(\mathbf{r}) = 1$ and $n(\mathbf --- where $\eta(\mathbf{r}, x)$ is the escape factor at a certain frequency. Generalizations and discussions of this theorem are given by Rybicki and Hummer (1983). The expression for the escape factor (4.39) should be equivalent to Eq. (4.38). The Kernel function $G(\mathbf{r}, \mathbf{r}')$ from Sec. 4.2 can be written as a divergence (Lawler *et al.* 1993) $$ G(\mathbf{r}, \mathbf{r}') = -\nabla_r \cdot \left[ \frac{\mathbf{e}_{\mathbf{r}, \mathbf{r}'}}{4\pi|\mathbf{r} - \mathbf{r}'|^2} T(|\mathbf{r} - \mathbf{r}'|) \right] + \delta^3(\mathbf{r}, \mathbf{r}') $$ (4.41) Inserting Eq. (4.41) into Eq. (4.39) and using Gauss' theorem, we indeed get Eq. (4.38). Before computing actual escape factors, let us first compute the transmission factor $T(\rho)$ for the two most important lineshapes: Doppler and Lorentz. We will find that in the case of high opacities, there are some simple approximations which are very useful for the computation of $\eta$. We start with the definition of $T(\rho)$, $$ T(\rho) = C_x \int_{-\infty}^{\infty} k(x) \exp[-\rho k(x)] dx $$ (4.42) For a Doppler profile, this is $$ T(\rho) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \exp(-x^2) \exp(-k_0 \rho e^{-x^2}) \mathrm{d}x $$ (4.43) With the substitution $x = (\ln(k_0\rho)/u)^{1/2}$, we get (Holstein 1947) $$ T(\rho) = \frac{1}{k_0 \rho \sqrt{\pi}} \int_0^{k_0 \rho} \frac{\exp(-u)}{\sqrt{\ln(k_0 \rho) - \ln(u)}} \mathrm{d}u $$ (4.44) For large opacities, i.e., for large $k_0\rho$, we can extend the upper limit of integration to infinity and set the square root in the integrand equal to $(\ln(k_0\rho))^{1/2}$. We arrive at a high-opacity, Doppler lineshape approximation for the transmission factor of $$ T(\rho) \approx \frac{1}{k_0 \rho \sqrt{\pi \ln(k_0 \rho)}}. $$ (4.45) For Doppler broadening in a slab, the high-opacity approximation was circumvented by Scherr (1971), who used a series expansion of $T(\rho)$ $$ \begin{aligned} T(\rho) &= \frac{1}{\sqrt{\pi} k_0 \rho} \int_0^{k_0 \rho} \frac{\exp(-u)}{\sqrt{\ln(k_0 \rho) - \ln(u)}} \mathrm{d}u \\ &= \frac{1}{\sqrt{\pi} k_0 \rho} \sum_{i=0}^{\infty} \frac{(-1)^i}{i!} \int_0^{k_0 \rho} \frac{u^i}{\sqrt{\ln(k_0 \rho) - \ln(u)}} \mathrm{d}u. \end{aligned} $$ (4.46) The integral on the right-hand side is equal to $(k_0\rho)^{i+1}[\pi/(i+1)]^{1/2}$, and $T(\rho)$ becomes $$ T(\rho) = \sum_{i=0}^{\infty} \frac{(-k_0 \rho)^i}{i!(i + 1)^{1/2}}. $$ (4.47) The expansion in Eq. (4.47) converges rapidly for small opacities. --- [FIGURE: The transmission factor T(ρ) for a Doppler lineshape: exact and high-opacity approximation.] FIG. 4.7. The transmission factor $T(\rho)$ for a Doppler lineshape: exact and high-opacity approximation. Analogously inserting into Eq. (4.42) for a Lorentzian lineshape, $$ T(\rho) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{1}{1+x^2} \exp\left(-\frac{k_0 \rho}{1+x^2}\right) \mathrm{d}x, \qquad (4.48) $$ and with the substitution $$ \frac{1}{1+x^2} = \frac{1 - \cos(u)}{2} \qquad (4.49) $$ we get $$ T(\rho) = \exp\left(-\frac{k_0 \rho}{2}\right) I_0\left(\frac{k_0 \rho}{2}\right). \qquad (4.50) $$ For large opacities, this becomes $$ T(\rho) = \frac{1}{\sqrt{\pi k_0 \rho}}, \qquad (4.51) $$ a very useful high-opacity approximation for the transmission factor for Lorentzian lineshapes. For a Voigt lineshape (again in the high-opacity case), Walsh (1959) derived an interpolation between the Doppler and Lorentz formulas (for all geometries). He starts out with a very simple approximation to the Voigt profile, $$ k(x) = k_0 \left[ \exp(-x^2) + \frac{a}{\sqrt{\pi}} \frac{1}{x^2} \right]. \qquad (4.52) $$ --- THE ESCAPE FACTOR [FIGURE: FIG. 4.8. The transmission factor T(ρ) for a Lorentzian lineshape: exact and high-opacity approximation.] Equation (4.52) holds within 20% for $|x| > 2$, i.e. in the wings of the line. Since we consider high opacities, the trapping is mostly determined by the wings, so that this representation of the lineshape is sufficient for first estimates. The next step is the computation of $T(\rho)$ $$T(\rho) = C_x \int_{-\infty}^{\infty} k(x) \exp(-k(x)\rho)\mathrm{d}x, \qquad (4.53)$$ where for $C_x$, we use the value for pure Doppler broadening, $C_x = 1/(k_0\sqrt{\pi})$. Inserting Eq. (4.52) into Eq. (4.53), we get $$T(\rho) = f1 + f2$$ $$f1 \approx \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \exp(-x^2) \exp\left[-k_0\rho \left(\exp(-x^2) + \frac{a}{\sqrt{\pi}x^2}\right)\right] \mathrm{d}x \qquad (4.54)$$ $$f2 \approx \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{1}{x^2} \exp\left[-k_0\rho \left(\exp(-x^2) + \frac{a}{\sqrt{\pi}x^2}\right)\right] \mathrm{d}x.$$ The integrals are evaluated by a Taylor expansion around $a = 0$ $$f1 = f1(0) + a \cdot \left. \frac{\partial f1}{\partial a} \right|_{a=0} + \dots \qquad (4.55)$$ where --- $$ \left. \frac{\partial^m f1}{\partial a^m} \right|_{a=0} = \frac{(-1)^m}{\sqrt{\pi}} \left( \frac{k_0 \rho}{\sqrt{\pi}} \right)^m \int_{-\infty}^{\infty} \frac{\exp(-x^2)}{x^{2m+1}} \exp(-k_0 \rho \exp(-x^2)) \, \mathrm{d}x. \qquad (4.56) $$ These integrals are evaluated in the same manner as Eq. (4.43). Inserting these expressions into Eq. (4.55), we get $$ f1 = \frac{1}{k_0 \rho \sqrt{\pi \ln(k_0 \rho)}} \left[ 1 - \frac{a k_0 \rho}{\sqrt{\pi \ln(k_0 \rho)}} + \frac{1}{2!} \left( \frac{a k_0 \rho}{\sqrt{\pi \ln(k_0 \rho)}} \right)^2 - \dots \right] \qquad (4.57) $$ For $f2$, the terms of the Taylor expansion are $$ \left. \frac{\partial^m f2}{\partial a^m} \right|_{a=0} = \frac{(-1)^m a}{\pi} \int_{-\infty}^{\infty} \left( \frac{k_0 \rho}{\sqrt{\pi}} \right)^m \frac{\exp(-k_0 \rho \exp(-x^2))}{x^{2(m+1)}} \, \mathrm{d}x. \qquad (4.58) $$ The expression $\exp(\dots)$ is approximated by a step function which is 0 for $|x| < [\ln(k_0 \rho)]^{1/2}$ and 1 elsewhere. Equation (4.58) then becomes $$ f2 = \frac{2a}{\pi \sqrt{\ln(k_0 \rho)}} \left[ 1 - \frac{1}{3} \frac{1}{1!} \frac{a k_0 \rho}{\sqrt{\pi \ln(k_0 \rho)}} + \frac{1}{5} \frac{1}{2!} \left( \frac{a k_0 \rho}{\sqrt{\pi \ln(k_0 \rho)}} \right)^2 - \dots \right] \qquad (4.59) $$ The term in brackets in Eq. (4.57) is the series representation of an exponential function. The bracket in Eq. (4.59) is the series representation of the error function. $T(\rho)$ can thus be written as $$ \begin{array}{c} T(\rho) = T^\mathrm{D} \exp\left[ -\pi \left( \frac{T^\mathrm{DL}}{2T^\mathrm{L}} \right)^2 \right] + T^\mathrm{L} \mathrm{erf}\left( \sqrt{\pi} \frac{T^\mathrm{DL}}{2T^\mathrm{L}} \right) \\[15pt] T^\mathrm{D} = \frac{1}{k_0 \rho \sqrt{\pi \ln(k_0 \rho)}}, \qquad T^\mathrm{L} = \sqrt{\frac{a}{\sqrt{\pi} k_0 \rho}}, \qquad T^\mathrm{DL} = \frac{2a}{\pi \sqrt{\ln(k_0 \rho)}}. \end{array} \qquad (4.60) $$ We thus conclude that the escape factor, $T(\rho)$, can be written as a combination of the transmission factors $T^\mathrm{D}$ (Doppler), $T^\mathrm{L}$ (Lorentz), and $T^\mathrm{DL}$ (Doppler absorption with Lorentz emission), The high-opacity approximations to the transmission factor play a central role in the derivation not only of the escape factors given below, but also for many approximate solutions of the Holstein equation—most notably Holstein's own classical computations, which we will encounter in Sec. 5.1. *** ### Insert 4.2 Escape factors for slab, cylinder, and sphere **Escape factor for the slab** Evaluating the definition of the escape factor, Eq. (4.38), for the slab geometry, we get (Irons 1979) $$ \eta(z) = \frac{1}{2} \int_0^{\pi/2} T\left( \frac{k_0(L/2 - z)}{|\cos \vartheta|} \right) \sin \vartheta \, \mathrm{d}\vartheta + \frac{1}{2} \int_{\pi/2}^{\pi} T\left( \frac{k_0(L/2 + z)}{|\cos \vartheta|} \right) \sin \vartheta \, \mathrm{d}\vartheta \qquad (4.61) $$ --- When we insert the high-opacity approximations for $T(\rho)$, we can perform the integration over $\vartheta$ analytically. For a Doppler profile at high opacities we get the approximation $$ \eta(z) \approx \frac{1}{4} T (k_0(L/2 - z)) + \frac{1}{4} T (k_0(L/2 + z)), $$ (4.62) and for a Lorentz profile at high opacities $$ \eta(z) \approx \frac{1}{3} T (k_0(L/2 - z)) + \frac{1}{3} T (k_0(L/2 + z)), $$ (4.63) For the slab, we can also derive a slightly different approximation that is valid at all opacities and additionally has a nice physical interpretation. We will see in Chapter 8 that we can define 'typical' photons that behave like the average over all photons (Eddington approximation). The direction of these 'typical' photons is at a 60 degree angle with the $z$-axis. A typical photon that is created at point $z$ and flies towards the wall at $L/2$ has to cover the distance $(L/2 - z)/\cos(60^\circ) = (L - 2z)$ before it reaches the wall. Its probability of escape is $T(k_0(L - 2z))$. A typical photon that flies to the wall at $-L/2$ has to cover $(L + 2z)$. Since half the photons are emitted to the left, and half to the right, the escape factor approximates to $$ \eta(z) \approx \frac{1}{2} T (k_0(L - 2z)) + \frac{1}{2} T (k_0(L + 2z)), $$ (4.64) At the centre of a slab, the escape factor becomes $$ \eta(0) \approx T(k_0L) $$ (4.65) Equation (4.64) is valid at all opacities. When we explicitly insert the high-opacity approximations, we see that equations (4.62), (4.63), and (4.64) give almost the same results. **For the cylinder**, there are the following high-opacity approximations for the escape factor for points on the axis: For a Doppler profile: $$ \eta(0) \approx \frac{\pi}{4} T(k_0R) $$ (4.66) For a Lorentzian profile: $$ \eta(0) \approx 0.874 T(k_0R) $$ (4.67) **For the sphere**, the general escape factor is given by Mancini *et al.* (1987) $$ \begin{aligned} \eta(r) = C_x \int k(x) \left( \frac{\exp(-k(x)R)}{2} \left[ (1 + k(x)R) \frac{\sinh (k(x)r)}{k(x)r} + \cosh (k(x)r) \right] + \right. \\ \left. + \frac{R^2 - r^2}{4rR} k(x)R \left[ \mathrm{Ei}_1 (k(x)(R + r)) - \mathrm{Ei}_1 (k(x)(R - r)) \right] \right) \mathrm{d}x \end{aligned} $$ (4.68) The function $\mathrm{Ei}_1$ is the exponential integral of first order. For the escape from the centre of the sphere, this reduces to $$ \eta(0) = T(k_0R) $$ (4.69) --- Escape factors for Stark broadening are discussed by Weisheit (1979) and Mancini *et al.* (1987); Puetter (1981) gives the escape factor for semi-infinite atmospheres. Escape factors for an inhomogeneous distribution of absorbers in a slab are discussed by Direktor *et al.* (1987). ### Insert 4.3 Averaged escape factors for slab, cylinder, and sphere We can define an **average escape factor** $\bar{\eta}$, which is the escape factor $\eta(\mathbf{r})$ averaged over the cell **for a certain spatial distribution of excited-state atoms**, $$ \bar{\eta} = \frac{\int_V n(\mathbf{r})\eta(\mathbf{r})\mathrm{d}\mathbf{r}}{\int_V n(\mathbf{r})\mathrm{d}\mathbf{r}} $$ (4.70) Explicit formulas for two spatial distributions are given below, (1) for the uniform distribution and (2) for the parabolic distribution. **(1) Averaged escape factors for a uniform distribution of excited atoms.** **For the slab geometry**, we insert for $\eta(\mathbf{r})$ from Eq. (4.61) and get $$ \bar{\eta} = \frac{1}{L} \int_{-L/2}^{L/2} \int_0^1 T\left(\frac{k_0(L/2 - z)}{\mu}\right) \mathrm{d}\mu \mathrm{d}z $$ (4.71) where $\mu = |\cos \vartheta|$. **For a Doppler lineshape at high opacities**, this becomes (Irons 1979) $$ \bar{\eta} \approx T(k_0L) \cdot \ln(k_0L) $$ (4.72) and **for a Lorentzian lineshape**, $$ \bar{\eta} \approx \frac{4}{3} T(k_0L). $$ (4.73) Fitting equations for all opacities for a uniform spatial distribution are given by Drawin and Emard (1973) **For the infinite cylinder** and **for a Doppler lineshape**, $$ \bar{\eta} \approx 2T(k_0R) \cdot \ln(k_0R) $$ (4.74) **for a Lorentzian lineshape**, $$ \bar{\eta} \approx 1.95T(k_0R). $$ (4.75) For a finite cylinder, the equations are given by Kapaces *et al.* (1985). --- **For a sphere,** $$ \bar{\eta} = C_x \int k(x) \frac{3}{4} (k(x)R)^{-3} \left[ (k(x)R)^2 - \frac{1}{2} + \left( k(x)R + \frac{1}{2} \right) \exp(-2k(x)R) \right] \mathrm{d}x $$ (4.76) **(2) Averaged escape factors for a parabolic distribution of excited atoms.** A parabolic spatial distribution is defined by $$ n(z) = \left[ 1 - \left( \frac{2z}{L} \right)^2 \right]^m $$ (4.77) where the factor $m > 0$ determines the 'steepness' of the distribution. With $m \to \infty$, we have a delta distribution, whereas with $m = 1$ we get a genuine parabola, which is similar to the excitation function in discharge lamps (Grossmann *et al.* 1986). We will see in the next chapter that $m = 0.25$ and $m = 0.5$ correspond to the shapes of the lowest-order modes of the Holstein equation at high opacities. **For high opacities**, Bezuglov *et al.* (1997) give an equation for the escape factor that is valid for arbitrary $m$ and for all three geometries $$ \bar{\eta} = T(k_0 R) \frac{\sqrt{\pi} \Gamma(\gamma + c_{\text{geo}}/2)}{2\Gamma(\gamma + 1.5)\Gamma(c_{\text{geo}}/2)} \frac{\Gamma(1 + m - 2\gamma)\Gamma(1 + m + c_{\text{geo}}/2)}{\Gamma(1 + m - \gamma)\Gamma(1 + m - \gamma + c_{\text{geo}}/2)} $$ (4.78) where $\Gamma$ is the Euler gamma function. The parameter $c_{\text{geo}}$ is set to 1, 2, or 3 for a slab, cylinder or sphere (for the slab, set $R = L/2$). It is assumed that the wings of the line behave like $|x|^{-\kappa}$. The parameter $\gamma$ determines the lineshape by $\gamma = (\kappa - 1)/2\kappa$. For a Doppler lineshape, $\kappa$ equals infinity, so that $\gamma = 1/2$. With a parabolic distribution, the difference in the escape factors for various values of $m$ is some constant. However, for a Doppler lineshape, a *uniform* spatial distribution has an escape factor that differs from all other escape factors by a factor $\ln(k_0 L)$ and thus increases with opacity. The assumption of such a uniform distribution has thus to be used with great care (Apruzese *et al.* 1976, 1977). **For low opacities**, the averaged escape factor can be computed from $$ \bar{\eta} = \frac{\Gamma(1 + m + c_{\text{geo}}/2)}{\Gamma(c_{\text{geo}}/2)} 2^{1+m} C_x \int R k^2(x) \int_{k(x)R}^{\infty} \frac{I_{m+c_{\text{geo}}/2}^B(u) K_{c_{\text{geo}}/2}^B(u)}{u^{m+2}} \mathrm{d}u \mathrm{d}x $$ (4.79) where $I^B$ and $K^B$ are modified Bessel functions. For the slab case, $R$ has to be replaced by $L/2$ in Eq. (4.79). Actually, Eq. (4.79) works for **all** opacities, but in practice will be applied only for low opacities, because at high opacities, the use of Eq. (4.78) allows faster computation. --- The method of describing the radiation trapping effects by using an 'effective lifetime' is also known as the 'first-order escape probability method'. As pointed out by Hummer and Rybicki (1982b) and Rybicki (1984, 1985), it is especially problematic near --- the boundaries of the vapour cell. To remedy this situation somewhat, the so-called 'second-order escape probability methods' have been proposed (Athay 1976), (Athay 1984), (Canfield and Ricchiazzi 1980), (Canfield and Puetter 1981), (Canfield *et al.* 1981), (Canfield *et al.* 1984), (Puetter *et al.* 1982), and (Frisch and Bardos 1981). In these methods, the equation of radiative transfer is reduced to a first-order differential equation for the *integrated intensity* $J$. This is also the quantity that occurs in the rate equations to describe the absorption–reemission process. Thus, the rate equation and the equation of radiative transfer are then only a system of coupled *differential* equations, where the unknowns $J$ and $n_2$ depend only on position. This is in strong contrast to the exact description, where we have a system of *integro-differential equations*, where the unknowns $I$ depend also on frequency and angle. --- # 5 ## MATHEMATICAL METHODS FOR THE HOLSTEIN EQUATION In the previous chapter, we have discussed various methods to describe radiation trapping. The most important one is the Holstein equation, $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau} n(\mathbf{r}, t) + \frac{1}{\tau} \int_V n(\mathbf{r}', t) G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}', $$ (5.1) In this chapter, we will discuss various methods for solving this equation. This chapter, as well as the following, is concentrated on the mathematical methods. The physical interpretation of the results will be given in Chapter 7. ### 5.1 The variational technique Holstein (1947) solved Eq. (5.1) by a variational technique. The symmetry of the Kernel function $G(\mathbf{r}, \mathbf{r}') = G(\mathbf{r}', \mathbf{r})$ permits this formulation of Eq. (5.1) as a variational problem. Since we are looking for the eigenfunctions of an integro-differential equation, application of the operator $\int n(\mathbf{r}, t) \mathrm{d}\mathbf{r}$ must result in an expression that is invariant to small changes in $n$. Multiplying Eq. (5.1) by this integral operator, we get $$ \tau \int n(\mathbf{r}, t) \frac{\partial n(\mathbf{r}, t)}{\partial t} \mathrm{d}\mathbf{r} = - \int n^2(\mathbf{r}, t) \mathrm{d}\mathbf{r} + \iint n(\mathbf{r}, t) n(\mathbf{r}', t) G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{r}', $$ (5.2) With the known form of the solution, $n(\mathbf{r}, t) = \psi(\mathbf{r}) \cdot \exp[-t/(\tau \cdot g)]$, this becomes $$ \frac{1}{g} = 1 - \frac{\iint G(\mathbf{r}, \mathbf{r}') \psi(\mathbf{r}) \psi(\mathbf{r}') \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{r}'}{\int \psi^2(\mathbf{r}) \mathrm{d}\mathbf{r}}. $$ (5.3) The escape probability of a photon is (see Sec. 4.5) $$ \eta(\mathbf{r}) = 1 - \int G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}'. $$ (5.4) Inserting this, Eq. (5.3) becomes, after some manipulation, $$ \frac{1}{g} = \frac{\int \psi^2(\mathbf{r}) \eta(\mathbf{r}) \mathrm{d}\mathbf{r} + \frac{1}{2} \iint G(\mathbf{r}, \mathbf{r}') [\psi(\mathbf{r}) - \psi(\mathbf{r}')]^2 \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{r}'}{\int \psi^2(\mathbf{r}) \mathrm{d}\mathbf{r}} $$ (5.5) --- Now we apply the Ritz variational procedure, which means that the eigenmodes $\psi(\mathbf{r})$ are approximated by a finite sum over some given basis functions $b_i^{\text{var}}(\mathbf{r})$ with unknown coefficients $a_i$, $$ \psi(\mathbf{r}) = \sum_i a_i \cdot b_i^{\text{var}}(\mathbf{r}). $$ (5.6) With this substitution, Eq. (5.5) becomes $$ \frac{1}{g} = \frac{\sum_k \sum_m a_k a_m A_{k,m}^{\text{var}}}{\sum_k \sum_m a_k a_m B_{k,m}^{\text{var}}} $$ (5.7) where $A_{k,m}^{\text{var}}$ and $B_{k,m}^{\text{var}}$ stand for $$ \begin{aligned} A_{k,m}^{\text{var}} &= \int b_k^{\text{var}}(\mathbf{r})b_m^{\text{var}}(\mathbf{r})\eta(\mathbf{r})\mathrm{d}\mathbf{r} + \\ &\quad \frac{1}{2} \iint \left(b_k^{\text{var}}(\mathbf{r}) - b_k^{\text{var}}(\mathbf{r}')\right) \cdot \left(b_m^{\text{var}}(\mathbf{r}) - b_m^{\text{var}}(\mathbf{r}')\right) G(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r}' \\ B_{k,m}^{\text{var}} &= \int b_k^{\text{var}}(\mathbf{r})b_m^{\text{var}}(\mathbf{r})\mathrm{d}\mathbf{r}. \end{aligned} $$ (5.8) If $\psi(\mathbf{r})$ is a solution of the Holstein equation, then the fraction on the right-hand side of Eq. (5.3) becomes a maximum with respect to a variation of $\psi(\mathbf{r})$ (Schwartz inequality). Consequently, $1/g$ (the left-hand side of Eq. (5.3)) becomes a minimum—since it is calculated as one minus the fraction that becomes a maximum. For Eq. (5.7) this means that $1/g$ should be a minimum upon variation of the coefficients $a_k$ and $a_m$, which means that $\partial(1/g)/\partial a_k = 0$, allowing the determination of the coefficients. Performing the differentiation, we obtain $$ \sum_m \left[ a_m A_{k,m}^{\text{var}} - \frac{B_{k,m}^{\text{var}}}{g} \right] = 0. $$ (5.9) Equation (5.9) only has a non-trivial solution if the determinant equals zero, $$ \begin{vmatrix} A_{0,0}^{\text{var}} - \frac{B_{0,0}^{\text{var}}}{g} & \cdots & \cdots & A_{0,m}^{\text{var}} - \frac{B_{0,m}^{\text{var}}}{g} \\ \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots \\ A_{m,0}^{\text{var}} - \frac{B_{m,0}^{\text{var}}}{g} & \cdots & \cdots & A_{m,m}^{\text{var}} - \frac{B_{m,m}^{\text{var}}}{g} \end{vmatrix} = 0 $$ (5.10) Note that $A_{k,m}^{\text{var}} = A_{m,k}^{\text{var}}$, $B_{k,m}^{\text{var}} = B_{m,k}^{\text{var}}$. **Solution of Eq. (5.10) yields the eigenvalues $g$.** --- The variational technique was first used by Holstein to derive **closed-form approximations for $g$ for the high-opacity slab case ($k_0L >> 1$).** **For a Doppler line**, $T(\rho)$ can be approximated as (see Sec. 4.5) $$ T(\rho) = \frac{1}{k_0\rho\sqrt{\pi \ln(k_0\rho)}}. \qquad (5.11) $$ The Kernel function then becomes $$ G(z, z') = \frac{1}{4k_0|z - z'|^2} \frac{1}{\sqrt{\pi \ln(k_0L/2)}}, \qquad (5.12) $$ and for the escape probability one gets $$ \eta(z) = \frac{1}{4k_0L\sqrt{\pi \ln(k_0L/2)}} \frac{1}{\frac{1}{4} - (z/L)^2}. \qquad (5.13) $$ We now introduce the normalized spatial coordinate $\hat{z} = 2z/L$ and insert Eqs (5.12) and (5.13) into Eq. (5.5). We thus get $$ \frac{1}{g'} = \frac{k_0L\sqrt{\pi \ln(k_0L/2)}}{g_0} = \frac{\int_{-1}^1 \frac{\psi^2(\hat{z})}{1 - \hat{z}^2} \mathrm{d}\hat{z} + \frac{1}{4} \int_{-1}^1 \int_{-1}^1 \frac{[\psi(\hat{z}) - \psi(\hat{z}')]^2}{(\hat{z} - \hat{z}')^2} \mathrm{d}\hat{z}\mathrm{d}\hat{z}'}{\int_{-1}^1 \psi^2(\hat{z})\mathrm{d}\hat{z}} \qquad (5.14) $$ Using these approximations also on the one-dimensional analogue of Eq. (5.8), we get $$ \begin{aligned} A_{k,m}^{\mathrm{var}} &= \int_{-1}^1 \frac{b_k^{\mathrm{var}}(\hat{z})b_m^{\mathrm{var}}(\hat{z})}{1 - \hat{z}^2} \mathrm{d}\hat{z} + \frac{1}{4} \int_{-1}^1 \int_{-1}^1 \frac{[b_k^{\mathrm{var}}(\hat{z}) - b_k^{\mathrm{var}}(\hat{z}')] \cdot [b_m^{\mathrm{var}}(\hat{z}) - b_m^{\mathrm{var}}(\hat{z}')]}{(\hat{z} - \hat{z}')^2} \mathrm{d}\hat{z}\mathrm{d}\hat{z}' \\ B_{k,m}^{\mathrm{var}} &= \int_{-1}^1 b_k^{\mathrm{var}}(\hat{z})b_m^{\mathrm{var}}(\hat{z})\mathrm{d}\hat{z}. \end{aligned} \qquad (5.15) $$ As basis functions $b_k^{\mathrm{var}}(\hat{z})$, Holstein used the normalized polynomials $$ b_0^{\mathrm{var}}(\hat{z}) = \frac{1}{2} \qquad b_1^{\mathrm{var}}(\hat{z}) = \frac{3}{4}(1 - \hat{z}^2). \qquad (5.16) $$ Inserting these into Eq. (5.15), the $A_{k,m}^{\mathrm{var}}$ and $B_{k,m}^{\mathrm{var}}$ can be computed easily. A problem occurs with $A_{0,0}^{\mathrm{var}}$, because it diverges when the asymptotic expressions are used. The exact expression is $$ A_{0,0}^{\mathrm{var}} = k_0 \frac{\sqrt{\pi \ln(k_0L/2)}}{2} \int_{-L/2}^{L/2} \eta(z)\mathrm{d}z \qquad (5.17) $$ However, this integral is nothing but the escape factor for a uniform spatial distribution of excited atoms (see Sec. 4.5). Inserting the high-opacity approximation for this escape factor, we get --- $$ A_{0,0}^{\text{var}} \approx \frac{1}{2} \ln(k_0 L) $$ (5.18) We then proceed to evaluate the determinant Eq. (5.10). If we use only a one-term 'sum' to represent $n(z)$, i.e. $n(z) = a_1 b_1^{\text{var}}(\hat{z})$ in Eq. (5.6), we get $1/g' = 15/8$. If we use a two-term sum $n(z) = a_0 b_0^{\text{var}}(z) + a_1 b_1^{\text{var}}(z)$, we get $$ \frac{1}{g'} = \frac{15}{8} \left( 1 - \frac{1/16}{\ln(k_0 L) - 5/4} \right), $$ (5.19) which differs by less than 2% from 15/8 for opacities $k_0 L$ larger than 5. We can thus use $1/g' = 15/8$ with good accuracy. We finally arrive at the high-opacity approximation for the lowest-order trapping factor in a slab and for a Doppler lineshape, $$ g_0 = \frac{1}{1.06} k_0 L \sqrt{\ln(k_0 L/2)}. $$ (5.20) The amplitudes of the basis functions (for the two-parameter representation) are then $$ a_0 = \frac{3/8}{\ln(k_0 L) - 3/2}, \qquad a_1 = 1 - a_0. $$ (5.21) **For the Lorentz distribution**, $T(\rho)$ can be approximated as (see Sec. 4.5) $$ T(\rho) \approx \frac{1}{\sqrt{\pi k_0 \rho}}. $$ (5.22) Proceeding analogously to the above derivation one finally gets for the trapping factor, $$ g_0 = \frac{\sqrt{\pi k_0 L}}{1.15}. $$ (5.23) These are physically very important results. For a Doppler lineshape, the trapping factor increases approximately linearly with the opacity, while for a Lorentz lineshape, it increases only with the square root of the opacity, see Fig. 5.1. We will see later that this also has important consequences for Voigt lineshapes. **For statistical broadening**, the interaction force between the atoms is proportional to $r^{-m}$, where $r$ is the distance between the colliding atoms, with $m = 3$ for resonant interaction and $m = 6$ for foreign-gas broadening (see also Chapter 2). The lineshape is then proportional to $$ k(x) \propto x^{-(1+3/m)} $$ (5.24) and the transmission factor $T(\rho)$ is given by $$ T(\rho) = \frac{C_{\text{stat}}}{(k_0 \rho)^{3/(3+m)}} $$ (5.25) --- THE VARIATIONAL TECHNIQUE [FIGURE: FIG. 5.1. The trapping factor g_0 in the Holstein approximation for Doppler and Lorentz lineshapes.] where $c_{\text{stat}}$ is a constant dependent on the strength of the interaction and on the particle density. The trapping factor can then be computed as $$ g_0 = \frac{(k_0L)^u}{3c_{\text{stat}}} \frac{(1 - u^2)(2 - u)(3 - u)(5 - u)}{10 - u - u^3 - u\sqrt{101 - 40u + 2u^2 + u^4}} \qquad (5.26) $$ where $u = 3/(3 + m)$. **For the cylinder**, the computations are considerably more complicated. As a first approximation, we eliminate for Doppler broadening the $\rho$-dependence of the logarithmic term in $T(\rho)$ by setting it equal to $\ln(k_0R)$. We can thus write the transmission factor for all lineshapes as $$ T(\rho) = \frac{c_{\text{stat}}}{(k_0\rho)^u} \qquad (5.27) $$ We have a cylindrical coordinate system $(r, \varphi, z)$ and define a unit cross-sectional area $\mathrm{d}\sigma_{\text{cyl}} = r \cdot \mathrm{d}r \cdot \mathrm{d}\varphi$. Next, we integrate the Kernel function $G(\mathbf{r}, \mathbf{r}')$ over $z$ $$ \int_{-\infty}^{\infty} G(\mathbf{r}, \mathbf{r}') \mathrm{d}z' = \frac{u c_{\text{stat}}}{4\pi k_0^u} \int_{-\infty}^{\infty} \frac{1}{\rho^{u+3}} \mathrm{d}z' \qquad (5.28) $$ where $$ \rho^2 = r^2 + r'^2 - 2rr'\cos(\varphi - \varphi') + (z - z')^2 = q^2 + (z - z')^2 \qquad (5.29) $$ Analytical integration yields $$ \int_{-\infty}^{\infty} G(\mathbf{r}, \mathbf{r}') \mathrm{d}z' = \frac{c1}{q^{u+2}}, \quad \text{where} \quad c1 = \frac{u c_{\text{stat}}}{2^{3+u}} \frac{(u+1)!}{[(u+1)/2]! k_0^u} \qquad (5.30) $$ --- MATHEMATICAL METHODS FOR THE HOLSTEIN EQUATION [FIGURE: FIG. 5.2. The trapping factor g_0 for statistical broadening, parameter c_stat = 0.5.] With this result, we can compute the escape factor $\eta(r)$ as $$ \eta(r) = \int_{\text{ext}} \frac{c1}{q^{u+2}} d\sigma_{\text{cyl}}' $$ (5.31) where $\sigma_{\text{cyl}}'$ is the unit element in the cylinder cross-section. The integration is done over the *exterior* of the cylinder. We can then write the variational formulation as (Eq. (5.5)) $$ \frac{1}{g} = \frac{\displaystyle \int_{\text{ext}} \psi^2(r) \frac{c1}{q^{u+2}} d\sigma_{\text{cyl}} d\sigma_{\text{cyl}}' + \frac{1}{2} \iint [\psi(r) - \psi(r')]^2 \frac{c1}{q^{u+2}} d\sigma_{\text{cyl}} d\sigma_{\text{cyl}}'}{\displaystyle \int \psi^2(r) d\sigma_{\text{cyl}}} $$ (5.32) As before, $1/g$ can then be computed from Eq. (5.7). As basis functions, we use the polynomials $$ \begin{aligned} b_0^{\text{var}}(r) &= 1 \\ b_1^{\text{var}}(r) &= 1 - (r/R)^2 \end{aligned} $$ (5.33) Evaluation of the $B_{k,m}^{\text{var}}$ matrix elements is straightforward; evaluation of the $A_{k,m}^{\text{var}}$ requires a lot of tedious computations that are described in detail in Appendix B of Holstein (1951). We finally get --- THE VARIATIONAL TECHNIQUE $$ \begin{aligned} A_{0,0}^{\text{var}} &= \frac{\pi^2}{2^{u+3}} \frac{C_{\text{stat}}}{R^{u-2}} \frac{(1+u)!(1-u)!}{[\frac{1}{2}(1+u)]!^2 [1-\frac{1}{2}u]!^2} \frac{2-u}{1-u} \frac{1}{k_0^u} & B_{0,0}^{\text{var}} &= \pi R^2 \\ A_{0,1}^{\text{var}} &= \frac{\pi^2}{2^{u+3}} \frac{C_{\text{stat}}}{R^{u-2}} \frac{(1+u)!(1-u)!}{[\frac{1}{2}(1+u)]!^2 [1-\frac{1}{2}u]!^2} \frac{4}{4-u} \frac{1}{k_0^u} & B_{0,1}^{\text{var}} &= \frac{\pi}{2} R^2 \\ A_{1,1}^{\text{var}} &= \frac{\pi^2}{2^{u+3}} \frac{C_{\text{stat}}}{R^{u-2}} \frac{(1+u)!(1-u)!}{ --- ## 5.2 Exact solutions in the high-opacity case ### 5.2.1 Van Trigt's solution In a series of papers, the Dutch theoretician C. van Trigt computed a solution of the Holstein equation that is exact in the limit of infinite opacity (van Trigt 1969–75), see also (van Trigt 1973). **For a Doppler line in a slab**, the eigenfunctions are $$ \psi_j(z) = \sqrt{1 - \left(\frac{2z}{L}\right)^2} \sum_{m=0}^{\infty} a_{m,j}^{\text{D}} U_m \left(\frac{2z}{L}\right), $$ (5.38) where the $U_m(z)$ are the Chebycheff polynomials of the second kind and the factors $a_{m,j}^{\text{D}}$ are the expansion coefficients. The coefficients $a_{m,j}^{\text{D}}$ for $m \neq j$ are quite small, which shows that the polynomials $U_j$ are good approximations to the eigenfunctions. The trapping factors are $$ g_j = \frac{1}{m_j^{\text{D}}} k_0 L \sqrt{\ln \left(\frac{k_0 L}{2}\right)} $$ (5.39) The values of the constants $m_j^{\text{D}}$ are given --- EXACT SOLUTIONS IN THE HIGH-OPACITY CASE * The equations are valid only in the limit of infinite opacity, while Holstein's results are valid for large opacities. Comparisons of Holstein's and van Trigt's eigenvalues with more accurate numerical computations show that in the practically occurring high-opacity cases ($20 \le k_0L \le 1000$), Holstein's approximations give better results than van Trigt's. * The eigenfunctions in Eq. (5.38) and (5.40) are—due to the infinite-opacity assumption—independent of the opacity, so that they deviate considerably from the eigenfunctions at finite opacities. * van Trigt treats Voigt and Lorentz profiles as identical. This is justified in the infinite-opacity approximation, because then photons can escape only in the wings (which are Lorentzian in a Voigt profile). This treatment is, however, *not* always justified for large but finite opacities. The main advantage of van Trigt's formulation, on the other hand, is that it also computes the higher-order eigenvalues and eigenfunctions in an analytical way. ### 5.2.2 The geometrical quantization technique A more general, but also more complicated method was developed by Asadullina *et al.* (1989). The method is called the 'geometrical quantization technique' (GQT). It is based on performing a Fourier transform of the Holstein equation and writing the result in the same way as the Schrödinger equation $$ \begin{gathered} H^{\text{H}}(\mathbf{r}, -\underline{j}\nabla_r)n(\mathbf{r}) = \frac{1}{g\tau}n(\mathbf{r}) \qquad H^{\text{H}}(\mathbf{r}, \mathbf{p}) = \frac{W(\mathbf{p})}{\tau} + Q(\mathbf{r}) \\ W(\mathbf{p}) = 1 - \frac{1}{|\mathbf{p}|} \int_{-\infty}^{\infty} C_x k(x)^2 \text{atan}\left(\frac{|\mathbf{p}|}{k(x)}\right) \text{d}x \end{gathered} $$ (5.42) where $H^{\text{H}}$ is the Hamiltonian, $Q(\mathbf{r})$ is the quenching rate at position $\mathbf{r}$, and $\mathbf{p}$ are the Fourier transform coordinates, which can be interpreted as a momentum. At first glance, it does not seem as if anything has been gained by this reformulation. However, since we now have the formulation of the Schrödinger equation, we can use all the sophisticated mathematical tools developed for just this application.[^8] The function $W$, which is analogous to the kinetic energy, is the Fourier transform of $1 - G$ over infinite space. However, reabsorption of photons can only happen in the vapour cell. To account for that, we set the quenching rate (which corresponds to the potential energy) to infinity outside the vapour cell, and to zero inside the cell. This means that Eq. (5.42) becomes the Schrödinger equation of a particle in a potential well. One solution technique for the Schrödinger equation is the WKB method, also known as the short-wavelength approximation. The approximation inherent in this method means that it is suited for high opacities. [^8]: Note, however, that this is a purely mathematical analogy. We are still dealing with the trapping problem by the Holstein equation, i.e. a semiclassical description. The GQT is *not* a correct quantum-mechanical description of trapping. --- Under this condition, the function $W(\mathbf{p})$ has, **for a Doppler lineshape**, the form (note a misprint in (Asadullina *et al.* 1989, Eq. 9)) $$ W(\mathbf{p}) = \frac{\sqrt{\pi}}{4} \frac{|\mathbf{p}|}{k_0} \frac{1}{\sqrt{\ln \left( \frac{k_0}{|\mathbf{p}|} \right)}} $$ (5.43) **For a Lorentzian lineshape**, the result is $$ W(\mathbf{p}) = \frac{\sqrt{2}}{3} \sqrt{\frac{|\mathbf{p}|}{k_0}} $$ (5.44) The quantization conditions giving the allowed values for $\mathbf{p}_j$ must be computed by very complicated topological methods. Once the $\mathbf{p}_j$ are known, the trapping factors can be computed easily by $1/g_j = V(\mathbf{p}_j)$. **For the plane-parallel slab**, the result is $$ p_j = \frac{\pi}{2L} (2j + 1 + \gamma) \qquad j = 0, 1, \dots $$ (5.45) where $\gamma = 0.5$ for Doppler lineshapes and $0.25$ for Lorentzian lineshapes. **For a cylinder**, $$ p_j = \frac{\pi}{2R} \left(2j + 1 + \frac{\gamma}{2}\right) \qquad j = 0, 1, \dots, $$ (5.46) and **for a sphere** $$ p_j = \frac{\pi}{2R} \left(2j + 1.5 + \frac{\gamma}{2}\right) \qquad j = 0, 1, \dots $$ (5.47) Generally, the GQT with the approximations in Eqs (5.43) and (5.44) suffers from the same drawbacks as van Trigt's solution. Furthermore, results are less accurate than van Trigt's. Its main advantage lies in the fact that there are ready-made results for two-and three-dimensional geometries (see Chapter 10). A related method was also proposed by Nagirner (1994–1996) for a cylinder. ### 5.2.3 The Fourier transform technique A somewhat related technique is the Fourier transform technique suggested by Alley (1983) and extended by Streater *et al.* (1988a). They consider the transfer equation in an infinitely extended medium. We then define the function $\tilde{G}$, which is related to the Fourier transform of the Kernel function $$ \tilde{G}(\mathbf{p}, x) = \frac{k(x)}{|\mathbf{p}|} \text{atan} \left( \frac{|\mathbf{p}|}{k(x)} \right) $$ (5.48) The transfer equation then becomes $$ \tilde{J}(\mathbf{p}, x, t) = \tilde{G}(\mathbf{p}, x)\tilde{S}(\mathbf{p}) $$ (5.49) where $\tilde{S}$ is the Fourier transform of the source function. --- The excited-state distribution can be computed from $$ \tilde{n}(\mathbf{p}, t) = \tilde{n}(\mathbf{p}, t = 0) \exp ( - A_{21} t \tilde{\eta}(\mathbf{p}) ) $$ where $$ \tilde{\eta}(\mathbf{p}) = 1 - \int C_x k(x) \tilde{G}(\mathbf{p}, x) dx $$ (5.50) The excited-state density in the spatial domain is then computed by an inverse transform. For a Cartesian coordinate system, we use the usual Fourier transform. For cylindrical geometries, the Hankel transform is the appropriate transform. (For the computation of the expansion coefficients, an efficient program is publicly available (Piessens 1982).) As mentioned above, one of the basic assumptions is that the medium is infinitely extended. For that reason, similar treatments were used in the first days of the astrophysical treatment of trapping (see e.g. Veklenko (1957, 1959b)). In chemical physics, this method is especially of interest when we deal with a vapour cell with a high opacity, which is excited, e.g., by a laser beam near the cell axis, and we want to compute the excited-state distribution shortly after the laser pulse. In that case, the number of photons that have completely escaped from the vapour is very small, in other words, most photons are reabsorbed within the vapour. In that case, it does not matter very much whether one assumes an infinitely extended medium or the actual cell geometry, so that we can put this method to use. There are also extensions for the case of partial frequency redistribution; these are described by Streater *et al.* (1988a) and Alley (1983). ## 5.3 The piecewise-constant approximation An exact analytical solution of the Holstein equation (at arbitrary opacities) is not possible even in one-dimensional geometries, so that we often have to take refuge in numerical methods. In this section, we describe a simple yet efficient method for the numerical solution of the Holstein equation. It is based on a piecewise-constant approximation of the eigenfunctions and will thus be referred to as the PCA approach. This method was first proposed by Biberman (1947) and was later extended by Phelps (1958) and Molisch *et al.* (1992a, 1993a, c, d). The basic principle is to divide the vapour cell into $N_r$ subregions with centres $\mathbf{r}_k$ and assume that $\psi(\mathbf{r})$ is constant within each subregion. This means that we approximate the eigenfunctions $\psi_j$ by a finite orthogonal system of functions $$ \psi(\mathbf{r}) \approx \sum_{m=0}^{N_r-1} \psi(\mathbf{r}_m) p_m(\mathbf{r}), $$ (5.51) where $p_m(\mathbf{r})$ is 1 inside the subregion $m$ and 0 elsewhere. Inserting Eq. (5.51) into the Holstein equation, Eq. (5.1), then multiplying by $p_k(\mathbf{r})$, integrating over the volume and using the orthonormality relations, we get a system of $N_r$ homogeneous linear algebraic equations with the unknowns $\psi(\mathbf{r}_k)$ --- $$ \psi(\mathbf{r}_k) \cdot \left(1 - \frac{1}{g}\right) - \sum_{m=0}^{N_r-1} \psi(\mathbf{r}_m) A_{k,m} = 0. $$ (5.52) The matrix elements $A_{k,m}$ are $$ A_{k,m} = \int_{\text{subregion } m} G(\mathbf{r}_k, \mathbf{r}')\mathrm{d}\mathbf{r}'. $$ (5.53) Inserting the Kernel function of Eqs (4.16)–(4.18) into equation (5.53), we get expressions for the $A_{k,m}$ that require the evaluation of threefold (for slab and sphere) or fourfold (for the cylinder) integrals. The vital point for the PCA computations is that by some straightforward but tedious manipulations and functional approximations, these expressions can be reduced to single or double integrals. The derivation of these simplified expressions can be found in Appendix B. For an infinite cylinder and pure Lorentz broadening, an alternative formulation of the $A_{k,m}$ elements can be found in (Pinhao 1994). For the sphere, an alternative formulation is derived by Schmid-Burgk (1975). By inserting the appropriate $A_{k,m}$ elements into Eq. (5.52), we get an algebraic eigenvalue problem $$ \psi(\mathbf{r}_k)\lambda = \sum \psi(\mathbf{r}_m)A_{k,m}. $$ (5.54) In the slab case, the matrix is symmetric (it is even a Toeplitz matrix), so that Eq. (5.54) can be solved by a Jacobi algorithm. In the cylinder and the sphere cases, the matrix is not symmetric (but is of course real) and Eq. (5.54) is solved by reduction to Hessenberg form and then by a QR algorithm (Press *et al.* 1993). Details of the computer evaluation are given in Appendix C. We assume for all computations that the subregions are spaced equidistantly. For the slab and the sphere cases, this leads to considerable savings in CPU time, because the $A_{k,m}$ elements then depend only on $|k - m|$ (and for the sphere additionally on $k + m$). Furthermore, non-equidistant discretization gives no significant improvement in accuracy and can lead to numerical problems in the computation of the eigenvalues. Basically, it is not necessary to use piecewise-constant $n(\mathbf{r})$ for the above method. One could use any orthonormal set of functions. However, only the piecewise-constant approximation gives the simple result of Eq. (5.53) for the $A_{k,m}$ matrix elements. When we want to use a more general set of basis functions $$ \psi(\mathbf{r}) = \sum_m a_m b_m(\mathbf{r}), $$ (5.55) the $A_{k,m}$ must be computed as $$ A_{k,m} = \iint G(\mathbf{r}, \mathbf{r}')b_m(\mathbf{r}')b_k(\mathbf{r})\mathrm{d}\mathbf{r}'\mathrm{d}\mathbf{r}. $$ (5.56) For the slab case, piecewise linear and piecewise quadratic functions have been proposed (Kalkofen 1974b), (Avrett and Kalkofen 1968), (Kurucz 1969). For very high --- [FIGURE: FIG. 5.3. Results of the PCA method: trapping factors for a Doppler lineshape for the first three modes $g_0^D$, $g_1^D$, and $g_2^D$.] opacities, quadratic representations are essential to get physically reasonable results (flux conservation), as is known from astrophysics (Kalkofen 1974b). For the opacity range occurring in laboratory situations, however, this problem is not relevant. Our computations with the PCA method gave results that agree well with the results of van Trigt, which are exact in the high-opacity limit. Thus, the increase in accuracy when using piecewise linear or quadratic functions is usually not worth the trouble of the more complicated $A_{k,m}$ computation. This is especially true if multiple integrals occurring in Eq. (5.56) cannot be reduced any more. A more complicated $A_{k,m}$ computation can cost *much* more computer time than a computation with piecewise constant $n(\mathbf{r})$ and slightly increased number of subregions, $N_r$, to get higher accuracy. Hearn (1966) uses Chebycheff polynomials as basis functions for the steady-state case and claims that 20th order polynomials are required at very high opacities. ## 5.4 The propagator function method Lawler, Parker, and Hitchon (Lawler *et al.* 1993), (Parker *et al.* 1993) have developed a new numerical method for the solution of the Holstein equation that incorporates some properties both of the PCA and of the multiple-scattering representation. As in the PCA method, the cell is divided into subregions and the probability that a photon that is emitted in one subregion is reabsorbed in a different subregion is computed. Essentially, this means that we compute the $A_{k,m}$ elements of the PCA method (which Lawler calls the propagators). However, the evaluation is somewhat different. --- The time derivative of the excited-state distribution is approximated as $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} \approx \frac{n(\mathbf{r}, t + \Delta t) - n(\mathbf{r}, t)}{\Delta t} $$ (5.57) so that the Holstein equation becomes $$ n(\mathbf{r}, t + \Delta t) = E(\mathbf{r}, t)\Delta t + n(\mathbf{r}, t) - \frac{\Delta t}{\tau} \left( n(\mathbf{r}, t) - \int_V n(\mathbf{r}', t)G(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}' \right). $$ (5.58) with the piecewise-constant approximation for $n(\mathbf{r})$, this becomes $$ n(\mathbf{r}_k, t + \Delta t) = n(\mathbf{r}_k, t) \left( 1 - \frac{\Delta t}{\tau} \right) + \frac{\Delta t}{\tau} \sum_m A_{k,m} n(\mathbf{r}_m, t) + E_k(t)\Delta t $$ (5.59) We then start out with a given initial distribution $n(r_k, 0)$ and use Eq. (5.59) to compute the distribution at time $\Delta t$. We get $n(r_k, \Delta t)$, which is then used on the right-hand side of Eq. (5.59) to compute $n(r_k, 2\Delta t)$, and so on. When we compare this method to the PCA, we see that it can be faster, because we have to perform no eigenvalue search, but only use summations (multiplication of the matrix $\mathbf{A}$ with the vector $n(\mathbf{r}_m)$). On the other hand, if we use a very fine temporal discretization, we have to perform a lot of these operations, which can also take some time. Furthermore, we have to do the computation of $n(r_k, i \cdot \Delta t)$ anew for each initial distribution we want to analyse (we can of course reuse the $A_{k,m}$ values). In this respect, the method is quite similar to a multiple-scattering solution, as we will see in Chapter 6. Generally, we believe that PCA is superior to PFM if the number of subcells is reasonably low (i.e. $< 300$) and repeated computations are required. On the other hand, PFM is well suited for computations including incomplete frequency redistribution (see Sec. 11.4) and saturation (see Chapter 13). A method that is very similar to the PFM method is the one proposed by Apruzese. For the steady state, his method is just the computation of the $A_{k,m}$ matrix elements by using an approximate expression for $A_{k,m}$, and solving the resulting system of equations. These approximate expressions are essentially analytical fits for the escape factor $\eta$, and saying that $A_{k,m} = \eta(\mathbf{r}_{\text{bound2}}) - (\mathbf{r}_{\text{bound1}})$, where bound1 and bound2 are the boundaries of cell $k$ closer and farther away from the cell of emission, respectively. This method is described in further detail in (Apruzese *et al.* 1980), (Apruzese 1981), (Duston and Davis 1981), (Duston *et al.* 1983), and (Apruzese 1985). Extensions to continuum radiation can be found in (Clark *et al.* 1995). Similarly, Avrett and Loeser (1984) compute integral operators by following a single ray through the vapour (i.e. making a one-point quadrature). ## 5.5 Other computation methods Apart from the methods described in the previous sections, there are several other approaches to solving trapping problems that are not covered in detail here. We just delineate the basic ideas—the interested reader is referred to the references given. --- ### Numerical methods: Gruschinske and Ueno (1971) converted the Holstein equation into a coupled set of differential equations. However, this seems to give no clear advantages, especially since we now have an initial values problem that has to be integrated from $0$ to $\infty$, probably causing numerical problems. Avrett and Hummer (1965) introduced the so-called 'kernel approximation method' for the computation of the steady-state. In this method, the Kernel and the excitation function are represented as a sum of exponential functions of $|z' - z|$. For such exponential functions, the solution of the Holstein equation can be written down exactly. Numerical values for the expansion coefficients in the slab case are given in the appendix of Avrett and Hummer (1965). Note that this expansion can only be approximate, since a sum of exponential functions cannot represent correctly the singularity of the Kernel function at $|z - z'| = 0$. Athay (1965), and Athay and Skumanich (1967) proposed the 'flux-divergence method', where the steady-state Holstein equation is rewritten in the basic form (see also (Athay 1972a), (Cannon 1985)) $$ \begin{aligned} 0 = \frac{C_x}{2} \int_{-\infty}^{\infty} \frac{\partial}{\partial z} \left[ \int_z^{L/2} k(x)\mathrm{Ei}_2 \left[ k(x)(z' - z) \right] n(z')\mathrm{d}z' - \right. \\ \left. - \int_{-L/2}^z k(x)\mathrm{Ei}_2 \left[ k(x)(z - z') \right] n(z')\mathrm{d}z' \right] \mathrm{d}x \end{aligned} $$ (5.60) This formulation has the advantage that the new *Kernel* does not have a singularity at $z = z'$. On the other hand, this singularity is removed by the analytical integration in the piecewise-constant method anyway. Jones (1973) and Jones and Skumanich (1973) solved multi-dimensional steady-state problems by an integral-equation technique that reduces the problem to an algebraic matrix problem. Spherical problems are treated with an integral technique by Kalkofen and Wehrse (1984). A different integral technique, which is mainly applicable when we have the source located in the centre of the sphere, is used by Schmid-Burgk and Scholz (1984). The influence of continua on steady-state problems is discussed by Hummer (1968a) and Elitzur and Ferland (1986). Quantum-mechanical descriptions with a density-matrix approach include the papers of D'yakonov and Perel (1965) (see also Chapter 12), Preobrazhenskii and Senina (1971), and Melnikov and Polivenko (1987) who include non-linear effects. ### Analytical methods: Exact solutions of the trapping problem have been derived by means of reducing the Holstein equation to a singular integral equation (Cannon 1985), (Abramov *et al.* 1969), (Bishnu and Das Gupta 1987), (Ivanov 1973), (Sobolev 1963), (Frisch and Frisch 1977, 1982), (Frisch and Froeschle 1977), (Ivanov 1994). Kernel functions for infinite space were found by Veklenko (1957, 1959b). A Fourier transform method, also using Kernel functions, is used by Cannon (1973c). Asymptotic expressions for the Kernel function in the slab are given by Hummer (1982b); for a line plus continuum by Hummer (1991). Solutions of the slab problem in terms of the Chandrasekhar $X$ and $Y$ functions are given by Heaslet and Warming (1968), Yelle (1988), Yelle and Wallace (1989), and Wallace and Yelle (1989); the invariance principle is applied to spherical geometries by Heaslet and Warming (1965). --- # 6 # METHODS FOR THE MULTIPLE-SCATTERING REPRESENTATION For the multiple-scattering (MS) representation, we have to solve for the probability that a photon escapes after suffering $i$ absorption/reemisison processes.[^9] For this computation, we have two possible approaches: Monte Carlo simulations, which have been used for 30 years, and analytical methods (Wiorkowski 1988), (Wiorkowski and Hartmann 1988), (Falecki *et al.* 1989, 1991). ## 6.1 Monte Carlo simulations A Monte Carlo (MC) simulation is a very general, purely numerical algorithm for the modelling of complex physical processes. In the case of radiation trapping, it means tracing one photon after the other on its path through the vapour (House and Avery 1969), (Avery and House 1968). The photon's emission frequency, direction, and path length before reabsorption are taken at random, according to the appropriate probability distributions. This means that the numbers are taken at random, but in such a way that they fulfil the statistics of the described physical process. In order to model a Doppler lineshape, e.g., the emission frequency must have a probability density function $\mathrm{pdf}(x) = \mathrm{const} \cdot \exp(-x^2)$. Since probabilities take values between 0 and 1, the constant $\mathrm{const}$ has to be chosen such that the integral over the pdf is normalized to 1. For our example of a Doppler lineshape, $$ \int_{-\infty}^{\infty} \exp(-x^2)\mathrm{d}x = \sqrt{\pi} \quad \Rightarrow \quad \mathrm{const} = 1/\sqrt{\pi}. \qquad (6.1) $$ Tracing a photon ends when it has left the cell. After tracing a lot of photons (typically $10^4$ for 1% accuracy in a plane-parallel slab), we can compute the set of probabilities $p_i$ that a photon has escaped from the vapour after exactly $i$ absorption–reemission processes. These probabilities $p_i$ constitute the output of the MC program. A Monte Carlo program to calculate the time-decay of some initial distribution $n(\mathbf{r}, 0)$ always follows roughly the same algorithm: 1) Select the place of the first photon emission at random from the initial distribution of excited atoms. The probability density function $\mathrm{pdf}(\mathbf{r}) = \mathrm{const} \cdot n(\mathbf{r}, 0)$. [^9]: We have already shown in Sec. 4.3 how to compute, e.g. the emergent radiation from these $p_i$. r r w C : C C : M h h f C u i M L C M L h a D D h e f L M h f c a f d d L C L L a C L L L d C --- 2) Select the frequency of the emitted photon from the lineshape distribution; pdf($x$) $= C_x \cdot k(x)$. 3) Select the direction of emission from an isotropic three-dimensional distribution. For a plane-parallel slab this means that the $\cos(\vartheta)$ has a uniform distribution, where $\vartheta$ is the angle between the $z$-axis and the photon's flight direction. 4) Select the length the photon covers before being reabsorbed from pdf($\rho$) = const $\cdot \exp(-k(x)\rho)$. 5) Since we know the point of emission, the distance between the points of emission and reabsorption, and the direction of the photon, we can compute the point of reabsorption. If the point of reabsorption lies within the vapour cell, we note the position of the newly excited atom, note that the photon has suffered one absorption–reemission process, and go back to step 2. If the point of reabsorption lies outside, the photon has left the vapour cell. We know that it had previously suffered $i$ absorption–reemission processes. Hence, we increase by one the number of photons that have escaped after $i$ absorptions and go back to step 1. 6) After we have observed a sufficient number, $N_{sim}$, of photons, we stop tracing and compute the list of probabilities $p_i$ that a photon has escaped from the vapour after exactly $i$ absorption–reemission processes. The probabilities $p_i$ are computed as the number of photons that have escaped after $i$ absorptions, divided by $N_{sim}$. At high opacities, a photon has to go through many absorption/reemission processes before it leaves the vapour. The Monte Carlo simulation then becomes quite slow. In order to alleviate this situation, a technique called 'weighting' can be used (House and Avery 1969), see also (Auer 1968), and (El-Samie Mostafa *et al.* 1977). In this technique, we emit photons not according to the true pdf (e.g. Doppler), but as some distorted pdf that is more biased towards the wings of the line. As a compensation, the photons emitted near the centre of the line have more 'weight'. Physically, we can interpret this technique the following way: we follow not single photons through the vapour, but 'bundles' of photons. If the frequency is in the line centre, we bundle a lot of photons together, and follow them all at once. If the frequency is in the wings, then we follow smaller bundles. However, in chemical physics, the opacities are rarely so large that the CPU-time savings allowed by this technique are worth the trouble; it is used, however, in astrophysics (Ferland and Netzer 1979), even though there are numerical problems (Meier and Lee 1978), (Roark *et al.* 1974). For the temporal behaviour of the **distribution** of the excited atoms, we have to divide the vapour cell into small subvolumes and to observe in each subvolume the number of photons that are absorbed there after $i$ absorptions and reemissions. However, to get good statistical averaging within **each subvolume**, we require a lot more photons in the MC simulation than we need for getting just the total number of excited-state atoms, $n^\Sigma(t)$, and the emergent radiation, $Y(t)$. --- ### Insert 6.1 Generation of random numbers with arbitrary statistics Most random number generators generate numbers $x$ with a uniform distribution $\text{pdf}(x) = 1$ for $0 < x < 1$. A different $\text{pdf}(x)$ can be realized in the following way: first, the probability density function $\text{pdf}(x)$ is integrated (and, if necessary, normalized) to give the cumulative distribution function $\text{cdf}(x)$ $$ \text{cdf}(x) = \int_{-\infty}^{x} \text{pdf}(u)du \Bigg/ \int_{-\infty}^{\infty} \text{pdf}(u)du. (6.2) $$ Note that the denominator in (6.2) should be 1 if the pdf has been properly normalized. We then take a random number 'random', ($0 < \text{random} < 1$), from the uniform-distribution random number generator and search for an $x$ so that $\text{cdf}(x) = \text{random}$. The so-found number $x$ is distributed according to the desired pdf. This is best understood by having a look at Fig. 6.1. Finding an $x$ such that $\text{cdf}(x) = \text{random}$ can be done very easily when $\text{cdf}(x)$ is an analytically invertible function, i.e. when we can find the inverse function. If this is not the case, $\text{cdf}(x)$ can be computed in advance at discrete points $x_i$. Then whenever we need a random number corresponding to $\text{cdf}(x)$, we search for the $x_i$ so that the uniformly distributed random number 'random' lies between $\text{cdf}(x_i)$ and $\text{cdf}(x_{i+1})$. This can be done with a very fast binary search. (A binary search is described in any standard textbook on computer programming, e.g. (Knuth 1973)). We then just have to make a linear interpolation between $x_i$ and $x_{i+1}$ to get a highly accurate approximation for $x$. [FIGURE: Graphs of pdf(x) and cdf(x) vs Normalized frequency x] FIG. 6.1. Generation of random numbers with arbitrary statistics. As an example, the desired pdf($x$) is the spectrum of a hyperfine-split spectral line. --- Some **shortcuts for often needed cases:** **For the Lorentzian pdf,** the cdf can easily be inverted. The cdf is $$ \text{cdf}(x) = \frac{1}{2} + \frac{\text{atan}(x)}{\pi} $$ (6.3) **For a Gaussian pdf,** we use a trick. First, we generate a random number $\text{random}_\text{R}$ that obeys Rayleigh statistics, $$ \text{random}_\text{R} = \sigma\sqrt{-2\ln(1 - \text{random})} $$ (6.4) where $\sigma$ is the variance of the wanted Gaussian distribution. Next we generate an independent random number $\text{random}_\text{U}$ that is uniformly distributed between 0 and $2\pi$, $$ \text{random}_\text{U} = 2\pi \cdot \text{random} $$ (6.5) and finally obtain two independent Gaussian random numbers as $$ \text{random}_\text{R} \cdot \cos(\text{random}_\text{U}) \quad \text{and} \quad \text{random}_\text{R} \cdot \sin(\text{random}_\text{U}). $$ (6.6) **For a Voigt pdf,** we can use the fact that the Voigt profile is defined as the convolution of a Doppler and a Lorentzian profile. We thus first generate a Gaussian random number $\text{random}_\text{G}$ with $\sigma = 1/\sqrt{2}$ and a uniformly [0, 1] distributed random number 'random' and get the Voigt random number (Lee 1974b) from $$ \text{random}_\text{V} = a \cdot \tan\left((\text{random} - \frac{1}{2})\pi\right) + \text{random}_\text{G} $$ (6.7) **For a (completely intermixed) hfs line,** we first 'throw the dice' to determine in which hfs component the photon is to be emitted, taking the relative strengths of the components into account. Then we simply determine the photon's shift from the component line centre according to the appropriate lineshape (Doppler, Lorentz or Voigt). --- ### Insert 6.2 Pitfalls in the interpretation of the results of a Monte Carlo simulation In (Klots and Anderson 1972), there is a Monte Carlo simulation of trapping assuming a Doppler lineshape. The output of this simulation is the probability $P(I)$ for a photon to remain inside the slab after $I$ absorptions. It is assumed that $P(I)$ has the same functional dependence on $I$ as the time decay $n(z, I \cdot \tau)$. Taking Eq. (4.31) for just one mode $j$, it follows that the probability that a photon in the $j$th mode escapes after exactly $i$ absorptions is $$ p_{ij} = \frac{1}{g_j} \left(1 - \frac{1}{g_j}\right)^{i-1}. $$ (6.8) With this result, $P_j(I)$ can be computed by summing the geometrical series $$ P_j(I) = 1 - \sum_{i=1}^I p_{ij} = \left(1 - \frac{1}{g_j}\right)^I \approx \exp\left(-\frac{I}{g_j}\right). $$ (6.9) --- [FIGURE: Comparison of the exact P_j(I) (solid) with the approximation, Eq. (6.9) (dashed).] Klots and Anderson (1972) use the last relation at high $g_j$, where the last relation actually holds. However, we warn that it cannot be applied at intermediate or low $g_j$, see Fig. 6.2. A further danger occurs when we want to compute higher-order trapping factors from MC simulations. This has been done in the literature by releasing all photons at the centre of the slab, and the resulting time distribution is approximated by the sum of two exponential functions. Expanding this initial spatial distribution (a delta function) into eigenfunctions of the Holstein equation, we see that many modes—certainly more than two—have significant amplitudes. --- MC simulations can also be used to compute the lowest-order trapping factor in a straightforward way. When the initial distribution of excited atoms is the eigenfunction, the average number of absorptions $\bar{p}$ is equal to the trapping factor. Thus, if we use the lowest-order eigenmode (as obtained from the solution of the Holstein equation) as initial distribution, the MC simulation allows for the computation of the trapping factor $g_0 = \bar{p} = \Sigma i \cdot p_i$. Although the MC computation of $g_0$ uses a result of the Holstein equation (the shape of the lowest-order mode), it is still an (almost) independent method, because the average number of absorptions differs very little from the trapping factor even if the initial distribution has only a passing resemblance to the eigenmode. For the slab geometry, this can be shown in the following way. We have seen in Sec. 4.5 that the two most extreme shapes of the eigenfunction we can imagine are the spatially uniform distribution and a Dirac delta function at $z = 0$. The trapping factor must always lie between the average number of absorptions for these two initial distributions. It is furthermore obvious that the difference between these two cases will be a maximum at high opacities—at very low opacities there is no trapping, and $g_0$ and $\bar{p}$ tend to 1. For the lowest-order eigenmode we get at high opacities (see Appendix D), $$ g_0 = \frac{k_0 L}{1.025 \sqrt{\ln \left( \frac{k_0 L}{2} \right)}}. $$ (6.10) We compare this result with the $\bar{p}$ obtained for the two extreme initial distributions. For very high opacities, $k_0 L$, and a Doppler line, van Trigt (1969–75) showed that --- $$ \bar{p} = \frac{2k_0L}{\sqrt{\pi}} \frac{\sqrt{\ln \left( \frac{k_0L}{2} \right)}}{\int_{-L/2}^{L/2} n(z, 0)dz} \int_{-L/2}^{L/2} n(z, 0)\sqrt{1 - \left(\frac{2z}{L}\right)^2} dz, $$ (6.11) so that for a spatially uniform distribution, $$ \bar{p}_U = \frac{\pi}{4} \frac{2k_0L}{\sqrt{\pi}} \sqrt{\ln \left( \frac{k_0L}{2} \right)} = 0.908g_0, $$ (6.12) and for the delta distribution, $$ \bar{p}_\delta = \frac{2k_0L}{\sqrt{\pi}} \sqrt{\ln \left( \frac{k_0L}{2} \right)} = 1.157g_0, $$ (6.13) The upper and lower limits, $\bar{p}_U$ and $\bar{p}_\delta$, are separated by a factor of only $\pi/4 = 0.79$. Hence, as long as the initial distribution in an MC simulation is chosen so that it at least resembles the lowest-order mode of the Holstein-solution, the average number of reabsorptions, $\bar{p}$, can safely be set equal to the lowest-order trapping factor $g_0$. A check on how well the initial distribution resembles the lowest-order eigenfunction can also be done by Eq. (4.31): when the initial distribution is the eigenfunction, then $g_0$ equals $\bar{p}$, the average number of reabsorptions, and the $p_i$ must fulfil $$ p_i = \frac{1}{\bar{p}} \left( 1 - \frac{1}{\bar{p}} \right)^{i-1}. $$ (6.14) The accuracy of the eigenfunction is indicated by how well Eq. (6.14) is fulfilled. An alternative approach is of course to discard all the $p_i$ with 'small' values of $i$ (i.e. at those times where higher-order modes still play a role) and compute the trapping factor as $(1 - 1/g_0) = p_{i+1}/p_i$ with $i$ very large (see also Sec. 6.2). The drawback of that method is that at these 'late times', the $p_i$ are rather small, so that the statistical uncertainties of the MC simulation have a larger influence than at 'early times', where we have large $p_i$. We thus trade off a systematic error (wrongly shaped initial distribution) against a random error (larger statistical uncertainty of the MC method). The decision of which method we want to use depends mainly on the prior information and the CPU time we have. Yet another method is to compute $Y(t)$ from the equations of Chapter 4.3, and to fit an exponential curve at 'late times'. The time constant of the exponential then gives the trapping factor $g_0$. When we compare the MC simulation to the solution of the Holstein equation, we see that each approach has specific advantages. An MC simulation is simply a computer simulation of the physical processes in the vapour on a microscopic scale, so that once --- we understand the physical processes in the vapour, it is quite easy to write an MC program. Even very complex processes can be straightforwardly implemented. A genuinely three-dimensional geometry requires just a small change in the computer code. With the Holstein equation approach, this would require much more complicated approaches, e.g. in the PCA method, we have to compute fivefold integrals, which cannot be evaluated analytically. It is also quite easy to incorporate partial frequency redistribution or polarization effects in the MC scheme. When the assumptions and idealizations listed in Sec. 4.2.3 are not fulfilled, an MC simulation provides an easy way to circumvent this difficulty. The advantage of the Holstein equation, on the other hand, is that once the equation is solved and the eigenvalues and eigenfunctions are known, any arbitrary initial distribution can be dealt with analytically—the Holstein equation has to be solved only once. Furthermore, an analytical representation is an advantage by itself, because it allows for an intuitive grasp of the effects of a change in one controlling parameter. We have seen in Chapter 5, however, that the solution of the Holstein equation is quite difficult, and generalizations are not easily made. Comparing the PFM (propagator function method) with MC, we can give no absolute conclusion about which is the better method. The MC simulation is easier to program, and generalizations to more complicated geometries can be done more easily. The PFM method is, however, faster once we have computed the propagators. Lawler also proposed a combination of the two methods, namely to compute the propagators by an MC simulation and then use them in a PFM computation of $n(r_k, i \cdot \Delta t)$. The propagators ($= A_{k,m}$ elements) computed by this method could of course also be used in a PCA computation. ## 6.2 Analytical solutions The analytical solution to the trapping in the MS representation (henceforth called the AMS method) is a kind of intermediate step between a Monte Carlo simulation and the methods of Chapter 5. On one hand, the solution is in terms of the $p_i$, exactly as in the case of the MC simulation. On the other hand, these $p_i$ are not computed by a purely numerical method, but as a solution of the Holstein equation. In a first step, we write the Holstein equation as $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau}n(\mathbf{r}, t) + \frac{1}{\tau}\Lambda n(\mathbf{r}, t) $$ (6.15) where the operator $\Lambda$ is defined so that $\Lambda n(\mathbf{r}, t) = \int n(\mathbf{r}', t)G(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}'$. Now we have seen in Chapter 4 that the emergent radiation can be written as the sum of terms that describe photons that escape after the first emission, the second emission, and so on: $$ Y(t) = \sum_i p_i Y_i(t) = \frac{1}{\tau} \sum_i p_i \left( \frac{t}{\tau} \right)^{i-1} \frac{\exp(-t/\tau)}{(i - 1)!}. $$ (6.16) In a completely analogous way, we can write the excited-state density as a sum of atoms that are created by first-generation photons, second-generation photons, etc. --- ANALYTICAL SOLUTIONS $$ n(\mathbf{r}, t) = \sum_{i=1}^{\infty} n_i(\mathbf{r}) \left(\frac{t}{\tau}\right)^{i-1} \frac{\exp(-t/\tau)}{(i-1)!}. $$ (6.17) When we now insert this relation into the Holstein equation, Eq. (6.15), we get a recurrence relation for the $n_i(\mathbf{r})$, $$ n_i(\mathbf{r}) = \Lambda n_{i-1}(\mathbf{r}) $$ (6.18) where $n_1(\mathbf{r})$ is of course the initial distribution of excited-state atoms. Formally, this means that we are computing the Neumann series of the integral equation (Preobrashesnkii and Senina 1971). Since the emergent radiation $Y(t)$ is related to the excited-state population as (see Chapter 4, Eqs 4.26 and 4.27) $$ Y(t) = -\frac{\partial}{\partial t} \int_V n(\mathbf{r}, t)\mathrm{d}\mathbf{r} $$ (6.19) the escape probabilities $p_i$ and the atom generations $n_i(\mathbf{r})$ are related by $$ p_i = \int_V (n_i(\mathbf{r}) - n_{i+1}(\mathbf{r}))\mathrm{d}\mathbf{r} $$ (6.20) However, the computation of the $n_i$ by means of this recursive relation is useful only for extremely simple examples. For each term in the series, we have to solve the convolution of the Kernel function with a certain spatial distribution $n(\mathbf{r})$. Let us assume that $n(\mathbf{r})$ can be represented as a piecewise-constant distribution with $N_r$ subdivisions (i.e. not too steep spatial variations of the distribution). Then, for each (time) step $i$, we have to compute $N_r^2$ integrals $\int G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}'$ over a subvolume. When we want to compute $N_t$ terms of the series of the $n_i(\mathbf{r})$, this means that we have to solve $N_r^2 \cdot N_t$ integrals. Of course, we can make this procedure more efficient: in the PCA method, we have approximated $\Lambda n(\mathbf{r}_k)$ as $\Sigma A_{k,m} n(\mathbf{r}_m)$, where the summation goes over $m$. We can do exactly the same thing here in the AMS; we thus then have to compute the $A_{k,m}$ only once and can compute the $n_i(\mathbf{r})$ by a simple summation. This closes the circle of interrelations: the $A_{k,m}$ elements are a basic element for both the PCA and the AMS method, the solution of the PCA method (eigenmodes) is related to the solution of the AMS ($p_i$) by the relation of Chapter 4. Finally, all these methods are closely related to the propagator function method (PFM). In the simple formulation shown above, we may need quite a lot of terms in the time series when the trapping is strong—so that there are many absorptions/reemissions. To improve that situation, Lai *et al.* (1993a, b) suggested a modification. It can be shown that in the limit of large $i$, $$ \frac{p_{i+1}}{p_i} \to \lambda_0 \quad \text{for} \quad i \to \infty $$ (6.21) where $\lambda_0 = (1 - 1/g_0)$ is the lowest-order eigenvalue of the Holstein equation. In other words, there is a value $ii$ so that $$ |p_i - p_{ii} \cdot \lambda_0^{i-ii}| < \epsilon \quad \text{for} \quad i > ii $$ (6.22) --- With this knowledge, we split the series so that $$ \sum_{i=1}^{\infty} \left(\frac{t}{\tau}\right)^{i-1} \frac{p_i}{(i-1)!} = \sum_{i=1}^{\infty} \left(\frac{t}{\tau}\right)^{i-1} \frac{p_i - p_{ii}\lambda_0^{i-ii}}{(i-1)!} + \frac{p_{ii}}{\lambda_0^{ii}} \sum_{i=1}^{\infty} \left(\frac{t}{\tau}\right)^{i-1} \frac{\lambda_0^i}{(i-1)!} $$ (6.23) The second series on the right-hand side determines the behaviour at late times (i.e. the lowest-order mode). In the first series on the right-hand side, we can replace the infinite upper limit by an upper limit of $ii$, since all higher terms are smaller than $\epsilon$ anyway. We thus have to compute only $ii$ values of the $p_i$. A similar relation holds for the $n_i(\mathbf{r})$. Thus, if we want to compute the late-time behaviour of the emergent radiation, we just have to compute so many terms that the ratio $p_{i-1}/p_i$ can be considered constant within the required accuracy. The required number of terms can even be estimated quite easily *a priori*. The ratio being constant means nothing else than that all higher-order modes have died out. Since the second-highest mode has a trapping factor of roughly one-third of the lowest-order mode, we require that $\exp[-2t/(g_0\tau)]$ must be much smaller than unity. This condition is achieved very quickly in a low-opacity vapour, while in a high-opacity vapour, this can take quite a long time. We note in passing that the conditions for the convergence of the ratio is somewhat similar to the condition of convergence of the $\Lambda$-iteration technique (see Sec. 7.5) for the computation of steady-state distributions (see also (Heering 1975)). This is no accident—the whole AMS could be seen as the transient analogue to the $\Lambda$-iteration technique. Lai *et al.* (1993b) also proposed an interesting new derivation of the multiple-scattering representation. If $\Lambda$ were just some real number, we could formally integrate Eq. (6.15) to get $$ n(\mathbf{r}, t) = \exp[-(1 - \Lambda)t/\tau]n(\mathbf{r}, 0) $$ (6.24) We can actually do the same also when $\Lambda$ is some (integro-differential) operator. The exponential of the operator is given by the series expansion of the exponential function $$ \exp(\Lambda) = \sum_{i=0}^{\infty} \frac{\Lambda^i}{i!} $$ (6.25) An application of the AMS to phase-modulation fluorometry, a special technique to measure atomic lifetimes, is described in Lai *et al.* (1994). A form very similar to the AMS was derived by Alley (1983). He interpreted the absorption–reemission processes as a kind of random walk (but taking the frequency redistribution into account). With this formalism, he derived a very general form of the Holstein equation, taking into account the partial frequency redistribution, motion of the atoms, and even the flight time of the photons (we will derive these effects later from a slightly different point of view). An interpretation of the absorption–reemission processes as a Markov chain is given by Katkovsky *et al.* (1980). A somewhat related approach that is also used for partial frequency redistribution is used by Bush and Chakrabarti (1995). --- # 7 # FITTING EQUATIONS AND PHYSICAL INTERPRETATION In the previous two chapters, we have investigated various methods to solve the Holstein equation. While we have derived closed-form solutions for the high-opacity case, a general solution is possible only by the numerical methods described. The results obtained for the trapping factors and modes can be fitted by some simple expressions, which we will describe in this chapter. We will also give graphical representations of the results, and compare results obtained by the various methods of Chapters 5 and 6. Finally, we will discuss the physical interpretation of the solutions to the Holstein equation. ## 7.1 Doppler and Lorentz lines Most closed-form approximations are either valid only at high opacities (van Trigt 1969–75), give only the ground mode (Mewe 1967), or have both limitations (Holstein 1947). To remedy this situation, we have developed closed-form analytical fitting equations that are valid at all opacities and are given in Appendix D for the first 20 (slab) or the first 10 (cylinder and sphere) modes. In a very large opacity range ($0 \leq k_0L \leq 1000$), the eigenfunctions can be approximated very well by sinusoidal functions (slab), Bessel functions (cylinder) or spherical Bessel functions (sphere). Due to these simple shapes, modal expansions can be done very easily. The trapping factors **for Doppler lineshapes** are of the form $$ g_j^\mathrm{D} = 1 + \frac{1}{m_j^\mathrm{D}} \cdot k_0L \sqrt{\ln\left(\frac{k_0L}{2} + e\right)} - \frac{c_{0,j}^\mathrm{D} k_0L \ln(k_0L) + c_{1,j}^\mathrm{D} k_0L + c_{2,j}^\mathrm{D} (k_0L)^2}{1 + c_{3,j}^\mathrm{D} k_0L + c_{4,j}^\mathrm{D} (k_0L)^2} \qquad (7.1) $$ and **for the Lorentz shape** $$ g_j^\mathrm{L} = \frac{1}{m_j^\mathrm{L}} \sqrt{\pi k_0L + (m_j^\mathrm{L})^2} - \frac{c_{0,j}^\mathrm{L} k_0L \ln(k_0L) + c_{1,j}^\mathrm{L} k_0L + c_{2,j}^\mathrm{L} (k_0L)^2}{1 + c_{3,j}^\mathrm{L} k_0L + c_{4,j}^\mathrm{L} (k_0L)^2} \qquad (7.2) $$ For the cylinder and sphere, $L$ has to be replaced by $R$. **At high opacities**, these equations tend to the same values as the van Trigt equations. For the constants $m_j$, we used van Trigt's values, when available. In the slab case, he gave the first five values explicitly. We computed the higher $m_j$ from the PCA with $k_0L = 10^{12}$ in the Doppler case and $k_0L = 10^6$ in the Lorentz case. For the Lorentz case, the numerical computations reproduced van Trigt's $m_j$ within 0.5%, and for the Doppler case within 2.5%. In the cylinder case, van Trigt gave the first few $m_j$ explicitly and the higher $m_j$ can be computed very accurately from the --- [FIGURE: FIG. 7.1. Lowest-order trapping factor $g_0^\mathrm{D}$ for a Doppler profile as a function of the opacity for slab, cylinder, and sphere geometry.] asymptotic relation Eq. (5.41). For the sphere, no analytical computations are available, so we computed all $m_j$ from the PCA with $k_0R = 10^{12}$ and $10^6$, respectively. In the small opacity limit, $k_0L \rightarrow 0$ ($k_0R \rightarrow 0$), the fitting equations correctly tend to 1. The rational terms on the right-hand side of Eqs. (7.1) and (7.2) are functional fits that do not interfere with the limiting behaviour in the high-opacity and low-opacity limits but have no further physical significance. The constants $c_{i,j}$ of these terms were least-square-fitted to the PCA results in the range $0 \leq k_0L \leq 1000$. The error in this range is $< 2\%$ for the slab, $< 2.5\%$ for Doppler shapes in spheres, $< 1.5\%$ for Lorentz shapes in spheres, $< 3\%$ for Doppler shapes in cylinders, and $< 2.5\%$ for Lorentz shapes in cylinders. At higher opacities in the Doppler case, the error for the ground mode is also within these limits but the error for higher-order modes can become larger (e.g., up to 12% for $k_0L = 10^4$ in a slab for the tenth mode), since even $10^4$ cannot be considered to lie in the 'high-opacity regime' for higher eigenmodes. We confined our least-square fitting, however, to the regime $k_0L \leq 1000$, because these opacities occur in the overwhelming majority of practical situations. Furthermore, at least some pressure broadening will usually occur at the vapour densities that result in these high opacities so that the optically thin wings of the spectral shape (which determine the trapping factors at high opacities) will be Lorentzian. In short, pure Doppler broadening at high opacities rarely occurs in practice. In the Lorentz case, the error at opacities $> 1000$ still lies within the limits given above. If the least-square fitting term is omitted, the error (in the slab case) is less than 20% for the ground mode for both lineshapes (see Fig. 7.2) but much larger for the higher-order modes. As mentioned above, the eigenfunctions can be approximated very well by simple functions for $k_0L \leq 1000$. --- DOPPLER AND LORENTZ LINES [FIGURE: Graph of Relative error vs Opacity k_0L with and without LSF-term] FIG. 7.2. Errors, as compared to PCA, of the approximations for $g_0^{\mathrm{D}}$ in the slab, with and without the least-square fitting term. **For the eigenfunctions in the slab** we have used the functions $$ \begin{aligned} \psi_j(z) &= \sqrt{\frac{2/L}{1 + \mathrm{si}(\zeta_j\pi)}} \cos\left(\zeta_j\pi\frac{z}{L}\right) \quad \text{for } j = \text{even}, \\ \psi_j(z) &= \sqrt{\frac{2/L}{1 - \mathrm{si}(\zeta_j\pi)}} \sin\left(\zeta_j\pi\frac{z}{L}\right) \quad \text{for } j = \text{odd}, \end{aligned} $$ (7.3) **For the cylinder case,** $$ \psi_j(r) = \frac{1}{R} \sqrt{\frac{2}{\left[J_0^{\mathrm{B}}(\zeta_j)\right]^2 + \left[J_1^{\mathrm{B}}(\zeta_j)\right]^2}} J_0^{\mathrm{B}}\left(\zeta_j \frac{r}{R}\right), $$ (7.4) where $J_0^{\mathrm{B}}$ and $J_1^{\mathrm{B}}$ are the zero- and first-order Bessel functions of the first kind. **For the sphere,** $$ \psi_j(r) = 2 \sqrt{\frac{\zeta_j}{R \left[2\zeta_j - \sin(2\zeta_j)\right]}} \frac{\sin\left(\zeta_j \frac{r}{R}\right)}{r}, $$ (7.5) which are normalized spherical Bessel functions of zero order. --- **The parameters $\zeta$ in the slab** case are given by $$ \zeta_0 = d_1 \left( \frac{k_0L + d_2}{k_0L + d_2d_3} \right)^{d_4} \qquad \zeta_1 = 1 + d_5 \left( \frac{k_0L + d_6}{k_0L + d_6d_7} \right)^{d_8} $$ (7.6) where the constants $d_1$ to $d_8$ are given in Appendix D, Table D.1.3. The higher-order modes $\zeta_j$ can be computed from the orthogonality relation $\int \psi_i(z)\psi_j(z)\mathrm{d}z = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta symbol, as $$ \begin{aligned} & \zeta_0 \tan (\zeta_0 \pi / 2) = \zeta_j \tan (\zeta_j \pi / 2) \quad \text{for even } j, \\ \text{and} \qquad & \\ & \zeta_1 \cot (\zeta_1 \pi / 2) = \zeta_j \cot (\zeta_j \pi / 2) \quad \text{for odd } j. \end{aligned} $$ (7.7) **For the cylinder and sphere**, $\zeta_0$ also has the functional form of Eq. (7.6), and the constants $d_i$ are given in Tables D.2.3 and D.3.3, respectively. The $\zeta_j$ for all higher-order modes ($j = 1, 2, \dots$) are given by the orthogonality relations for the cylinder $$ \zeta_j \frac{J_1^\mathrm{B}(\zeta_j)}{J_0^\mathrm{B}(\zeta_j)} = \zeta_0 \frac{J_1^\mathrm{B}(\zeta_0)}{J_0^\mathrm{B}(\zeta_0)} $$ (7.8) and for the sphere as $$ \zeta_0 \cot(\zeta_0) = \zeta_j \cot(\zeta_j) $$ (7.9) The constants $d_i$ of tables D.1.3, D.2.3, and D.3.3 were determined by least square fitting the modal shapes of Eqs. (7.3)–(7.5) to the PCA solution of the lowest-order mode. The criterion for the fit was to keep the number of photons in the lowest-order mode equal, i.e. to approximate the PCA eigenfunction $\psi_0$ by a cosine function $\psi_0'$ of the same area (we describe here only the procedure for even modes in the slab, all other modes can be treated completely analogously). The number of photons in a mode for a given initial distribution is proportional to the expansion coefficient times the area of the eigenfunction. Thus, for an arbitrary function $f(z)$, we must have $$ \int f(z)\psi_0(z)\mathrm{d}z \cdot \int \psi_0(z)\mathrm{d}z = \int f(z)\psi_0'(z)\mathrm{d}z \cdot \int \psi_0'(z)\mathrm{d}z $$ (7.10) As this must be fulfilled for arbitrary $f(z)$, we can write $$ \psi_0(z) \cdot \int \psi_0(z)\mathrm{d}z = \psi_0'(z) \int \psi_0'(z)\mathrm{d}z. $$ (7.11) With the approximate eigenfunctions, this equality will not hold exactly. The aim of our least-square procedure was to minimize the quadratic difference between the two terms in Eq. (7.11). Figures 7.4 and 7.5 show that the analytical fits give very good approximations in a large range of opacities. Figure 7.4 also indicates that van Trigt's eigenfunctions for the --- DOPPLER AND LORENTZ LINES [FIGURE: Graph of parameters \zeta_0, \zeta_1 vs Opacity k_0L] FIG. 7.3. The parameters $\zeta_0$ and $\zeta_1$ of the cosine approximation to the eigenfunction $\psi(z)$, for the slab case, Doppler lineshape. Both the PCA result and the rational approximation are drawn, but the traces nearly coincide. [FIGURE: Graph of Eigenfunction \psi_0^D(z) vs z] FIG. 7.4. The eigenfunction $\psi_0(z)$ for opacity $k_0L = 5$ in a slab, Doppler lineshape. The cosine fit and the PCA solution nearly coincide. --- [FIGURE: FIG. 7.5. The eigenfunction $\psi_0(r)$ for $k_0R = 50$ in a sphere, Doppler lineshape. The spherical Bessel fit and the PCA solution nearly coincide.] slab, though exact in the limit of infinite opacities, deviate considerably from the true eigenfunctions in the low to moderately high opacity regime. Even at $k_0L = 1000$, the cosine approximation is closer to the true eigenfunction than van Trigt's curve. Furthermore, the cosine approximation combined with the orthogonality relations of Eq. (7.7) gives good approximations to the higher-order eigenfunctions, see Fig. 7.6. There is a considerable number of fitting formulas for $g_0$ in the literature (Fujimoto 1979), (Fill 1988), (Geddes and McCullough 1993), (Fujimoto *et al.* 1984), but to our knowledge, only Mewe (1967) gave the accuracy of his result (better than 25%). Expressions for the transmission factor in the low-opacity case were suggested, e.g., by Bassyouni (1982) (some papers tend to obscure the difference between $g_0$, the transmission factor, and the escape factor). Several papers express $g_0$ at low opacities as $1 + k_0L \cdot \text{const}$, where const is some (often arbitrarily chosen) constant (Klose and Voigt 1977), (Lawrence 1968), (Lawrence and Liszt 1969), (Simon *et al.* 1994). We will see in Chapter 8 that this is quite reasonable for opacities smaller than three. Scherr (1971) has tabulated his results for the Doppler case and is, to our knowledge, the only one who also tabulated fits for the spatial ground mode at all opacities. Walsh (1957) and Holstein (1947) gave results for the high-opacity case which they obtained by a variational technique (Sec. 5.1). The results of Scherr and Walsh have the drawback that they exist only for the Doppler case. Furthermore, they are problematic from a theoretical point of view. Both used a three-parameter polynomial approximation to the eigenfunction and got results that resemble a Gaussian bell-shape. Scherr also tabulated his approximations by giving the parameters of this Gaussian shape. This form differs considerably from the true shape. We can thus conclude that the two-parameter --- [FIGURE: Graph of the fourth-order eigenfunction $\psi_4^D(z)$ for $k_0L = 5$ in a slab, Doppler line-shape, comparing PCA, Cosine fit, and van Trigt.] FIG. 7.6. The fourth-order eigenfunction $\psi_4(z)$ for $k_0L = 5$ in a slab, Doppler line-shape. computations of Holstein give better results than the three-parameter computations. From a more practical point of view, Scherr's eigenfunctions are inconvenient for modal expansions, because even simple initial distributions result in integrals that can be solved only numerically. **In the cylinder case**, most authors use a variational procedure (like Holstein). Golubovskii and Lyagushchenko (1975, 1976) and Golubovskii *et al.* (1971) used a high-opacity approximation for $T(\rho)$ to compute the $A_{k,m}$ matrix with $N_r = 15$ and tabulated the $A_{k,m}$. For the lowest-order mode, their results do not differ from Holstein's. Note that in (Golubovskii and Lyagushchenko 1976), wrong equations for the matrix elements are given; however, the numerical $A_{k,m}$ values are correct. Payne and Cook (1970) did similar computations in the high-opacity regime, assuming that the initial distribution of excited atoms is a Dirac function in the axis of the cylinder, $n(r, 0) = \delta(r)$. Their results are similar to those of van Trigt. Figure 7.7 compares $g_0$ obtained from the fitting equations to the results of other methods. Excellent agreement with the variational results of Asadullina *et al.* (1987) is evident, while the results computed via high-opacity approximations deviate considerably even at quite high opacities. **For spherical geometries** under laboratory conditions, the only solution (apart from Molisch *et al.* (1992a)) we are aware of is that of Scherr (1971). He tabulates $g_0^D$ and $\psi_0^D$ for some opacities $k_0R < 30$. Cuperman *et al.* (1963) consider only the steady state and give no specific results. One interesting aspect is that in a sphere, there is a larger probability that photons are reemitted outward than inward (i.e. towards smaller radii), simply because of the geometry, while in a slab, there is equal probability that they are reemitted inward or outward; this gives a first impression why the trapping factor is lower in spheres than in slabs (Kunasz and Hummer 1974a). Actually, this should also be true for a cylinder; however, there remains always the question whether one should compare a slab of width $L$ or of $2L$ with a cylinder of radius $R$ (if we used a slab with $2L$, the above considerations would be correct). --- [FIGURE: FIG. 7.7. The trapping factor $g_0^D$ in a cylinder as a function of opacity: the fitting equations are compared to the high-opacity approximation, and to results of Asadullina et al. (1987) (marks).] Let us now turn to the physical interpretation of all these results. For this, we have to distinguish between high-opacity and low-opacity regions. **For the high-opacity** case an extremely useful interpretation can be derived from the work of Zanstra (Irons 1979). He states that only photons in the wings have a reasonable chance of escaping from the high-opacity vapour. We thus have to find the frequencies $x_{\text{esc}}$ for which $$ k_0 k(x_{\text{esc}}) L \leq 1. $$ (7.12) The escape factor is then approximately $$ \eta = C_x \int_{x_{\text{esc}}} k(x)dx $$ (7.13) and $g$ is set equal to $1/\eta$. This implies that photons escape essentially only if they have such a frequency that the opacity they 'see' is small. A Doppler shape has a very steep descent. When opacity increases, the 'area' of the wings, $C_x \int k(x)dx$ decreases very quickly. With a Lorentzian profile, the wings of the line descend only slowly. Even at high opacities, photons can easily manage to come into the wings of the line and thus still have a good chance to escape. This leads to the fact that for a Doppler lineshape, the trapping factor increases approximately linearly with the opacity, while for a Lorentz lineshape, it increases only with the square root of the opacity. --- Actually, we have two competing (and intertwined) processes that lead to an escape of photons. On one hand, we have the photons getting into the wings, and thus having the chance to escape in one step. The basic probability for this process is rather low, but decreases comparatively slowly as the opacity increases (linearly for the Doppler shape, with a square-root dependence for the Lorentz shape). On the other hand, line-centre photons can 'diffuse' through the vapour with a mean-free path $1/k_0$. At low opacities, this allows escape after just a few absorptions/reemissions. However, the distance that a photon can cover in that way decreases *quadratically* as the line-centre opacity increases. It is thus hardly surprising that at line-centre opacities larger than about 10, the escape via the wings of the line becomes dominant. **For low opacities**, all lineshapes are approximately equal. The escape of photons is dominated by the spatial diffusion of the line-centre photons. The essential parameter for this situation is thus the line-centre opacity–it determines how often a photon that stays at the line centre is absorbed before escaping the vapour. The process of photons moving to the wings of the line (which has a rather low probability) does not play an important role, since the number of these reemissions is rather small. One interesting fact that follows from the high-opacity behaviour of a Lorentz line is the following: if the pressure broadening of the Lorentzian lines is due to foreign-gas broadening, then the trapping factor increases also with the square root of the particle density. If, however, the pressure broadening is due to self-broadening, then the trapping factor becomes independent of the opacity (just as long as it is much larger than unity). Due to the Ladenburg relation (see Chapter 2), it depends only on the frequency $\nu_0$ and on the vapour cell dimensions (Wieme and Mortier 1973), (Turner 1965), so that for a slab $$g_0 = \frac{1}{0.211} \cdot \left(\frac{\nu_0}{c}L\right)^{0.5}$$ (7.14) A similar relation for noble gas mixtures is discussed by Igarashi *et al.* (1995). ## 7.2 Voigt lines In many cases, broadening is not pure Doppler or pure Lorentzian but a mixture of these two, so that the lineshape is a Voigt profile (see also Sec. 2.2.4). For such profiles, we would also like to have closed-form equations for the trapping factors. We know that $g$ must lie between $g^{\mathrm{D}}$ and $g^{\mathrm{L}}$; we just require the appropriate interpolation. We have seen in Sec. 4.5 that the transmission factor $T(\rho)$ can be written as a combination of the transmission factors $T^{\mathrm{D}}$ (Doppler), $T^{\mathrm{L}}$ (Lorentz), and $T^{\mathrm{DL}}$ (Doppler absorption with Lorentz emission). Walsh (1959) concluded that also the trapping factor $g$ can be written in such a form. He states that the constant $m_0$ is always close to unity for all shapes and gives the following interpolations for $g$: $$\frac{1}{g} = \frac{\exp\left[-(g^{\mathrm{L}}/g^{\mathrm{DL}})^2\right]}{g^{\mathrm{D}}} + \frac{\mathrm{erf}\left(g^{\mathrm{L}}/g^{\mathrm{DL}}\right)}{g^{\mathrm{L}}}.$$ (7.15) --- However, this equation breaks down at low opacities. A similar approximation of Zollweg *et al.* (1981) is claimed to be valid at all opacities but gives errors up to 30% in the intermediate-opacity region. To remedy this situation, we have derived a new interpolation equation $$ \frac{1}{g^{\text{V}}(k_0 R)} = \frac{1}{g^{\text{D}}(k_0 R)} \exp\left[-g_{\text{A}}^2 (k_0 R)\right] \cdot \left( 1 - \frac{\sqrt{\pi}}{2} \frac{g_{\text{A}} (k_0 R)}{(1 + k_0 R/m_j^{\text{D}})^2} \right) + $$ $$ \frac{\text{erf}\left[g_{\text{A}} (k_0 R)\right]}{g^{\text{L}}\left[k_0 R/(a\sqrt{\pi})\right] g_{\text{C}} (k_0 R)}, $$ $$ g_{\text{A}} (k_0 R) = g^{\text{L}}\left[k_0 R/(a\sqrt{\pi})\right] / g_{\text{B}} (k_0 R), \qquad (7.16) $$ $$ g_{\text{B}} (k_0 R) = 1 + \frac{1}{a}\sqrt{\pi \ln(k_0 R + e)}m_j^{\text{L}}/m_j^{\text{D}}, $$ $$ g_{\text{C}} (k_0 R) = 1 - \frac{1.5}{a + 1} \left[ \frac{k_0 R}{k_0 R + c1 \cdot m_j^{\text{D}}/m_j^{\text{L}}} - \frac{k_0 R}{k_0 R + c2 \cdot m_j^{\text{D}}/m_j^{\text{L}}} \right]. $$ where $k_0 R$ denotes the opacity for a pure Doppler line, and $$ \begin{matrix} c1 = 5 & c2 = 8 & \text{for the slab,} \\ c1 = 4 & c2 = 12 & \text{for the cylinder,} \\ c1 = 5 & c2 = 12 & \text{for the sphere.} \end{matrix} \qquad (7.17) $$ Note that the dependence of the trapping factors $g$ on the modal index $j$ has been omitted for readability. Here, $g_{\text{C}}$ is a purely empirical factor that corrects the behaviour at intermediate opacities and has no influence on the limiting behaviour. In the limit of high opacities, the Walsh approximation, Eq. (7.15), is recovered. While Eq. (7.15) gives $g < 1$ for small opacities even when the exact $g^{\text{D}}$ and $g^{\text{L}}$ (and not the high-opacity approximations) are used, the factor in brackets multiplying $g^{\text{D}}$ in Eq. (7.16) eliminates this problem almost completely. For the lowest-order mode, Eq. (7.16) gives 8% accuracy for the cylinder, 10% for the sphere, and 12% for the slab, tested over a wide range of parameters by comparison with MC simulation and with data from Asadullina *et al.* (1987). To our knowledge, Eq. (7.16) is the only interpolation formula that also covers higher-order modes. For higher-order modes, the accuracy is estimated conservatively to be better than 20% for all three geometries (tested to be better than 15% for $a = 0.1, 0.3,$ and $1$). The functional form of the eigenfunctions is nearly unchanged, only the parameters $\zeta$ change according to $$ \zeta^{\text{V}} = \zeta^{\text{D}} - (\zeta^{\text{D}} - \zeta^{\text{L}}) \frac{a k_0 R}{a k_0 R + 2}. \qquad (7.18) $$ This is a mathematical fit without a physical interpretation. Figure 7.8 compares the interpolation formula with MC simulation results at an intermediate opacity. --- Figures 7.9 and 7.10 show some trapping factors for Voigt profiles in the slab, obtained with the PCA method. For a Voigt lineshape, similar physical interpretations as for the Doppler and Lorentz lines can be given. It depends on the Voigt parameter when the opacity dependence changes from the linear (Doppler-like) to the square-root (Lorentz-like) regime. When the 'escape frequency' $x_{\text{esc}}$ is already in the Lorentzian wing of the line, then the functional dependence of the trapping factor on the opacity will be Lorentz-like. ## 7.3 Hyperfine split lines Many lines occurring in practice are not single lines but are hyperfine-split (we also include isotope splitting in this context (Holstein *et al.* 1952)). The splitting can be neglected if the distance between the hyperfine lines is much less than the Doppler (or Lorentz) width. If, however, the splitting is at least of the same order of magnitude, hfs must explicitly be taken into account. In the case of hfs lines, the assumption of complete frequency redistribution, CFR, becomes especially critical. CFR now also implies that the hfs component in which the photon is reemitted is independent of the hfs component in which the photon was absorbed. In other words, the atom has to suffer at least one hfs-changing collision during its natural lifetime. This condition is less often fulfilled than the validity of CFR *within* an hfs line. The criterion is that $$ v_{\text{rel}} \cdot \sigma_{\text{mix}} \cdot N_{\text{pert}} \cdot \tau > 1. $$ (7.19) [FIGURE: Plot of Trapping factor vs Voigt parameter a comparing MC simulation and Interpolation formula] FIG. 7.8. Trapping factor $g_0^{\text{V}}$ in a slab at the actual line-centre opacity $k_0'L = 20$, as a function of the Voigt parameter $a$: interpolation formula compared to MC simulation. --- [FIGURE: Lowest-order trapping factor g_0^V in a slab as a function of the actual centre-of-line opacity k_0'L] FIG. 7.9. Lowest-order trapping factor $g_0^\text{V}$ in a slab as a function of the actual centre-of-line opacity $k_0'L$ for different Voigt parameters, calculated with the PCA method. [FIGURE: First-order trapping factor g_1^V in a slab as a function of the actual centre-of-line opacity k_0'L] FIG. 7.10. First-order trapping factor $g_1^\text{V}$ in a slab as a function of the actual centre-of-line opacity $k_0'L$ for different Voigt parameters, calculated with the PCA method . --- where $N_{\text{pert}}$ is the density of perturbers (atoms of the same kind or foreign-gas atoms), $v_{\text{rel}}$ is the relative velocity, and $\sigma_{\text{mix}}$ is the hfs intermixing cross-section (for a discussion, see also Wang *et al.* (1996). The problem with Eq. (7.19) is that these cross-sections are rarely known. For want of a better model, CFR is usually assumed even when $\sigma$ is unknown—and sometimes even when it is *known* that Eq. (7.19) is not fulfilled. A formulation that is also valid if Eq. (7.19) is not fulfilled is given below. For an hfs-split line, the absorption coefficient can be written as $$ k(x) = \sum_i w_i k(x - \delta_i)^{\text{unsplit}}, $$ (7.20) where the $w_i$ are the relative amplitudes (normalized so that $\sum w_i = 1$) and $\delta_i$ are the frequency separations of the lines. Note that $C_x$ is independent of the hfs. If the intermixing of the levels is not complete, we must use a generalized equation, assuming partial overlap and partial intermixing of the lines. The excited-state density in one state is decreased by natural decay and by collisional transfer to other hfs lines. It is increased by collisional transfer from other hfs states and by reabsorption of photons from *all* hfs components. The generalized Holstein equation is then (we drop the dependence of $n$ and $G$ on $\mathbf{r}$) $$ \frac{\partial n_i}{\partial t} = -A_{il} n_i - \sum_{j \neq i} C_{ij} n_i + \sum_{j \neq i} C_{ji} n_j + \sum_j A_{jl} \int G^{ij} n_j \text{d}\mathbf{r}' $$ (7.21) where the Kernel function $G^{ij}$ is the probability that a photon created when an excited atom in state $j$ decayed, is absorbed to state $i$. Explicitly, this Kernel function is $$ G^{ij}(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi\rho^2} \int_{-\infty}^{\infty} C_x k(x - \delta_j)^{\text{unsplit}} w_i k(x - \delta_i)^{\text{unsplit}} \exp{[-k(x)\rho]} \, \text{d}x $$ (7.22) The emission probability for a photon with frequency $x$ that is created when an excited atom in hfs-state $j$ decays is given by $C_x k(x - \delta_j)^{\text{unsplit}}$. The absorption probability into state $i$ is $w_i k(x - \delta_i)^{\text{unsplit}}$. The probability of transversing a distance $\rho$ without being absorbed is given by the exponential factor. A similar equation (though less detailed) was derived by Holstein *et al.* (1952). The situation is practically the same when there is also isotope splitting; see also Ostertag *et al.* (1991). The escape factors for overlapping lines is discussed by Elitzur and Netzer (1985). An easy and physically reasonable **approximation for high opacities** is offered by the method of Zanstra for computing the probability that a photon gets into the wings, see Sec. 7.1. **For completely separated lines**, Bezuglov *et al.* (1987) proposed the following equation $$ \frac{1}{g} = \frac{g'_1}{g'_1 + g'_2} \frac{1}{g_{\text{unsplit}} \left( \frac{g'_1}{g'_1 + g'_2} k_0 L \right)} + \frac{g'_2}{g'_1 + g'_2} \frac{1}{g_{\text{unsplit}} \left( \frac{g'_2}{g'_1 + g'_2} k_0 L \right)} $$ (7.23) --- [FIGURE: FIG. 7.11. Trapping factor $g_0$ in the slab for an hfs-split line with two components of equal strength, Doppler lineshape, for three different line separations $\delta$ and $g_0(k_0/2)$ for an unsplit line.] where $g$ is the trapping factor and $g'_1$ and $g'_2$ are the statistical weights of the two lower states under consideration. **For low and intermediate opacities**, and arbitrary separation of the lines, no closed-form approximations exist. We have either to use the Milne approximation (see Chapter 8) or full numerical computations. Van Trigt (1968) computed series representations of the integrated absorption, i.e. the equivalent linewidth, for hyperfine-split Voigt lines. Since we saw in Chapter 4 that this equivalent width is related to $T(\rho)$, his series representations can be used to compute $T$, and thus approximations to the trapping factor. A fitting formula for the special case of a Hg-Ar mixture was given by Wani (1990). Walsh (1959) computed analytical high-opacity approximations for completely intermixed hfs-split Voigt lines under the assumption of equal strength and spacing of the lines. Rons (1996) approximately computed the excited-state distribution in terms of the distribution that would occur for zero splitting. Figure 7.11 shows the lowest-order trapping factor (for a Doppler line) as a function of the opacity $k_0L$. The trace for $\delta = 0$ shows the trapping factor when there is no splitting, the trace $\delta = 1$ is for the case that there are two lines separated by one Doppler width. At high opacities (near $k_0L = 1000$), $g_0$ does not differ from the case of no splitting. This is so because in the middle between the two components, the escape probability is still very low (due to the small splitting and the large opacity), so that the only good chance of escape for the photons is in the 'left' wing of the lower line --- [FIGURE: FIG. 7.12. Normalized trapping factor $\hat{g}_0$ in the slab for an hfs-split line with components of equal strength, Doppler lineshape, as a function of the hfs-separation $\delta$ for low and high opacities.] and in the ‘right’ wing of the upper line. At lower opacities, the trapping factor is more like $g_0(k_0L/2)$. The trapping factor is also similar to $g_0(k_0L/2)$ when the splitting of the lines is very large. For a Doppler lineshape, a splitting of three Doppler widths is already quite large, so that the traces for $\delta = 3$ and for $g_0(k_0L/2)$ almost coincide. In Fig. 7.12, we see the trapping factor (normalized to $g_0$ of the unsplit line) as a function of the splitting $\delta$ in Doppler widths. Again, we see that for high opacities, a larger splitting is needed to reduce the trapping factor than at low opacities. An interesting case occurs when we have a single ground state and a doublet in the first resonance state. There, we have two limiting cases. When there is no intermixing between the hfs (or fine-structure) levels, and the lineshapes of the two resonance lines do not overlap, we can compute the excited-state densities separately, and immediately obtain a solution. For the resonance levels of an alkali atom, for example, the opacity of the D2 line from the $s$ to the $p_{3/2}$ state is about twice the opacity of the D1 line from the $s$ to the $p_{1/2}$ state. In steady state, the excited-state density of the $p_{3/2}$ state will thus be considerably larger (Avrett 1966), (Seiwert 1956) and the effective decay times will strongly differ. In the second limiting case, strong intermixing, the excited-state densities of the two levels will be in the ratio prescribed by detailed balancing, and their decay times will be the same. For the trapping process the two levels are essentially just one level, and have to be treated jointly. Since intermixing changes the shape of the spectral line (which is --- part of the Kernel function), nothing but a full solution of the Holstein equation with the new lineshape gives the correct results. One application of the theory of radiation trapping with one common upper level is to check whether there is noticeable trapping at all. When there is no trapping, then the emergent intensities from the lines must be in the ratio dictated by the statistical weights. If, however, one line is trapped more strongly than the other (as is usually the case for fine- or hyperfine components), this ratio is distorted (Wood *et al.* 1987), (Kastner and Kastner 1990). A further interesting result of hfs is that the escape factor technique can break down completely. Let us consider the case where the upper state is a doublet, and the two resonance lines overlap partially. Irons (1980a) showed that the escape factor technique only works if the ratio of the excited-state distributions is constant and in the ratio of the statistical weights. However, this is fulfilled usually only for strong intermixing. The fact that hfs-splitting reduces radiation trapping also has important practical consequences. In 1952, Holstein *et al.* (1952) observed that the contamination of a Hg$^{198}$ sample with as little as 0.5% Hg$^{200}$ could lead to considerable reduction of the radiation trapping. More than 30 years later, Anderson *et al.* (1985) and Grossman *et al.* (1986) realized that this fact can be used to increase the efficiency of discharge lamps by adding rare Hg isotopes to natural Hg. Since radiation trapping decreases the efficiency of the lamps, reduction of the trapping by adding these rare isotopes is beneficial (see also Chapter 18). A similar effect can also be achieved by the application of a magnetic field, in this case, the splitting of the lines due to the Zeeman effect leads to additional 'escape channels', and thus to a higher efficiency of the lamps (Sommerer 1993). ## 7.4 Higher-order modes We have already seen that strictly speaking, the excited-state density and the emergent radiation can only be computed by using an appropriate number of modes in the solution of the Holstein equation. In this section, we want to stress again the importance of incorporating the **higher-order modes**. In practice, this is often circumvented by one of four methods: (1) Observing the output signal only at late times $t$, when the higher-order modes have died out (Matland 1953). This sacrifices measurement accuracy, since the SNR worsens with time. (2) Assuming that only the ground mode exists, which introduces errors of unknown magnitude (Kowalczyk *et al.* 1985), (Yanson *et al.* 1989). (3) In some papers, where only the central region of a vessel is excited by a laser, the authors assume an imaginary vessel whose diameter is set equal to the diameter of the laser beam. This procedure can lead to errors of orders of magnitude, since the correct approach is to expand the beam into the eigenmodes of the real vessel,[^10] (Jahreiss and Huber 1983). [^10]: (Krebs and Schaerer 1981); the error of their approach was pointed out by Huennekens and Gallagher (1983b). --- [FIGURE: Graph of Output signal vs Normalized time t / \tau showing Ground mode only and Modal expansion and MC simulation] FIG. 7.13. Temporal decay of the emergent radiation in a slab for a Doppler profile with $k_0L = 10$: a three-term modal expansion already gives near perfect coincidence with an MC simulation. [FIGURE: Graph of Output signal vs Normalized time t / \tau showing Ground mode only and Modal expansion and MC simulation] FIG. 7.14. Temporal decay of the emergent radiation in a cylinder, axially excited by a laser beam for a Doppler profile with $k_0R = 20$: a 10-term modal expansion is needed to give good agreement with an MC simulation. --- (4) In a large number of papers, only the escape factor is evaluated. As mentioned in Sec. 4.5, such computations are, however, only valid if we have *a priori knowledge* of the distribution of the excited atoms (Irons 1979), (Preobrazhenskii and Senina 1971). This approach is thus often useful in the study of steady-state plasmas (McWirther 1965) or of electrical discharges, for which the distribution of excited atoms is determined by known processes, but not for problems where trapping is the dominant process. All these methods have serious drawbacks; the only correct approach is a modal expansion. One usually needs the first 3–10 eigenfunctions and eigenmodes. Figures 7.13 and 7.14 demonstrate why it is necessary to incorporate higher-order modes. In Fig. 7.13, a uniform initial distribution in a slab decays, and the temporal behaviour of the emergent radiation is shown. A three-term eigenmode expansion gives almost perfect agreement with an MC simulation, while considerable deviations result when assuming that only the ground mode exists. Figure 7.14 shows an even more drastic example. An atomic vapour in a cylinder is excited by a quite narrow laser beam. Here, we use a 10-term modal expansion to get good agreement with the MC simulation. Using only the ground mode gives completely wrong results for small times $t$. ## 7.5 Steady-state solutions of the Holstein equation The steady-state Holstein equation $$E(\mathbf{r}) = \frac{1}{\tau}n(\mathbf{r}) - \frac{1}{\tau}\int_V n(\mathbf{r}')G(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}',$$ (7.24) can be solved basically by two methods: (i) **by a direct numerical approach**: For this method, we compute $$E_k = \tau \int_{\text{subregion } k} E(\mathbf{r})\mathrm{d}\mathbf{r} \Bigg/ \int_{\text{subregion } k} \mathrm{d}\mathbf{r},$$ (7.25) i.e. we approximate the excitation function by a piecewise constant function. Equation (7.24) is then approximately $$E_k = n(r_k) - \sum_{m=0}^{N_r-1} A_{k,m}n(r_m),$$ (7.26) where the elements $A_{k,m}$ are the same as in the time-dependent case. The algebraic system of equations, Eq. (7.24), can then be solved by standard methods. Of course, a similar procedure could also be used by applying Holstein's variational technique. This method has also been used in astrophysics (Finn 1968), (Avrett 1971), (Finn and Jefferies 1968a), and in plasma research (Hearn 1963). --- [FIGURE: FIG. 7.15. Steady state distribution and lowest-order mode (dashed) for uniform and delta excitation in a slab.] FIG. 7.15. Steady state distribution and lowest-order mode (dashed) for uniform and delta excitation in a slab. Lineshape is Doppler at opacity $k_0L = 10$. The results are normalized. (ii) **via the eigenfunctions:** When the eigenvalues and eigenfunctions are known (e.g. from the fitting equations), another method is more advantageous. We expand the excitation function into a (Fourier) series of the eigenfunctions $$ E(\mathbf{r}) = \sum_j \alpha_j^{\mathrm{exc}} \psi_j(\mathbf{r}) $$ (7.27) and also express the desired solution as such a series $$ n(\mathbf{r}) = \sum_j \alpha_j^{\mathrm{st}} \psi_j(\mathbf{r}). $$ (7.28) The expansion coefficients of the excitation, $\alpha^{\mathrm{exc}}$, are known, and we look for the expansion coefficients of the solution, $\alpha^{\mathrm{st}}$. We insert Eqs (7.27) and (7.28) into Eq. (7.24), $$ \tau \sum_j \alpha_j^{\mathrm{exc}} \psi_j(\mathbf{r}) = \sum_j \alpha_j^{\mathrm{st}} \psi_j(\mathbf{r}) - \sum_j \alpha_j^{\mathrm{st}} \int G(\mathbf{r}, \mathbf{r}') \psi_j(\mathbf{r}') \mathrm{d}\mathbf{r}'. $$ (7.29) From the solution of the time-dependent equation, we know that $$ \int G(\mathbf{r}, \mathbf{r}') \psi_j(\mathbf{r}') \mathrm{d}\mathbf{r}' = \left( 1 - \frac{1}{g_j} \right) \psi_j(\mathbf{r}). $$ (7.30) By inserting Eq. (7.30) into Eq. (7.29), multiplying by $\psi_k(\mathbf{r})$ and integrating over the cell, we get $$ \alpha_k^{\mathrm{st}} = \tau \alpha_k^{\mathrm{exc}} g_k. $$ (7.31) In Fig. 7.15a, we see the steady-state distribution and the lowest-order mode for a uniformly excited vapour in a slab. The shapes of the lowest-order mode and of the --- steady-state distribution are very similar. Figure 7.15b shows the same for a delta excitation at the centre of the slab. For the delta-excitation, there is still an appreciable difference between the full distribution and the lowest-order mode. The situation becomes especially critical when we have an externally irradiated vapour. A large part of the radiation is absorbed near the input window, so that the excitation function is very different from the lowest-order mode. In that case, the correct expansion of the excitation function and consideration of many higher-order modes can considerably influence the result (van Trigt 1969–76), (Perrin and Broekhuizen 1987). Scholz *et al.* (1996) have observed another interesting physical effect occurring in steady state. They excite the vapour with a small-diameter laser beam, and observe the excited-state distribution as a function of the particle density. At very low densities, they observe of course a distribution that has the same shape as the laser beam, because no reabsorption of the fluorescence happens. As the density increases, the excited-state distribution becomes first wider, but at high opacities becomes narrower again. This has been both predicted by computations (with a method similar to the piecewise-constant method described above) and observed experimentally. We see that the computation of the steady-state distribution (under our assumptions) is quite simple once the time-dependent decay problem is solved. This is why we have put the main emphasis in Chapters 4–8 on the time-dependent case. In Chapter 9, we will review methods that are suitable for the steady-state solutions of the transfer- plus rate-equation; this approach is not based on a time-dependent solution. The steady-state distribution can also be computed from the MS representation (Ma and Lai 1994). Starting with the steady-state Holstein equation, we can write the excited-state density as $$n(\mathbf{r}) = \frac{1}{A_{21} - A_{21}\Lambda} E(\mathbf{r}) = \frac{1}{A_{21}} \sum_{i=0}^{\infty} \Lambda^i E(\mathbf{r})$$ (7.32) We have already shown in Chapter 6 how to efficiently compute the expression $\Lambda^i E(\mathbf{r})$ for large $i$. With these expressions, we then also know the excited-state distribution, from which we can compute the emergent radiation and the effective quenching. We finally point out a method known as $\Lambda$-iteration in astrophysics (Cannon 1985). Let us write the steady-state Holstein equation formally as $$n(\mathbf{r}) = \Lambda n(\mathbf{r}) + \tau E(\mathbf{r})$$ (7.33) where $\Lambda$ is the operator $\int G(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}'$. The solution $n(\mathbf{r})$ is computed in an iterative way. We start out by assuming $n_0(\mathbf{r}) = \tau \cdot E(\mathbf{r})$, and compute each higher iteration step as $$n_{i+1}(\mathbf{r}) = \Lambda n_i(\mathbf{r}) + \tau E(\mathbf{r})$$ (7.34) However, the convergence of this method is extremely slow. We need at least $\bar{p}$ iterations, where $\bar{p}$ is the average number of absorptions/reemissions. If there are also --- quenching processes, then $\bar{p}$ (and thus the number of iterations) is decreased accordingly. Properties can be improved somewhat by using a Newton–Raphson algorithm to accelerate the convergence. Still, at higher opacities the number of computations is unacceptably high, especially since the exact solution can be obtained by a single matrix inversion. **For a high-opacity vapour in a cylindrical geometry**, Bezuglov and Tsyganov (1985) gave explicit equations (which require, however, one numerical integration) for the excited-state density. They also derived expressions for the ratio $n_2/n$, which is useful for the determination of collision cross-sections. The same situation is also analysed by Zollweg (1986). One important parameter especially **for plasmas** is the ‘thermalization length’ $L_{\text{th}}$. This is defined as the distance between the point of creation and the point of destruction of a ‘typical’ photon. By destruction, we mean here either that the photon escapes from the vapour or that the excited atom is quenched non-radiatively. The probability for such a destruction per absorption/reemission process is denoted as $p_{\text{destr}}$. It is strongly related to the mean number of absorption/reemission processes that a photon undergoes. For the most important lineshapes, the thermalization length in units of the optical density is (Rybicki and Hummer 1968), (Hummer and Stewart 1966) $$ L_{\text{th}} = \begin{cases} p_{\text{destr}}^{-1} & \text{Doppler} \\ a p_{\text{destr}}^{-2} & \text{Voigt} \\ p_{\text{destr}}^{-2} & \text{Lorentz} \\ p_{\text{destr}}^{-3} & \text{van der Waals} \\ p_{\text{destr}}^{-0.5} & \text{coherent} \end{cases} $$ (7.35) **The mean number of absorption/reemission processes** is discussed by Hummer (1964) and Athay and Skumanich (1971), see also Chapter 6. **Further details and scaling laws** can be found in the papers of Frisch (1982a); comparison with exact results is done by Frisch (1982b). Extensions for the inclusion of continua can be found in (Hummer and Rybicki 1970) and (Frisch 1980b). The dependence of the escape factor on the thermalization length is computed by Otsuka (1979). Generalizations to partial frequency redistribution (see also Chapter 8) can be found in (Frisch and Bardos 1981), (Basko 1981), and (Frisch 1980b). An alternative derivation of the scaling laws by a mathematical procedure called the ‘renormalization group approach’ is given by Bell *et al.* (1978). --- ## 7.6 The emergent spectrum Up to now, we have dealt only with the excited-state distribution and, by extension, with the total number of emerging photons, but not with the lineshape of the emergent radiation. In many applications in chemical physics, we are, however, interested in the angular and spectral distribution of the emergent radiation, i.e. in the intensity $I (\mathbf{r}^{\text{bound}}, \mathbf{\Omega}, \nu, t)$ where $\mathbf{r}^{\text{bound}}$ is a point on the boundary of the cell (Eicher and Allen 1984). The problem is rather simple once the excited-state distribution $n(\mathbf{r}, t)$ is known. Each photon emitted by an excited atom that lies along a path $\mathbf{s}$ has a probability of $C_x k(x)$ of being emitted at a certain frequency and a probability of $\exp(-k(x)|\mathbf{s}|)$ of reaching the boundary unabsorbed. The path $\mathbf{s}$ is determined by the considered point, $\mathbf{r}^{\text{bound}}$, and the considered direction $\mathbf{\Omega}$. Since the distribution of emitters along the path $\mathbf{s}$ is known, we just have to integrate over this path to get the emergent radiation. In the time-decay slab case, this can be written as (Chandrasekhar 1952) $$ I(z = \pm \frac{L}{2}, x, \mu --- THE EMERGENT SPECTRUM [FIGURE: FIG. 7.16. The forward and backward scattering factors, h_f (dashed) and h_b (dotted), compared to the lowest-order trapping factor g_0 (solid). Optical depth is denoted by \tau_0. From Van Volkenburgh and Carrington (1971).] can escape quite easily. In the forward direction, they ‘see’ a much larger opacity, and thus need a larger number of absorption/reemission processes to get through the whole vapour to the front of the cell. In terms of the eigenmodes, there are a lot of higher-order modes (the excitation function is a far cry from being similar to the lowest-order mode). For positive directions $\mu$, this means that the higher modes increase the average number of reabsorptions; for backward directions, it decreases this number. For the other geometries, the emergent intensity can be computed using the same principle, but the integration along the path $s$ is usually more difficult. Though these computations can become somewhat tedious, they are just an exercise in basic geometry and analysis. Solutions for the piecewise-constant approximations to $n(\mathbf{r})$ in a cylinder and in a sphere are, e.g., implemented in Rad-Trap2 (see Appendix C). Sometimes, there is also interest in the luminosity of an object (i.e. what is seen by a distant observer). For the slab case, this is just the radiation emerging with $\mu = 1$. For other objects, we have to take the appropriate averages over the rays that go towards the observer. For three-dimensional objects, the luminosity is discussed by Rybicki and Hummer (1983). For a cylinder, analysis of the angular distribution of the radiation with different excitation functions is also done by Monte Carlo simulations by Doughty (1995). The computation of the emergent lineshape as influenced by trapping is especially important in the measurement of line-broadening coefficients and in the determination of satellites and other far-wing peculiarities. These phenomena are often observed to de- --- FITTING EQUATIONS AND PHYSICAL INTERPRETATION [FIGURE: FIG. 7.17. Spectrum of the emergent radiation in a slab, Doppler lineshape.] termine the interaction potentials of colliding atoms (Drummond and Gallagher 1974). The following two figures, 7.17 and 7.18, show the shape of the spectrum emerging from a slab. We see that at low opacities, the spectrum looks pretty much like the absorption lineshape (i.e. Doppler or Lorentz). At higher opacities, photons in the line centre can no longer escape very well, while those in the wings can escape. We thus see the characteristic self-reversal (the dip at the line centre). This phenomenon has been studied for a long time (see e.g. the review by Cowan and Dieke (1948)), but only radiation trapping theory can give a quantitative explanation of the observed phenomena. All spectra shown are for steady-state excitation. If we excite the slab uniformly, there is an appreciable number of excited atoms near the cell wall. The photons emitted by these atoms have a good chance of escaping even when they are near a line centre, so that self-reversal is less pronounced than for a delta excitation in the middle of the slab. In the latter case, there are hardly any excited atoms near the cell wall. We also see that the spectrum emerging at a $60^\circ$ angle shows stronger self-reversal than the normally emerging spectrum. This is intuitively clear, since the photons emerging at an angle have to cover a longer distance than those that emerge normally to the surface. Finally, we see that the self-reversal is more pronounced for a Doppler lineshape than for a Lorentz lineshape. The real problem lies however, in the case where we know the emergent radiation and want to find the excited-state distribution (i.e. what is done in emission spectroscopy (Kunze 1986b)). The situation is most difficult when the parameters of the lineshape—e.g., the Doppler width—vary in the vapour cell. Such a situation rarely occurs in chemical physics, but can happen in a discharge lamp or a plasma. For this case, Bartel's theory must be used (Karabourniotis 1986). Otherwise, the simpler Cowan–Diecke model can be used (Cowan and Diecke 1948). Finally, let us discuss the computation of emerging spectra by MC simulations. Of --- [FIGURE: FIG. 7.18. Spectrum of the emergent radiation in a slab, Lorentz lineshape.] course, it would be possible to count the number of photons emerging from the vessel with a certain direction and being in a certain frequency range. However, this method is very inefficient; especially when we observe a small spatial angle, we throw away the overwhelming majority of photons. To avoid this, it is best to first compute the excited-state distribution in the cell (either just the steady-state distribution, or the $n_i(\mathbf{r})$, the distribution of atoms that emit photons that have previously suffered $i$ absorptions). From this, the emergent spectra can be computed analytically, and with far less statistical uncertainty. The computation of the emergent spectra is also used for plasma diagnostics (Griem 1974), (Tallents 1980). However, there are a lot of free parameters, and one either has to make many *a priori* assumptions, or to measure most parameters independently. It seems impossible to extract all parameters just from the spectra. --- # 8 # THE MILNE AND EDDINGTON APPROXIMATIONS ## 8.1 The original Milne theory One of the very first treatments of radiation trapping was the theory developed by Milne (1926). He reduced the trapping problem to a differential equation. It is shown below that this is mathematically incorrect and that the Milne theory gives results that actually can be wrong by orders of magnitude for large opacities. However, for small opacities ($k_0L < 30$, see Sec. 8.4), the Milne theory gives acceptable accuracy, and the differential equation is much easier to treat than the (Holstein) integral equation. For these reasons, the Milne theory is still in use for the computation of trapping at low opacities (Kibble *et al.* 1967), (Blickensderfer *et al.* 1976), (Garver *et al.* 1982). Milne starts his derivation with the Ladenburg relation, which links the integral over the absorption coefficient to the properties of the absorbers, $$ \int k(\nu)\mathrm{d}\nu = \frac{B_{12}h\nu}{4\pi}N $$ (8.1) He approximates the integral as $$ \int k(\nu)\mathrm{d}\nu = \bar{k}\Delta\nu, $$ (8.2) which means that he replaces the lineshape $k(\nu)$ by a box-shaped line of height $\bar{k}$ and width $\Delta\nu$. Already it should now be clear that this theory can by no means be applicable at high opacities, where the line wings are crucial for the trapping behaviour. With this approximation for the shape of the absorption coefficient, the equation of radiative transfer (compare to Eq. (4.32)) reduces to $$ (\mathbf{\Omega}\nabla)I = -\bar{k} \left[ I - \frac{n}{N} \frac{A_{21}}{B_{12}} \right] $$ (8.3) The rate equation for the excited state atoms is (compare to Eq. (4.35)) $$ \frac{\partial n}{\partial t} = -\frac{n}{\tau} + N B_{12} \frac{1}{4\pi} \int I \mathrm{d}\Omega. $$ (8.4) Specialized to the slab geometry, the transfer equation and the rate equation, Eqs. (8.3) and (8.4), become 3BEJBUJPO5SBQQJOHJO"UPNJC7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- THE ORIGINAL MILNE THEORY $$ \cos \vartheta \frac{\mathrm{d} I(z, \vartheta, t)}{\mathrm{d} z} = -\bar{k} \left[ I(z, \vartheta, t) - \frac{n(z, t)}{N} \frac{A_{21}}{B_{12}} \right] $$ (8.5) and $$ \frac{\partial n(z, t)}{\partial t} = -\frac{n(z, t)}{\tau} + N B_{12} \frac{1}{2} \int_0^\pi I(z, \vartheta, t) \sin \vartheta \, \mathrm{d}\vartheta. $$ (8.6) We now introduce the total radiation flux density (analogously to (Cayless 1986))[^11] $$ F^\Sigma(z, t) = \int_0^{2\pi} \int_0^\pi I(z, \vartheta, t) \cos \vartheta \sin \vartheta \, \mathrm{d}\vartheta \, \mathrm{d}\varphi = 2\pi \int_0^\pi I(z, \vartheta, t) \cos \vartheta \sin \vartheta \, \mathrm{d}\vartheta $$ (8.7) and the forward and backward fluxes $$ F^f(z, t) = 2\pi \int_0^{\pi/2} I(z, \vartheta, t) \cos \vartheta \sin \vartheta \, \mathrm{d}\vartheta \qquad F^b(z, t) = 2\pi \int_{\pi/2}^\pi I(z, \vartheta, t) \cos \vartheta \sin \vartheta \, \mathrm{d}\vartheta. $$ (8.8) Multiplying the transfer equation, Eq. (8.5), by $\sin(\vartheta)$, integrating from $0$ to $\pi$, and using the rate equation, Eq. (8.6), we get $$ \frac{1}{2\pi} \frac{\partial F^\Sigma(z, t)}{\partial z} = -2 \frac{\bar{k}}{B_{12} N} \frac{\partial n(z, t)}{\partial t}. $$ (8.9) We now define $$ \bar{I}^f(z, t) = 2\pi \int_0^{\pi/2} I(z, \vartheta, t) \sin \vartheta \, \mathrm{d}\vartheta \qquad \bar{I}^b(z, t) = 2\pi \int_{\pi/2}^\pi I(z, \vartheta, t) \sin \vartheta \, \mathrm{d}\vartheta. $$ (8.10) These quantities are used in the so-called Schuster–Schwarzschild approximation for the fluxes $$ F^f(z, t) \approx \frac{1}{2} \bar{I}^f(z, t) \qquad F^b(z, t) \approx -\frac{1}{2} \bar{I}^b(z, t). $$ (8.11) This is also known as the two-stream approximation; a physical interpretation will be given below. Multiplying the transfer equation, Eq. (8.5), by $\sin \vartheta$, integrating over an angle from $0$ to $\pi/2$ and from $\pi/2$ to $\pi$, respectively, we get the transfer equation in terms of the approximate radiation fluxes, $$ \begin{aligned} \frac{1}{2} \frac{\mathrm{d} \bar{I}^f(z, t)}{\mathrm{d} z} &= -\bar{k} \left[ \bar{I}^f(z, t) - \frac{2\pi n(z, t)}{N} \frac{A_{21}}{B_{12}} \right], \\ \frac{1}{2} \frac{\mathrm{d} \bar{I}^b(z, t)}{\mathrm{d} z} &= +\bar{k} \left[ \bar{I}^b(z, t) - \frac{2\pi n(z, t)}{N} \frac{A_{21}}{B_{12}} \right] \end{aligned} $$ (8.12) [^11]: Note that the constant in front of the integral is defined differently by various authors; Milne and Chandrasekhar denote as $\pi \cdot F$ what is $F^\Sigma$ in our notation. --- THE MILNE AND EDDINGTON APPROXIMATIONS Analogously for the rate equation, Eq. (8.6), $$ \frac{\partial n(z, t)}{\partial t} = -\frac{n(z, t)}{\tau} + N B_{12} \frac{1}{4\pi} \left[ \bar{I}^f(z, t) + \bar{I}^b(z, t) \right] \qquad (8.13) $$ Finally, the total flux, Eq. (8.7), becomes in the two-stream approximation $$ F^\Sigma(z, t) = \frac{1}{2} \left[ \bar{I}^f(z, t) - \bar{I}^b(z, t) \right]. \qquad (8.14) $$ Introducing the sum of the forward and backward intensities, $$ \bar{I}^\Sigma(z, t) = \bar{I}^f(z, t) + \bar{I}^b(z, t), \qquad (8.15) $$ the rate equation, Eq. (8.13), reads $$ \frac{\partial n(z, t)}{\partial t} = -\frac{n(z, t)}{\tau} + N B_{12} \frac{1}{4\pi} \bar{I}^\Sigma(z, t) \qquad (8.16) $$ and the transfer equation, Eq. (8.12), again becomes a single equation, $$ \frac{1}{2} \frac{\partial \bar{I}^\Sigma(z, t)}{\partial z} = -2\bar{k} F^\Sigma(z, t). \qquad (8.17) $$ Finally, the sought-for result, the excited state density, $n(z, t)$, can be computed by combining Eq. (8.9), and the rate and transfer equations, Eqs. (8.16) and (8.17). This is done simply by differentiating Eq. (8.17) with respect to $z$, substituting Eq. (8.9), and substituting the resulting expressions for $\partial^2 \bar{I}^\Sigma / \partial z^2$ into the doubly differentiated Eq. (8.16). This yields $$ \frac{\partial^2}{\partial z^2} \left[ n(z, t) + \tau \frac{\partial n(z, t)}{\partial t} \right] = 4\bar{k}^2 \tau \frac{\partial n(z, t)}{\partial t}. \qquad (8.18) $$ The differential equation (8.18) is the Milne equation for the excited-state atoms in the slab geometry. Blickensderfer *et al.* (1976 ) generalized it (without giving a derivation) to $$ \textbf{the general Milne equation} \quad \nabla^2 \left[ n(\mathbf{r}, t) + \tau \frac{\partial n(\mathbf{r}, t)}{\partial t} \right] = 4\bar{k}^2 \tau \frac{\partial n(\mathbf{r}, t)}{\partial t}. \qquad (8.19) $$ The Milne equation has the form of a diffusion equation and can be solved by elementary means. The solutions are—as you might have guessed—of the now well-known modal form $$ n(\mathbf{r}, t) = \sum_j \alpha_j \psi_j(\mathbf{r}) \exp \left[ -t / (g_j \tau) \right], \qquad (8.20) $$ where the eigenfunctions for the slab are --- THE ANGLE APPROXIMATION $$ \psi_j(z) = \begin{cases} \sqrt{\frac{2/L}{1 + \text{si}(\zeta_j\pi)}} \cos \left(\zeta_j\pi \frac{z}{L}\right), & \text{for even } j \\ \sqrt{\frac{2/L}{1 - \text{si}(\zeta_j\pi)}} \sin \left(\zeta_j\pi \frac{z}{L}\right), & \text{for odd } j \end{cases} $$ (8.21) Inserting the solution, Eq. (8.20) and Eq. (8.21), back into the Milne equation, Eq. (8.18), we get the interrelation between the trapping factors $g_j$ and the parameters for the shape of the modes, $\zeta_j$, namely $$ g_j = 1 + \left( \frac{2\bar{k}L}{\zeta_j \pi} \right)^2. $$ (8.22) The parameters are determined from the boundary conditions of the time-decay problem, which are that no flux is incident at the slab boundaries $z = \pm L/2$. In terms of the Schuster–Schwarzschild approximation, Eq. (8.11), this means that the forward or backward fluxes, respectively, must vanish at the boundaries: $$ \begin{aligned} \bar{F}^{\text{b}}(+L/2, t) &= -\bar{I}^{\Sigma}(+L/2, t) + 2F^{\Sigma}(+L/2, t) = 0 \\ \bar{F}^{\text{f}}(-L/2, t) &= \phantom{-}\bar{I}^{\Sigma}(-L/2, t) + 2F^{\Sigma}(-L/2, t) = 0 \end{aligned} $$ (8.23) By inserting Eqs (8.9) and (8.11) into the boundary conditions, Eq. (8.23), and using the general solution, Eq. (8.20), the parameters $\zeta_j$ are finally determined: $$ \begin{aligned} \tan \left( \frac{\pi}{2} \zeta_j \right) &= +\frac{2\bar{k}L}{\pi \zeta_j} \qquad \text{for even } j \\ \cot \left( \frac{\pi}{2} \zeta_j \right) &= -\frac{2\bar{k}L}{\pi \zeta_j} \qquad \text{for odd } j. \end{aligned} $$ (8.24) ## 8.2 The angle approximation We now take a closer look at the Schuster–Schwarzschild approximation. It states that $$ \begin{aligned} \int_0^{\pi/2} I(\vartheta) \cos \vartheta \sin \vartheta \, \text{d}\vartheta &\approx \frac{1}{2} \int_0^{\pi/2} I(\vartheta) \sin \vartheta \, \text{d}\vartheta \\ \int_{\pi/2}^{\pi} I(\vartheta) \cos \vartheta \sin \vartheta \, \text{d}\vartheta &\approx -\frac{1}{2} \int_{\pi/2}^{\pi} I(\vartheta) \sin \vartheta \, \text{d}\vartheta. \end{aligned} $$ (8.25) ### Interpretation 1 The approximation becomes exact in the case that the intensity $I$ is independent of angle $\vartheta$, i.e. for isotropic radiation, and in the case that $$ I(\vartheta) = c1 \cdot \delta \left( \vartheta - \frac{\pi}{3} \right) + c2 \cdot \delta \left( \vartheta - \frac{4\pi}{3} \right). $$ (8.26) Equation (8.26) physically means that the stream of photons from all directions is replaced by two streams of 'typical' photons that all move in the directions $\vartheta = \pm 60^\circ$. --- The Schuster–Schwarzschild approximation, Eq. (8.25), is thus also known as the ‘two-stream approximation’. Experimental investigations of the anisotropy of the radiation have been performed by Streater and Cooper (1988). ### Interpretation 2 A second interpretation of this angle approximation can be found when the integrals over the angles are replaced by quadrature sums (Chandrasekhar 1952) $$ \int_{-1}^{1} f(\mu) d\mu = \sum_{i=1}^{N_\mu} b_i f(\mu_i). $$ (8.27) When we truncate the sum after two terms, $N_\mu = 2$, and use $\mu_i = \cos \vartheta_i = \pm 1/2$ as abscissae for the evaluation, we again have the Schuster–Schwarzschild approximation. When we set the abscissae for a Gaussian quadrature, $\mu_i = \pm 1/\sqrt{3}$, we get an equivalent formulation. Note, however that the term $4\bar{k}^2$ occurring in the Milne equation is replaced by $3\bar{k}^2$. A generalization towards more accuracy is easily obtained by including more terms in the summation, $N_\mu > 2$. In that case, we inadvertently switch over to another method—we get a discrete-ordinate solution with $N_\mu$-term quadrature. ### Interpretation 3 Finally, we can give yet another interpretation, --- THE ANGLE APPROXIMATION [FIGURE: FIG. 8.1. Trapping factor $g_0$ as computed by the Milne theory with various higher-order (order $N_\mu$) quadrature sums as compared to the result of a Monte Carlo simulation for a Doppler lineshape at opacity $k_0L = 0.1$.] We suspected that the Schuster–Schwarzschild approximation is the culprit and proved this by performing $N_\mu$-term Gaussian quadrature computations at very low opacities (Molisch *et al.* 1992c). Figure 8.1 shows the result. For very large $N_\mu$, the trapping factor $g_0$ tends to the correct value. Hence the non-vanishing error at low opacities stems from the Schuster–Schwarzschild approximation, which means performing only an $N_\mu = 2$ quadrature sum. For the steady-state case, the results of Avrett and Hummer (1965) indicate that the error in the excited-state distribution by using the Eddington approximation (not the Schuster–Schwarzschild) is smaller than 15%. A discussion of the most probable Eddington factors can be found in (Minerbo 1978). The Eddington approximation for spheres is discussed by Simmoneau (1976, 1978a, b) and Hummer (1984); see also the next section. In time-dependent problems, the use of the Eddington approximation can lead to unphysical results when the gradients of the flux density $J$ are large. In that case, propagation velocities in excess of the speed of light may occur. In order to combat these problems, so-called 'flux-limiters' are introduced. There is a considerable number of such flux-limiters, see e.g. Mihalas and Weaver (1982), Pomraning (1981, 1982, 1983, 1984, 1986) and Levermore (1984). From Eqs (8.22) and (8.24) it follows that the trapping factor $g_0$ goes like $1 + \bar{k}L$ at very low opacities, $\bar{k}L \ll 1$ (using the Schuster–Schwarzschild approximation). More detailed computations show, however, that the actual behaviour is $1 + \bar{k}L \cdot \ln(\bar{k}L)$. When we use the Eddington approximation in the Holstein equation, we are spared the integration of the Kernel function over the variable $\mu$. The Kernel function then reads (Finn 1968) $$ G(|z - z'|) = \frac{\sqrt{3}}{2} C_x \int_{-\infty}^{\infty} k(x)^2 \exp \left( -\sqrt{3}k(x)|z - z'| \right) \mathrm{d}x \qquad (8.30) $$ --- The savings for the solution of the Holstein equation are, however, not very large, because the integration over angle can be done analytically, anyway. The evaluation of the resulting exponential integrals does not take much longer than the evaluation of the exponential function in Eq. (8.30). ## 8.3 The generalized Eddington approximation Giovanelli (1959, 1963) derived a generalized approximation to the (frequency-independent) equation of radiative transfer that also accounts for an inhomogeneous absorption coefficient. He starts out with the transfer equation $$ \frac{\mathrm{d}I}{\mathrm{d}s} = -k[I - S]. $$ (8.31) In rectangular coordinates, $$ \sin \vartheta \cos \varphi \frac{\partial I}{\partial x} + \sin \vartheta \sin \varphi \frac{\partial I}{\partial y} + \cos \vartheta \frac{\partial I}{\partial z} = -k[I - S], $$ (8.32) where $\varphi$ and $\vartheta$ are the angles between $I$ and the $x$ and $z$ axes, respectively. The intensity $I$ is now expanded into a series of spherical harmonics (with directions $\mu = \cos \vartheta$) $$ I = \sum_{i=0}^{\infty} \left\{ I_i P_i(\mu) + \sum_{j=1}^{i} \left[ a_i^j \cos(j\varphi) + b_i^j \sin(j\varphi) \right] P_i^j(\mu) \right\}, $$ (8.33) where the functions $P_i(\mu)$ are the Legendre polynomials and $P_i^j(\mu)$ are the associated Legendre functions. The parameters $I_i$, $a_i^j$ and $b_i^j$ are the expansion coefficients. Substituting the expansion Eq. (8.33) into the transfer equation, Eq. (8.32), and integrating over $\varphi$, he gets $$ \mu \frac{\partial}{\partial z} \sum_{i=0}^{\infty} I_i P_i(\mu) + \frac{1}{2} \sin \vartheta \sum_{i=1}^{\infty} \left[ \frac{\partial a_i^1}{\partial x} + \frac{\partial b_i^1}{\partial y} \right] P_i^1(\mu) = -k \left[ \sum_{i=0}^{\infty} I_i P_i(\mu) - S \right]. $$ (8.34) He then multiplies by the polynomials $P_j(\mu)$ and integrates from $-1$ to $1$: $$ \begin{aligned} & \frac{\partial}{\partial z} \sum_{i=0}^{\infty} I_i \left( \int_{-1}^{1} P_i(\mu) P_j(\mu) \mu \mathrm{d}\mu \right) + \\ & + \frac{1}{2} \frac{\partial}{\partial x} \sum_{i=1}^{\infty} a_i^1 \left( \int_{-1}^{1} P_i^1(\mu) P_j^0(\mu) \sqrt{1 - \mu^2} \mathrm{d}\mu \right) + \\ & + \frac{1}{2} \frac{\partial}{\partial y} \sum_{i=1}^{\infty} b_i^1 \left( \int_{-1}^{1} P_i^1(\mu) P_j^0(\mu) \sqrt{1 - \mu^2} \mathrm{d}\mu \right) = \\ & = -k \left[ \sum_{i=0}^{\infty} I_i \left( \int_{-1}^{1} P_i(\mu) P_j(\mu) \mathrm{d}\mu \right) - S \left( \int_{-1}^{1} P_j(\mu) \mathrm{d}\mu \right) \right]. \end{aligned} $$ (8.35) --- From the theory of Legendre functions, the following results are now used (with $i \ge j$) $$ \begin{aligned} \int_{-1}^1 P_i(\mu) d\mu &= \begin{cases} 2, & i = 0 \\ 0, & i > 0 \end{cases} \\ \int_{-1}^1 P_i(\mu) P_j(\mu) \mu d\mu &= \begin{cases} \frac{2i}{(2i)^2 - 1}, & i = j + 1 \\ 0, & \text{else} \end{cases} \\ \frac{2i + 1}{2} \int_{-1}^1 P_i(\mu) P_j(\mu) d\mu &= \begin{cases} 1, & i = j \\ 0, & \text{else} \end{cases} \\ \int_{-1}^1 P_i^1(\mu) P_j^0(\mu) \sqrt{1 - \mu^2} d\mu &= \begin{cases} \frac{2(j + 2)!}{j!(2j + 1)(2j + 3)}, & i = j + 1 \\ 0, & \text{else} \end{cases} \end{aligned} \qquad (8.36) $$ With $j = 0$, we have --- $$ \frac{1}{3}\nabla^2 J = \frac{1}{3k}\nabla J \nabla k + k^2(J - S). $$ (8.43) Equation (8.43) is **the generalized Eddington approximation** to the equation of radiative transfer in three dimensions, including the effect of an inhomogeneous distribution of absorbers. ## 8.4 The frequency approximation The previous two sections have discussed the angle approximation both in the Milne theory and in its own right—in that case, called the Eddington approximation. A much more restrictive approximation in the Milne theory is the replacement of the actual lineshape by the ‘equivalent’ box-shaped line. We first repeat Holstein’s argument why this approximation, which (together with the angle approximation) allows the reduction to a diffusion type equation, cannot give correct results (Holstein 1947). The Milne equation is a diffusion type equation, which means that we have to define a mean free path. For an arbitrary lineshape, this mean free path must be averaged over frequency. The mean free path is defined as $$ \text{mfp} = - \int_0^\infty \rho \frac{\partial T}{\partial \rho} \mathrm{d}\rho, $$ (8.44) where $T(\rho, x)$ is the probability of traversing the distance $\rho$ without being reabsorbed, see Section 4.2. Inserting the expression for $T$, and averaging over all frequencies, we get $$ \text{mfp} = - \int_{-\infty}^\infty \int_0^\infty C_x k(x) \frac{\partial [\exp(-k(x)\rho)]}{\partial \rho} \rho \mathrm{d}\rho \mathrm{d}x = C_x \int_{-\infty}^\infty \mathrm{d}x, $$ (8.45) which diverges. Since the mean free path is infinite, its definition is not meaningful, and representation of the trapping problem by a diffusion equation is not possible. This is a consequence of the assumption of complete frequency redistribution. One could now argue that the above derivation says nothing about partial frequency redistribution. However, van Trigt showed that the diffusion equation description is *only* possible for complete frequency coherence in the laboratory rest frame; see also (Vasilev and Kogan 1967). In that case, we would have to replace the factor $C_x k(x)$ in Eq. (8.45) by $\delta(x - x_0)$, where $x_0$ is the frequency of the ‘initial’ photon. With that, $\text{mfp} = 1/k(x_0)$, as we expect. That said, we now proceed to consider what is the best definition of an equivalent opacity. Several definitions have been proposed, e.g. by Zemansky (1927, 1930, 1932) and Kenty (1932). The most popular one is the one by Samson (1932), see also (Blickensderfer *et al.* 1976), $$ \exp(-\bar{k}L) = C_x \int k(x) \exp[-k(x)L] \mathrm{d}x $$ (8.46) The idea behind this definition of an equivalent opacity $\bar{k}L$ is, for photons with normal incidence on the slab, to keep constant the number of photons that pass through --- the slab without being absorbed. With this definition, the Milne theory gives results for the lowest-order trapping factor $g_0$ that are accurate within 10% for opacities $k_0L < 4$. This is already quite valuable, but we would like to have a theory that bridges the gap to the high-opacity approximations, i.e. one would wish to extend the validity up to opacities of about $k_0L = 20$. Motivated by this consideration, we have proposed a new definition for the equivalent absorption coefficient $\bar{k}$. It has been shown by Kenty (1932) that the mean square free path is more meaningful for trapping problems than the mean free path. This fact led the way in our search for a suitable definition of a mean photon path length $L_m$ to be used instead of the cell dimension $L$ in the definition of an equivalent opacity, Eq. (8.46). We found an astonishingly good expression for this mean path length, $$ L_m^2 = C_x \int_{-\infty}^{\infty} \frac{k(x)}{[1/L]^2 + [0.18 \cdot k(x)]^2} \mathrm{d}x. \qquad (8.47) $$ This definition combines the idea of using mean squares with the idea that a photon's path is delimited either by absorption or by the cell boundaries. Hence, Eq. (8.47) defines a kind of harmonic mean between the reabsorption length and the geometric dimensions determining the escape. The weight factor 0.18 is rather arbitrary and was chosen because it gives good results for a large range of lineshapes and opacities. We can now use this mean path length $L_m$ instead of the dimension $L$ in the equivalent opacity definition, Eq. (8.46). With this new definition, the error in the trapping factor $g_0$ remains below 10% for opacities $k_0L < 30$ for a Lorentz lineshape and for $k_0L < 50$ for a Doppler shape, see Fig. 8.2. We also tested Voigt lineshapes for a large range of parameters $a$ and found that the error was less than in the Lorentz case. In Fig. 8.3, a very general lineshape (three partially overlapping Voigt hfs components of different magnitudes) was used, and again 10% accuracy up to opacity $k_0L = 30$ was achieved. In Fig. 8.4, we show the lowest-order trapping factor for a realistic example, $^{87}$rubidium vapour at about 80$^\circ$C with 4.5 mbar xenon. The hyperfine structure of $^{87}$Rb is shown in Fig. 8.5. We again see that for opacities smaller than 30, the (modified) Milne theory gives good results. We must emphasize that this definition of the equivalent opacity is rather arbitrary. We proposed it because it is simple, is physically reasonable, gives good results (in the slab case) up to opacities that can already be treated with high-opacity approximations, and is valid for general forms of the absorption coefficient. The definition of an equivalent opacity can become arbitrarily complicated when one attempts to achieve very high accuracy (for the Doppler case, see e.g. Kenty (1932)). In this case, we might just as well solve the integral equation. The Milne equation can thus be used as a straightforward approximation for simple cases. However, one has to be very careful when treating general problems. Effects of, e.g. inhomogeneous absorber distributions on the accuracy of the approximation are currently unknown; it would be thus dangerous to simply infer validity of the Milne equation at low opacities also for those cases. --- [FIGURE: Log-log plot of Trapping factor g_0^D vs Opacity k_0L] FIG. 8.2. Trapping factor $g_0^D$ for a Doppler profile in a slab. Comparison of MC-simulation with results from the Milne theory with Samson's $\bar{k}$, and with the Milne theory with the new $\bar{k}$. [FIGURE: Log-log plot of Trapping factor g_0^V vs Opacity k_0'L] FIG. 8.3. Trapping factor $g_0^V$ for a three-component Voigt lineshape: $k_0'k(x) = \mathrm{const} \cdot [k^V(x) + 0.4k^V(x - 6) + 0.7k^V(x + 3)]$. Voigt parameter $a = 0.3$, the maximum absorption coefficient is denoted as $k_0'$. Comparison of MC-results to Milne with the new $\bar{k}$. --- THE FREQUENCY APPROXIMATION [FIGURE: Graph of Trapping factor g_0 vs Opacity k_0L] FIG. 8.4. Trapping factor $g_0$ for the 780 nm Rb line with 4.5 mbar xenon buffer gas at 80 °C, Doppler FWHM = 0.55 GHz, Voigt parameter $a = 0.1$. Comparison of MC-simulation with results from the Milne theory with Samson's $\bar{k}$, and with the Milne theory with the new $\bar{k}$. [FIGURE: Diagram of 87Rb D2-Resonance Line hfs components] FIG. 8.5. Relative amplitudes and frequencies of the hfs components of the absorption coefficient of the 780 nm resonance line of $^{87}$Rb. --- # 9 # MATHEMATICAL METHODS FOR THE TRANSFER EQUATION ## 9.1 Discrete ordinate solution The discrete-ordinate method was proposed first for solving the equation of radiative transfer by Chandrasekhar (1952). It has been applied to trapping especially for the computation of steady-state problems in slabs in the transfer- plus rate-equation formulation. We start out with the basic equations $$ \mu \frac{d I(x, \mu, z)}{d z} = -k(x) \left[ I(x, \mu, z) - \frac{A_{21}}{B_{12}} \frac{n(z)}{N} \right] $$ (9.1) $$ n(z) = \frac{N B_{12}}{A_{21}} \frac{1}{2} \int_{-1}^{1} \int_{-\infty}^{\infty} I(x, \mu, z) C_x k(x) \mathrm{d}x \mathrm{d}\mu, $$ (9.2) where we denote $0.5 C_x \int \int I \cdot k(x) \mathrm{d}x \mathrm{d}\mu$ as $\bar{J}$. Following the approach of Hummer and Kunasz (1976), we then split the intensity into two components, namely $I_{\text{ext}}$, which describes the photons that have never been scattered, and $I_{\text{t}}$, which describes the photons that were scattered at least once, so that $I = I_{\text{ext}} + I_{\text{t}}$. The diffuse (i.e. scattered) field is the solution of Eqs (9.1) and (9.2) with no incident radiation in the slab. The boundary conditions are thus $$ \begin{array}{c} I_{\text{t}}(x, -\mu, +L/2) = 0 \\ I_{\text{t}}(x, +\mu, -L/2) = 0 \end{array} $$ (9.3) The occurring integrals are approximated by Gaussian quadrature $$ \begin{aligned} C_x \int_{-\infty}^{\infty} k(x) f(x) \mathrm{d}x &\approx \sum_{-N_x}^{N_x} a_i f(x_i) \\ \int_{-1}^{1} f(\mu) \mathrm{d}\mu &\approx \sum_{-N_\mu}^{N_\mu} b_i f(\mu_i) \end{aligned} $$ (9.4) where $a_i$ and $b_i$ are the weights and $x_i$ and $\mu_i$ the abscissae of the quadrature. The normalization conditions $$ \sum a_i = \frac{1}{2} \sum b_i = 1 $$ (9.5) must be fulfilled. 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- We then solve the homogeneous equation (i.e. with no external field) by looking for solutions of the form $$ \begin{aligned} I_{i,j} &= c_{i,j} \cdot \exp \left( -\gamma k_0 \left( z + \frac{L}{2} \right) \right) \\ n_j &= d_j \cdot \exp \left( -\gamma k_0 \left( z + \frac{L}{2} \right) \right), \end{aligned} $$ (9.6) where $c_{i,j}$, $d_j$, and $\gamma$ are constants. Inserting Eqs (9.4)–(9.6) into the equation of radiative transfer, we obtain the characteristic equation for $\gamma$, $$ \frac{1}{2} \sum_i \sum_j \frac{a_i b_j}{1 + \frac{\mu_j \gamma}{C_x k(x_i)}} - 1 = 0 $$ (9.7) The roots of Eq. (9.7) determine the solution of the homogeneous problem. For the particular solution, the external field is expressed as $$ \bar{J}^{\text{ext}}(z) = \sum_l J_l \exp \left( -q_l k_0 \left( z + \frac{L}{2} \right) \right) $$ (9.8) and inserted into the equation of radiative transfer. The factors $q_l$ are known constants. By applying the boundary conditions, the general solution is then found. This method is treated in more detail in Sec. 10.4.3; see also (Kunasz and Kunasz 1975). One troubling property of the discrete-ordinate technique is that coefficients are determined by boundary conditions at both boundaries, which (for high-opacity slabs) can lead to numerically bad behaviour. This problem is not that serious in laboratory applications (more so in astrophysics), furthermore Schmidt and Wehrse (1987) showed how to eliminate it by appropriate matrix manipulations. In their formulation, the discrete-ordinate method also shows relations to the discrete-space theory (see below). **Further numerical aspects** are discussed by Wehrse (1985), Stamnes *et al.* (1988), and Nakajima and Tanaka (1986). The numerical properties of the closely related $S_N$-method (and also the discrete-ordinate technique itself) are discussed by Lathrop and Carlson (1967, 1971). Gouttebrouze (1990) showed that the discrete ordinate method can also be applied to cylinders. It was applied furthermore in many of the early astrophysical papers on radiation trapping, e.g. (Thomas 1957) and (Jefferies 1960). **An alternative formulation** was devised by Rybicki and Usher (1966). The main problem in the usual discrete-ordinate method is that we have two coupled exponentially increasing and decreasing terms, whose amplitudes are fixed by the two boundary conditions. Due to the increasing terms, the problem is numerically badly behaved. The idea behind the Rybicki–Usher transformation is now to transform the problem into a one-point boundary value problem, where we have only the decreasing terms. This transform results in a generalized Riccatti differential equation. The elements of the transformation matrix have to be found as the solution of differential equations. While this method is simpler than the invariant-embedding method (which is popular in --- scattering theory), it has gained little following—a competing method, the Feautrier technique,[^12] is very popular (see Sec. 9.2). Also **the discrete-space theory** (Peraiah 1984) bears some relation to this technique. It is not in widespread use for laboratory radiation trapping problems—still, it is a nice mathematical idea, so that we briefly outline the thoughts. This theory was originally devised to solve usual scattering processes (Grant and Hunt 1968, 1969a, b). Let us assume a plane-parallel geometry. We first consider an 'elementary cell' and see that the radiation emerging from the cell top consists of (i) the radiation reflected by the top of the elementary cell, (ii) the radiation incident on the elementary cell bottom and being transmitted, and (iii) the radiation generated within the elementary cell. We next define two-element vectors, containing the radiation in the positive and negative directions at points $z_i$. The matrix elements are determined by the properties of the elementary cell. The radiation at point $z_1$ and $z_N$ (so that we have $N$ elementary cells between the two points) is related by a matrix that is built up from the elementary matrices by an operation called the 'star-product'. One can also go to the limit of infinitely thin elementary cells and derive a differential equation for the intensities. Related papers are (Waterman 1981) and (Peraiah and Wehrse 1978). The spherical case is treated by Grant and Peraiah (1972), Peraiah (1973), Peraiha and Varghese (1985), Peraiah (1987), and Wehrse and Kalkofen (1985). Polarization is covered by Nagendra and Peraiah (1984, 1985). ## 9.2 The Feautrier technique The discrete ordinate solution is a solution method for the transfer equation that can easily be applied both to steady-state and to transient problems. The techniques that we will present in this chapter are intended mainly for the steady-state problems. Strangely enough, they are not well-suited for the classical steady-state problem treated in this chapter. On the contrary, we believe that the integral equation method described previously is more stable, more accurate, and easier to implement. In addition, these integral equation techniques are easily applied to finding the eigenvalues, i.e. to solving the transient problem in a rather straightforward way.[^13] The methods that we describe in this section (essentially the Feautrier technique and its derivatives) are, however, the basis for important methods for treating *non-linear* [^12]: The Feautrier method, which we will encounter in the next section, converts the problem into a single second-order differential equation with two boundary conditions, which is also numerically well behaved. [^13]: This is a somewhat personal view. In the astrophysical literature, the discussion between advocates of the integral equation method and the differential equation method has been raging for decades. Every month or so, somebody presents a modification to one of these techniques and claims that the new modified method gives absolute superiority. Our view that the integral equation technique (especially the piecewise-constant technique) is superior *for standard steady-state problems of chemical physics* is based on the following point. For the computation of the $A_{k,m}$ elements, we can use standard 'canned' integration routines with adaptive choice of the frequency integration points. For the differential equation techniques we have to prescribe the discrete frequencies at which we evaluate the lineshape, which requires a lot of prior knowledge of the solution. For the *standard transient problems*, the integral equation technique also seems preferable. Finding the eigenvalues of the $A_{k,m}$ values is much simpler than of the matrices occurring in the Feautrier technique, simply because the latter are *much* larger. The only possiblity of using the Feautrier technique for transient problems is thus a finite differencing in the time domain, similar to the propagator function method PFM of Chapter 5; the comments about PFM made there apply in this case, too. --- transient or steady-state problems, and problems with *partial frequency redistribution*. The usefulness of these techniques will become obvious in Chapters 11–14. Reviews of the Feautrier technique and its derivatives can be found in (Auer 1971), (Hummer and Rybicki 1971b), and (Mihalas 1984). The following treatment deals with the plane-parallel slab case, so that the describing equations are (see Chapter 4.4) $$ \mu \frac{\partial I(z, \mu, \nu)}{\partial z} = -k(x)[I(z, \mu, \nu) - S(z)] $$ (9.9) where $S$ is the source function, $(A_{21}/B_{12}) \cdot (n(z)/N)$, and is related to the average intensity by $$ S(z) = \frac{1}{2} C_\nu \int_{-\infty}^{\infty} \int_{-1}^{1} I(z, \mu, \nu)k(\nu)\mathrm{d}\mu \mathrm{d}\nu + E^{\mathrm{S}}(z) $$ (9.10) where $E^{\mathrm{S}}$ is the excitation function in units of the source function (i.e. its unit is an intensity, not a particle density per second). ### 9.2.1 The basic Feautrier technique When we try to solve a radiation trapping problem in the simplest (differential) formulation, we would basically require the solution of a first-order differential equation, with one boundary condition. However, this is numerically very problematic. As we have only one boundary condition, there are exponentially increasing terms, so that a small error is amplified as we progress through the slab. Solutions obtained by this method are thus often meaningless. It is preferable to have a two-point boundary value problem, where the second boundary condition puts a limit on the amplitude of the increasing term, thus giving much better numerical stability. The Feautrier technique (Feautrier 1964) achieves this transformation from the usual transfer equation to a second-order differential equation by introducing the variables $$ \begin{aligned} \Theta(z, x, \mu) &= \frac{1}{2}[I(z, x, \mu) + I(z, x, -\mu)] \quad \mu > 0 \\ \Gamma(z, x, \mu) &= \frac{1}{2}[I(z, x, \mu) - I(z, x, -\mu)] \end{aligned} $$ (9.11) We then write down one transfer equation for radiation in direction $\mu$, and one for direction $-\mu$. We add and subtract these two equations, and get a coupled system of first-order equations for $\Theta$ and $\Gamma$: $$ \begin{aligned} \frac{\mu}{k(x)} \frac{\partial \Gamma(z, x, \mu)}{\partial z} &= -\Theta(z, x, \mu) + S(z) \\ \frac{\mu}{k(x)} \frac{\partial \Theta(z, x, \mu)}{\partial z} &= -\Gamma(z, x, \mu) \end{aligned} $$ (9.12) Inserting the second into the first equation, the transfer equation is transformed to --- 144 MATHEMATICAL METHODS FOR THE TRANSFER EQUATION $$ \begin{aligned} \left( \frac{\mu}{k(x)} \frac{\partial}{\partial z} \right) \left( \frac{\mu}{k(x)} \frac{\partial}{\partial z} \right) \Theta(z, x, \mu) &= \Theta(z, x, \mu) - S(z) = \\ &= \Theta(z, x, \mu) - C_x \int_{-\infty}^{\infty} k(x) \int_0^1 \Theta(z, x, \mu') d\mu' dx - E^S(z) \end{aligned} $$ (9.13) The boundary conditions are a slightly modified form of Eq. (9.12). They are of the Cauchy type: $$ \begin{aligned} \left. \frac{\mu}{k(x)} \frac{\partial \Theta(z, x, \mu)}{\partial z} \right|_{z=+L/2} &= -\Theta(L/2, x, \mu) + I_{\text{inc}}(L/2, x, -\mu) \\ \left. \frac{\mu}{k(x)} \frac{\partial \Theta(z, x, \mu)}{\partial z} \right|_{z=-L/2} &= +\Theta(-L/2, x, \mu) - I_{\text{inc}}(-L/2, x, \mu) \end{aligned} $$ (9.14) The integro-differential equation (9.13) and the boundary conditions (9.14) completely specify the problem in the Feautrier formulation. The solution is found by finite differencing. There are several possibilities to specify the first- and second-order derivatives. One example is the formulation given in Appendix E. (The equations in Appendix E are for the general case of a finite cylinder; the plane-parallel slab is just a special case). Other possibilities are described by Cannon (1985, pp. 137–139). All these formulations, however, approximate the second derivative as $$ \begin{aligned} \left. \left( \frac{\mu_j}{k(x_i)} \frac{\partial}{\partial z} \right) \left( \frac{\mu_j}{k(x_i)} \frac{\partial}{\partial z} \right) \Theta(z, x_i, \mu_j) \right|_{z=z_k} = \mu_j^2 \bigg( &a_{k,i} \Theta_{k-1,i,j} + b_{k,i} \Theta_{k,i,j} \\ &+ c_{k,i} \Theta_{k+1,i,j} \bigg) \end{aligned} $$ (9.15) where the actual value of the constants $a$, $b$, $c$ depends on the chosen finite-difference formulation. For the computation of the source function, we use the quadrature formula $$ \left. C_x \int_{-\infty}^{\infty} k(x) \int_0^1 \Theta(z, x, \mu) d\mu dx \right|_{z=z_k} \approx \sum_{i=1}^{N_x} \sum_{j=1}^{N_\mu} p_i q_j \Theta_{k,i,j} $$ (9.16) where $p$ and $q$ are related to the quadrature weights. In the simplest case of equidistant discretization and trapezoidal quadrature, $p_i = C_x k(x_i)(x_i - x_{i-1})$, and $q_j = \mu_j - \mu_{j-1}$. We now write all $\Theta$ at a certain spatial point $z_k$ into a vector $\mathbf{\Theta_k}$ --- $$ \mathbf{\Theta}_k = \begin{pmatrix} \Theta_{k,1,1} \\ \Theta_{k,2,1} \\ \cdot \\ \cdot \\ \Theta_{k,N_x,1} \\ \Theta_{k,1,2} \\ \cdot \\ \cdot \\ \Theta_{k,N_x,N_\mu} \end{pmatrix} $$ (9.17) and define the matrices $$ \mathbf{A}_k = \begin{pmatrix} a_{k,1}\mu_1^2 & & & & & 0 \\ & \cdot & & & & \\ & & a_{k,N_x}\mu_1^2 & & & \\ & & & a_{k,1}\mu_2^2 & & \\ & 0 & & & \cdot & \\ & & & & & a_{k,N_x}\mu_{N_\mu}^2 \end{pmatrix} $$ (9.18) $$ \mathbf{C}_k = \begin{pmatrix} c_{k,1}\mu_1^2 & & & & & 0 \\ & \cdot & & & & \\ & & c_{k,N_x}\mu_1^2 & & & \\ & & & c_{k,1}\mu_2^2 & & \\ & 0 & & & \cdot & \\ & & & & & c_{k,N_x}\mu_{N_\mu}^2 \end{pmatrix} $$ (9.19) $$ \mathbf{P}_k = \begin{pmatrix} p_1 q_1 & p_2 q_1 & \cdot & \cdot & p_{N_x} q_{N_\mu} \\ \cdot & & & & \cdot \\ \cdot & & & & \cdot \\ p_1 q_1 & & \cdot & \cdot & p_{N_x} q_{N_\mu} \end{pmatrix} $$ (9.20) The matrix $\mathbf{B}_k$ is then defined as $$ \mathbf{B}_k = \mathbf{1} - \mathbf{P}_k + \mathbf{A}_k + \mathbf{C}_k $$ (9.21) We furthermore define a vector $\mathbf{E}_k$ that has the value $E^S(z_k)$ in every element. The Feautrier problem can thus be written as a matrix equation $$ -\mathbf{A}_k \mathbf{\Theta}_{k-1} + \mathbf{B}_k \mathbf{\Theta}_k - \mathbf{C}_k \mathbf{\Theta}_{k+1} = \mathbf{E}_k $$ (9.22) --- MATHEMATICAL METHODS FOR THE TRANSFER EQUATION The boundary conditions are $$ \begin{matrix} \mathbf{B}_1 \mathbf{\Theta}_1 - \mathbf{C}_1 \mathbf{\Theta}_2 = \mathbf{E}_1 \\ -\mathbf{A}_{N_z} \mathbf{\Theta}_{N_z-1} + \mathbf{B}_{N_z} \mathbf{\Theta}_{N_z} = \mathbf{E}_{N_z}, \end{matrix} $$ (9.23) where $$ \mathbf{B}_1 = \mathbf{1} + \mathbf{C}_1, \qquad \mathbf{C}_1 = \frac{1}{z_2 - z_1} \begin{pmatrix} \frac{\mu_1}{k(x_1)} & & & \\ & \frac{\mu_1}{k(x_2)} & & \\ & & \cdot & \\ & & & \frac{\mu_{N_\mu}}{k(x_{N_x})} \end{pmatrix}, $$ $$ \mathbf{E}_1 = \begin{pmatrix} I_{inc}(x_1, \mu_1) \\ I_{inc}(x_2, \mu_1) \\ \cdot \\ \cdot \\ I_{inc}(x_{N_x}, \mu_{N_\mu}) \end{pmatrix} $$ (9.24) and $$ \mathbf{B}_{N_x} = \mathbf{1} + \mathbf{A}_{N_z}, \qquad \mathbf{A}_{N_z} = \frac{1}{z_{N_z} - z_{N_z-1}} \begin{pmatrix} \frac{\mu_1}{k(x_1)} & & & \\ & \frac{\mu_1}{k(x_2)} & & \\ & & \cdot & \\ & & & \frac{\mu_{N_\mu}}{k(x_{N_x})} \end{pmatrix}, $$ $$ \mathbf{E}_{N_z} = \begin{pmatrix} I_{inc}(x_1, \mu_1) \\ I_{inc}(x_2, \mu_1) \\ \cdot \\ \cdot \\ I_{inc}(x_{N_x}, \mu_{N_\mu}) \end{pmatrix} $$ (9.25) The brute-force approach for solving the discretized Feautrier equation (9.22) would be to simultaneously solve the $N_z \cdot N_\mu \cdot N_x$ linear equations. However, such an approach is incredibly inefficient. The huge matrix that contains all the $\mathbf{A}_k, \mathbf{B}_k, \dots$ is largely sparse, so that special techniques for the evaluation of sparse-matrix equations can be used. This is in strong contrast to the integral-equation approach. In the integral equation, the $A_{k,m}$ matrix is full, and while there are some symmetry properties that can be --- exploited (e.g., the Toeplitz structure for the plane-parallel slab case), the basic numerical effort is proportional to $N_z^3$ (i.e. the third power of the matrix size). For the Feautrier equation, we need to exploit the special structure of the matrix. The system of linear equations (9.22) has a block tri-diagonal structure that can be exploited for an efficient computation in the original Feautrier method. We make forward-elimination and back-substitution, starting out by computing the matrices $\mathbf{U}$ and $\mathbf{V}$ at the lower boundary $$ \mathbf{U}_1 = \mathbf{B}_1^{-1} \mathbf{E}_1 \qquad \mathbf{V}_1 = \mathbf{B}_1^{-1} \mathbf{C}_1 $$ (9.26) Furthermore, we compute the matrices $\mathbf{U}$ and $\mathbf{V}$ at a general position given by $$ \begin{aligned} \mathbf{U}_k &= (\mathbf{B}_k - \mathbf{A}_k \mathbf{V}_{k-1})^{-1} (\mathbf{E}_k + \mathbf{A}_k \mathbf{U}_{k-1}) \\ \mathbf{V}_k &= (\mathbf{B}_k - \mathbf{A}_k \mathbf{V}_{k-1})^{-1} \mathbf{C}_k \end{aligned} $$ (9.27) At the upper boundary, the boundary condition is $$ \mathbf{V}_{N_z} = \mathbf{0} \qquad \mathbf{U}_{N_z} = \mathbf{\Theta}_{N_z} $$ (9.28) This immediately gives $\mathbf{\Theta}$ at the upper boundary. At the other points, we get $\mathbf{\Theta}$ by back-substitution $$ \mathbf{\Theta}_k = \mathbf{U}_k + \mathbf{V}_k \mathbf{\Theta}_{k+1} $$ (9.29) The elimination scheme given above is the 'classical' one, originally by Feautrier, and used in a huge number of papers. However, a numerically better behaved scheme is obtained by introducing the auxiliary variables (Rybicki and Hummer 1991) $$ \mathbf{H}_k = -\mathbf{A}_k + \mathbf{B}_k - \mathbf{C}_k \quad \text{and} \quad \mathbf{F}_k = \mathbf{V}_k^{-1} - \mathbf{1} $$ (9.30) so that the elimination scheme becomes $$ \begin{aligned} \mathbf{F}_1 &= \frac{\mathbf{H}_1}{\mathbf{C}_1} & \mathbf{U}_1 &= \mathbf{B}_1^{-1} \mathbf{E}_1 \\ \mathbf{F}_k &= \left( \mathbf{H}_k + \frac{\mathbf{A}_k \mathbf{F}_{k-1}}{\mathbf{1} + \mathbf{F}_{k-1}} \right) \mathbf{C}_k^{-1} & \mathbf{U}_k &= \frac{\mathbf{E}_k + \mathbf{A}_k \mathbf{U}_{k-1}}{\mathbf{C}_k (\mathbf{1} + \mathbf{F}_k)} \\ \mathbf{\Theta}_{N_z+1} &= \mathbf{0} & \mathbf{\Theta}_k &= (\mathbf{1} + \mathbf{F}_k)^{-1} \mathbf{\Theta}_{k+1} + \mathbf{U}_k \end{aligned} $$ (9.31) The largest part of the numerical effort is the inversion of the $N_z$ matrices of size $N_x \cdot N_\mu$. Since these matrices are not diagonal, the numerical effort for each of these inversions is proportional to $(N_x \cdot N_\mu)^3$. All in all, the numerical effort is about $N_z \cdot (N_x \cdot N_\mu)^3$. Some further numerical niceties of solving the system (especially concerning roundoff errors) can be found in (Nordlund 1982). A lot of effort has been devoted to improving the basic algorithm described in this subsection. In the subsequent subsections, we will discuss boundary conditions, finite-differencing equations, Rybicki's reorganization, and splitting algorithms. For these --- rather numerical points, we will not give the full derivations but just state the final results that are necessary to implement them in a computer program. Other points of research were the choice of the quadrature points and weights (Hummer and Rybicki 1967), and the stability of the Feautrier technique. These aspects are also discussed by Cannon (1985). The variable Eddington factor technique will be described in Sec. 9.3. ### 9.2.2 Modified finite-differencing equations The equations transforming the differential operators into difference operators that are given in every textbook on numerical analysis provide second order accuracy (also true for the equations described in Appendix E). However, it is possible to derive equations that have fourth-order accuracy while being no more complicated than the usual ones. This important improvement was invented by Auer (1976), see also (Auer 1984). Without going into the details of the derivation, we have to find matrices $\mathbf{A^a}$, $\mathbf{B^a}$, and $\mathbf{C^a}$, which are defined as $$ \begin{aligned} \mathbf{A_k^a} &= \mathbf{A_k} + (\mathbf{1} - \mathbf{P_{k-1}}) \mathbf{D1_k} \\ \mathbf{B_k^a} &= \mathbf{B_k} + (\mathbf{1} - \mathbf{P_k}) \mathbf{D2_k} \\ \mathbf{C_k^a} &= \mathbf{C_k} + (\mathbf{1} - \mathbf{P_{k+1}}) \mathbf{D3_k} \\ \mathbf{E_k^a} &= \mathbf{E_k} - \mathbf{D1_k} \mathbf{E_{k-1}} - \mathbf{D2_k} \mathbf{E_k} - \mathbf{D3_k} \mathbf{E_{k+1}} \end{aligned} $$ (9.32) The auxiliary matrices $\mathbf{D1_k}$, $\mathbf{D2_k}$, and $\mathbf{D3_k}$ are of diagonal form, where each element on the diagonal has the value $$ \begin{aligned} (D1_k) &= \frac{1}{6} \frac{\xi_{k+1} - \xi_k}{\xi_k - \xi_{k-1}} \cdot \frac{\xi_{k+1} - \xi_k}{\xi_{k+1} - \xi_{k-1}} - \frac{1}{6} \\ (D3_k) &= \frac{1}{6} \frac{\xi_k - \xi_{k-1}}{\xi_{k+1} - \xi_k} \cdot \frac{\xi_k - \xi_{k-1}}{\xi_{k+1} - \xi_{k-1}} - \frac{1}{6} \\ (D2_k) &= -(D1_k) - (D3_k) \end{aligned} $$ (9.33) assuming that the lineshape does not depend on position. The optical density is denoted by $\xi$, $$ \xi = \int_0^z k_0(u) du $$ (9.34) The only difference with respect to complexity is that the matrices $\mathbf{A^a}$ and $\mathbf{C^a}$ are not diagonal matrices. However, we have to invert a matrix that contains the full matrix $\mathbf{B}$ in any case, so that the 'fullness' of $\mathbf{A^a}$ and $\mathbf{C^a}$ does not play a role with respect to CPU time. As mentioned above, Auer's formulation gives fourth-order accuracy. The only drawback is that it might be more sensitive with respect to badly-chosen discretization points $z_k$. A discussion of the finite- differencing equations for spherical geometries is given by Auer (1984) and Leung (1976). The accuracy of the scheme was analysed by Mohan Rao *et al.* (1995), who also compared it to the discrete-space theory. --- THE VARIABLE EDDINGTON FACTOR TECHNIQUE ### 9.2.3 *Choice of the boundary conditions* The boundary conditions described in subsection 9.2.1 give only first-order accuracy. It is, however, possible, to obtain second-order accuracy without significantly increasing the computational effort (Auer 1967), see also (Auer and Mihalas 1968b). For this, we need the matrices $\mathbf{C}_1^\mathrm{b}$, $\mathbf{B}_1^\mathrm{b}$, and $\mathbf{D}_1^\mathrm{b}$, which are defined as $$ \mathbf{C}_1^\mathrm{b} = \frac{2}{C_x^2 (\xi_2 - \xi_1)^2} \begin{pmatrix} \left( \frac{\mu_1}{k(x_1)} \right)^2 & & 0 \\ & \ddots & \\ 0 & & \left( \frac{\mu_{N_\mu}}{k(x_{N_x})} \right)^2 \end{pmatrix} $$ (9.35) $$ \mathbf{B}_1^\mathrm{b} = \mathbf{1} + \mathbf{C}_1^\mathrm{b} + \mathbf{D}_1^\mathrm{b} - \mathbf{P}_1 $$ (9.36) $$ \mathbf{D}_1^\mathrm{b} --- $$ \frac{\partial H(z, x)}{\partial z} = -k(x) \left( J(z, x) - S(z) \right) $$ (9.41) We then multiply the transfer equation with the direction $\mu$, and again integrate over all $\mu$ to get $$ \frac{\partial K(z, x)}{\partial z} = -k(x) H(z, x) $$ (9.42) The two integrated versions of the transfer equation, Eqs. (9.41) and (9.42) can be combined, $$ \frac{\partial^2 K(z, x)}{\partial z^2} = k^2(x) \left( J(z, x) - S(z) \right) $$ (9.43) Next, we introduce the Eddington factors, which give the relative importance of the higher-order angular momenta of the intensity (see also (Atanackovic-Vukmanovic and Simonneau 1994) and (Simonneau and Atanackovic-Vukmanovic 1996)) $$ f_K(z, x) = \frac{K(z, x)}{J(z, x)} \qquad f_H(z, x) = \frac{\int_0^1 \theta(z, x, \mu) \mu \mathrm{d}\mu}{J(z, x)} $$ (9.44) Equation (9.43) then becomes a differential equation for the angle-averaged intensity, $$ \frac{\partial^2 f_K(z, x) J(z, x)}{\partial z^2} = k^2(x) \left( J(z, x) - S(z) \right) $$ (9.45) The boundary conditions are $$ \begin{aligned} \left. \frac{\partial f_K(z, x) J(z, x)}{\partial z} \right|_{z=-\frac{L}{2}} &= -k(x) \left( -f_H \left(-\frac{L}{2}, x\right) J \left(-\frac{L}{2}, x\right) + \int_0^1 \mu I_{\mathrm{inc}} \left(-\frac{L}{2}, x, +\mu\right) \mathrm{d}\mu \right) \\ \left. \frac{\partial f_K(z, x) J(z, x)}{\partial z} \right|_{z=+\frac{L}{2}} &= -k(x) \left( +f_H \left(+\frac{L}{2}, x\right) J \left(+\frac{L}{2}, x\right) - \int_0^1 \mu I_{\mathrm{inc}} \left(+\frac{L}{2}, x, -\mu\right) \mathrm{d}\mu \right) \end{aligned} $$ (9.46) At first glance it seems that we have gained nothing with that formulation, because the Eddington factors $f_K$ and $f_H$ are unknown. However, we can compute them by a simple iterative procedure. We start out with the guess values $f_K = 1/3$ and $f_H = 1/\sqrt{3}$ everywhere in the slab. We then solve the differential equation Eq. (9.45) with the Cauchy-type boundary conditions, Eq. (9.46). Equation (9.45) is of exactly the same type as the usual Feautrier equation, so that we can use the techniques that were already discussed above. The output of this ‘Feautrier’ solution is the source function, respectively particle density. With this source function, we can solve the original equation of radiative transfer, Eq. (9.40) to get the intensity $I(z, x, \mu)$ for *arbitrary directions* $\mu$. We can then easily compute the moments $J$, $H$, and $K$ of the intensity and thus the improved Eddington factors $f_H(z, x)$ and $f_K(z, x)$. With these factors, we repeat the procedure, starting by again solving Eq. (9.45) to get an improved source function and the next values for the Eddington factors. This procedure is continued until convergence is achieved. --- THE VARIABLE EDDINGTON FACTOR TECHNIQUE [FIGURE: Convergence of the variable Eddington factor technique as a function of optical depth] FIG. 9.1. Convergence of the variable Eddington factor technique as a function of optical depth $\xi$. Five iterations are plotted, but three overlap almost completely. Slab geometry, opacity $k_0L = 100$, steady-state, normalized quenching rate $Q/A_{21} = 10^{-4}$, thermal excitation. From (Auer 1984). The convergence of this iteration is very fast. Usually, two to three iterations are sufficient so that the change in the excited-state densities between subsequent iterations is smaller than, say, 1%. An example is given in Fig. 9.1. **Insert 9.1 Equations for the Eddington factor iteration in the spherical and cylindrical geometries.** **For the spherical geometry,** the *transfer equation* is $$ \mu \frac{\partial I(r, x, \mu)}{\partial r} + \frac{1 - \mu^2}{r} \frac{\partial I(r, x, \mu)}{\partial \mu} = -k(x) (I(r, x, \mu) - S(r)) $$ (9.47) and the *modified Feautrier equation* is $$ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{1}{q(r, x)} \frac{\partial q(r, x) f_K(r, x) J(r, x)}{\partial r} \right) = k^2(x) (J(r, x) - S(r)) $$ (9.48) where the function $q(r, x)$ is given by $$ q(r, x) = \exp \left[ \int_0^r \frac{1}{u} \left( 3 - \frac{1}{f_K(u, x)} \right) du \right] $$ (9.49) and the *boundary conditions* are $$ \begin{aligned} \left. \frac{\partial f_K(r, x) q(r, x) J(r, x)}{\partial r} \right|_{r=R} &= k(x) q(R, x) \left( \int_0^1 I_{\text{inc}}(R, x, -\mu) \mu d\mu - f_H(R, x) J(R, x) \right) \\ \left. \frac{\partial f_K(r, x) q(r, x) J(r, x)}{\partial r} \right|_{r=0} &= 0 \end{aligned} $$ (9.50) --- MATHEMATICAL METHODS FOR THE TRANSFER EQUATION A space-dependent absorption coefficient $k_0(r)$ poses no additional difficulties; see also (Hummer and Rybicki 1971a). The *intensities* $I(r, x, \mu)$ are computed from the linear ray equation $$ \mu \frac{\partial}{\partial r} \left( \mu \frac{\partial \Theta(r, x, \mu)}{\partial r} \right) = k^2(x) (\Theta(r, x, \mu) - S(r)) \qquad (9.51) $$ along rays parallel to the $z$-axis, where $\mu = [1 - (d_z/r)]^{1/2}$, and $d_z$ is the distance from the $z$-axis. The boundary conditions are $$ \begin{aligned} \mu \frac{\partial \Theta}{\partial r} &= k(x) [I_{\text{inc}} - \Theta] && \text{at } r = R \\ \mu \frac{\partial \Theta}{\partial r} &= 0 && \text{at } r = d_z \end{aligned} \qquad (9.52) $$ Once the intensities are known, the Eddington factors are computed from $$ f_K(r, x) = \frac{\int_0^1 \Theta(r, x, \mu) \mu^2 \mathrm{d}\mu}{\int_0^1 \Theta(r, x, \mu) \mathrm{d}\mu} \qquad f_H(r, x) = \frac{\int_0^1 \Theta(r, x, \mu) \mu \mathrm{d}\mu}{\int_0^1 \Theta(r, x, \mu) \mathrm{d}\mu} \qquad (9.53) $$ Good *starting values* for the iteration are $f_K = 1/3$, $f_H = 1/2$, and $q = 1$. Numerical aspects of these integrations are discussed by Auer (1971), Leung (1975), Leung (1976), Spagna and Leung (1987), Hummer *et al.* (1973), Hummer and Rybicki (1971a), Kunasz and Hummer (1974a), and Mihalas and Mihalas (1984). A computer program that solves this problem is publicly available (Spagna and Leung 1983). **For the infinite cylinder,** the *transfer equation* is $$ \sin(\vartheta) \left( \cos(\varphi) \frac{\partial I(r, x, \vartheta, \varphi)}{\partial r} - \frac{\sin(\vartheta)}{r} \frac{\partial I(r, x, \vartheta, \varphi)}{\partial \varphi} \right) = k(x) (I(r, x, \vartheta, \varphi) - S(r)) \qquad (9.54) $$ Introducing the Eddington factors, it becomes the *modified Feautrier equation* $$ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{1}{q(r, x)} \frac{\partial q(r, x) f_K(r, x) J(r, x)}{\partial z} \right) = k^2(x) (J(r, x) - S(r)) \qquad (9.55) $$ where the function $q(r, x)$ is defined by $$ \begin{aligned} q(r, x) &= \exp \left( \int_0^r \frac{2 f_K(u, x) - f1(u, x) / J(r, x)}{u \cdot f_K(u, x)} \mathrm{d}u \right), \\ f1(r, x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I(r, x, \vartheta, \varphi) \sin^3(\vartheta) \mathrm{d}\vartheta \mathrm{d}\varphi \end{aligned} \qquad (9.56) $$ and the *boundary conditions* are given in Eq. (9.63). --- The exact *intensities* $I(r, x, \vartheta, \varphi)$ are as usual obtained from solving the linear ray equation, and the moments $J$, $H$, and $K$ are computed from $$ \begin{aligned} J(r, x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I(r, x, \vartheta, \varphi) \sin \vartheta \, d\vartheta \, d\varphi \\ H(r, x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I(r, x, \vartheta, \varphi) (\sin \vartheta \cos \varphi)^1 \sin \vartheta \, d\vartheta \, d\varphi \\ K(r, x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I(r, x, \vartheta, \varphi) (\sin \vartheta \cos \varphi)^2 \sin \vartheta \, d\vartheta \, d\varphi \end{aligned} $$ (9.57) The starting value for $f_K$ is 1/3; for $q$ it is $q = 1$. The case of **the finite cylinder** is discussed in detail by Spagna and Leung (1987). It is essentially just more of the same, but since it is an important practical geometry, we list here the equations for the *transfer equation* in the Eddington approximation (to simplify notation, we drop all dependencies on spatial coordinates) $$ \begin{aligned} \frac{\partial}{\partial z} \left[ \frac{1}{k(x)} \frac{\partial f_{zz}(x) J(x)}{\partial z} + \frac{1}{r k(x)} \frac{\partial f_{rz}(x) J(x)}{\partial r} \right] &+ \\ + \frac{1}{r} \frac{\partial}{\partial r} \left[ \frac{r}{k(x)} \frac{\partial f_{rz}(x) J(x)}{\partial z} + \frac{r}{k(x) q(x)} \frac{\partial f_{rr}(x) q(x) J(x)}{\partial r} \right] &= k(x) [J(x) - S(x)] \end{aligned} $$ (9.58) where $$ q(x) = \exp \left[ \int_0^r \left( 1 - \frac{f_{\varphi\varphi}(r', z, x)}{f_{rr}(r', z, x)} \right) \frac{dr'}{r'} \right] $$ (9.59) where the spatial dependence of $f_{\varphi\varphi}$ and $f_{zz}$ was written down explicitly to show which variable to integrate over. The variable *Eddington* factors are defined as $$ \begin{aligned} f_{zz}(x) &= \frac{K_{zz}(x)}{J(x)} & f_{\varphi\varphi}(x) &= \frac{K_{\varphi\varphi}(x)}{J(x)} \\ f_{rr}(x) &= \frac{K_{rr}(x)}{J(x)} & f_{rz}(x) &= \frac{K_{rz}(x)}{J(x)} \end{aligned} $$ (9.60) where the moments $J$, $H$, and $K$ are computed from the definitions $$ \begin{aligned} J(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin \vartheta \, d\vartheta \, d\varphi & K_{rr}(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin^3 \vartheta \cos^2 \varphi \, d\vartheta \, d\varphi \\ H_r(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin^2 \vartheta \cos \varphi \, d\vartheta \, d\varphi & K_{zz}(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin \vartheta \cos^2 \vartheta \, d\vartheta \, d\varphi \\ H_z(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin \vartheta \cos \vartheta \, d\vartheta \, d\varphi & K_{\varphi\varphi}(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin^3 \vartheta \sin^2 \varphi \, d\vartheta \, d\varphi \\ H_\varphi(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin^2 \vartheta \sin \varphi \, d\vartheta \, d\varphi & K_{rz}(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^\pi I \sin^2 \vartheta \cos \vartheta \cos \varphi \, d\vartheta \, d\varphi \end{aligned} $$ (9.61) --- 154 MATHEMATICAL METHODS FOR THE TRANSFER EQUATION Intensity $I$ is short for $I(x, \varphi, \vartheta)$. For isotropic radiation, the values for the Eddington factors are $f_{rr} = f_{zz} = f_{\varphi\varphi} = 1/3$, $f_{rz} = 0$. These are also good *starting values* for the iteration. For the *boundary conditions*, we first define the Eddington factors $$ \begin{aligned} f_r(x) &= \frac{H_r^+(x) + H_r^-(x)}{J(x)} & f_z(x) &= \frac{H_z^+(x) + H_z^-(x)}{J(x)} \\ H_r^+(x) &= \frac{1}{4\pi} \int_{-\pi/2}^{\pi/2} \int_0^\pi I \sin^2 \vartheta \cos \varphi d\vartheta d\varphi & H_z^+(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_0^{\pi/2} I \sin \vartheta \cos \vartheta d\vartheta d\varphi \\ H_r^-(x) &= \frac{1}{4\pi} \int_{\pi/2}^{3\pi/2} \int_0^\pi I \sin^2 \vartheta \cos \varphi d\vartheta d\varphi & H_z^-(x) &= \frac{1}{4\pi} \int_0^{2\pi} \int_{\pi/2}^\pi I \sin \vartheta \cos \vartheta d\vartheta d\varphi \end{aligned} $$ (9.62) The incident intensity is usually prescribed, so that only one of the two flux pairs $H^\pm$ is unknown. The *boundary conditions* for the Eddington approximation are then $$ \begin{aligned} \frac{\partial f_{rz}(x)J(x)}{\partial z} + \frac{1}{q(x)} \frac{\partial f_{rr}q(x)J(x)}{\partial r} &= \begin{cases} -k(x) \left[ f_r(x)J(x) - 2H_r^-(x) \right] & \text{at } r = R \\ \hfill 0 \hfill & \text{at } r = 0 \end{cases} \\ \frac{\partial f_{zz}(x)J(x)}{\partial z} + \frac{1}{r} \frac{\partial r f_{rz}J(x)}{\partial r} &= \begin{cases} -k(x) \left[ f_z(x)J(x) - 2H_z^-(x) \right] & \text{at } z = L/2 \\ \hfill 0 \hfill & \text{at } z = 0 \end{cases} \end{aligned} $$ (9.63) where we assumed symmetry with respect to the $z = 0$ plane. Again, for a known source function, *the intensities* are computed from the ray equations with the prescribed boundary conditions. --- ## 9.4 Modified Feautrier approaches ### 9.4.1 The Rybicki reorganization In one-dimensional geometries, it is quite common that the number of spatial discretization points is on the order of 20 (for rather low accuracy) to 80, while the number of frequency discretization points can easily reach 100 and more (especially for hyperfine-split lines and other complicated line structures). In that case, it is advantageous to arrange the Feautrier matrix in a slightly different way (Rybicki 1971). In the original Feautrier technique, we have grouped all matrix entries belonging to a certain *spatial* discretization point together into one submatrix. The numerical effort for the solution of the system of linear equations was then $N_z \cdot (N_x \cdot N_\mu)^3$. In the Rybicki scheme, we group all entries belonging to one *angle-frequency* discretization point. We will see below that the numerical effort is $N_z^2 \cdot (N_x \cdot N_\mu)$. Obviously, this can lead to vast savings in computer time. --- The basic structure of the reorganized matrix looks as follows $$ \begin{pmatrix} \mathbf{T}_1^\mathbf{R} & & & & & \mathbf{U}_1^\mathbf{R} \\ & \mathbf{T}_2^\mathbf{R} & & & & \mathbf{U}_2^\mathbf{R} \\ & & \mathbf{T}_3^\mathbf{R} & & & \mathbf{U}_3^\mathbf{R} \\ & & & \cdot & & \cdot \\ & & & & \mathbf{T}_{N_x \cdot N_\mu}^\mathbf{R} & \mathbf{U}_{N_x \cdot N_\mu}^\mathbf{R} \\ \mathbf{V}_1^\mathbf{R} & \mathbf{V}_2^\mathbf{R} & \mathbf{V}_3^\mathbf{R} & \cdot & \mathbf{V}_{N_x \cdot N_\mu}^\mathbf{R} & -\mathbf{1} \end{pmatrix} \begin{pmatrix} \mathbf{\Theta}_1^\mathbf{R} \\ \mathbf{\Theta}_2^\mathbf{R} \\ \mathbf{\Theta}_3^\mathbf{R} \\ \cdot \\ \mathbf{\Theta}_{N_x \cdot N_\mu}^\mathbf{R} \\ \mathbf{J}^\mathbf{R} \end{pmatrix} = \begin{pmatrix} \mathbf{E}_1^\mathbf{R} \\ \mathbf{E}_2^\mathbf{R} \\ \mathbf{E}_3^\mathbf{R} \\ \cdot \\ \mathbf{E}_{N_x \cdot N_\mu}^\mathbf{R} \\ \mathbf{0} \end{pmatrix} $$ (9.64) where $\mathbf{\Theta}_1^\mathbf{R}$ now contains $\Theta_1, \Theta_2, \dots \Theta_{Nz}$, and $\mathbf{E}_1^\mathbf{R}, \dots$ is the excitation at frequency $x_1$ (note that at the boundary points, this is the sum of radiative excitation and inner sources). The last 0 in the right-hand side is a zero only for the standard transfer equation. When we have non-radiative processes, then it becomes a diagonal matrix. The matrices $\mathbf{V}^\mathbf{R}$ and $\mathbf{U}^\mathbf{R}$ are diagonal matrices. $\mathbf{J}^\mathbf{R}$ contains the total flux, and the matrices $\mathbf{T}^\mathbf{R}$ finally are tri-diagonal matrices containing the difference operator (just as in the Feautrier technique). Since Eq. (9.64) is only a rearrangement of terms, the detailed values for the matrix elements can easily be obtained from Sec. 9.2. The solution of this system of equations can be done quite efficiently. We multiply each row by $\mathbf{V}_i^\mathbf{R} \cdot (\mathbf{T}_i^\mathbf{R})^{-1}$ and subtract it from the last row. This brings Eq. (9.64) into diagonal form: $$ \begin{pmatrix} \mathbf{T}_1^\mathbf{R} & & & & & \mathbf{U}_1^\mathbf{R} \\ & \mathbf{T}_2^\mathbf{R} & & & & \mathbf{U}_2^\mathbf{R} \\ & & \mathbf{T}_3^\mathbf{R} & & & \mathbf{U}_3^\mathbf{R} \\ & & & \cdot & & \cdot \\ & & & & \mathbf{T}_{N_x \cdot N_\mu}^\mathbf{R} & \mathbf{U}_{N_x \cdot N_\mu}^\mathbf{R} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdot & \mathbf{0} & \mathbf{D1}^\mathbf{R} \end{pmatrix} \begin{pmatrix} \mathbf{\Theta}_1^\mathbf{R} \\ \mathbf{\Theta}_2^\mathbf{R} \\ \mathbf{\Theta}_3^\mathbf{R} \\ \cdot \\ \mathbf{\Theta}_{N_x \cdot N_\mu}^\mathbf{R} \\ \mathbf{J}^\mathbf{R} \end{pmatrix} = \begin{pmatrix} \mathbf{E}_1^\mathbf{R} \\ \mathbf{E}_2^\mathbf{R} \\ \mathbf{E}_3^\mathbf{R} \\ \cdot \\ \mathbf{E}_{N_x \cdot N_\mu}^\mathbf{R} \\ \mathbf{D2}^\mathbf{R} \end{pmatrix} $$ (9.65) where $$ \begin{aligned} \mathbf{D1}^\mathbf{R} &= -\mathbf{1} - \sum \mathbf{V}_i^\mathbf{R} (\mathbf{T}_i^\mathbf{R})^{-1} \mathbf{U}_i^\mathbf{R} \\ \mathbf{D2}^\mathbf{R} &= \phantom{-}\mathbf{0} - \sum \mathbf{V}_i^\mathbf{R} (\mathbf{T}_i^\mathbf{R})^{-1} \mathbf{E}_i^\mathbf{R} \end{aligned} $$ (9.66) This allows the explicit computation of $\mathbf{J}^\mathbf{R}$, and thus of the source function (i --- in subsection 9.2.1—the only difference is that there the elements were matrices, while here they are scalars. The numerical effort for each of the inversions of $\mathbf{T}_i^\mathbf{R}$ is thus $N_z^2$, while the effort for solving for $\mathbf{J}^\mathbf{R}$ (solving the equation $\mathbf{D1}^\mathbf{R} \cdot \mathbf{J}^\mathbf{R} = \mathbf{D2}^\mathbf{R}$) is $N_z^3$. The total numerical effort is thus $N_x N_\mu N_z^2 + N_z^3$. For the case that we have many frequency and angle discretization points, this constitutes a large saving in CPU time. Actually, the numerical effort is the same as for the integral equation method. The Rybicki scheme is only useful when the assumption of complete frequency redistribution is fulfilled. For partial frequency redistribution, the block-matrix structure is destroyed, and we have to solve a non-sparse matrix of size $(N_x N_\mu N_z)$, i.e. the brute-force approach described at the beginning of this section. The original Feautrier technique, on the other hand, has no such problems. There, the PFR just causes the *submatrices* to have a more complicated structure, but the *block-matrix* structure is retained. However, a modified Rybicki scheme that avoids most of these problems has been proposed by Hubeny (1985b). Finally, a few words concerning the choice of **the frequency points for a high opacity vapour**. The simplest way of choosing the frequency discretization points is to make equidistant discretization of the frequency axis. The smallest and largest frequencies must fulfil two criteria: (i) the opacity at these frequencies must be much smaller than unity—note that, e.g., in a plane-parallel slab, this means that $k(x_1)L/\mu_{N_\mu} \ll 1$, if we want to model the radiation in direction $\mu_{N_\mu}$ correctly—and (ii) the integral from $-\infty$ to $x_1$ plus the integral from $x_{N_x}$ to $\infty$ over the frequency-dependent escape factor must be much smaller than the frequency-integrated escape factor. This condition can be explained as follows. The frequencies at which the vapour is optically thin (opacity on the order of unity) are responsible for the largest part of the radiation transport and are thus the most important ones. In order to correctly model this escape, we must have discretization points in almost the whole frequency region that leads to escape. This second condition can mean quite often that we have to choose very large values for $x_{N_x}$ and $x_1$. In order to reduce this problem, we can give an integration weight to the highest and lowest frequencies that is not $k(x_{N_x})$, but $$ \frac{1}{\Delta x} \int_{x_{N_x} - \Delta x/2}^{\infty} k(x)\mathrm{d}x $$ (9.67) On the other hand, the discretization points should be so closely spaced that the escape probability does not change significantly within each interval. In the wings of the line, half the FWHM seems to be the maximum acceptable spacing. Near the line centre, the distance could be larger—one just has to take care that the normalization of the lineshape is retained correctly, i.e. that $C_x$ is computed with sufficiently high accuracy. As mentioned above, we do not have these problems in the integral equation technique. There, we can use ‘canned’ integration routines from commercial or public domain software packages that automatically choose the best frequency discretization. ### 9.4.2 *The core saturation method* Especially at high opacities, the matrices for the Feautrier method can become quite large, so that the CPU time requirements are high. Rybicki (1972) proposed an approximate method, the ‘core saturation method’, that reduces these problems somewhat (see also (Rybicki 1984)). --- The essential idea is that in the core of the line, photons have a very small probability of escape, $\exp(-k_0L)$, so the whole transport occurs in the wings. If the escape probability is very small, the angle-averaged intensity $J(x)$ at the line centre is almost identical to the source function $S$ (this can be seen by inspection of the transfer equation). The core saturation method now defines a 'border frequency' $x_{\text{core}}$ and assumes that at all frequencies smaller than $x_{\text{core}}$, the approximation $J \approx S$ is strictly valid. When we say 'at all frequencies smaller than $x_{\text{core}}$', we mean at all frequencies so that $k(x) > k(x_{\text{core}})$. This corresponds to the 'smaller' positive frequency of a single unsplit line—we just use this expression to simplify notation. The 'wing' frequencies are treated by the usual Feautrier technique. When $x_{\text{core}}$ is chosen very large, the method is rather inaccurate. When it is chosen very small, we have practically no saving as compared to the normal Feautrier technique. Since this is an approximate technique, it is a good basis for iterations. Mathematically speaking, this idea is expressed as $$ J = \int_{x < x_{\text{core}}} C_x k(x) J(x) \mathrm{d}x + \int_{x > x_{\text{core}}} C_x k(x) J(x) \mathrm{d}x \approx (1 - \eta)S + J_{\text{wings}} $$ (9.68) since the integral of $C_x k(x)$ over the line core is approximately one minus the escape probability (see also Sec. 4.5). This is the intensity $J$ that has to be used in the rate equations to describe the reabsorption of the radiation. The core saturation method is applied in the algorithm of Scharmer (see Sec. 13.3), and in (Flannery *et al.* 1979). It is generalized to two-dimensional problems by Stenholm (1977). ### 9.4.3 The implicit integral method Another clever method to reduce the computational burden is the 'implicit integral technique' of Simonneau and Crivellari (1993). The basic idea of this method is to divide the slab into substripes. The relations between the 'emerging' and 'incoming' intensities of each substripe are written by means of a reflection matrix (this is also similar to the discrete-space theory, see Sec. 9.1). By means of a forward-elimination, back-substitution scheme, the intensities are then computed without the need for any matrix inversion. This results in a scheme that has computational effort only proportional to $N_z$. An explicit 'recipe' for the implementation of this algorithm is given by Simonneau and Crivellari (1993, Sec. 4.2). ### 9.4.4 Splitting algorithms The problem in the Feautrier method is the solution of a large system of linear algebraic equations, $\mathbf{Mx} = \mathbf{y}$, where the matrix $\mathbf{M}$ has a special structure. This system can be solved either directly, as in the previous subsections, or iteratively. The double-splitting algorithm proposed by Klein *et al.* (1989) uses the following procedure: --- $$ \mathbf{x}(m + 1/2) = \mathbf{A}_1^{-1} (\mathbf{y} - \mathbf{B}_1\mathbf{x}(m)) \, , \quad \mathbf{x}(m + 1) = \mathbf{A}_2^{-1} (\mathbf{y} - \mathbf{B}_2\mathbf{x}(m + 1/2)) \, , $$ (9.69) $$ \mathbf{M} = \mathbf{A}_1 + \mathbf{B}_1 = \mathbf{A}_2 + \mathbf{B}_2 $$ where $m$ denotes the $m$th iteration step. $\mathbf{A}_1$ describes the spatial transport operator (i.e. the three diagonals coming from the $\nabla \mathbf{I}$ terms) plus the last row that describes $\bar{\mathbf{J}} = \Sigma \mathbf{J}_\nu$. Matrix $\mathbf{B}_2$ contains just the highest and the lowest diagonal of the matrix. $\mathbf{A}_2$ and $\mathbf{B}_1$ are the respective complements to give the original matrix $\mathbf{M}$. Convergence can be enhanced by various acceleration schemes; see (Klein *et al.* 1989 for details). A similar method was used by Mihalas *et al.* (1982). They suggested a successive over-relaxation technique and Gauss-Seidel iteration (see e.g. (Golub and van Loan 1982) for a description of these techniques). A similar approach was also suggested by Trujilli Bueno and Fabiani Bendicho (1995). The splitting techniques are also strongly related to the operator perturbation methods described in Chapter 13. Such iterative techniques are especially useful when the solution of the Feautrier technique is only part of a larger iteration method. In that case, the solution of the Feautrier equation need not be very accurate (since it will be changed in the next iteration step anyway), so that we need only a few iterations, and thus save a lot of CPU time. ### 9.4.5 *Quadrature perturbation* The main computational effort in all the above procedures lies in the inversion of the matrices, whose size is determined by the number of quadrature (i.e. discretization) points. The quadrature perturbation technique (QPT) is based on solving the whole problem with a small number of points and iteratively refining the solutions (Cannon 1973a, b). This technique is a special case of the 'operator perturbation technique', which will be treated in detail in Chapter 13, and thus all discussion is relegated to this later time. We want to stress, however, that operator perturbations are extremely useful and highly recommended techniques, so that the reader should have a glance at Sec. 13.3.2 before deciding to solve a problem by the usual Feautrier technique. ### 9.4.6 *The discontinuous finite-element (DFE) method* The DFE method is used for the formal solution of the transfer equation (i.e. with known source function). For full radiation trapping problems, it is only useful as part of an iteration procedure, see Chapter 13. We again use the 'optical-depth' representation along the considered line, so that the transfer equation is $$ \frac{\mathrm{d}I}{\mathrm{d}\xi} = -I + S $$ (9.70) We then represent the intensity as $$ I(\xi) = I_k^+ u_k^+(\xi) + I_{k+1}^- u_{k+1}^-(\xi) $$ (9.71) where the $I^\pm$ are the unknowns and the functions $u^\pm$ are --- $$ u_k^+(\xi) = \frac{\xi_{k+1} - \xi}{\xi_{k+1} - \xi_k} \qquad u_{k+1}^-(\xi) = \frac{\xi - \xi_k}{\xi_{k+1} - \xi_k} $$ (9.72) Multiplying by $u^\pm$, and integrating over the interval $[\xi_k, \xi_{k+1}]$, we get a $2 \times 2$ system of equations $$ \begin{aligned} \frac{I_{k+1}^- + I_k^+ - 2I_k^-}{\Delta\xi} &= \frac{2}{3} (S_k - I_k^+) + \frac{1}{3} (S_{k+1} - I_{k+1}^-) \\ \frac{I_{k+1}^- - I_k^+}{\Delta\xi} &= \frac{1}{3} (S_k - I_k^+) + \frac{2}{3} (S_{k+1} - I_{k+1}^-) \end{aligned} $$ (9.73) where $\Delta\xi = \xi_{k+1} - \xi_k$ We see that on the right-hand side, the matrix $$ \begin{pmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} \end{pmatrix} $$ (9.74) multiplies the $(S - I)$ terms. From the theory of such finite-element systems, it is now known that stability is better when we simply replace this matrix by the $2 \times 2$ unit matrix. When we then eliminate $I^+$ from the resulting equations, we get a recurrence relation for $I^-$ $$ (\Delta\xi^2 + 2\Delta\xi + 2) I_{k+1}^- - 2I_k^- = \Delta\xi S_k + \Delta\xi(\Delta\xi + 1)S_{k+1} $$ (9.75) and $I^+$ can be computed from $$ (\Delta\xi^2 + 2\Delta\xi + 2) I_k^+ - 2(\Delta\xi + 1)I_k^- = \Delta\xi(\Delta\xi + 1)S_k - \Delta\xi S_{k+1} $$ (9.76) When we have the incident radiation prescribed at one boundary, $k = 0$, we can then step along the considered path, computing the $I^-$ from the recursive relations, and the $I^+$ from the $I^-$. When the radiation is prescribed at the other boundary, $k = N$, we just have to reverse notation. --- # 10 # SIMPLE GENERALIZATIONS In this chapter, we will deal only with rather straightforward generalizations, which still cause enormous mathematical problems. For the cases treated here, the solutions still take the form of a modal expansion—so no change in this respect: $$n(\mathbf{r}, t) = \sum_j \alpha_j \psi_j(\mathbf{r}) \exp\left(-\frac{t}{g_j \tau}\right)$$ (10.1) This class of problems includes multi-level atoms, reflecting cell walls, particle diffusion, and inhomogeneous distributions of absorbers. Added *conceptual* difficulties arise when CFR is not valid, or when non-linear effects come into play. These problems are relegated to Chapters 11–13. ## 10.1 Branching and quenching The simplest generalization of the Holstein equation is the inclusion of foreign-gas quenching and the treatment of multi-level atoms with only one trapped transition. Consider the (synthetic) level scheme in Fig. 10.1. Only level {1}, the ground state, is populated appreciably. In level {2}, there is a certain initial distribution of excited-state atoms, $n(z, 0)$. These atoms decay with a probability $\beta$ to state {1}. Since the ground state {1} is heavily populated, transition {2}–{1} is trapped. With probability $(1 - \beta)$, atoms in level {2} decay to states {3} and {4}. On these decay paths, the emitted photons have no chance of being reabsorbed, since the levels {3} and {4} are not populated appreciably. We are no longer concerned with atoms in these states. Whatever happens to them will not change the populations in levels {1} and {2}. Usually, they will simply decay to the ground state via their (trapped) radiative transitions. That [FIGURE: FIG. 10.1. Energy level scheme with branching.] --- does not mean that the ground state population is noticeably increased, since we are still assuming linearity, i.e. only a small fraction of atoms may be in excited states. The population in state {2} will thus decrease by natural decay, by self-quenching, and by foreign gas quenching—we allow for the presence of a foreign gas. Population will increase by reabsorption of photons that were created by a radiative decay {2} $\rightarrow$ {1}. The Holstein equation then becomes $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau} n(\mathbf{r}, t) - Q n(\mathbf{r}, t) + \frac{\beta}{\tau} \int_V n(\mathbf{r}', t) G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}', $$ (10.2) where $Q$ is the quenching rate. Equation (10.2) is equivalent to multiplying the integral operator in the basic Holstein equation by some constant, so that the eigenfunctions remain the same and the eigenvalues are just scaled by a constant. Equation (10.2) can be rewritten as $$ \left( 1 - \frac{1}{g} + Q\tau \right) \frac{1}{\beta} n(\mathbf{r}, t) = \int_V n(\mathbf{r}', t) G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}'. $$ (10.3) Using Eqs. (10.2) and $(1 - 1/g^{\text{simple}}) = \int G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}'$, we get $$ \left( 1 - \frac{1}{g} + Q\tau \right) = \beta \left( 1 - \frac{1}{g_j^{\text{simple}}} \right). $$ (10.4) where $g_j^{\text{simple}}$ is the solution of the standard Holstein equation of the two-level scheme {2}–{1}. For the case that we have only the lowest-order mode and no quenching, this becomes the intuitively clear result (Klose 1969, 1975a) $$ \frac{1}{\tau_{\text{eff}}} = \frac{A_{21}}{g_0^{\text{simple}}} + \sum_{\substack{\text{untrapped} \\ \text{transitions}}} A_{2i} $$ (10.5) This can be used to solve trapping problems with branching in a very simple way (instead of solving the complete Holstein equation, as has been done in the literature). Consider the steady-state problem in a slab with spatially uniform excitation, high opacity (e.g. opacity $k_0 L = 1000$) and branching ratio smaller than unity, e.g. $\beta = 0.9$. Due to the branching, all terms $(1 - 1/g_j)$ have practically the same value 0.9, since the terms $(1 - 1/g_j^{\text{simple}})$ are very close to one due to the high opacity. Thus all trapping factors $g_j$ are approximately 10. From Sec. 7.5 it follows that the expansion coefficients of the steady-state distribution, $\alpha_j^{\text{st}}$, are proportional to the expansion coefficients of the excitation, $\alpha_j^{\text{exc}}$ times the trapping factors $g_j$, so that $n(z)$ is uniform across the slab. Generally, we can state that in the case of quenching or branching, the spatial steady-state distribution will be almost identical to the excitation function, which means in the above case of a uniform excitation, that $n(z)$ is constant. In the case of a sharply peaked excitation function, $n(z)$ is sharply peaked as well, $n(z) \approx \delta(z)$. The above derivation --- is valid for all multi-level atoms where no more than one trapped transition emanates from each level. When trapping of the considered transition is very strong—so that $A_{\text{trapped trans}}/g_0 << A_{\text{untrapped trans}}$, then the trapped transition often effectively ceases to exist, and can be disregarded for further computations. This is often a great simplification of computations in multi-level problems (Bates *et al.* 1962). Quenching, plus the effects of transient external excitation, can also be treated by the multiple scattering representation (Lai *et al.* 1993b). Again, we start out with the Holstein equation in operator form, $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = - [A_{21} + Q - A_{21}\Lambda] n(\mathbf{r}, t) + E(\mathbf{r}, t) $$ (10.6) which gives $$ n(\mathbf{r}, t) = \exp{[-(A_{21} + Q - A_{21}\Lambda)t]} n(\mathbf{r}, 0) + \int_0^t \exp{[-(A_{21} + Q - A_{21}\Lambda)t']} E(\mathbf{r}, t - t')dt' $$ (10.7) Late-time behaviour, and the like, can be computed as outlined in Sec. 6.2. Note that also with quenching, higher-order modes must be considered. Usually, there is no simple mono-exponential decay. In steady-state, quenching can no longer be treated by the simple Stern–Volmer equation, because the transient behaviour is no longer a simple exponential. The excited-state density can be written as $$ n(\mathbf{r}) = \frac{1}{A + Q - A\Lambda} E(\mathbf{r}) = \frac{1}{A + Q} \sum_{i=0}^{\infty} \left( \frac{A}{A + Q} \right)^i \Lambda^i E(\mathbf{r}) $$ (10.8) The applicability of the 'effective-lifetime' approximation to trapping with quenching was also analysed by Bezuglov *et al.* (1980). ## 10.2 Three-level atoms A more complicated situation arises when we have two levels with appreciable population density and a third level that can decay to either of those levels. This is, e.g., the case for mercury, thallium and barium atoms when their metastable state is populated appreciably, see Fig. 10.2. This type of three-level scheme is also known as the $\Lambda$-configuration. A state-$c$ atom decays with probability $\beta$ to the ground state $a$ or with $(1 - \beta)$ to the metastable state $b$, emitting an $ac$ or $bc$ photon, respectively. Photons $ac$ or $bc$ have a good chance of being reabsorbed, which restarts the process by the creation of another state-$c$ atom. The game will be repeated until an $ac$ or $bc$ photon leaves the vapour cell. For this process, the Holstein equation becomes --- [FIGURE: FIG. 10.2. Energy level scheme considered in the three-level trapping problem.] $$ \begin{aligned} \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau}n(\mathbf{r}, t) &+ \frac{\beta}{\tau} \int_V n(\mathbf{r}', t)G^{ac}(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}' + \\ &+ \frac{1-\beta}{\tau} \int_V n(\mathbf{r}', t)G^{bc}(\mathbf{r}, \mathbf{r}')\mathrm{d}\mathbf{r}', \end{aligned} \qquad (10.9) $$ where $G^{ac}$ is the Kernel function for an $ac$ photon and $G^{bc}$ for a $bc$ photon. Equation (10.9) could be solved by the numerical methods described in Chapter 5. However, making use of the linearity of the Holstein equation, we have developed a new method that saves considerable computer time (Molisch *et al.* 1992b). Solutions to Eq. (10.9) still have the form of Eq. (10.1): $$ n(\mathbf{r}, t) = \sum_j \alpha_j \psi_j(\mathbf{r}) \exp(-t/(g_j \tau)), \qquad g_j = \frac{1}{1 - \lambda_j}. \qquad (10.10) $$ Denoting the integral operator $\int \mathrm{d}\mathbf{r}'G(\mathbf{r}, \mathbf{r}')$ as usual by the symbol $\Lambda$, we are thus looking for solutions of the equation $$ \lambda^c \psi^c(\mathbf{r}) = \beta \Lambda^{ac} \psi^c(\mathbf{r}) + (1 - \beta) \Lambda^{bc} \psi^c(\mathbf{r}), \qquad (10.11) $$ where superscript $c$ denotes the coupled system. We now introduce the expansions $$ \psi^c(\mathbf{r}) = \sum_k c_k^c \psi_k^{ac}(\mathbf{r}) \qquad \begin{array}{l} \psi_i^{ac}(\mathbf{r}) = \sum_k c_{i,k}^{ac} \psi_k^{bc}(\mathbf{r}) \\ \psi_i^{bc}(\mathbf{r}) = \sum_k c_{i,k}^{bc} \psi_k^{ac}(\mathbf{r}) \end{array} \qquad (10.12) $$ --- The coefficients $c^{ac}$ and $c^{bc}$ are known and the coefficients $c^c$ are as yet unknown. Inserting Eq. (10.12) into Eq. (10.11), we get $$ \lambda^c \sum_k c_k^c \psi_k^{ac}(\mathbf{r}) = \beta \Lambda^{ac} \left( \sum_k c_k^c \psi_k^{ac}(\mathbf{r}) \right) + (1 - \beta) \Lambda^{bc} \left( \sum_k c_k^c \sum_l c_{k,l}^{ac} \psi_l^{bc}(\mathbf{r}) \right). $$ (10.13) We now introduce the known solutions of the two-level problems $$ \begin{aligned} \Lambda^{ac} \psi_j^{ac}(\mathbf{r}) &= \lambda_j^{ac} \psi_j^{ac}(\mathbf{r}) \\ \Lambda^{bc} \psi_j^{bc}(\mathbf{r}) &= \lambda_j^{bc} \psi_j^{bc}(\mathbf{r}), \end{aligned} $$ (10.14) and Eq. (10.13) becomes $$ \lambda^c \sum_k c_k^c \psi_k^{ac}(\mathbf{r}) = \beta \sum_k c_k^c \lambda_k^{ac} \psi_k^{ac}(\mathbf{r}) + (1 - \beta) \sum_k c_k^c \sum_l c_{k,l}^{ac} \lambda_l^{bc} \psi_l^{bc}(\mathbf{r}). $$ (10.15) Multiplying by $\psi_m^{ac}(\mathbf{r})$ and integrating over the cell, we get the solution of the coupled problem from $$ \lambda^c c_m^c = \beta \lambda_m^{ac} c_m^c + (1 - \beta) \sum_k \sum_l c_k^c c_{k,l}^{ac} \lambda_l^{bc} c_{l,m}^{bc}. $$ (10.16) Basically, Eq. (10.16) is an infinite system of linear homogeneous equations, which we truncate after $N$ terms. We thus have to find the $N$ eigenvalues $\lambda_i^c$ and eigenvectors $c_{i,k}^c$ of this system. In practice, $N = 3\text{--}6$ will usually give sufficient accuracy. The decay constants of the coupled system are $g_i^c = 1/(1 - \lambda_i^c)$, and the coefficients $c_{i,m}^c$ determine the coupled eigenfunctions $\psi_i^c(\mathbf{r})$ via $$ \psi_i^c(\mathbf{r}) = \sum_k c_{i,k}^c \psi_k^{ac}(\mathbf{r}). $$ (10.17) We can also find out how many $ac$ photons leave the vapour cell. The distribution of state-$c$-atoms at any time is $$ n(\mathbf{r}, t) = \sum_j \alpha_j^c \psi_j^c(\mathbf{r}) \exp \left( - \frac{t}{g_j^c \tau} \right), $$ (10.18) where $\alpha_j^c$ are the expansion coefficients of the initial distribution into the modes of the coupled system, $\psi_j^c$. Usually, it is convenient to compute the coefficients $\alpha_j^c$ as $$ \alpha_j^c = \sum_m c_{j,m}^c \alpha_m^{ac}. $$ (10.19) Inserting Eq. (10.17) into Eq. (10.18), the number of $ac$ photons that are created at a point $\mathbf{r}$ within the time interval $[t, t + \mathrm{d}t]$ is --- $$ \frac{\beta}{\tau} \sum_j \alpha_j^c \sum_k c_{j,k}^c \psi_k^{ac}(\mathbf{r}) \exp\left(-\frac{t}{g_j^c \tau}\right) \mathrm{d}t. $$ (10.20) The escape probability of an $ac$ photon belonging to the $k$th mode of the $ac$ transition is $1/g_k^{ac}$, see Chapter 4. Thus the number of photons that are created between $t$ and $t + \mathrm{d}t$, and leave the slab without reabsorption is $$ \frac{\beta}{\tau} \sum_j \alpha_j^c \sum_k c_{j,k}^c \int \psi_k^{ac}(\mathbf{r}) \mathrm{d}\mathbf{r} \exp\left(-\frac{t}{g_j^c \tau}\right) \mathrm{d}t \frac{1}{g_k^{ac}}. $$ (10.21) Equation (10.21) gives the temporal behaviour of the emergent $ac$ radiation. When we integrate this over time and compare it to the number of state-$c$ atoms at time zero, we finally get the fraction of state-$c$ atoms that emits $ac$ photons from the cell: $$ \beta \frac{\sum_j g_j^c \alpha_j^c \sum_k c_{j,k}^c \int \psi_k^{ac}(\mathbf{r}) \mathrm{d}\mathbf{r} \frac{1}{g_k^{ac}}}{\sum_j \alpha_j^c \sum_k c_{j,k}^c \int \psi_k^{ac}(\mathbf{r}) \mathrm{d}\mathbf{r}}. $$ (10.22) The presented method has decoupled the trapping effects of the transitions $ac$ and $bc$. In contrast to the direct solution of Eq. (10.9), where the solution depends on two parameters ($ac$ and $bc$ opacities), this approach allows one to solve the two one-parameter basic Holstein equations for the $ac$ and $bc$ transitions separately, and to combine the results via Eq. (10.16). This allows for a considerably more efficient computation of trapping effects. When we want to know the solutions of $N^{ac}N^{bc}$ combinations of $ac$ and $bc$ opacities, this approach requires only $N^{ac} + N^{bc}$ solutions of the Holstein equation instead of $N^{ac}N^{bc}$ solutions of the generalized equation (10.9). Furthermore, we can use the closed-form approximations for the solutions of the Holstein equation of Chapters 5 and 7, while no such approximations exist for Eq. (10.9). Figure 10.3 shows the geometry and the modes for a cylindrical vapour cell where the top is reflecting for $ac$ photons, the bottom is reflecting for $bc$ photons, and the side walls are completely reflecting. This geometry is not as far fetched as it might seem—it is the cell we used in experiments for a thallium atomic line filter (Oehry *et al.* 1991). The ‘in’ in Figure 10.3 was on transition $bc$ (green light of 535 nm), ‘out’ was on transition $ac$ (UV emission at 378 nm). The mirrored cylindrical cell is equivalent to a plane-parallel slab of length $2L$ with no reflecting walls, where the centre of the equivalent slab for the $ac$-transition is at $z = +L/2$ (i.e. at the $ac$-reflecting wall of the original cell), and the centre of the equivalent slab for the $bc$ transition is at $z = -L/2$. Correspondingly, the lowest-order modes for the $ac$ and $bc$ transitions have their maxima at different positions. The shape of the coupled lowest-order mode, $\psi_0^c$, is also shown on the right. --- [FIGURE: FIG. 10.3. a) Geometry of mirrored cylindrical vapour cell, b) equivalent slab, and c) the modes $\psi_0^{ac}$ (for $2k_0^{ac}L = 10$), $\psi_0^{bc}$ (for $2k_0^{bc}L = 4$), and the coupled solution $\psi_0^c$.] The modal combination approach can also be used for steady-state computations. In the past, the $A_{k,m}$ elements for all transitions were simply written into a large matrix (Finn 1968), (Finn and Jefferies 1968b), (Avrett and Kalkofen 1968). Since the numerical effort for the inversion of a matrix is proportional to the third power of the matrix size, the solution for a three-level system means eight times the effort of solving for a two-level system. By using the modal combination technique, this effort can be reduced considerably. The approximate treatment of multiplets with the Feautrier technique is discussed by Takeda (1989). The approach presented above is easily generalized to multi-level systems. However, due to the selection rules of atomic transitions, we rarely encounter such systems, while three-level systems frequently occur in practice. ## 10.3 Reflecting walls ### 10.3.1 Diffusely reflecting walls Another case of practical interest is a vapour cell with partially reflecting walls. For diffusely reflecting walls, Weinstein (1962) gave the generalization of the Holstein equation as $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau}n(\mathbf{r}, t) + \frac{1}{\tau} \int_V n(\mathbf{r}', t)G^{\Xi}(\mathbf{r}, \mathbf{r}')d\mathbf{r}'. \qquad (10.23) $$ where $G^{\Xi}$ is the Kernel function including the reflecting walls. A general equation for $G^{\Xi}(\mathbf{r}, \mathbf{r}')$ is given by Weinstein (1962). Insert 7.1 gives the Kernel functions for slab, cylinder, and sphere with diffusely reflecting walls. With these expressions for the Kernel function, the generalized Holstein equation (10.23) can be solved by standard means. --- Ingold made numerical computations based on Weinstein's Kernel function for the slab case (Ingold 1968, 1970). However, he uses a variational procedure that gives quite inaccurate results at high opacities (up to 10% error). --- ### Insert 10.1 Kernel functions for diffusely reflecting walls **For the slab, with the diffuse reflection coefficient $\Xi$** $$ \begin{aligned} G^\Xi(z, z') &= \frac{C_x}{2} \int k^2(x) \left[ \text{Ei}_1 (k(x)|z - z'|) + \frac{4\Xi}{1 - 2\Xi \text{Ei}_3 (k(x)L)} f1(z, x) f1(z', x) \right] \text{d}x \\ f1(z, x) &= \frac{1}{2} \left[ \text{Ei}_2 (k(x)|L/2 - z|) + \text{Ei}_2 (k(x)|L/2 + z|) \right]. \end{aligned} $$ (10.24) For different reflectivity of the two walls, the Kernel function is given by (Colbert and Wexler 1993). **For the cylinder,** $$ \begin{aligned} G^\Xi(r, --- Weinstein also gave approximations for slightly reflecting walls. **For the slab,** $$ \frac{\Delta g_0}{\Xi g_0} = \frac{\ln(k_0L/2)}{\ln(k_0L)} \frac{0.345 + 1.52u + u^2 \ln(k_0L)}{1 + \frac{5}{2}u + \frac{15}{8}u^2}, \quad \text{where} \quad u = \frac{1}{4 \left[ \ln(k_0L) - \frac{15}{8} \right]} $$ (10.27) and $\Delta g$ is the difference in the trapping factors between the cases of reflectivity $= \Xi$ and reflectivity $= 0$. **For the cylinder**, similar computations give (Weinstein 1970) $$ \frac{\Delta g_0}{\Xi g_0} = \frac{0.239 + 1.28u + \frac{15}{8}u^2 \ln(k_0R)}{1 + 3u + 3u^2}, \quad \text{where} \quad u = \frac{2}{15 \left[ \ln(k_0R) - \frac{8}{5} \right]} $$ (10.28) For the opacity range $k_0L = 20\text{--}1000$, these results are claimed to give $\pm 20\%$ accuracy. ### 10.3.2 *Specularly reflecting walls* For specularly reflecting walls, we have to include the contribution of the 'mirror volumes'. Photons that are reflected by specularly reflecting walls behave as if they would come from a 'mirror source', i.e. the image of the original source mirrored to behind the cell wall. In that case the Holstein equation becomes (Tkachuk 1963) $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau}n(\mathbf{r}, t) + \frac{1}{\tau} \sum_{m=0}^{\infty} \Xi^m \int_{V_m} n(\mathbf{r}'_m, t)G(\mathbf{r}, \mathbf{r}'_m)\mathrm{d}\mathbf{r}'_m, $$ (10.29) where $\mathbf{r}_m$ is the $m$th mirror image of $\mathbf{r}$. The sum on the right-hand side can be written as a power series in $(1 - \Xi)$. **For the slab** with equal reflectivity at both walls, we can sum this series analytically. The Holstein equation then reads $$ \frac{\partial n(z, t)}{\partial t} = -\frac{1}{\tau}n(z, t) + \frac{1}{\tau} \int_{-L/2}^{L/2} n(z', t)G^\Xi(z, z')\mathrm{d}z' $$ (10.30) where $G^\Xi = G + G^a$, with $G$ denoting the usual Kernel function, and $$ G^a(z, z') = \Xi \int_{-\infty}^{\infty} C_x k^2(x) \int_1^\infty \frac{\exp\left(-k(x)Lu\right) \cosh\left[k(x)(z + z')u\right]}{1 - \Xi \exp\left(-k(x)Lu\right)} \frac{\mathrm{d}u}{u}. $$ (10.31) This result is a special case of the derivation for two non-uniformly reflecting walls that will be given below. --- When an analytical summation of the series is not possible, e.g., **in different geometries**, we can usually find a good approximation for high reflectivity by neglecting higher-order terms in the series of $(1 - \Xi)^m$. In the case of low wall reflectance, we get a good approximation for the lowest-order mode when we increase the dimensions of the vessel by $1/(1 - \Xi)$ and solve the resulting Holstein equation (Menon and Nolle 1971). The escape factor for a cylinder with specularly reflecting walls is given by Bezuglov *et al.* (1985) (note that there is a typing error in the original source, (N. N. Bezuglov, private comm.)). $$ \eta(r) = \frac{C_x}{4\pi} \int_{-\infty}^{\infty} \int_0^{2\pi} \int_0^\pi k(x) \exp\left[-k(x)\sec\vartheta \left(r \cos\varphi + \sqrt{R^2 - r^2 \sin^2\varphi}\right)\right] \frac{1 - \Xi}{1 - \Xi \exp\left[-k(x)\sec\vartheta 2\sqrt{R^2 - r^2 \sin^2\varphi}\right]} \sin\vartheta \mathrm{d}\vartheta \mathrm{d}\varphi \mathrm{d}x $$ (10.32) The effective lifetime for a slab, a cylinder, and a sphere (in the limit of high opacities) was computed by Bezuglov (1984). Molisch *et al.* (1994a) have derived an exact solution to the Holstein equation in a slab geometry under quite general boundary conditions. The reflection coefficients $\Xi_+$ and $\Xi_-$ (at $+L/2$ and $-L/2$, respectively) can depend on the angle $u = 1/\cos(\vartheta)$ and the frequency $x$: $\Xi_+ = \Xi_+(x, u)$ and $\Xi_- = \Xi_-(x, u)$. First, note that $u$ is not changed by a specular reflection. Photons that are emitted at $z'$ towards the wall at $z = L/2$, are reflected once, and are then reabsorbed at $z$, have to cover the distance $u \cdot (L/2 - z') + u \cdot (L/2 - z)$ so that the Kernel function for these photons is $$ G^{\text{one refl}}(z, z') = \frac{C_x}{2} \int_1^\infty \int_{-\infty}^\infty \Xi_+(x, u) \frac{k^2(x)}{u} \exp\left[-k(x)(L - z - z')u\right] \mathrm{d}x \mathrm{d}u. $$ (10.33) Similar terms can be written for photons that are reflected once at the other wall at $z = -L/2$, for photons that are reflected once at $+L/2$ and once at $-L/2$, twice at $+L/2$ and once at $-L/2$, and so on. These Kernel functions are then all added up. Properly arranged, we get four infinite series that can be summed up analytically. We thus get --- $$ \begin{aligned} G^{\text{refl}} (z, z') &= G^{\text{no refl}}(|z - z'|) + \\ &+ \frac{C_x}{2} \int_1^\infty \int_{-\infty}^\infty \frac{k^2(x)}{u} \frac{\Xi_+(x, u)\Xi_-(x, u) \exp(-k(x)(2L + z - z')u)}{1 - \Xi_+(x, u)\Xi_-(x, u)\exp(-2k(x)Lu)} \mathrm{d}x \mathrm{d}u \\ &+ \frac{C_x}{2} \int_1^\infty \int_{-\infty}^\infty \frac{k^2(x)}{u} \frac{\Xi_+(x, u)\Xi_-(x, u) \exp(-k(x)(2L - z + z')u)}{1 - \Xi_+(x, u)\Xi_-(x, u)\exp(-2k(x)Lu)} \mathrm{d}x \mathrm{d}u \\ &+ \frac{C_x}{2} \int_1^\infty \int_{-\infty}^\infty \frac{k^2(x)}{u} \frac{\Xi_+(x, u) \exp(-k(x)(L - z - z')u)}{1 - \Xi_+(x, u --- [FIGURE: FIG. 10.4. Trapping factor $g_0$ for specularly reflecting walls as a function of $\Xi_+$ and for three different values of $\Xi_-$, for a Doppler lineshape with opacity $k_0L = 5$.] In a slab with non-reflecting walls and with a Doppler lineshape, the trapping factors for the higher-order modes are related to the trapping factor for the lowest-order mode roughly by $g_j = 1 + (g_0 - 1) \cdot m_0/m_j$ (the relation $g_j/g_0 = m_0/m_j$ is strictly valid in the limit of infinite $g_0$), where $m_0 = 1.025$, $m_1 = 2.4$ and $m_2 = 3.9$ (see Appendix D). One could now think that this relation would stay the same in a slab with reflecting walls. Figure 10.5 shows the trapping factors $g_0$ and $g_1$ when $\Xi_+ = 1$ and $\Xi_-$ is varied and compares them to the results of this relation. We see that the relation breaks down completely as $\Xi_-$ is increased—as a matter of fact, the higher-order trapping factors are almost independent of the reflection coefficient. This is somewhat surprising at first glance, but can be explained by physical reasoning. At low opacity and high reflection coefficients, the excited-state distribution tends to become practically uniform within a very short time. Due to the low opacity, the photons can pass through the vapour for long distances, and will be reflected several times before reabsorption. The point of reabsorption is then almost independent of the point of emission, and the points of reabsorption will be distributed uniformly throughout the slab. Even when the initial distribution is sharply peaked at the centre of the slab, we will get a constant excited-state distribution (which is roughly the shape of the lowest-order mode, see the left side of Fig.10.6) within a time that is determined by the decay time of the higher-order modes for a slab of width $k_0L$, independent of any wall reflections. Thus, $g_j$ must be practically independent of $\Xi$. The average number of reabsorptions (which is approximately equal to $g_0$) is determined by both the vapour opacity and the wall reflection coefficient, so that $g_0$ strongly depends on wall reflections. We also note that the *area* of the higher-order modes (which is proportional to the total number of --- [FIGURE: Graph of Trapping factor g vs Reflectivity \Xi_-] FIG. 10.5. Trapping factors $g_0$ and $g_1$ in a slab with one wall ideally mirrored, as a function of the reflectivity $\Xi_-$ of the other wall. As a comparison, the relation $g_1^* = 1 + (g_0 - 1) \cdot m_0/m_1$ is shown (Doppler lineshape, opacity $k_0L = 1$ ). photons in a mode) is almost zero in the case that both $\Xi_+$ and $\Xi_-$ are very large, see Fig. 10.6. Hence, the higher-order modes have no influence on the temporal behaviour of the number of photons emerging from the slab. When the two reflection coefficients differ, they can have a larger influence on the higher-order modes. The higher-order trapping factors vary just as strongly (or weakly) as in the case that both walls are highly reflecting, but the variation of $g_0$ for only one highly reflecting wall is much smaller, so that the relative influence of the higher-order modes becomes larger. For opacity $k_0L = 1$, and reflectivities $\Xi_+ = 1$ and $\Xi_- = 0$, we get $g_2 = 1.18$, while for $\Xi_+ = \Xi_- = 0.995$, we get $g_2 = 1.19$ (as compared to $1.16$ when the walls are transparent, respectively non-reflecting), even though $g_0$ in the first case is only $3.02$, while it is $149.2$ in the latter case. In addition, as the right side of Fig. 10.6 shows, the area of the higher-order modes is much larger than in the case of two highly reflecting walls so that now they have a considerable influence on the temporal behaviour of the number of photons escaping from the slab. The presented approach permits the inclusion of frequency-dependent reflection coefficients. However, the linewidth of a resonance line is usually only several GHz, and the optical properties of reflecting materials will rarely change within such a small frequency range. When more than one resonant transition is present, the excited-state distributions can be computed either by a PCA approach (Colbert and Wexler 1993) or by the modal combination technique (Sec. 10.2). In both cases the results for the single lines (either the Kernel functions or the trapping factors and eigenmodes) are combined and the wall reflectivities are assumed to be frequency-independent within *each* resonant line. --- [FIGURE: FIG. 10.6. Eigenmodes $\psi(z)$ for reflectivities $\Xi_+ = \Xi_- = 0.99$ (left), and $\Xi_+ = 1, \Xi_- = 0$ (right). Doppler lineshape with $k_0L = 1$.] More importantly, the reflection coefficient may depend on the angle of incidence. With a metallic mirror, the reflection coefficient is almost independent of the angle of incidence. This can be used to somewhat simplify the expressions for the $A_{k,m}$ matrix elements. In that case, $\Upsilon_i^c/\Xi_+ = \Upsilon_i^d/\Xi_-$, so that we have $2N_r$ integrals less to compute. As will be shown, one can approximate mirrored walls by diffusely reflecting walls with tolerable error, but since the formalism of Eq. (10.35) allows one to compute specular reflection faster than diffuse reflection, there is no reason for such an approximation. The number of integrals that have to be evaluated for diffusely reflecting walls is proportional to $N_r^2$, while for specularly reflecting walls, it is proportional to $N_r$. For multilayer dielectric mirrors and for metallic mirrors coated with dielectric layers, the reflection coefficient may significantly depend on the angle of incidence. Accurate results can be obtained by a full evaluation of Eq. (10.35). We can now compare the results for diffusely and specularly reflecting walls. Figure 10.7 shows the relative difference in $g_0$ as a function of wall reflectivity $\Xi$ for low, intermediate, and high opacities and a Doppler lineshape. The difference is rather small. It is largest in the region of comparatively low opacities and becomes significantly smaller at very high opacities. For very low opacities and simultaneously very high reflection coefficients, so that $g_0 \approx 1.5-1.8$, the difference can become up to 8% for a Doppler lineshape and 10% for a Lorentz lineshape. For opacities $k_0L > 0.5$, it has a maximum value of 3% for a Doppler lineshape and 6% for a Lorentzian lineshape. Thus the often-used method of approximating specularly reflecting walls by diffusely reflecting walls can be justified. This is also of great interest for other geometries, where it is much more difficult to treat specularly reflecting walls, and Weinstein's results for diffusely reflecting walls may stay the only available mathematical treatment. Tkachuk (1963) also gave a very useful approximation for the steady state at high opacities. For a mirrored enclosure with reflectance $\Xi$, the difference between the actual steady-state distribution $n^\Xi(\mathbf{r})$ and the distribution that would result if the mirrors were perfect $n^{\text{Perfect}}(\mathbf{r})$ is --- $$ \frac{n^{\text{Perfect}}(\mathbf{r}) - n^{\Xi}(\mathbf{r})}{n^{\text{Perfect}}(\mathbf{r})} = \frac{1 - \Xi}{2} \left[ 1 - \frac{E(\mathbf{r})}{n^{\text{Perfect}}(\mathbf{r})} \right], $$ (10.36) where $E(\mathbf{r})$ is the distribution of sources. Tkachuk derived this result for a semi-infinite slab ($-\infty < y < 0$, $-L/2 < z < L/2$) and inferred its general validity. Reflecting walls are much easier to include in the equation of radiative transfer—it is sufficient to set the boundary conditions appropriately. For the slab with specularly reflecting walls, $I(L/2, \mu) = \Xi I(L/2, -\mu)$, while for diffusely reflecting walls, $I(L/2, \mu) = \Xi \int I(L/2, -\mu)\mathrm{d}\mu; \ \mu > 0$. ## 10.4 Particle diffusion ### 10.4.1 Formulation and direct solution The assumption that the excited atoms do not move is a good approximation in many, but by no means all practical cases. In the following, the effect of particle diffusion is considered explicitly. A fully exact treatment including particle movement makes the trapping problem much more complicated. Specifically, we will see in the next chapter that the absorption coefficient of an atom is coupled to its velocity. Thus, if we wanted to set up the exact equation for trapping including particle movement, we would need the coupled kinetic equations for particles and photons. For the steady state, this system of equations was derived by Oxenius (1985), Borsenberger *et al.* (1985–1987), Oxenius (1986, 1990). Solution of these equations gives the velocity distribution of the excited-state atoms. Within reasonable CPU time, iteration is the only path to a [FIGURE: Graph showing relative differences in the trapping factors g_0 between diffusely and specularly reflecting walls against Reflectivity Xi] FIG. 10.7. Relative differences in the trapping factors $g_0$ between diffusely and specularly reflecting walls. Doppler lineshape; same reflectivity at both walls, $\Xi = \Xi_+ = \Xi_-$. --- solution. Direct solution of the corresponding transient eigenvalue problem seems to be rather costly. Similar systems of equations were also derived by McIntyre and Fowler (1961), McIntyre and D'Arcy (1973), and Cipolla and Morse (1979), see also Chapter 16, (Oxenius 1979), (Veklenko 1959a). An exact solution in semi-infinite space was given by Lagarkow and Medvedeva (1977). Steady-state solutions are given by Gornyi and Matisov (1984). We restrict ourselves to the case that an excited atom suffers several velocity-changing collisions during each natural lifetime. More specifically, the number of elastic collisions per natural lifetime, divided by one plus the number of inelastic collisions, must be larger than unity. We furthermore require that the mean free path of the atoms must be much smaller than the mean free path of the photons at line centre. Under these conditions, particle diffusion can be included in the rate equation of the Holstein equation by simply introducing an additional term $D\nabla^2n(\mathbf{r}, t)$ for the decay of excited atoms (Düchs and Oxenius 1977), (Düchs *et al.* 1978): $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau}n(\mathbf{r}, t) + D\nabla^2n(\mathbf{r}, t) + \frac{1}{\tau} \int G(\mathbf{r}, \mathbf{r}')n(\mathbf{r}', t)d\mathbf{r}', \qquad (10.37) $$ The boundary conditions need some closer inspection. Near the cell wall, some atoms are travelling towards the wall and the others are coming from the wall. The atoms leaving the wall have all been quenched—except for the rather rare cases that the walls are covered, e.g., by paraffin to prevent wall quenching. Excited atoms travelling towards the wall have not yet suffered a wall collision and are not yet quenched. Hence, the excited state population just at the walls, $n(\pm L/2, t)$, does not equal zero. We have absorbing boundary conditions. It can be shown (Morse and Feshbach 1953) that these absorbing boundary conditions can be approximated very well by a Cauchy-type boundary condition, $$ n\left(\pm\frac{L}{2}, t\right) = 0.71\bar{\mu} \left| \frac{\partial n(z, t)}{\partial z} \right|_{z=\pm L/2}, \qquad (10.38) $$ where $\bar{\mu}$ is the mean free path of the diffusion. The mean free path is related to the diffusion constant by $D = \bar{\mu} \cdot \bar{v}$, where $\bar{v}$ is the mean relative velocity of the colliding atoms. In most practical cases, the right-hand side of Eq. (10.38) is so small that the Dirichlet boundary condition $n(\pm L/2, t) = 0$ can be used without significant error. This condition is often used in the literature without further discussion (see e.g. (Tambovtsev 1968), (Aleksandrov *et al.* 1978)). However, the validity of this approximation must be checked a posteriori by examining the right-hand side of Eq. (10.38). The boundary conditions for cases where the metastables are not quenched at the wall, but partially reflected, are discussed by Stewart (1994). The equations (10.37) and (10.38) can be solved by standard numerical techniques. Generally, the numerical solution of Eq. (10.37) is numerically *much* worse behaved than the PCA solution of the standard Holstein equation. **At very high opacity**, the eigenfunctions of the Holstein equation are very similar to the eigenfunctions of the diffusion equation (in the slab case, $\psi_j(z) \approx \cos[(j + 1)\pi z/L]$ for even $j$). In that case, we get (McGregor 1987), (Phelps and McCoubrey 1960), (Phelps 1959) --- PARTICLE DIFFUSION $$ \frac{A_{21}}{g_j^{\text{diff}}} = \frac{A_{21}}{g_j^{\text{nodiff}}} + \frac{1}{\tau_j^{\text{diff}}} $$ (10.39) where $\tau_j^{\text{diff}}$ is the decay time constant for the $j$th eigenmode of the diffusion equation. In the high opacity limit, (Tambovtsev 1968) gives a variational solution for the cylinder. However, his equation is based on a three-term polynomial for the variational procedure and assumes high opacities. The accuracy is thus not much better than for the simple estimate Eq. (10.39). Düchs and Oxenius (1977), and Düchs *et al.* (1978) used a **purely numerical technique** for solving Eq. (10.37) in a slab. They replaced the integral by a sum and the $\nabla^2$ term by differences. The resulting system of algebraic equations can be solved by standard means. This approach has, however, serious drawbacks. In order to approximate $\nabla^2$ correctly, we have to represent $n(z)$ within each substripe by a quadratic polynomial. The same approximation has of course to be used in the integral. Computation of integrals of the type $$ \int_{z_m-\Delta/2}^{z_m+\Delta/2} (a + bz' + cz'^2)G(z, z')\mathrm{d}z $$ (10.40) is much more difficult than the computation of the $A_{k,m}$ matrix elements of Chapter 5. This problem can be somewhat alleviated by using a piecewise linear (instead of quadratic) approximation for $n(z)$. We stress that it is *not* mathematically justified, but still works for sufficiently fine discretization. Düchs used a piecewise-constant approximation; this is also not justified mathematically and works only because he assumes Neumann boundary conditions. For the Dirichlet boundary conditions that are usually applied, this approach would give $n(z) = 0$ in the whole substripe closest to the boundary. With a linear or quadratic approximation, we have $n = 0$ only at $z = \pm L/2$. When the effect of diffusion is small, $\tau^{\text{diff}} \gg g\tau$, another major problem arises. In that case, wall quenching occurs only in a thin layer close to the boundary. The population $n(z)$ varies very little in the middle of the cell and has a sharp drop close to the cell walls. This implies that we either need non-equidistant discretization or a very large number of discretization points. The former often leads to numerical instabilities, while the latter requires much CPU time. ### 10.4.2 *The modal combination technique* The problems with the direct numerical solution of the trapping-plus-diffusion equation (10.37) enticed us to develop a new method (Molisch *et al.* 1994b), which is quite similar to our treatment of three-level atoms. The basic idea is the same, namely to appropriately combine the solutions of the two 'pure' problems, in this case pure trapping and pure diffusion. The following derivation is done for the slab but is easily generalized to other geometries. For pure trapping, i.e. with diffusion constant $D$ equal zero, the solution of Eq. (10.37) is now an old friend, $$ n^{\text{T}}(z, t) = \sum_j \alpha_j^{\text{T}} \psi_j^{\text{T}}(z) \exp \left( -t \frac{A_{21}}{g_j^{\text{T}}} \right) \qquad \qquad g_j^{\text{T}} = \frac{1}{1 - \lambda_j^{\text{T}}} $$ (10.41) For diffusion alone, we have the very similar looking solution (for the even modes) --- $$ n^{\text{diff}}(z, t) = \sum_j \alpha_j^{\text{diff}} \psi_j^{\text{diff}}(z) \exp\left(-\frac{t}{\tau_j^{\text{diff}}}\right) \qquad \lambda_j^{\text{diff}} = -\frac{1}{\tau_j^{\text{diff}}} = -D \left(\frac{(j + 1)\pi}{L}\right)^2 \qquad (10.42) $$ The solution modes of the coupled problem are put up as $$ \psi^{\text{c}}(z) = \sum_k c_k^{\text{c}} \psi_k^{\text{b}}(z), \qquad (10.43) $$ where the functions $\psi_k^{\text{b}}(z)$ are a set of orthonormal basis functions. We now expand the basis functions themselves in terms of the diffusion and the trapping modes, $$ \psi_i^{\text{b}}(z) = \sum_k c_{i,k}^{\text{diff}} \psi_k^{\text{diff}}(z) \qquad \psi_i^{\text{b}}(z) = \sum_k c_{i,k}^{\text{T}} \psi_k^{\text{T}}(z), \qquad (10.44) $$ and insert Eqs (10.43) and (10.44) into Eq. (10.37) to get $$ \lambda^{\text{c}} \sum_k c_k^{\text{c}} \psi_k^{\text{b}}(z) = \Lambda^{\text{diff}} \sum_k c_k^{\text{c}} \sum_l c_{k,l}^{\text{diff}} \psi_l^{\text{diff}}(z) + \Lambda^{\text{T}} \sum_k c_k^{\text{c}} \sum_l c_{k,l}^{\text{T}} \psi_l^{\text{T}}(z), \qquad (10.45) $$ where the superscripts c, diff, and T stand for the coupled problem, pure diffusion, and pure trapping. Proceeding analogously to the three-level problem, we get $$ \lambda^{\text{c}} c_m^{\text{c}} = \sum_k \sum_l c_k^{\text{c}} \left( c_{k,l}^{\text{diff}} \lambda_l^{\text{diff}} c_{m,l}^{\text{diff}} + c_{k,l}^{\text{T}} \lambda_l^{\text{T}} c_{m,l}^{\text{T}} \right). \qquad (10.46) $$ Equation (10.46) is a standard eigenvalue problem that can be solved by the usual methods. This approach to combined trapping and diffusion has the following advantages: * When the solutions for the separate trapping and diffusion problems are already known, which is usually the case, we need to solve only the eigenvalue problem Eq. (10.46). For good accuracy, the number of terms included in the sums usually has to be 3 to 10. The solution of the diffusion equation is always known. For the solution of the Holstein equation, we can use the methods of Chapter 5. * When the solution of the Holstein equation is not known, it can be computed by numerical methods. The PCA method is also a direct numerical solution of the Holstein equation with the integral replaced by a sum. However, the numerical solution of the Holstein equation is numerically better behaved and faster convergent than the direct solution of Eq. (10.37). * If we wish to solve the combined problem for $m$ values of the diffusion constant $D$, we only have to solve the Holstein equation once, and Eq. (10.46) $m$ times. The choice of the basis functions $\psi^{\text{b}}$ is of crucial importance for the efficiency of the method. The closer the basis function resembles the actual shape of the modes, $\psi^{\text{c}}$, the fewer terms will be required in Eq. (10.46) for a given accuracy. Furthermore, the basis functions should already fulfil the boundary conditions. For a low-opacity vapour and a small diffusion constant, $\psi_0^{\text{c}}(z)$ will be rather flat in the middle of the slab and show a steep gradient near the cell walls. It is obvious --- that neither the solution of the Holstein equation nor of the diffusion equation is a good choice. The pure trapping modes, $\psi^{\text{T}}(z)$, do not even fulfil the boundary conditions. This an important difference to the three-level trapping problem, where the solutions of either of the two single problems could be used as basis functions—the choice of basis functions was rather uncritical since the coupled modes of the three-level problem were smooth functions. We have seen in Chapter 7 that the trapping modes $\psi_j^{\text{T}}(z)$ can be approximated very well by $K \cdot \cos(\pi \zeta_j^{\text{T}} z/L)$, where $\zeta^{\text{T}}$ are tabulated functions of the opacity and of the lineshape. (We deal here only with even modes, odd modes can be treated completely analogously). For the basis functions $\psi^{\text{b}}$ we can use $$ \psi_j^{\text{b}}(z) = c_{\text{diff}} \cdot \left[ \cos(\pi \zeta_j^{\text{diff}} z/L) - f1(\zeta^{\text{diff}}, \varsigma_j) \cosh(\pi \varsigma_j z/L) \right] $$ (10.47) where: for Dirichlet boundary conditions, $\psi(\pm L/2) = 0$ $$ f1(\zeta_j^{\text{diff}}, \varsigma_j) = \frac{\cos(\pi \zeta_j^{\text{diff}}/2)}{\cosh(\pi \varsigma_j/2)} $$ (10.48) and for Cauchy boundary conditions, see Eq. (10.38) $$ f1(\zeta_j^{\text{diff}}, \varsigma_j) = \frac{\cos(\pi \zeta_j^{\text{diff}}/2) + 0.71 \frac{\bar{\mu}}{L} \pi \zeta_j^{\text{diff}} \sin(\pi \zeta_j^{\text{diff}}/2)}{\cosh(\pi \varsigma_j/2) - 0.71 \frac{\bar{\mu}}{L} \pi \varsigma_j \sinh(\pi \varsigma_j/2)}. $$ (10.49) with the parameters $$ \zeta_0^{\text{diff}} = \zeta_0^{\text{T}} \qquad \varsigma_j = 1.5 \sqrt{\frac{A_{21}}{D g_j^{\text{T}}}} \frac{L}{(j+1)\pi}. $$ (10.50) The remaining parameters $\zeta_j$ can be computed from the orthonormality relation $$ \int_{-L/2}^{L/2} \psi_i(z)\psi_j(z)\text{d}z = 0 \quad \text{for } i \neq j. $$ (10.51) For actual implementation in a computer program, it is best to evaluate the integral in Eq. (10.51) analytically. This yields the implicit equation for the $\zeta_j$ parameters $$ \begin{aligned} 0 &= \frac{1}{2} \left[ \frac{\sin \left(\pi (\zeta_0^{\text{diff}} - \zeta_j^{\text{diff}})/2\right)}{\zeta_0^{\text{diff}} - \zeta_j^{\text{diff}}} + \frac{\sin \left(\pi (\zeta_0^{\text{diff}} + \zeta_j^{\text{diff}})/2\right)}{\zeta_0^{\text{diff}} + \zeta_j^{\text{diff}}} \right] + \\ &\quad + \frac{1}{2} f1(\zeta_j^{\text{diff}}, \varsigma_j) f1(\zeta_0^{\text{diff}}, \varsigma_0) \left[ \frac{\sinh \left(\pi (\varsigma_0 + \varsigma_j)/2\right)}{\varsigma_0 + \varsigma_j} + \frac{\sinh \left(\pi (\varsigma_0 - \varsigma_j)/2\right)}{\varsigma_0 - \varsigma_j} \right] - \\ &\quad - \frac{f1(\zeta_0^{\text{diff}}, \varsigma_0)}{(\zeta_j^{\text{diff}})^2 + (\varsigma_0)^2} \left[ \varsigma_0 \sinh(\pi \varsigma_0/2) \cos(\pi \zeta_j^{\text{diff}}/2) + \zeta_j^{\text{diff}} \cosh(\pi \varsigma_0/2) \sin(\pi \zeta_j^{\text{diff}}/2) \right] - \\ &\quad - \frac{f1(\zeta_j^{\text{diff}}, \varsigma_j)}{(\zeta_0^{\text{diff}})^2 + (\varsigma_j)^2} \left[ \varsigma_j \sinh(\pi \varsigma_j/2) \cos(\pi \zeta_0^{\text{diff}}/2) + \zeta_0^{\text{diff}} \cosh(\pi \varsigma_j/2) \sin(\pi \zeta_0^{\text{diff}}/2) \right]. \end{aligned} $$ (10.52) --- [FIGURE: Graph showing Ground modes $\psi_0(z)$ vs z, comparing different basis functions and solutions] FIG. 10.8. Comparison of basis functions and solutions for a Doppler lineshape, $k_0 = 1.3$, $L = 1$, $A_{21} = 1$, and diffusion constant $D = 10^{-3}$. Using Eq. (10.52) instead of Eq. (10.51) reduces the run time for the root search by more than an order of magnitude. A good starting point for the root search is usually $\zeta_j = j + 0.5$. The normalization constant $c_{\mathrm{diff}}$ is determined from the equation $$ \int_{-L/2}^{L/2} \psi_j(z)^2 \mathrm{d}z = 1. $$ (10.53) Analytical evaluation of this integral yields $$ \frac{1}{c_{\mathrm{diff}}^2} = \frac{L}{2} \left[ 1 + \frac{\sin (\pi \zeta_j^{\mathrm{diff}})}{\pi \zeta_j^{\mathrm{diff}}} \right] + \frac{L}{2} f1(\zeta_j^{\mathrm{diff}}, \varsigma_j)^2 \left[ 1 + \frac{\sinh (\pi \varsigma_j)}{\pi \varsigma_j} \right] - \frac{4L}{\pi} \frac{f1(\zeta_j^{\mathrm{diff}}, \varsigma_j)}{(\zeta_j^{\mathrm{diff}})^2 + (\varsigma_j)^2} \left[ \varsigma_j \sinh(\pi \varsigma_j/2) \cos(\pi \zeta_j^{\mathrm{diff}}/2) + \zeta_j^{\mathrm{diff}} \cosh(\pi \varsigma_j/2) \sin(\pi \zeta_j^{\mathrm{diff}}/2) \right]. $$ (10.54) Figure 10.8 shows the basis function $\psi_0^b(z)$, for $k_0 = 1.3$, $L = 1$, $A_{21} = 1$, and $D = 10^{-3}$, and compares it with the resulting $\psi_0^c$, as computed by the numerical solution of Eq. (10.37) and as computed with Eq. (10.46). We see that $\psi_0^b$ shows a good resemblance to $\psi_0^c$, while the diffusion ground mode $\psi_0^{\mathrm{diff}}$ differs considerably. The ten-term modal solution practically coincides with the direct numerical solution but required much less CPU time. Figure 10.9 shows the trapping factor $g_0^c$ as a function of the number of modes used --- [FIGURE: FIG. 10.9. Trapping factor $g_0^c$ as a function of the number of modes used in Eq. (10.46).] in Eq. (10.46) when we use as basis functions (i) the basis functions Eq. (10.47) and (ii) the eigenfunctions of the diffusion equation. With the functions from Eq. (10.47), we need fewer terms for good accuracy. It must be emphasized that the diffusion modes are bad basis functions only when $\lambda^{\text{T}} \gg \lambda^{\text{diff}}$. Otherwise, using the diffusion modes will give good results. We now apply the above method to compute the percentage of atoms whose excitation is destroyed by wall collisions. Proceeding analogously to Sec. 10.2, the fraction of quenched atoms is $$ p_{\text{quench}} = 1 - \frac{\sum_j \alpha_j^c g_j^c \sum_k c_{j,k}^c \sum_l \frac{c_{k,l}^{\text{T}}}{g_l^{\text{T}}} \int \psi_l^{\text{T}}(z)\mathrm{d}z}{\sum_j \alpha_j^c \int \psi_j^c(z)\mathrm{d}z} \qquad (10.55) $$ where the factors $\alpha_j^c$ are the expansion coefficients of the initial distribution of excited atoms into the modes of the solution, $\psi_j^c$. Figure 10.10 shows $p_{\text{quench}}$ as a function of the diffusion constant for a Lorentzian lineshape with opacity $k_0L = 4$ for uniform initial distribution; we used 10-term modal expansions. Note that for $D = 0.035$ (i.e. when $\lambda_0^{\text{diff}} = \lambda_0^{\text{T}}$), $p_{\text{quench}}$ equals 60%, and not 50%, as one would guess intuitively. This is due to the existence of the higher-order modes, which have a slightly smaller radiative decay time but a *much* smaller diffusion lifetime. --- [FIGURE: FIG. 10.10. Percentage of wall-quenched atoms as a function of the diffusion constant D, for a Lorentz lineshape, k_0 = 4, L = 1, A_{21} = 1.] Finally, a very simple but useful estimate was given by (Chung 1987). Wall collisions will quench 1/6 of the atoms that are in a layer of a width that an atom with average velocity can move during one lifetime. The accuracy of this estimate is only 50%, but it gives a first idea whether wall quenching can be neglected. ### 10.4.3 *The discrete-ordinate technique* Another possibility of including particle diffusion is to add the particle transport to the rate equation of the excited atoms and to solve the resulting system of rate equation plus transfer equation, e.g., by the discrete-ordinate technique. This approach was chosen by Hummer and Kunasz (1976). They consider the situation where we have only a single type of atom in the vapour cell, and the excitation can change from one atom to another during one natural lifetime by resonant excitation exchange. This situation is, however, quite similar to a vapour cell with a buffer gas, where the velocity- (or direction-) changing collisions with buffer gas atoms are roughly equivalent to the transfer of excitation to another atom that has a different velocity (see also Chapter 3). The subsequent derivation makes the usual simplifying assumptions (apart from the presence of particle diffusion and reflecting walls) and deals with the plane-parallel slab case. We start out by defining an excited-state density $n(\mathbf{\Omega}, z)$, the density of excited atoms that have velocity vectors in the direction $\mathbf{\Omega}$. As a simplification, we assume that all atoms have the same speed $\bar{v}$. The rate at which excited atoms with velocity vector in direction $\mathrm{d}\mathbf{\Omega}$ are produced by transfer of excitation from other excited atoms is --- $$ C_{22} \int n(\boldsymbol{\Omega}', z)\mathrm{d}\boldsymbol{\Omega}' \frac{\mathrm{d}\boldsymbol{\Omega}}{4\pi} $$ (10.56) where $C_{22} = N \cdot \bar{v}_{\text{rel}} \cdot \sigma$ is the rate at which excitation transfer happens ($\sigma$ is the cross-section for the transfer, and $\bar{v}_{\text{rel}}$ the relative velocity between the two colliding atoms). To set up the rate equations, we need the following terms: The rate at which excited atoms are removed from $\mathrm{d}\boldsymbol{\Omega}$ $$ C_{22}n(\boldsymbol{\Omega}, z)\mathrm{d}\boldsymbol{\Omega} $$ (10.57) The transport operator for the excited atoms $$ \bar{\mathbf{v}} \cdot (\boldsymbol{\Omega} \cdot \nabla)n(\boldsymbol{\Omega}, z) $$ (10.58) The de-excitation rate $$ n(\boldsymbol{\Omega}, z) (A_{21} + C_{21}(z)) \mathrm{d}\boldsymbol{\Omega} $$ (10.59) The radiative and collisional excitation rate $$ N (B_{12}\bar{J}(z) + C_{12}(z)) \frac{\mathrm{d}\boldsymbol{\Omega}}{4\pi} $$ (10.60) where $C_{12}$ is the non-radiative rate from state 1 to state 2, and $$ \bar{J}(z) = \frac{1}{4\pi} \int_{4\pi} \int_{-\infty}^{\infty} C_x k(x) I(x, \boldsymbol{\Omega}, z)\mathrm{d}x\mathrm{d}\boldsymbol{\Omega} = \frac{1}{2} \int_{-1}^{1} \int_{-\infty}^{\infty} C_x k(x) I(x, \mu, z)\mathrm{d}x\mathrm{d}\mu $$ (10.61) For the slab case, we have axial symmetry, so that $\mathrm{d}\boldsymbol{\Omega} = 2\pi \mathrm{d}\mu$, and the rate equation becomes $$ -\bar{v}\mu \frac{\partial n(\mu, z)}{\partial z} = n(\mu, z) (A_{21} + C_{21} + C_{22}) - \frac{1}{2}C_{22} \int_{-1}^{1} n(\mu, z)\mathrm{d}\mu - N \frac{B_{12}\bar{J} + C_{12}}{4\pi} $$ (10.62) Equation (10.62), together with the equation of radiation transfer, can be solved by the discrete-ordinate technique described in Sec. 9.1. The details of this solution are discussed below in a rather tedious derivation—the rest of this subsection may safely be skipped without disturbing the continuity. There is also a publicly available computer program (Kunasz and Kunasz 1975) that implements this solution, see Appendix C. At this point, we will deviate from our usual notation and choice of geometry and adhere strictly to the notation of Hummer and Kunasz. The reason for this is that a very useful (and publicly available) software package, the program SLAB3 by Kunasz and Kunasz (1975) is based on the solutions of Hummer and Kunasz. When the reader is using SLAB3, it would be very confusing to have the computer program in one notation and our theoretical explanations in a different notation. We thus chose to accept the small inconsistencies in the notation of our book. --- In this notation, the slab extends from $0$ to $L$. The direction $\mu$ is negative for increasing $z$. We introduce the abbreviations $$ \begin{aligned} \Omega &= \frac{4\pi A_{21}}{N B_{12}} & \omega &= \frac{C_{22}}{C_{22} + A_{21} + C_{21}} \\ \bar{n}_2 &= \frac{1}{2} \int_{-1}^1 n \mathrm{d}\mu & \rho &= \frac{\bar{v}/C_x}{C_{22} + A_{21} + C_{21}} \\ \lambda(x) &= \frac{1}{C_x k(x)} & S &= \frac{N}{4\pi} \frac{C_{12} + B_{12}\bar{J}}{C_{22} + A_{21} + C_{21}} = \chi + \psi \bar{J} \\ \tau &= k(0)z & & \end{aligned} $$ (10.63) The rate equation then becomes $$ \rho \mu \frac{\mathrm{d}n_2}{\mathrm{d}\tau} = n_2 - \omega \bar{n}_2 - S $$ (10.64) The equation of radiative transfer is, as usual, $$ \lambda \mu \frac{\mathrm{d}I}{\mathrm{d}\tau} = I - \bar{n}_2 \Omega $$ (10.65) The intensity $I$ is written as the sum of the intensity due to those photons that have never been absorbed at all, $I^{\mathrm{ext}}$, plus the 'diffuse' intensity $I^{\mathrm{d}}$, due to photons that have been absorbed and reemitted at least once. The diffuse intensity satisfies the system of equations $$ \begin{aligned} \lambda \mu \frac{\mathrm{d}I^{\mathrm{d}}}{\mathrm{d}\tau} &= I^{\mathrm{d}} - \bar{n}_2 \Omega \\ \rho \mu \frac{\mathrm{d}n_2}{\mathrm{d}\tau} &= n_2 - \omega \bar{n}_2 - \psi \bar{J}^{\mathrm{d}} - (\chi + \psi \bar{J}^{\mathrm{ext}}) \end{aligned} $$ (10.66) The boundary conditions for the diffuse reflectance are $$ \begin{aligned} I^{\mathrm{d}}(x, -\mu, 0) &= R_{\mathrm{r}} I^{\mathrm{d}}(x, +\mu, 0) & n_2(-\mu, 0) &= R_{\mathrm{a}} n_2(+\mu, 0) \\ I^{\mathrm{d}}(x, +\mu, T) &= R_{\mathrm{r}} I^{\mathrm{d}}(x, -\mu, T) & n_2(+\mu, T) &= R_{\mathrm{a}} n_2(-\mu, T) \end{aligned} $$ (10.67) where $R_{\mathrm{r}}$ is the reflection coefficient for the radiation, and $R_{\mathrm{a}}$ is the reflection coefficient for the excited atoms, and $T$ is short for $k(0)L$. When all excited atoms hitting the cell wall are quenched, as is usually the case, then $R_{\mathrm{a}} = 0$. The externally incident radiation (at $z = 0$, respectively $\tau = 0$, see Eq. (10.63)) is assumed to be a collimated beam and can be written as $$ I^{\mathrm{ext}}(x, \mu, \alpha, \tau) = f(x)\delta(\alpha - \alpha_0)\delta(\mu - \mu_0) \exp{[-\tau \Phi(x)/|\mu_0|]} $$ (10.68) where $\Phi(x) = C_x k(x)$, $\mu_0$ is the angle between the incident beam and the $z$-axis, and $\alpha_0$ is the angle with an arbitrarily chosen axis in the plane normal to the $z$-axis. If the input radiation is reflected at $\tau = T$ (or is reflected several times), these contributions have to be added in Eq. (10.68). The spectral shape of the incident radiation, $f(x)$, can be arbitrarily chosen, and then $\bar{J}^{\mathrm{ext}}$ can be computed from Eqs (10.68) and (10.61). --- PARTICLE DIFFUSION For a monochromatic input beam of frequency $x_0$, one gets $$ \bar{J}^{\text{ext}} = \frac{I_0 \Phi(x_0)}{4\pi \Delta \nu_D} \exp (-\tau \Phi(x_0)/|\mu_0|) \qquad (10.69) $$ and for white light $$ \bar{J}^{\text{ext}} = \frac{I_0}{4\pi \Delta \nu_D} \int_{-\infty}^{\infty} \Phi(x) \exp (-\tau \Phi(x)/|\mu_0|) \,\mathrm{d}x \qquad (10.70) $$ where $\Delta \nu_D$ is the Doppler FWHM. Equations (10.66)–(10.67) and (10.69) or alternatively (10.70) fully specify the problem. They are solved by the discrete-ordinate technique that was already described in Chapter 9.1. The integrals are approximated by (Gaussian) quadrature sums. The quadrature points are chosen in such a way that $$ \begin{array}{lll} x_i = -x_{-i} & a_i = a_{-i} & i = 1, 2, \dots, n \\ \mu_j = -\mu_{-j} & b_j = b_{-j} & j = 1, 2, \ --- $$ \left[ \omega + \Omega \psi S_1(k^2) \right] S_2(k^2) - 1 = 0 \qquad (10.76) $$ with the abbreviations $$ \begin{aligned} S_1(k^2) &= \frac{1}{2} \sum_{i=-n}^n \sum_{j=-m}^m \frac{a_i b_j}{1 + \lambda_i \mu_j k} = 2 \sum_{i=1}^n \sum_{j=1}^m \frac{a_i b_j}{1 - \lambda_i^2 \mu_j^2 k^2} \\ S_2(k^2) &= \frac{1}{2} \sum_{j=-m}^m \frac{b_j}{1 + \rho \mu_j k} = \sum_{j=1}^m \frac{b_j}{1 - \rho^2 \mu_j^2 k^2} \end{aligned} \qquad (10.77) $$ There are $2 \cdot n \cdot (m + 1) = 2p$ roots of the characteristic equation. We denote the roots by $k_\alpha$, where $\alpha = 1, 2, \dots p$ and $k_\alpha = -k_{-\alpha}$. It can be shown that all roots are real. The coefficients $g$ and $h$ are now written as $$ g_{ij}^\alpha = \frac{\Omega A}{1 + k_\alpha \lambda_i \mu_j} \quad \text{and} \quad h_j^\alpha = \frac{A}{1 + k_\alpha \rho \mu_j} \frac{1}{S_2(k_\alpha^2)} \qquad (10.78) $$ In the next step, we compute the particular solution. $\bar{J}^{\text{ext}}$ is written in the form $$ \bar{J}^{\text{ext}} = \sum_{l=1}^N J_l \exp(-q_l \tau) \qquad (10.79) $$ where $J_l$ and $q_l$ are known constants. The particular solution is $$ \begin{aligned} I_{ij}^{\text{part}}(\tau) &= \Omega \chi D + \Omega \psi \sum_{l=1}^N J_l \Theta_l D \frac{\exp(-q_l \tau)}{1 + \lambda_i \mu_j q_l} \quad \text{and} \\ n_j^{\text{part}}(\tau) &= \phantom{\Omega} \chi D + \phantom{\Omega} \psi \sum_{l=1}^N J_l D_l \frac{\exp(-q_l \tau)}{1 + \rho \mu_j q_l} \end{aligned} \qquad (10.80) $$ where $$ \begin{aligned} D &= \frac{1}{1 - \omega - \psi \Omega} & \Phi_l &= 2 \sum_{i=1}^n \sum_{j=1}^m \frac{a_i b_j}{1 - (\lambda_i \mu_j q_l)^2} \\ D_l &= \frac{1}{1 - \Theta_l (\omega + \psi \Omega \Phi_l)} & \Theta_l &= \sum_{j=1}^m \frac{b_j}{1 - (\rho \mu_j q_l)^2} \end{aligned} \qquad (10.81) $$ Finally, we write down the general solution and determine the integration constants. The general solution has the form $$ \begin{aligned} I_{ij} &= \Omega \sum_{\alpha=1}^p \left[ \frac{L_\alpha \exp(-k_\alpha \tau)}{1 + \lambda_i \mu_j k_\alpha} + \frac{M_\alpha \exp(-k_\alpha (T - \tau))}{1 - \lambda_i \mu_j k_\alpha} \right] + \Omega \chi D + \\ &\quad + \Omega \psi \sum_{l=1}^N \frac{J_l \Theta_l D_l \exp(-q_l \tau)}{1 + \lambda_i \mu_j q_l} \\ n_j(\tau) &= \sum_{\alpha=1}^p \frac{1}{S_2(k_\alpha^2)} \left[ \frac{L_\alpha \exp(-k_\alpha \tau)}{1 + \rho \mu_j k_\alpha} + \frac{M_\alpha \exp(-k_\alpha (T - \tau))}{1 - \rho \mu_j k_\alpha} \right] + \chi D + \\ &\quad + \psi \sum_{l=1}^N \frac{J_l D_l \exp(-q_l \tau)}{1 + \rho \mu_j q_l} \end{aligned} \qquad (10.82) $$ --- The $2p$ integration constants $L_\alpha$ and $M_\alpha$ are now determined from the boundary conditions $$ \begin{aligned} &\sum_{\alpha=1}^p (L_\alpha U_{\alpha ij} + M_\alpha U_{\alpha ij}') = H_{ij} \\ &\quad H_{ij} = (R_r - 1)\chi D + \psi \sum_{l=1}^N J_l \Theta_l D_l \left[ \frac{R_r}{1 + \lambda_i \mu_j q_l} - \frac{1}{1 - \lambda_i \mu_j q_l} \right] \\ &\sum_{\alpha=1}^p (L_\alpha U_{\alpha ij}' + M_\alpha U_{\alpha ij}) = H_{ij}' \\ &\quad H_{ij}' = (R_r - 1)\chi D + \psi \sum_{l=1}^N J_l \Theta_l D_l \exp(-q_l T) \left[ \frac{R_r}{1 - \lambda_i \mu_j q_l} - \frac{1}{1 + \lambda_i \mu_j q_l} \right] \\ &\sum_{\alpha=1}^p (L_\alpha V_{\alpha j} + M_\alpha V_{\alpha j}') = K_j \\ &\quad K_ --- $$ \begin{aligned} I(0) &= \frac{\Phi(x)}{\mu} \Omega \int_0^T \frac{N_2(\tau')}{4\pi} \exp(-\tau'\Phi(x)/\mu)\mathrm{d}\tau' & \mu \ge 0 \\ I(T) &= \frac{\Phi(x)}{|\mu|} \Omega \int_0^T \frac{N_2(\tau')}{4\pi} \exp(-(T - \tau')\Phi(x)/|\mu|)\mathrm{d}\tau' & \mu \le 0 \end{aligned} $$ (10.86) The occurring integrals can be evaluated analytically. Hummer and Kunasz used this method to compute some special steady-state distributions in a high opacity vapour. The general computer code is available from the CPC program library and is briefly described in Appendix C. ### 10.4.4 Results Fujimoto and Phelps (1982) and Zajonc and Phelps (1981) made a series of experiments to check the validity of the computations and the effects predicted by these computations. Essentially, they shone a laser into a vapour cell filled with sodium vapour, and observed the absorption plus the emergent radiation at the cell bottom and at the cell top. When the sodium density is very high, a large part of the radiation is absorbed close to the input window and there is a large probability that it is quenched. Figure 10.11 shows the computed excited-state density as a function of the position. We see considerable deviations from the boundary condition $n(-L/2) = 0$. Using the parameters of this experiment, the mean free path is on the order of $10^{-4}$ m. Since the type of excitation enforces a rather high concentration of atoms near the walls, the gradient $\partial n/\partial z$ is large, so that the left-hand side of Eq. (10.38) cannot be neglected. When the laser is detuned from the line centre, then atoms farther away from the input window are also excited. This results first in a larger average excited-state density (fewer atoms are quenched) and secondly, in better validity of the Dirichlet boundary condition $n(-L/2) = 0$. (Note, however, that both the modal combination technique and the discrete-ordinate technique can deal with the Cauchy boundary condition.) The effect of the particle diffusion on the total number of excited atoms, $\int n(z)\mathrm{d}z$, is noticeable only for very high opacities and for no detuning of the input radiation. The effect in the spectrum of the radiation emerging at the cell bottom (i.e. back towards the laser) is much stronger and noticeable even for very small diffusion constants. Particle diffusion leads to a stronger self-reversal than would normally be observed. The physical reason for this behaviour is the following. Without diffusion, we have excited-state atoms also very close to the boundary—remember the shapes of the eigenmodes depicted in Chapter 7. These atoms can emit line-centre photons that can actually escape from the vapour. All line-centre photons emitted deeper in the vapour have no chance of escape. When now wall quenching destroys the excited atoms near the wall, there are no more line-centre photons that can escape. This implies stronger self-reversal. Figure 10.12 shows the emergent lineshape for various values of the diffusion constant. For comparison, when the cell wall would reflect the excited-state atoms instead of quenching them, self-reversal would be *less* pronounced than without diffusion. The solution of the Feautrier equation when there is particle diffusion is described by Cannon and Cram (1974). It does not differ significantly from the usual Feautrier method. --- [FIGURE: Plot of excited-state density n(6P_{1/2}) vs distance from input window for detunings \Delta = 0 and \Delta = 4GHz] FIG. 10.11. Excited-state density (Na, $6\text{P}_{1/2}$) as a function of distance from the input window. Excitation by a monochromatic laser with detuning $\Delta$ from the centre of the $6\text{S}_{1/2} \rightarrow 6\text{P}_{3/2}$ transition, incident flux is $10\text{W}/\text{cm}^2$; hfs is included. Na density $7\cdot 10^{14}\text{ cm}^{-3}$. From (Zajonc and Phelps 1981). ## 10.5 Inhomogeneous distribution of absorbers Up to now, we have assumed that the density of absorbing atoms is uniform across the cell. This is a good assumption when the lower level is the ground state and no metastable levels are involved. When the lower level is a metastable level, then—due to diffusion and wall quenching—the absorbers are distributed inhomogeneously across the cell so that the absorption coefficient also becomes inhomogeneous, $k_0 = k_0(\mathbf{r})$. **For the slab**, this complication can be eliminated easily by introducing the optical depth $\xi(z)$ $$ \xi(z) = \int_0^z k_0(u)du --- SIMPLE GENERALIZATIONS [FIGURE: Graph of Relative fluorescent intensity vs Laser detuning] FIG. 10.12. Frequency-integrated intensity on the Na $6\text{S}_{1/2} \rightarrow 6\text{P}_{3/2}$ transition in the backward direction (normalized to the value of a perfectly diffuse surface) as a function of detuning of the exciting laser. Na density: $10^{15} \text{ cm}^{-3}$. Dashed: no wall quenching. The traces D$\cdot$0.01 and D$\cdot$100 show the effect of a change in the diffusion coefficient. From (Zajonc and Phelps 1981) The path $S_{\text{int}}$ goes from point $(r, 0, 0)$ to point $(r', \varphi, z)$, where $r, \varphi$, and $z$ are the coordinates in a cylindrical coordinate system (see Fig. 10.13 for a sketch of the geometry). We again use the PCA technique. The $A_{k,m}$ matrix elements are $$ A_{k,m} = \int_{r_m-\Delta/2}^{r_m+\Delta/2} G(r, r')r'dr' = $$ $$ = \frac{\hat{C}_x}{2\pi} \int_{r_m-\Delta/2}^{r_m+\Delta/2} \int_0^\pi \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{k_0(r)\hat{k}^2(x)}{z^2 + r^2 + r'^2 - 2rr'\cos(\varphi)} \exp\left[-\hat{k}(x) \int_{S_{\text{int}}} k_0(s)ds\right] dx dz d\varphi \, r'dr'. \quad (10.89) $$ We see that each $A_{k,m}$ element is a fivefold integral. Direct evaluation of these expressions is out of the question, because CPU-time requirements would become prohibitive. On the other hand, the methods described in Appendix B, which reduce the $A_{k,m}$ elements to single or double integrals, cannot be applied, because they are only valid for a homogeneous distribution of absorbers. For the $A_{k,m}$ elements with $k \neq m$, we approximate the integrals over $r$ by the areas of rectangles $$ \int_{r_m-\Delta/2}^{r_m+\Delta/2} f(r')dr' \approx \Delta \cdot f(r_m). \quad (10.90) $$ --- INHOMOGENEOUS DISTRIBUTION OF ABSORBERS [FIGURE: FIG. 10.13. Geometry for calculating the trapping factor in a cylinder with an inhomogeneous distribution of absorbers.] The integral over $\varphi$ is computed as a $N_{\varphi}$-point quadrature sum with abscissa points $\varphi_j$ and weights $a_j$. The abscissa points can, e.g., be chosen equidistantly or for a Gaussian quadrature. After many manipulations, Eq. (10.89) becomes $$ A_{k,m} = \frac{\hat{C}_x}{\pi} \int_{-\infty}^{\infty} \hat{k}^2(x) \sum_j \frac{k_0(r_k)r_m \Delta \cdot a_j}{\sqrt{r_k^2 + r_m^2 - 2r_k r_m \cos(\varphi_j)}} \mathrm{Ki}_1 \left[ \hat{k}(x) \int_{s_{xy}} k_0(s)\mathrm{d}s \right] \mathrm{d}x, \qquad (10.91) $$ where $s_{xy}$ is the projection of the paths on to the $z = 0$ plane. The function $\mathrm{Ki}_1$ is the first repeated integral of $K_0^B$, the modified Bessel function of zero order, second kind. There are series expansions and polynomial approximations (Abramowitz and Stegun 1965) which allow fast and efficient computation of $\mathrm{Ki}_1$. At the beginning of the computations, we calculate the path length pieces $s_{\mathrm{int}}(k, m, j, m1)$, i.e. the length of the path from point $(r_k, 0, 0)$ to point $(r_m, \varphi_j, 0)$ that lies in the subregion $m1$. These path length pieces have to be computed only once, since they are given by the geometry. During the computations, the current optical path length is then computed as --- $$ \int_{s_{xy}} k_0(s) ds = \sum_{m1} s_{\text{int}}(k, m, j, m1) k_0(r_{m1}) $$ (10.92) We have thus reduced the fivefold integrals of Eq. (10.89) to single integrals. The approximation Eq. (10.90) works very well if $k \neq m$, but shows large inaccuracies for $k = m$. We also note a singularity of the integrand for $k = m, \varphi = 0$. The $A_{m,m}$ elements thus have to be computed in a different way. We first compute the escape factor $\eta(r_m)$, i.e. the probability that a photon emitted at radius $r_m$ leaves the cylinder without being reabsorbed. From (Irons 1979), the escape factor is $$ \eta(r_k) = \frac{\hat{C}_x}{\pi} \int_{-\infty}^{\infty} \int_0^\pi \int_0^{\pi/2} \hat{k}(x) \exp\left(-\frac{\hat{k}(x)}{\cos(\vartheta)} \int_{s_{xy}} k_0(s) ds\right) \cos(\vartheta) d\vartheta d\chi dx $$ (10.93) We make the substitution $u = 1/\cos\vartheta$, and express $\chi$ by $r_k$ and $\varphi$ (see Fig. 10.13). The escape factor can then be expressed as $$ \eta(r_k) = \frac{\hat{C}_x}{\pi} \int_{-\infty}^{\infty} \sum_j \hat{k}(x) \text{Ki}_2 \left( \hat{k}(x) \int_{s_{xy}} k_0(s) ds \right) a_j \frac{R^2 - r_k R \cos(\varphi_j)}{R^2 + r_k^2 - 2 R r_k \cos(\varphi_j)} dx $$ (10.94) where the optical path length is computed in the same way as above. The function $\text{Ki}_2$ is the second repeated integral of $K_0^B$ (Abramowitz and Stegun 1965). In the piecewise-constant technique for the cylinder, the Kernel $G$ is related to the $A_{k,m}$ elements by $$ A_{k,m} = G(r_k, r_m) r_m. $$ (10.95) From Sec. 4.5, the escape factor is related to the Kernel function by $$ \eta(r') = 1 - \int G(r, r') r dr $$ (10.96) so that the escape factor is related to the $A_{k,m}$ elements by $$ \eta(r_m) = 1 - \sum_k A_{k,m} \frac{r_k}{r_m}. $$ (10.97) Since we know $\eta(r_m)$ and $A_{k,m}$ for $k \neq m$, we can now compute the element $A_{m,m}$ from Eq. (10.97). However, care must be taken in the actual numerical evaluation at very low opacities. If the number of angle points is too low in comparison to the number of subregions, negative values of $A_{m,m}$ are possible. Comparisons with other algorithms for constant $k_0(r) = k_0$ show that the accuracy of this technique is better than 8%, while the CPU time is roughly comparable with those methods that work only for constant $k_0$. For the special case of a Lorentzian profile at high opacities, Bezuglov and Golubovskii (1981) gave analytical results for the $A_{k,m}$ elements. The results consist of the --- elements for the homogeneous case, plus a correction factor related to the hypergeometric function. **For spheres**, exactly the same procedure can be used. For the case that the lower-state density obeys a power law, and we consider the steady-state problem, Kunasz and Hummer (1974a) give solutions. ## 10.6 Two- and three-dimensional geometries Obviously, laboratory vapour cells can never be truly one-dimensional, even though the assumption of one-dimensionality often introduces only small errors. There are some cases where the assumption leads to such large errors that it cannot be sustained anymore, so that two- or three- dimensional computations must be done. There are several ways to make such computations: (i) The most straightforward is certainly a Monte Carlo simulation. The changes in the code that are necessary to incorporate multi-dimensional problems are trivial. However, program run-time increases considerably, since we have added a degree of freedom, and we need more photons to get a good statistical averaging over all degrees of freedom. (ii) A second possibility is to solve the Holstein equation for the multi-dimensional geometry, e.g. by the piecewise constant approximation. This leads, however, to an enormous increase in CPU time. On one hand, the computation of the matrix elements becomes more complicated, because the analytical tricks in Appendix B can no longer be used. For the finite cylinder, we get, e.g., triple integrals, and we have to evaluate $N_r \cdot N_z$ of them. The inversion of the $A_{k,m}$ matrix also takes much longer. A related method is the integral-operator technique of Jones and Skumanich (1973) and Jones (1973). (iii) A third possibility is to solve the equation of radiative transfer. As we have seen before, this is usually a good idea only for steady-state problems. A formulation of the Feautrier technique in two dimensions was originally derived by Cannon (1970) and is also reproduced in (Cannon 1985). Evaluation of this equation in a somewhat different formulation (including Rybicki reorganization) was done by Mihalas *et al.* (1978). However, both formulations are not very efficient. It is preferable to use either the variable Eddington factor formulation described in Sec. 9.3, or to use the operator perturbation technique described in Chapter 13. (iv) Escape factor computations (e.g. (Jones 1971)) suffer from the usual problems of escape factors, see Sec. 4.5. In the following, we will concentrate on the physical effects and on approximate equations. For the numerical methods, we refer to the above-mentioned sections later in the text. ### 10.6.1 *The finite cylinder* The most important two-dimensional geometry is the finite cylinder. Actually, it is of course three-dimensional, but as long as it is invariant with respect to rotation along --- [FIGURE: Graph of lowest-order trapping factor vs L/R for various effective opacities] FIG. 10.14. Decay time constant of the lowest-order mode, $g_0$, as a function of cylinder aspect ratio $L/R$ for various effective opacities $kL_{\text{eff}} = k_0/((1/L)^2 + (1/R)^2)^{1/2}$. the angle $\varphi$, it is two-dimensional mathematically. This situation always occurs when we shine a laser along the axis of a cylindrical cell. It is common in the literature to approximate that situation by using an infinite-cylinder geometry. However, we have to be extremely careful whether the necessary assumptions for this simplification are actually fulfilled. If we consider just the lowest-order mode, one-dimensional approximations are usually valid. Simulations show that the lowest-order trapping factor depends very little on the geometry as long as we keep the expression $$ kL_{\text{eff}} = \frac{k_0}{\sqrt{\frac{1}{L^2} + \frac{1}{R^2}}} $$ (10.98) constant. The expression $kL_{\text{eff}}$ can be interpreted as the 'effective' opacity of the vapour cell. Figure 10.14 shows the dependence of the trapping factor $g_0$ on the ratio $L/R$. We see that the results are always within 20% of the limiting cases of the plane-parallel slab or the infinite cylinder. We thus reach the important conclusion that for the lowest-order mode, we can use the formulas for the plane-parallel slab or for the infinite cylinder if we just replace $k_0L$ by $kL_{\text{eff}}$. Accuracy will be better than 20%. When we send a narrow laser beam into a vapour cell with a high-opacity vapour, then the direct excitation is concentrated near the entry point ($r = 0, z = -L/2$). This means that we have a very high percentage of photons in the higher modes, and we will have to wait a long time until they have all decayed to insignificant levels. --- When checking whether a situation can be approximated as one-dimensional, it is *not* sufficient to examine the geometry. We also have to check the initial distribution of excited atoms. As mentioned in Chapter 4, we require for the infinite cylinder in the axial direction that the length over which the excited-state distribution changes significantly must be much larger than the cell radius—and of course the vapour cell length must be much larger than the cell radius. Otherwise, two-dimensional computations have to be used. Let us consider the example of a vapour cell with Doppler lineshape, $k_0 = 5\text{ cm}^{-1}$, $R = 1\text{ cm}$, and $L = 3\text{ cm}$, where the laser is tuned to the line centre. If we considered just the geometric dimensions of the vapour cell, then the cell would be equivalent to an infinite cylinder, and since the laser beam is thin, excited atoms would be concentrated near the cylinder axis. However, since the atoms are initially concentrated near the entry point (roughly within 1/5 cm of the cell bottom), the reemitted photons mainly ‘see’ the cell top. In the beginning of the decay process, the behaviour will thus be rather like in a plane-parallel slab with opacity $k_0 L = 15$, with atoms concentrated near one slab surface, with an initial distribution of $n(z, 0) \approx \exp[(-L/2 - z) \cdot k_0]$. At late times, the temporal behaviour is determined by the lowest-order mode of the infinite cylinder with $k_0 R = 5$. One correct mathematical description would be to expand the initial distribution into the eigenmodes of the finite cylinder. However, computation of the modes of the finite cylinder is very difficult and time-consuming, while there are accurate fitting equations for the trapping factors and modes in a plane-parallel slab and infinite cylinder. We thus propose to use the following equations for the eigenmodes and eigenvalues $$ \psi_{j_z, j_r}^{\text{finite cyl}}(r, z) = \psi_{j_z}^{\text{slab}}(z)\psi_{j_r}^{\text{cyl}}(r) $$ $$ g_{j_z, j_r}^{\text{finite cyl}} = 1 + \frac{1}{\frac{1}{g_{j_z}^{\text{slab}} - 1} + \frac{1}{g_{j_r}^{\text{cyl}} - 1}} $$ (10.99) Equation (10.99) gives the correct limiting behaviour for low opacities ($g \to 1$ for $k_0 \to 0$), for very flat cylinders (slab solution), and for very long cylinders (infinite cylinder). Figure 10.15a shows the total number of excited atoms, $n^\Sigma(t)$, for the mentioned example, computed with Eq. (10.99), and with the infinite cylinder approximation. Numerical solutions for the true 2D-geometry are also shown for comparison. For both Eq. (10.99) and the infinite cylinder approximation, ten modes for the modal expansion were used. We see that our expectation with respect to the early and late-time behaviour is fulfilled, and that Eq. (10.99) gives much better results than the infinite cylinder approximation (actually, the deviation of the proposed expansion stays within the accuracy of the numerical method used). Figure 10.15b shows the same geometry, but with an opacity $k_0 R = 0.5$. We see that in this case the agreement between the infinite-cylinder computations and numerical computations is good. This is due to the fact that the distance of the 1/e-decrease of the initial distribution in the axial direction is equal to 2 cm, --- [FIGURE: Graphs showing integrated upper-state density vs time for a finite cylinder] FIG. 10.15. The integrated density of excited atoms, $n^\Sigma(t)$, in a finite cylinder of radius $R = 1\text{ cm}$, length $L = 3\text{ cm}$, and with absorption coefficient $k_0 = 5.0\text{ cm}^{-1}$ (left) and $k_0 = 0.5\text{ cm}^{-1}$ (right). The results of a direct numerical solution are compared to the proposed modal expansion and to the infinite cylinder approximation. which is larger than the cell radius. Thus, the infinite cylinder approximation is fulfilled quite well. Bezuglov (1982) and Asadullina *et al.* (1989) have derived similar equations for the trapping factors by the ‘geometrical quantization technique (GQT)’ (see Sec. 5.2). As always in the GQT, the trapping factors are computed from the allowed values of the Fourier transform variables $p_j$ as $1/g = W(p_j)$, where the function $W$ is defined in Sec. 5.2 for Doppler and Lorentz lineshapes. The allowed values for the finite cylinder are $$ p_{j_r, j_z} = \sqrt{p_{j_r}^2 + p_{j_z}^2}, \quad \text{with} \quad \begin{aligned} p_{j_r} &= \frac{\pi}{2R} \left( 2j_r + 1 + \frac{\gamma}{2} \right) & j_r &= 0, 1, \dots \\ p_{j_z} &= \frac{\pi}{2L} \left( 2j_z + 1 + \gamma \right) & j_z &= 0, 1, \dots \end{aligned} \qquad (10.100) $$ Bezuglov (1977) also computed the trapping factors by the piecewise-constant method. The equations for the $A_{k,m}$ matrix elements are the same as the basic equations for the infinite cylinder case. Only the integration in the $z$-direction extends not from $-\infty$ to $\infty$, but from $z_k$ to $z_{k+1}$. He also notes that by averaging over $r_k$, the $A_{k,m}$ matrix can be made symmetric, and that this additional integration can be done analytically. Bulyshev *et al.* (1978) computed the lowest-order trapping factors by MC simulations and gave fitting factors. Heddle and Samuel (1970) came to a similar conclusion as we above, namely that the trapping factor is determined by $R$ for $L > 2R$ and by $L$ otherwise; this was found from experimental investigations. --- [FIGURE: Graph of Geometry constant $m_0^D$ vs Radius of inner cylinder] FIG. 10.16. Value of $m_0^D$ for a hollow cylinder, Doppler Lineshape. From (Denisov and Preobrazhenskii 1983). For steady-state problems, a Feautrier solution with a variable Eddington factor is often a good choice. The appropriate equations were given in Insert 9.1. ### 10.6.2 *The parallelepiped* For the parallelepiped, the GQT gives the following values for the allowed values of $p_k$ $$ p_{j_x, j_y, j_z} = \sqrt{p_{j_x}^2 + p_{j_y}^2 + p_{j_z}^2}, \quad \text{with} \quad \begin{aligned} p_{j_x} &= \frac{\pi}{2L_x} (2j_x + 1 + \gamma) \quad j_x = 0, 1, \dots \\ p_{j_y} &= \frac{\pi}{2L_y} (2j_y + 1 + \gamma) \quad j_y = 0, 1, \dots \\ p_{j_z} &= \frac{\pi}{2L_z} (2j_z + 1 + \gamma) \quad j_z = 0, 1, \dots \end{aligned} $$ (10.101) Of course, an interpolation formula similar to that for the finite cylinder could also be used for the parallelepiped, where solutions for the $x$, $y$, and $z$ directions are simply slab solutions with the appropriate opacity. The two-dimensional case is also treated by Mihalas *et al.* (1978). The conclusions are intuitively clear: the escape probability is largest near the corners, and thus the source function is smallest there. Of course, this is also true for three-dimensional parallelepipeds. Expressions for the $A_{k,m}$ matrix elements, especially for the case of a Lorentzian line at high opacities, are given by Seiwert (1956). ### 10.6.3 *The torus and the hollow cylinder* For geometries that are not convex, the situation becomes much more complicated. The Holstein equation cannot be written down in the usual way, because it becomes almost impossible to account for photons that are emitted at one point in the vapour, escape --- [FIGURE: Value of $m_0^D$ and $m_0^L$ for a hollow cylinder] FIG. 10.17. Value of $m_0^D$ and $m_0^L$ for a hollow cylinder: $R_1$ is the radius of the outer cylinder, and $R_2$ is the radius of the inner cylinder. From (Denisov and Preobrazhenskii 1983) [FIGURE: Values of $m_0^D$ and $m_0^L$ for a torus] FIG. 10.18. Values of $m_0^D$ and $m_0^L$ for a torus. $R_1$ is the radius of the tube, $R_2$ is the distance from the centre to the inner point of the tube. From (Denisov and Preobrazhenskii 1983). --- through one boundary, enter the cell again at some other point, and are reabsorbed there. Practical examples for such geometries are the torus and hollow cylinders. These configurations can have applications with plasmas (torus) and with discharge lamps (hollow cylinder). Monte Carlo simulations are usually the most convenient way to treat these geometries. Denisov and Preobrazhenskii (1983) have proposed an approximate method that spares us the repeated computations usually associated with MC simulations. We know that at high opacities and with a Doppler lineshape, the trapping factor is given by $$ g_0^{\mathrm{D}} = \frac{1}{m_0^{\mathrm{D}}} k_0 L \sqrt{\ln(k_0 L)} \qquad (10.102) $$ where $m_0^{\mathrm{D}}$ is a constant that depends on the geometry. The idea is now to make the MC simulation only for a single opacity, in order to compute $m_0^{\mathrm{D}}$, and then to use Eq. (10.102) to extrapolate to all other opacities. Results for the hollow cylinder are given in graphical form in Figs 10.16 and 10.17. For the torus, results are given in Fig. 10.18. Here, $R_1$ is the radius of the 'bent' cylinder, and $R_2$ is the distance from the origin to the innermost point of the torus. Other multidimensional problems include a slab that has a (periodic) fine-structure—there might, e.g., be columns with different material properties embedded into the slab. This problem is of interest only in astrophysics, see (Avrett and Loeser 1971), (Kneer 1981). --- # 11 ## PARTIAL FREQUENCY REDISTRIBUTION Up to now, we have generously assumed that the emission frequency of a photon is completely independent of the frequency of the absorbed photon, so that there is complete frequency redistribution, CFR. This assumption is often, but not always fulfilled in practical situations. In this chapter, we analyse under what conditions CFR is valid, and what effects we have to expect when the emitted photon at least partly ‘remembers’, on which frequency it was absorbed—partial frequency redistribution, PFR. In the first subsection, we will deal with general considerations and show under which conditions PFR comes into play. In the subsequent sections, we will look at the mathematical formalism to solve trapping problems with PFR. The last section deals with the influence of large-scale particle flow. ## 11.1 The physical picture of PFR The first papers on radiation trapping assumed that the absorbed and reemitted frequencies are exactly the same, i.e. complete frequency coherence. Kenty (1932) was the first in the atomic physics literature to explicitly include the frequency redistribution due to the Doppler effect in his approximate computations. Holstein (1947) used the assumption of CFR to derive his famous equation. However, Holstein also realized that CFR is *not* always valid and gave a discussion about the influence of PFR on the trapping factor. Henyey, and Spitzer (1944), were the first in the astrophysical literature to point out the importance of frequency redistribution, and later Thomas (1957) assumed CFR to derive the rate-equation plus transfer-equation formulation that is equivalent to the Holstein equation, see Sec. 4.4. Actually, the problem of PFR is much more important in astrophysics than under laboratory conditions, and during the following years, astrophysicists did pioneering work on PFR. They defined the ‘redistribution function’ $R(x, x')$, i.e. the probability that a photon of frequency $x'$ is absorbed and a photon of frequency $x$ is emitted.[^14] With this redistribution function, it was now possible to give, at least theoretically, an exact description of the trapping problem under conditions of PFR—the generalization of the Holstein equation under PFR. However, this generalized Holstein equation is so complicated that various approximations are required if it is to be solved for practical cases. Numerous computation methods have been proposed, and will continue to be [^14]: In this chapter, we will often deal with probabilities, so it is convenient to define a **normalized absorption profile**, $\Phi(\nu) = C_\nu k(\nu)$. 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PM JTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- proposed, since the advent of ever more powerful computers allows for more and more accurate computations. For the computation of the redistribution function, we have to proceed in two steps. In a first step, we must determine the redistribution function in the rest frame of the atom. If this is to be done from first principles, it means a quite complicated quantum-mechanical exercise, (Burnett *et al.* 1980), (Burnett and Cooper 1980a, b), (Omont *et al.* 1972). We will quote here just the final results. Once these atomic-rest-frame redistribution functions are known, we transform them to the laboratory rest frame. This transformation takes the velocity vector of the absorbing atom into account. Finally, we average over the velocity distribution of the absorbing atoms, which is assumed to be Maxwellian (this assumption will be discussed later in this section). Strictly speaking, the redistribution function not only depends on the frequency, but also on angle, i.e. the reemission is not isotropic. The redistribution function then has to be written as $R(x, x', \mathbf{e}_{\text{n}}, \mathbf{e}'_{\text{n}})$, which means the probability that a photon from direction $\mathbf{e}'_{\text{n}}$ with frequency $x'$ is absorbed, and reemitted into direction $\mathbf{e}_{\text{n}}$ with frequency $x$. The angular redistribution is important mainly for polarization effects, which we will discuss in Chapter 12. When polarization is of no interest, it is common to use the angle-averaged redistribution function $R(x, x')$, even though it is not exact. ### 11.1.1 *Redistribution functions in the atomic rest frame* As stated above, we first have to determine the redistribution function in the atomic rest frame (ARF). In the rest frame, we can factor the redistribution function: $R(\nu, \nu', \mathbf{e}_{\text{n}}, \mathbf{e}'_{\text{n}}) = \Phi^{\text{ARF}}(\nu') \cdot p(\nu, \nu') \cdot \Gamma(\mathbf{e}_{\text{n}}, \mathbf{e}'_{\text{n}})$. The function $\Phi^{\text{ARF}}(\nu')$ denotes the probability that a photon of frequency $\nu'$ is absorbed. This probability is independent of the direction of the incident photon when there is no preferred direction on the atomic scale. The next term, $p(\nu, \nu')$, is the probability that a photon of frequency $\nu$ is emitted on condition that a photon of frequency $\nu'$ has previously been absorbed. Similarly, $\Gamma(\mathbf{e}_{\text{n}}, \mathbf{e}'_{\text{n}})$ is the conditional probability that a photon is scattered from direction $\mathbf{e}'_{\text{n}}$ into direction $\mathbf{e}_{\text{n}}$. This probability might e.g. be the radiation pattern of an electric dipole, where the dipole axis is determined by the incident direction; it is this term that is of most interest for the polarization effects of Chapter 12. From this physical picture, it is clear that the directional pattern $\Gamma$ must be independent of the frequency. Being probabilities, the functions $\Phi^{\text{ARF}}$, $p$, and $\Gamma$ must fulfil the normalization conditions $$ \int_{-\infty}^{\infty} \Phi^{\text{ARF}}(\nu')\text{d}\nu' = 1, \qquad \int_{-\infty}^{\infty} p(\nu, \nu')\text{d}\nu' = \int_{-\infty}^{\infty} p(\nu, \nu')\text{d}\nu = 1, \quad \text{and} $$ $$ \frac{1}{4\pi} \int_{4\pi} \Gamma(\mathbf{e}_{\text{n}}, \mathbf{e}'_{\text{n}})\text{d}\Omega = \frac{1}{4\pi} \int_{4\pi} \Gamma(\mathbf{e}_{\text{n}}, \mathbf{e}'_{\text{n}})\text{d}\Omega' = 1. \qquad (11.1) $$ Let us first consider the situation from a semi-classical point of view. When the atom absorbs a photon of frequency $\nu$ and energy $h\nu$, conservation of energy dictates --- that the emission frequency is unchanged, so that we have complete coherence between the absorbed and reemitted frequency. However, this semiclassical picture has to be used with caution. As explained by Jefferies (1968), a monochromatic scattering process cannot be divided into an absorption process and a separate reemission process, we have to consider it as one single process. Otherwise, we would have to know the times when these separate processes occur, and—due to the uncertainty principle—any measurement at these times would broaden the line. When the incident radiation is a very short pulse (which implies a large linewidth), then we do not have this problem, since the time of absorption is well determined. The situation becomes even more intricate when not only the upper level, but also the lower level shows natural broadening (i.e. when the lower level is not the ground state). We now state the functions $\Phi^{\mathrm{ARF}}(\nu')$, the absorption probability, and $p(\nu, \nu')$, the conditional probability of emission at $\nu$ after absorption at $\nu'$, for four important cases. **(i) Coherent scattering with both levels sharp.** This is equivalent to neglecting natural (and collisional) broadening of both states. Absorption and reemission can happen only on the frequency of the infinitely sharp transition, so that $\Phi^{\mathrm{ARF}}(\nu') = \delta(\nu' - \nu_0)$, and $p(\nu, \nu') = \delta(\nu - \nu')$, which means that there is absolutely no redistribution. Because of ever present natural broadening, this situation cannot arise in reality. Still, it is an interesting case, because it gives the limiting behaviour for pure Doppler broadening. **(ii) Coherent scattering with sharp lower level and radiatively broadened upper level.** This situation occurs when the lower level is the ground state (and is approximately true if it is a metastable state) and there are no collisional processes. Due to the natural broadening, see Chapter 2, the absorption profile $\Phi^{\mathrm{ARF}}(\nu')$ is a Lorentzian $$ \Phi^{\mathrm{ARF}}(\nu') = L(\nu') = \frac{\Delta\nu^n}{2\pi} \frac{1}{(\nu' - \nu_0)^2 + (\Delta\nu^n/2)^2} \qquad (11.2) $$ Since we have no collisional processes, energy conservation applies (in a semiclassical picture), and the reemitted frequency must be the same as the absorbed, $p(\nu, \nu') = \delta(\nu - \nu')$. So there is again no redistribution in frequency. **(iii) Complete redistribution with a sharp lower level and the upper level both collisionally and radiatively broadened** Both natural and collisional broadening lead to a Lorentzian line. The linewidth is just the sum of the widths of natural and of collisional broadening, and $\Phi^{\mathrm{ARF}}(\nu') = L(\nu')$. The frequent collisions lead to complete redistribution, so that the emission profile does not depend on the absorbed frequency, $p(\nu, \nu') = L(\nu)$. This limit is strictly valid where the collision width is much larger than the natural width, although it is also used as an approximation in other situations. --- (iv) **Subordinate lines** We assume that both the lower and the upper levels are radiatively broadened. The redistribution function for that case has been subject to considerable discussion in the literature. Weisskopf and Wooley derived one form, while Heitler in his classical book derived a different form that was widely used in the astrophysical literature up to the 1970s (note that it also forms the basis of Hummer's (1962) $R_{\text{IV}}$ function). Omont *et al.* (1972) showed that actually Wooley's equation is the correct one, so that (Heinzel 1981) $$ \begin{aligned} \Phi^{\text{ARF}}&(\nu') p(\nu, \nu') = \\ &= \frac{\Delta\nu_u \Delta\nu_l^2 (2\Delta\nu_l + \Delta\nu_u)}{4\pi^2 \left[ (\nu - \nu')^2 + \Delta\nu_l^2 \right]} \left[ (\nu - \nu_0)^2 + \frac{1}{4}(\Delta\nu_u + \Delta\nu_l)^2 \right]^{-1} \cdot \\ &\quad \cdot \left[ (\nu' - \nu_0)^2 + \frac{1}{4}(\Delta\nu_u + \Delta\nu_l)^2 \right]^{-1} + \\ &\quad + \frac{\Delta\nu_u \Delta\nu_l}{4\pi^2 \left[ (\nu - \nu')^2 + \Delta\nu_l^2 \right]} \left( \left[ (\nu - \nu_0)^2 + \frac{1}{4}(\Delta\nu_u + \Delta\nu_l)^2 \right]^{-1} + \right. \\ &\quad \left. + \left[ (\nu' - \nu_0)^2 + \frac{1}{4}(\Delta\nu_u + \Delta\nu_l)^2 \right]^{-1} \right) + \\ &\quad + \frac{\Delta\nu_l^2}{4\pi^2} \left[ (\nu - \nu_0)^2 + \frac{1}{4}(\Delta\nu_u + \Delta\nu_l)^2 \right]^{-1} \left[ (\nu' - \nu_0)^2 + \frac{1}{4}(\Delta\nu_u + \Delta\nu_l)^2 \right]^{-1} \end{aligned} $$ (11.3) where $\Delta\nu_u$ and $\Delta\nu_l$ are the natural widths of the upper and lower levels, respectively. The function has two peaks, one at $\nu = \nu'$, and the other at $\nu = \nu_0$. It has become common to denote this redistribution function by the index $V$ to distinguish it from the (incorrect) $R_{\text{IV}}$ based on Heitler's treatment. Further discussions, including added collisional broadening, can be found in (McKenna 1980), and (Heinzel and Hubeny 1982). When the absorption frequency $\nu'$ is far in the wings, the redistribution function can be approximated as $$ \Phi^{\text{ARF}}(\nu') p(\nu, \nu') = \Phi(\nu') \left[ \frac{\Delta\nu_u}{\Delta\nu_u + \Delta\nu_l} \delta(\nu - \nu') + \frac{\Delta\nu_l}{\Delta\nu_u + \Delta\nu_l} \Phi(\nu) \right] $$ (11.4) For that case, the redistribution function thus consists of a coherent and a redistributed part. Finally, we can also consider branching transitions, i.e. absorption of a photon at one wavelength, and emission at another. What we need to know is, when a photon is absorbed on transition $a$ with offset $\nu' - \nu_0^a$ from the line centre, at which frequency $\nu - \nu_0^b$ will it be emitted on transition $b$. These generalized redistribution functions are discussed by Hubeny (1982). Angle-averaged forms of this redistribution function will also be given in Sec. 11.1.8. Quantum-mechanical approaches to the general redistribution problem (sometimes also including stimulated emission) were made, e.g. by Burnett *et al.* (1980), Burnett and Cooper (1980a, --- b), Yelnik *et al.* (1981), Roussel-Dupre (1983), Mollow (1970, 1975, 1976, 1977), Huber (1969), Cooper and Ballagh (1978), and Kunze (1986a). A comparison of the semiclassical results of Heinzel, Hubeny, and coworkers with the quantum-mechanical computations of Burnett, Cooper, and coworkers can be found in (Hubeny and Cooper 1986). These aspects are also treated in the reviews of (Linsky 1985), and (Hubeny 1985a). The effect of magnetic fields on the redistribution function is discussed by Omont *et al.* (1973), and Landi degli'Innocenti (1983, 1984). The redistribution function in a hot dense plasma under the influence of the Stark effect is treated by Bulyshev *et al.* (1995). Redistribution for trapping in solids is described by Levinson (1978). ### 11.1.2 Redistribution functions in the laboratory rest frame We now consider an atom with velocity vector $\mathbf{v}$. Transformation of rest-frame frequencies $\nu_{\text{ARF}}$ to laboratory-frame frequencies $\nu_{\text{lab}}$ is given by the law of Doppler shift, $$ \nu_{\text{lab}} = \nu_{\text{ARF}} - \nu_0 \frac{\mathbf{v} \cdot \mathbf{e}_{\mathbf{n}}}{c} $$ (11.5) The redistribution function in the laboratory frame can thus be written as $$ R(\nu, \nu', \mathbf{e}_{\mathbf{n}}, \mathbf{e}'_{\mathbf{n}}) = \Phi^{\text{ARF}} \left( \nu' - \nu_0 \frac{\mathbf{v} \cdot \mathbf{e}'_{\mathbf{n}}}{c} \right) p \left( \nu - \nu_0 \frac{\mathbf{v} \cdot \mathbf{e}_{\mathbf{n}}}{c}, \nu' - \nu_0 \frac{\mathbf{v} \cdot \mathbf{e}'_{\mathbf{n}}}{c} \right) \Gamma \left( \mathbf{e}_{\mathbf{n}}, \mathbf{e}'_{\mathbf{n}} \right) $$ (11.6) In this formulation, the angular and the spatial redistribution get mixed, because $\mathbf{e}_{\mathbf{n}}$ and $\mathbf{e}'_{\mathbf{n}}$ enter in $\Phi^{\text{ARF}}$, $p$, and $\Gamma$. We now have to average this redistribution function over the velocity distribution of the absorbing atoms. The velocity distribution is assumed to be Maxwellian. When the absorbing atoms are ground-state atoms, this assumption will be reasonable—as long as we have no saturation effects, like Bennet holes, see Chapter 13. For subordinate lines, however, this assumption will often not be fulfilled, since in this case, also the absorbing state is an excited state and it was probably created by either narrow-band optical excitation (by a laser) or by collisional excitation. In the former case, absorbing atoms will tend to be in a velocity group corresponding to the detuning of the laser from the excitation-line centre. In the latter case of collisional excitation, the short-wavelength or the long-wavelength wing will be favoured for endothermic or exothermic processes. However, computation of the velocity distribution jointly with the atomic densities is often too complicated, so that Maxwellian distributions are used even against better knowledge. This aspect is discussed, e.g., by Oxenius (1965), Heinzel and Hubeny (1983), and Hubeny *et al.* (1983), and in Chapter 13. Averaging the redistribution function over the Maxwellian velocity distribution gives (Mihalas 1978) $$ \begin{aligned} R(\nu, \nu', \mathbf{e}_{\mathbf{n}}, \mathbf{e}'_{\mathbf{n}}) = \frac{1}{\pi} \Gamma(\mathbf{e}_{\mathbf{n}}, \mathbf{e}'_{\mathbf{n}}) \int_{-\infty}^{\infty} \exp(-u_1^2) \Phi^{\text{ARF}}(\nu' - w u_1) \cdot \\ \cdot \int_{-\infty}^{\infty} \exp(-u_2^2) p \left( \nu - w(u_1 \cos \vartheta + u_2 \sin \vartheta), \nu' - w u_1 \right) \mathrm{d}u_2 \mathrm{d}u_1 \end{aligned} $$ (11.7) --- THE PHYSICAL PICTURE OF PFR where $w = (2k_B T / m_{\text{atom}})^{0.5} \nu_0 / c$, and $\vartheta$ is the angle between the vectors $\mathbf{e_n}$ and $\mathbf{e_n'}$. We now just have to insert the functions $\Phi^{\text{ARF}}$, $p$, and $\Gamma$ to get the various angle-dependent frequency redistribution functions. When we finally switch to normalized frequencies $x$ we get **for pure Doppler broadening** $$ R_I(x, x', \mathbf{e_n}, \mathbf{e_n'}) = \frac{\Gamma(\mathbf{e_n}, \mathbf{e_n'})}{\pi \sin \vartheta} \exp\left(-x^2 - (x' - x \cos \vartheta)^2 \csc^2 \vartheta\right) $$ (11.8) **for Doppler plus natural broadening** $$ R_{II}(\cdots) = \frac{\Gamma(\mathbf{e_n}, \mathbf{e_n'})}{\pi \sin \vartheta} \exp\left[-\frac{1}{2}(x - x')^2 \csc^2(\vartheta/2)\right] H\left[a \sec(\vartheta/2), \frac{1}{2}(x + x') \sec(\vartheta/2)\right] $$ (11.9) where, as usual, --- ### 11.1.3 Angle-averaged redistribution functions For most purposes, the angular dependence of the redistribution function makes the solution of the Holstein equation too complicated. To simplify things, it is common to average the redistribution function over the vectors $\mathbf{e_n}$ and $\mathbf{e_n'}$ so that $$ R(\nu, \nu') = \frac{1}{4\pi} \int_{4\pi} R(\nu, \nu', \mathbf{e_n}, \mathbf{e_n'}) \mathrm{d}\Omega \qquad (11.12) $$ The angle-averaged redistribution function $R(\nu, \nu')$ fulfills the normalization condition (formally from Eq. (11.1)) $$ \iint R(\nu, \nu') \mathrm{d}\nu \mathrm{d}\nu' = 1 \qquad (11.13) $$ Furthermore, an absorbed photon of frequency $\nu'$ must be emitted at *some* frequency $-\infty < \nu < \infty$, $$ \int R(\nu, \nu') \mathrm{d}\nu = \Phi(\nu') \qquad (11.14) $$ and an emitted photon must have been absorbed at some frequency $$ \int R(\nu, \nu') \mathrm{d}\nu' = \Psi^{\text{white}}(\nu) \qquad (11.15) $$ where $\Psi^{\text{white}}$ is the profile upon natural excitation, i.e. the emission profile that occurs after excitation of the atom with white (frequency-independent) light. Actual evaluation of Eq. (11.12) is, however, not the most efficient method (except for $R_V$, where direct numerical integration seems the only way, see (Heinzel and Hubeny 1983)). For the other cases, one first sets up the angle-averaged redistribution function for a certain velocity $\mathbf{\hat{v}} = \mathbf{v}[m_{\text{atom}}/2k_{\text{B}}T]^{0.5}$. Generally, this equation can be written as (Hummer 1962, Mihalas 1978) $$ R(\nu, \nu', \hat{v}) = \frac{1}{2} \int_{-1}^{1} \Phi^{\text{ARF}}(\nu' - w\mu'\hat{v}) \int_{-1}^{1} \Gamma(\mu, \mu') p(\nu - w\mu\hat{v}, \nu' - w\mu'\hat{v}) \mathrm{d}\mu \mathrm{d}\mu' \qquad (11.16) $$ where $\mu = \mathbf{e_n} \cdot \mathbf{e_n'} = \cos \vartheta$. --- **For incoherent scattering**, we have to average the general expression Eq. (11.16) over the Maxwellian velocity distribution to get $$ R(\nu, \nu') = \frac{4}{\sqrt{\pi}} \int_0^\infty R(\nu, \nu', u)u^2 \exp(-u^2)\mathrm{d}u $$ (11.19) These general equations can now be used to derive the redistribution functions for several special cases—these will be given and discussed in the subsequent subsections. We will restrict ourselves to isotropic reemission, since the angle-averaged redistribution function for dipole scattering is very similar to $R(\nu, \nu')$ of isotropic scattering. When required, the dipole (or other anisotropic emission) redistribution functions can be derived from the general procedure described above. ### 11.1.4 *Pure Doppler broadening* Pure Doppler broadening (in the laboratory rest frame) corresponds to complete coherence in the atomic rest frame—case I in the discussion above. Let us first get a physical picture of the frequency redistribution in that case. As said, there is complete frequency coherence in the rest frame of the atom. However, the frequency seen by an observer (i.e in the laboratory rest frame) is $\nu_0 \cdot (1 - \mathbf{v} \cdot \mathbf{e}'_\mathbf{n}/c)$ for the absorbed frequency, while it is $\nu_0 \cdot (1 - \mathbf{v} \cdot \mathbf{e}_\mathbf{n}/c)$ for the reemitted frequency. The direction of reemission is completely random, so that the observed reemitted frequency is different from the observed absorbed frequency (it would be the same only when $\mathbf{e}_\mathbf{n} = \mathbf{e}'_\mathbf{n}$). This does *not* imply that it is independent of the absorbed frequency. Imagine that we send in a photon with normalized frequency $\nu_1$, which is far in the wings of the Doppler shape. *When* this photon is absorbed in the vapour, this means that the absorbing atom must have a very high velocity. The atom still has the high velocity upon reemission of the photon so there is a high probability that this results in an emission frequency $\nu_2$ that is also in the wings. Certainly this probability will be larger than the probability for reemission at this frequency under conditions of CFR, $\exp(-\nu_2^2)$. This picture qualitatively explains why there is a correlation between the absorbed and the reemitted frequency for the case of pure Doppler broadening. The angle-averaged redistribution function (with frequency $x$ in Doppler units, see Chapter 2) is $$ R_I(x, x') = \frac{1}{2}\mathrm{erfc} \left[ \max(|x|, |x'|) \right] $$ (11.20) The function $R_I(x, x')/\Phi(x')$ is depicted in Fig. 11.1. CFR would correspond to a Gaussian bell shape; this condition is clearly not fulfilled. In the multiple-scattering picture, it is also useful to consider the redistribution function of photons that are scattered $i$ times, $R_i(x, x_0)$. Quite generally, $$ R_i(x, x_0) = \int_{-\infty}^\infty R_1(x, x')R_{i-1}(x', x_0)\mathrm{d}x' $$ (11.21) --- [FIGURE: FIG. 11.1. Redistribution function for a pure Doppler line.] For pure Doppler broadening, this can be evaluated analytically to give (Ivanov and Schneeweis 1976): $$ R_i(x, x_0) = \frac{1}{\sqrt{\pi}} \sum_{k=0}^{\infty} \frac{1}{2^{2k}(2k + 1)^i(2k)!} \exp(-x^2)\text{He}_{2k}(x)\text{He}_{2k}(x_0) $$ (11.22) where the functions $\text{He}_k$ are the Hermite polynomials (Abramowitz and Stegun 1965). Figure 11.2 shows this redistribution function for $x_0 = 1$ and $x_0 = 2$. We see that after about three scatterings, the redistribution function has a Doppler lineshape, i.e. the same form as for CFR. It is a widespread opinion that for pure Doppler broadening, CFR is valid, and thus the comparatively simple computation methods presented in Part II are sufficient. Various authors have checked this assumption. Holstein (1947) discussed the problem qualitatively, and argued that the assumption should be fulfilled quite well. For steady-state problems and high opacities (and the assumption of zero natural broadening), Hearn (1964) showed that in a slab, the assumption of CFR gives less than 10% error in the excited-state distribution (for cylinders and spheres, see (Holt 1976)). Van Trigt (1976) showed that in the limit of infinite opacity, the trapping factors for the pure Doppler line with PFR are the same as for CFR, only the lineshape changes. A similar result was also obtained by Frisch (1980a), who considered the asymptotic behaviour of the trapping. Molisch *et al.* (1992a) analysed the lowest-order trapping factor in a plane-parallel slab by a Monte Carlo simulation. They found that for $k_0L < 20$ (and an input frequency $\nu_{\text{in}}$ so that $k(\nu_{\text{in}})L > 1$), there is a slight increase in the trapping factor $g_0$ (as predicted by Holstein), but the difference to the CFR case is smaller than 10%. --- THE PHYSICAL PICTURE OF PFR [FIGURE: Normalized redistribution function of photons] FIG. 11.2. $R_i(x, x_0)/\Phi(x_0)$, the normalized redistribution function of photons that are scattered $i$ times. Initial frequency $x_0 = 1$ (left) and $x_0 = 2$ (right). From (Ivanov and Schneeweis 1976). [FIGURE: Relative error in g_0 vs Opacity k_0L] FIG. 11.3. Relative difference $(g_{0,\text{CFR}} - g_{0,\text{exact}})/g_{0,\text{exact}}$ between the trapping factors computed with the exact redistribution function for a pure Doppler line with the assumption of CFR. Note that Molisch *et al.* (1995a) give the wrong sign of the error. Recently, Molisch *et al.* (1995a) have shown that the error in the lowest-order trapping factor in a slab is always smaller than 10%, see Fig. 11.3. Note, however, that in the high-opacity case, the assumption of zero-natural broadening is usually not justified (see below). Summarizing, it is common to assume that the Doppler effect leads to CFR, and the --- [FIGURE: FIG. 11.4. Redistribution function for Doppler plus natural broadening with Voigt parameter a = 0.05, compared to the JW approximation (dashed).] error in $g_0$ due to this assumption is smaller than 10%. If, however, the initial distribution is mainly at large velocities (e.g. when the vapour was excited by a laser that is tuned to the wings of the line), then at early times the decay will be considerably faster than for CFR. Note that in laboratory situations of high opacity, the excitation is nearly always on the line wing, since only this light penetrates deep inside the vapour. ### 11.1.5 Doppler plus natural broadening The Doppler width is usually much larger than the natural width. However, the Doppler lineshape decreases very fast with frequency, like $\exp(-x^2)$, while natural broadening decreases only slowly, like $1/(1+x^2)$. Hence the far wings of the line are almost entirely due to natural broadening. When the opacity is so high that photons can only easily escape in the far wings, i.e. when the 'escape frequency' $\nu_{\text{esc}}$, the frequency where $k(\nu_{\text{esc}})L = 1$, is far from line centre, we have to include natural broadening in the above considerations. The redistribution function becomes $$ R_{II}(x, x') = \frac{1}{\pi\sqrt{\pi}} \int_{|x-x'|/2}^{\infty} \exp(-u^2) [\mathrm{atan}(\mathrm{v}_{\min}) - \mathrm{atan}(\mathrm{v}_{\max})] \, \mathrm{d}u, $$ $$ \mathrm{v}_{\min} = [u + \min (|x|, |x'|)] / a \qquad (11.23) $$ $$ \mathrm{v}_{\max} = [\max (|x|, |x'|) - u] / a $$ Equation (11.23) describes the laboratory-frame redistribution when we have Lorentzian absorption and complete coherence at reemission in the atomic rest frame. Figure 11.4 shows the redistribution function $R_{II}(x, x')/\Phi(x')$ for a line with Voigt parameter $a = 0.05$. Figure 11.4 can be explained in the following way. We have seen before that for $a = 0$, photons can be absorbed only by atoms with the correct velocity; the reemis- --- sion is isotropic and leads to redistribution as described above. Since we now have also natural broadening, also photons with the 'wrong' velocity can absorb photons. This is especially true in the line wings. There, we have very few atoms that have the correct velocity to absorb a photon at the line centre of the natural width. Rather, the predominant absorption will be in the 'natural' wings of atoms with a small velocity. This 'natural' frequency is more or less retained when the photon is reemitted. The probability distribution of $x$ in the far wings is thus roughly of the shape $R_I(x, x')/\Phi(x')$, centred at $x'$, see the trace for $x' = 4$ in Fig. 11.4. We can also state that a little differently: in the line centre (which is determined by Doppler broadening), redistribution due to the Doppler effect is quite strong. In the wings, however, the difference in frequency between the absorbed and reemitted frequency can be no more than about one Doppler width, so that we have partial redistribution. If the frequency $x_{\mathrm{esc}}$, at which photons can escape easily, is $m$ Doppler widths from the line centre, it takes a photon at *least* $m$ absorption–reemission processes to get to this frequency. This is in marked contrast to CFR, where a photon can get to this frequency within *one* absorption–reemission process. It is quite difficult to evaluate a generalized Holstein equation that includes the redistribution function $R_{II}$. Jefferies and White (1960) thus developed an approximation for $R_{II}$ that is in widespread use, the so-called Jefferies–White approximation, or JW-approximation (see also (Jefferies 1968)). They divided $R(x, x')$ into a part that is completely coherent and a part that is completely redistributed $$ R(x, x') = \left\{ \mathrm{FC}(x')\delta(x - x') + \left[ 1 - \mathrm{FC}(x') \right] \Phi^{\mathrm{D}}(x) \right\} \Phi(x') $$ (11.24) The fraction $\mathrm{FC}(x')$ of *de facto* coherently scattered photons can be deduced from Eq. (11.23). It is very small for $x' < x_V$, then rises steeply, and is close to 1 for $x' > x_V$. The frequency $x_V$ is given by the condition $$ \exp \left(-x_V^2\right) = \frac{a}{x_V^2 \sqrt{\pi}} $$ (11.25) For small Voigt parameters $a$ (as assumed, only due to natural broadening), this is given to a good approximation by (Post 1986) $$ x_V^2 \approx \ln \left[ \frac{\sqrt{\pi}}{a} \ln \left( \frac{\sqrt{\pi}}{a} \ln \frac{\sqrt{\pi}}{a} \right) \right] $$ (11.26) The function FC can be approximated by (Post 1986) $$ \mathrm{FC}(x) = 1 - \frac{\Phi^{\mathrm{D}}(x)}{\Phi^{\mathrm{V}}(x)} $$ (11.27) where $\Phi^{\mathrm{D}}(x)$ is a pure Doppler lineshape and $\Phi^{\mathrm{V}}(x)$ is the considered Voigt lineshape. However, it is common to approximate FC by a step function with the step at $x = x_V$. --- Physically, this means that the line core, which is determined by the Doppler shape, is modelled with complete redistribution, while the natural wings are modelled with complete frequency coherence. This approximation ignores the fact that photons in the wings can change their frequency by about one Doppler width within one absorption/reemission process. The mean frequency shift is about $0.7 \cdot \Delta \nu^{\mathrm{D}}$. The repeated shifts constitute a kind of 'random walk' or 'diffusion' in frequency space. Photons can thus reach the 'escape frequency' $x_{\mathrm{esc}}$ in the natural wings, but this effect is not included in the JW-approximation. Hence, the JW-approximation will tend to overestimate the trapping factor. An alternative definition would replace the term $\delta(x - x')$ by $\Phi^{\mathrm{D}}(x - x')$ so that $$R(x, x') = \left\{ \mathrm{FC}(x')\Phi^{\mathrm{D}}(x - x') + \left[ 1 - \mathrm{FC}(x') \right] \Phi^{\mathrm{D}}(x) \right\} \Phi^{\mathrm{V}}(x') \qquad (11.28)$$ We will see below that this approximation gives too low values for $g$. This is due to the fact that the redistribution function is skewed towards the line centre (Frisch 1985), as can be seen from Fig. 11.4. Kneer (1975) suggested a variation of the original JW-approximation (i.e. with complete coherence in the wings) that is symmetric with respect to $(x, x')$, claiming good agreement with exact computations, however, the frequency 'diffusion' in the wings is still not fulfilled. Osterbrock (1962) suggested that for 'wing' photons (which he defined somewhat arbitrarily as photons with frequency offset $|x| > 3.5$), one can approximate the redistribution function by a discrete diffusion process, where the frequency is shifted by $1/\sqrt{2}$ at each scattering. The 'skewedness' of the redistribution function could then be accounted for by setting the probability of scattering towards the line centre to $0.5 \cdot (1 + \sqrt{2}/|x'|)$, and to $0.5 \cdot (1 - \sqrt{2}/|x'|)$ for moving away from the line centre. Other, more complicated JW-like approximations were proposed by Gayley (1992a, b, 1993). They included also Doppler diffusion in an approximate way, but actual computations are so complicated that little seems to be gained compared to the actual redistribution function. Figure 11.5 shows lowest-order trapping factors in a slab for Voigt parameters $a = 0.01$, $0.05$, and $0.3$ as a function of the opacity $k_0 L$. The trapping factors $g_0$ are computed with the exact redistribution function, with the Jefferies–White (JW) approximation, with the JW-approximation with the $\Phi^{\mathrm{D}}$ term, and with the CFR approximation. We see that for opacities $k_0 L > 100$, the difference between the approximations and the exact solution can become appreciable. At $k_0 L = 1000$, the JW-approximation gives results that can be wrong by an order of magnitude. We also see that the JW-approximation gives trapping factors $g_0$ that are higher than for a pure Doppler line (which completely ignores natural broadening). This can be explained as follows. In the JW-approximation, photons can get into the wings only by Doppler redistribution, $\Phi^{\mathrm{D}}(x)$, centred at frequency zero. If photons get far out into the wings, they can be reabsorbed by the 'natural wings'. Subsequently, they stay at that frequency. They thus have the same chance of getting into the wings as photons in a pure Doppler line, but have a higher chance of being reabsorbed once they are in the wings. --- THE PHYSICAL PICTURE OF PFR [FIGURE: Lowest order trapping factor vs Opacity for three Voigt parameters] FIG. 11.5. Lowest order trapping factor $g_0$ in a slab for Doppler plus natural broadening, for three Voigt parameters. Even with rather small Voigt parameters, we can get considerable errors with the JW-approximation at large opacities. We thus conclude that using the JW-approximation is admissible only for opacities $k_0L < 100\text{--}500$. At higher opacities, we have to use the exact redistribution function, or a better approximation scheme must be developed. Anderson *et al.* (1995) have recently shown with Monte Carlo simulations that for the Xe resonance line, the JW-approximation gives wrong trapping factors. Vermeersch and Wieme (1991) also discuss the inadequacy of the JW-approximation and give pres- --- sure ranges of the noble gases where partial frequency redistribution must be taken into account. In conclusion, partial redistribution must be taken into account in cases where natural broadening is important—we will give quantitative criteria below. The JW-approximation is only applicable for low Voigt parameters and for opacities that are not too high, otherwise, the exact redistribution function must be used. ### 11.1.6 Doppler plus collisional broadening When an atom collides during its natural lifetime between absorption and reemission with another atom, its energy is changed (randomized) and the coherence between absorption and emission frequency is destroyed. In its rest frame, the reemission frequency becomes completely independent of the absorption frequency. (The collision may be with an atom of the same or of a different kind.) However, randomization of energy in the atomic rest frame does not imply CFR, which is defined in the laboratory rest frame. Hummer (1962) derived the redistribution function $$ \begin{aligned} R_{III}(x, x') = \frac{1}{\pi^{5/2}} \int_0^\infty \mathrm{e}^{-u^2} & \left[ \mathrm{atan}\left(\frac{x+u}{a}\right) - \mathrm{atan}\left(\frac{x-u}{a}\right) \right] \\ & \cdot \left[ \mathrm{atan}\left(\frac{x'+u}{a}\right) - \mathrm{atan}\left(\frac{x'-u}{a}\right) \right] \mathrm{d}u \end{aligned} $$ (11.29) In the far wings, $R_{III}(x, x')$ asymptotically tends to complete redistribution. As a matter of fact, it is quite customary to assume that collisions lead to randomization in the laboratory rest frame and thus to CFR, even though the above equation shows that this is not strictly valid (Vardavas 1976b). Finn (1967) claims that the error in the source functions for steady-state may become larger than 20% in plane-parallel geometries; for spherical geometries, Singh (1994) showed that the error is much smaller. For the limit of large opacity, Frisch (1980a) showed that the behaviour is the same as for CFR. These considerations are valid when the excitation stays with the same atom and the atom does not change its velocity between absorption and reemission. There are also collisions where the excited atom transfers the excitation to a passing ground state atom of the same kind or where the atom changes its direction of flight—not just a fly-by through the other atom's electrostatic potential, but a closer hit. In that case, the assumption that collisions lead to energy randomization in the laboratory rest frame is strictly valid. Quite generally, if the velocity vector of the excited atom changes during the natural lifetime, **for coherence in the rest frame of the atoms**, i.e. for zero natural linewidth (Rees and Reichel 1968) $$ R(x, x') = \Phi^{\mathrm{D}}(x') f_1 \left( \frac{x_{\max} - x_{\min}}{2} \right), \quad x_{\max} = \max(x, x') \quad x_{\min} = \min(x, x') $$ (11.30) --- **for natural broadening**, then $$ R(x, x') = \Phi(x') \exp(-u^2) \, a \, f1\left(\frac{a}{2}(x_{\max} - x_{\min})\right) $$ $$ f1(u) = \frac{\exp(-u^2)}{\sqrt{\pi}} - u \mathrm{erfc}(u) $$ (11.31) ### 11.1.7 *Doppler, natural, plus collisional broadening* For Doppler, natural, plus collisional broadening, the redistribution function becomes (Payne *et al.* 1974), (Cooper *et al.* 1982) $$ R(x, x') = P_{\text{coll}} R_{III}(x, x') + (1 - P_{\text{coll}}) R_{II}(x, x') $$ (11.32) $R_{III}$ is approximated by complete redistribution, $$ R_{III}(x, x') = \Phi(x) \cdot \Phi(x') $$ (11.33) For both natural broadening and collisions, Zanstra (1941) showed that the ratio of the number of randomized over coherent photons is $$ P_{\text{coll}} = \frac{\Delta \nu^{\text{coll}}}{\Delta \nu^{\text{coll}} + \Delta \nu^n} $$ (11.34) When there are also inelastic collisions, leading to a sufficient frequency shift $\Delta \nu^{\text{I}}$ to broaden the line, the fraction of redistributed photons is $P_{\text{coll}} = \Delta \nu^{\text{coll}} / (\Delta \nu^{\text{coll}} + \Delta \nu^n + \Delta \nu^{\text{I}})$, because only elastic collisions lead to a 'reshuffling' of the frequency (Mihalas 1978, p. 415). Lombardi *et al.* (1985) also discuss these aspects for subordinate lines. It is possible to further simplify Eq. (11.32) if we use the JW-approximation for $R_{II}$. Let us now again check where the JW-approximation will introduce errors. At very low opacities, the trapping factor $g_0$ is solely determined by the (Doppler) core, so that redistribution in the wings plays no role; the error will thus be negligible. The natural wings only come into play when $x_{\text{esc}} > x_V$. In that case, the JW-approximation introduces an error, as discussed above. At very high densities, self-broadening of the line will set in, so that $P_{\text{coll}}$ will increase, and we finally get complete redistribution. The range of the intermediate opacity region depends both on the atomic density and on the size of the experimental vessel. The onset of self-broadening is determined solely by the atomic density, while the escape frequency $x_{\text{esc}}$ is determined by the opacity (i.e. atomic density times vessel dimensions). Post (1986) states that the error is largest when the trapping factor due to the photons emitted after decorrelating collisions is about equal to the trapping factor due to the Doppler redistribution—errors can become 30% and more. This discussion is of course only valid when we have only one atomic species in the vapour cell. If we add a sufficient amount of some buffer gas (e.g. a noble gas), we can have collisional redistribution at all opacities, so that CFR is always valid. --- Payne *et al.* also gave criteria for the validity of CFR $$ x_{\text{esc}}^{\text{D}} < x_V - 0.5 \quad \text{or} \quad P_c > 0.7, \quad \text{where} \quad x_{\text{esc}}^{\text{D}} = \sqrt{\ln(k_0/2)} $$ (11.35) These conditions are easily interpreted. CFR is valid when the lineshape is determined by Doppler and not by natural broadening at the frequency where the opacity has become equal to one, $k(x_{\text{esc}})L = 1$. Note that for CFR to be valid, only *one* of the conditions of Eq. (11.35) has to be fulfilled (see also Romberg and Kunze (1988) and the response by Huennekens and Colbert (1989)). The redistribution function for collisional broadening far from the line centre is discussed by Voslamber and Yelnik (1978), Nienhuis and Schuller (1977), and Cooper (1979). Numerical results for the redistribution in the case of impact broadening (i.e. close to the line centre) are given by Carrington *et al.* (1973). The redistribution for coupled hydrogen lines ($\text{Ly}\alpha$, $\text{Ly}\beta$, $\text{H}\alpha$) is discussed by Cooper *et al.* (1989). ### 11.1.8 Branching transitions Yet another redistribution function is valid for branching transitions. Let us again consider the three-level system depicted in Sec. 10.2. At first glance, it is astonishing that even when an $ac$ photon is absorbed, the frequency of a subsequently reemitted $bc$ photon shows some frequency correlation to the absorbed $ac$ frequency. For an explanation, let us assume that the $ac$ photon was very far in the wings of the line. This means that with a large probability, the absorbing atom has a large velocity component in the direction of the incident photon. It is thus highly probable that the reemission will also be in the wings of the $bc$ line. Essentially, this is just the Doppler redistribution effect encountered in redistribution cases (ii) and (iii). Complications arise, however, from the fact that now we have different Doppler widths in the two transitions. The actual redistribution function was derived by Milkey *et al.* (1975a) $$ R_{VI}(\nu, \nu') = \frac{\pi^{-3/2}}{\Delta\nu_{\text{D}}^{ac} \Delta\nu_{\text{D}}^{bc}} \int_{u_{\text{min}}}^{\infty --- ### 11.1.9 The Holstein equation with PFR We can now summarize the **recommended approximations for redistribution**: | | for the broadening mechanism | we assume as redistribution | | :--- | :--- | :--- | | **(i)** | pure natural broadening | complete frequency coherence | | **(ii)** | pure Doppler broadening | complete frequency redistribution, CFR | | **(iii)** | Doppler plus natural broadening | for a very small Voigt parameter, we assume CFR in the core and complete coherence in the wings of the line; for a large Voigt parameter (larger than 0.1) and high opacity (larger than 300), the exact redistribution function must be used. | | **(iv)** | collisional broadening larger than natural broadening | complete frequency redistribution, CFR | | **(v)** | collisional broadening smaller than natural broadening | the redistribution function given by Eq. (11.32) | Algorithms for the efficient evaluation of the redistribution functions were proposed by Adams *et al.* (1971) and Reichel and Vardavas (1975). McKenna (1980) criticized the angle-dependent redistribution functions in (Reichel and Vardavas 1975) and suggested alternative equations. A review of computation methods for the redistribution functions themselves is given in Heinzel (1985). When we have finally found the appropriate redistribution function, we can derive the generalized Holstein equation (Payne *et al.* 1974). We introduce the frequency-dependent excited-state density $n(\mathbf{r}, t, x)$, i.e. the density of excited-state atoms that will emit a photon with frequency $x$. The Holstein equation for these atoms reads $$ \frac{\partial n(\mathbf{r}, x, t)}{\partial t} = -\frac{1}{\tau}n(\mathbf{r}, x, t) + \frac{1}{\tau C_x} \int_{-\infty}^{\infty} \int_V R(x, x') \frac{\exp[-k(x')|\mathbf{r} - \mathbf{r}'|]}{4\pi|\mathbf{r} - \mathbf{r}'|^2} n(\mathbf{r}', x', t) d\mathbf{r}' dx' \quad (11.37) $$ This equation can be solved by various methods. One possible approach was developed by Payne *et al.* (1974), and is especially suited for the early-time behaviour of the vapour. Further possibilities to solve Eq. (11.37), which will be described in the subsequent sections, are the variational approach, the propagator function method, the eigenvalue method, and the Monte Carlo simulation. Equivalently, we can also formulate and solve the transfer equation including partial frequency redistribution, see Sec. 11.6. There has been some controversy over whether Eq. (11.32) in conjunction with Eq. (11.37) is a correct formulation. Although it is in widespread use in the literature, it is not strictly correct when natural broadening plays an important role. Streater *et al.* (1988b) pointed out that the --- coherent scattering process is essentially instantaneous and thus Eq. (11.32) is incorrect. On the other hand, one must not think that 'instantaneous scattering' really applies in a semiclassical sense to photons that completely retain their frequencies--such a picture is forbidden by the uncertainty relation. A strict formulation (for the case of no collisional processes) is given by the equation $$ \begin{aligned} n(\mathbf{r}, x, t) &= n(\mathbf{r}, x, 0) \exp(-A_{21}t) + \int_0^t E(\mathbf{r}, x, t') \exp[-A_{21}(t - t')] \, \mathrm{d}t' + \\ &\quad \frac{1}{C_x} \int_0^t \int_{-\infty}^\infty \int_V R(x, x', t - t') \frac{\exp[-k(x')|\mathbf{r} - \mathbf{r}'|]}{4\pi|\mathbf{r} - \mathbf{r}'|^2} n(\mathbf{r}', x', t) \mathrm{d}\mathbf{r}' \mathrm{d}x' \mathrm{d}t', \end{aligned} $$ where $$ \begin{aligned} R(x, x', t - t') &= A_{21} \left\{ \exp[-A_{21}(t - t')] \Phi( --- In the general case of PFR, it is often convenient to consider only ‘the absolutely lowest-order’ mode in the space–frequency domain. Our discussions show that such a mode will exist if there is a reasonable amount of redistribution. This ‘reasonable amount’ was found by Post (1986) to be the condition $$ P_{\text{coll}} + (1 - P_{\text{coll}})\eta (k(x_V)L) \gg \frac{1}{g_0} $$ (11.40) In the case of complete coherence, it is not meaningful to speak of a lowest-order mode. ## 11.2 Variational solution In his landmark paper, Post (1986) gave a variational solution of the generalized Holstein equation, including partial redistribution. Following his derivation, we will first describe the general procedure and then specialize to the plane-parallel slab and to the infinite cylinder. We start out by defining the **local de-excitation rate** $\gamma(\mathbf{r}, t)$ $$ \gamma(\mathbf{r}, t) = -\frac{1}{n(\mathbf{r}, t)} \frac{\partial}{\partial t}n(\mathbf{r}, t), $$ where $$ n(\mathbf{r}, t) = \int_{-\infty}^{\infty} n(\mathbf{r}, x, t)\mathrm{d}x $$ (11.41) Throughout the derivation it is assumed that the opacity, and thus the trapping factor, is large. This makes sense, since the effects of partial frequency redistribution can only be seen at high opacities, anyway. We furthermore define a **normalized emission line profile**, $\hat{\Psi}(\mathbf{r}, x, t)$ $$ \hat{\Psi}(\mathbf{r}, x, t) = \frac{n(\mathbf{r}, x, t)}{n(\mathbf{r}, t)} $$ (11.42) and the **escape function** $\eta(\mathbf{r}, x, t)$ $$ \eta(\mathbf{r}, x, t) = 1 - \int_V k(x)\frac{\exp(-k(x)|\mathbf{r} - \mathbf{r}'|)}{4\pi|\mathbf{r} - \mathbf{r}'|^2} \frac{n(\mathbf{r}', x, t)}{n(\mathbf{r}, x, t)}\mathrm{d}\mathbf{r}' $$ (11.43) The function $\eta$ is the net escape factor of photons with frequency $x$ from the volume element $\mathrm{d}\mathbf{r}$ at position $\mathbf{r}$. We then integrate Eq. (11.37) over frequency, and insert the above definitions to get for the de-excitation rate $$ \gamma(\mathbf{r}, t) = \frac{1}{\tau} \int_{-\infty}^{\infty} \hat{\Psi}(\mathbf{r}, x, t)\eta(\mathbf{r}, x, t)\mathrm{d}x $$ (11.44) The redistribution function $R(x, x')$ consists of one part that is due to the collisional redistribution, and one part that is redistributed according to the rules for Doppler plus --- r}' - \mathbf{r}|\right]}{4\pi|\mathbf{r}' - \mathbf{r}|^2} \text{d}\mathbf{r}'\text{d}x' $$ $$ f3 = P_{\text{coll}} \iint_V k(x')\Phi^{\text{V}}(x) \frac{n(\mathbf{r}', x', t)}{n(\mathbf{r}, t)} \frac{\exp \left[-k(x')|\mathbf{r}' - \mathbf{r}|\right]}{4\pi|\mathbf{r}' - \mathbf{r}|^2} \text{d}\mathbf{r}'\text{d}x' \qquad (11.46) $$ Wait, in $f1$, $f2$, $f3$, the integral is `\iint_V`. The `V` is a subscript on the double integral. Also, the `d` in `dr'dx'` is upright in the image. I will use `\text{d}`. Let's check the spacing. `\text{d}\mathbf{r}'\text{d}x'` is fine. Eq 11.47: $$ f1 = (1 - P_{\text{coll}})\text{FC}(x) (1 - \eta(\mathbf{r}, x, t)) \hat{\Psi}(\mathbf{r}, x, t) $$ $$ f2 = (1 - P_{\ --- VARIATIONAL SOLUTION $$ \frac{\partial}{\partial t} \hat{\Psi}(\mathbf{r}, x, t) = -\alpha(\mathbf{r}, x, t) \hat{\Psi}(\mathbf{r}, x, t) + \frac{1}{\tau} (1 - P_{\text{coll}}) \Phi^D(x) [1 - \tau \gamma(\mathbf{r}, t) - \varepsilon(\mathbf{r}, t)] + \\ + \frac{1}{\tau} P_{\text{coll}} \Phi^V(x) [1 - \tau \gamma(\mathbf{r}, t)] $$ (11.49) where $\Phi^D(x)$ is the normalized Doppler lineshape. This constitutes a differential equation for the normalized emission line profile. We have introduced the spectral relaxation rate $\alpha(\mathbf{r}, x, t)$ $$ \alpha(\mathbf{r}, x, t) = \frac{1}{\tau} \left[ 1 - \mathrm{FC}(x) (1 - P_{\text{coll}}) (1 - \eta(\mathbf{r}, x, t)) - \tau \gamma(\mathbf{r}, t) \right] $$ (11.50) and a correction factor $$ \varepsilon(\mathbf{r}, t) = \int_{-\infty}^{\infty} \hat{\Psi}(\mathbf{r}, x, t) (1 - \eta(\mathbf{r}, x, t)) \mathrm{FC}(x) \mathrm{d}x $$ (11.51) We can conclude from this equation that the correction factor is quite small, $\varepsilon \ll 1$. At line centre, the fraction of redistributed photons, $\mathrm{FC}(x)$, is very small, whereas in the line wings, escape is easy and the escape function $\eta(\mathbf{r}, x, t)$ is close to one, so that $(1 - \eta(\mathbf{r}, x, t))$ is very small. It can be shown that for opacities $k(x)L > 1$, the correction factor $\varepsilon$ is on the order of $a/(\pi x_V)$. Furthermore, at high opacities, also the term $\tau \cdot \gamma(\mathbf{r}, t)$ in Eq. (11.50) is small. So these contributions can be neglected, and Eq. (11.49) becomes $$ \frac{\partial}{\partial t} \hat{\Psi}(\mathbf{r}, x, t) = -\alpha(\mathbf{r}, x, t) \hat{\Psi}(\mathbf{r}, x, t) + \frac{1}{\tau} (1 - P_{\text{coll}}) \Phi^D(x) + \frac{1}{\tau} P_{\text{coll}} \Phi^V(x) $$ (11.52) Equation (11.52) is now the simplified differential equation for $\hat{\Psi}(\mathbf{r}, x, t)$. The second and third terms are the source terms; due to the above approximations, they are independent of time. We see in Eq. (11.50) that both in the line core, where we have complete Doppler redistribution, and in the far wings, where the photons can escape in one step, the spectral relaxation rate $\alpha(\mathbf{r}, x, t)$ equals $1/\tau$. The relaxation rate becomes smallest in the range $x_c < x < x_{\text{esc}}$. Post also states that in his computations, there is a separate spatial mode for each frequency in the wings. This is due to the JW approximation, which assumes complete frequency coherence in the wings. According to our discussion in Sec. 11.1, the eigenmodes in the space–frequency domain are then $\psi'(\mathbf{r})\delta(x - x_i)$. We will see later that this situation can be described by a diffusion equation (Milne theory). Post then proceeds to assume that there is enough redistribution to justify the definition of a ‘lowest-order mode’, and furthermore that the spectral relaxation rate $\alpha(\mathbf{r}, x, t)$ is much larger than the relaxation rate of the atoms $\gamma(\mathbf{r}, t)$. This means that the spectrum can adapt itself almost instantaneously to the changes in the spatial profile. The emission spectrum --- $\hat{\Psi}(\mathbf{r}, x, t)$ thus becomes time independent. Neglecting the term $\gamma(\mathbf{r}, t) \cdot \tau$ in the equation for $\alpha(\mathbf{r}, x, t)$, we get an approximate equation for the spectrum $\hat{\Psi}(\mathbf{r}, x)$ $$ \hat{\Psi}(\mathbf{r}, x) = \frac{(1 - P_{\mathrm{coll}})\Phi^{\mathrm{D}}(x) + P_{\mathrm{coll}}\Phi^{\mathrm{V}}(x)}{1 - (1 - P_{\mathrm{coll}})\mathrm{FC}(x)(1 - \eta(\mathbf{r}, x))} $$ (11.53) We see that in the limit of complete frequency redistribution, the spectrum becomes $\Phi^{\mathrm{V}}(x)$, as must be the case—admittedly a rather complicated way to compute the CFR case. For a hyperfine structure with complete intermixing of the hfs components (see Sec. 7.3 for a discussion of this assumption), $\hat{\Psi}$ is given by $$ \hat{\Psi}(\mathbf{r}, x) = \frac{\sum_i (1 - P_{\mathrm{coll}_i})w_i \Phi^{\mathrm{D}}(x - \delta_i) + \sum_i P_{\mathrm{coll}_i} w_i \Phi^{\mathrm{V}}(x - \delta_i)}{1 - \sum_i (1 - P_{\mathrm{coll}_i})\mathrm{FC}_i(x) (1 - \eta(\mathbf{r}, x)) \frac{k(x - \delta_i)}{\sum_i k(x - \delta_i)}} $$ (11.54) We have thus computed the emission profile $\hat{\Psi}(\mathbf{r}, x)$. In the next step, we need the escape factor $\eta(\mathbf{r}, x, t)$. With that, we can compute the decay factor from $$ \gamma(\mathbf{r}) = \int \hat{\Psi}(\mathbf{r}, x)\eta(\mathbf{r}, x)\mathrm{d}x $$ (11.55) In the following insert, the above theory is applied to two special cases, the infinite cylinder and the plane-parallel slab. --- **Insert 11.1: Escape factors for PFR in the infinite cylinder and in the slab geometries.** **For the cylinder**, we must find an approximation for the (frequency dependent) spatial excitation profile $\hat{n}(\mathbf{r}, x) = n(\mathbf{r}, x)/n(0, x)$. We have already seen in Chapter 7 that in the case of CFR, the shapes of the lowest order modes for Doppler and for Lorentz broadening are very similar. Both are proportional to $J_0^B(\zeta_0 \cdot r/R)$, where $\zeta_0$ depends on the opacity but is always somewhat smaller than 2.405 (first null of the Bessel function). In the core of the line, we have Doppler redistribution, so that the shape will be well described by the CFR Doppler shape. In the wings, the shape will be somewhat broader, but not by much. For complete coherence in the wings, the spatial shape must be the solution to a kind of diffusion equation at the considered frequency $x$ (see also Chapter 8, the Milne equation). However, the highest frequency we have to consider is the frequency where the opacity becomes equal to one, $k(x)R = 1$. At that frequency, the solution to the Milne equation (or even to the CFR Holstein equation for that opacity) is not much broader than the lowest-order Doppler mode at high opacities ($\zeta_0$ at $k_0 R = 1$ is 1.7, compared to 2.2 at high opacities). So we have good reason to make the approximation of a frequency-independent excitation profile, $\hat{n}(\mathbf{r}, x) \approx \hat{n}(\mathbf{r})$. --- In the next step, we compute the escape function $\eta$ on the axis of the cylinder for the given spatial profile $\hat{n}(\mathbf{r})$, $$ \eta(x, 0) = 1 - \int_V k(x) \frac{\exp(-k(x)|\mathbf{r}'|)}{4\pi |\mathbf{r}'|^2} \hat{n}(\mathbf{r}') \mathrm{d}\mathbf{r}' \qquad (11.56) $$ In a spherical coordinate system, this becomes $$ \eta(x, 0) = 1 - \int_0^{\pi/2} \sin \vartheta \int_0^{1/\sin \vartheta} k(x) \exp \left(-k(x)r'\right) \hat{n} \left(r' \sin(\vartheta)\right) \mathrm{d}r' \mathrm{d}\vartheta \qquad (11.57) $$ We now expand $\hat{n}(\mathbf{r})$ into a power series, $$ \hat{n}(r) = \sum_{n=\text{even}} a_n (1 - r^n) \qquad (11.58) $$ where the coefficients for the different profiles are: | | $a_2$ | $a_4$ | $a_6$ | $a_8$ | | :--- | :--- | :--- | :--- | :--- | | Lorentz profile, $\hat{n}^L(r)$ | 0.8983 | 0.3028 | $-1.6008$ | 1.3997 | | Doppler profile, $\hat{n}^D(r)$ | 1.0712 | $-0.0109$ | $-0.7878$ | 0.7275 | | Diffusion profile, $J_0^B(r)$ | 1.4450 | $-0.5212$ | 0.0799 | $-0.0037$ | and the escape function $\eta(x, 0)$ becomes $$ \begin{aligned} \eta(x, 0) = \sum_{n=\text{even}} a_n \eta_n(x), \qquad \eta_n(x) &= \int_0^{\pi/2} \Upsilon_n \left( \frac{k(x)R}{\sin \vartheta} \right) \sin \vartheta \, \mathrm{d}\vartheta \\ \Upsilon_n(u) &= \frac{n!}{u^n} - \sum_{k=1}^n \frac{n!}{(n - k)!} \frac{\exp(-u)}{u^k} \end{aligned} \qquad (11.59) $$ For small opacity $k(x)R$, it is advantageous to compute $\Upsilon_n(u)$ as $$ \Upsilon_n(u) = \sum_{i=0}^\infty (-1)^i u^i \sum_{k=0}^{n-1} \frac{(-1)^k n!}{(n - k - 1)! (i + k + 1)!} \qquad (11.60) $$ For large opacity $k(x)R$, the escape function can be computed as $$ \eta(x, 0) = \frac{4}{3} \frac{a_2}{k^2(x)R^2} \qquad (11.61) $$ With these expressions, the decay factor $\gamma$ can be computed from Eq. (11.53). **For the plane-parallel slab** and a Lorentzian lineshape for impact broadening, the escape function is (Colbert and Huennekens 1990) $$ \eta(x, 0) = \frac{1}{1 + 1.98 \left(k(x)L/2\right)^2} \left[ 1 - \frac{0.331 k(x)L/2}{0.185 + \left(k(x)L/2\right)^2} + \frac{5.85 \left(k(x)L/2\right)^2}{\left(9.95 + \left(k(x)L/2\right)^2\right)^2} \right] \qquad (11.62) $$ This allows a closed-form evaluation of the trapping factors. --- [FIGURE: FIG. 11.6. Lowest-order trapping factor of the Hg 185 nm line as a function of Hg density in a cylindrical cell, $R = 12.5$ mm. From Post (1986).] An alternative lies in using the escape factors of Sec. 4.5, Insert 4.3, with $m = (\kappa - 1)/2\kappa$. It is trivial to get the frequency dependent escape factors from Eqs. (4.78) and (4.79). --- After all these mathematical formalisms, we now consider the physical results of Post's computations. He applied his theory to the computation of trapping in a mercury vapour for atomic densities of $10^{13} < n < 10^{16} \text{ cm}^{-3}$. The trapping factor is shown in Fig. 11.6. It goes through a pronounced maximum and tends to a constant value at high densities. This can be explained by the following physical picture. At rather low densities, the wings are due to natural broadening, so that they display complete frequency coherence. Photons thus have practically no chance to get into the wings. In a very rough approximation the line behaves like a pure Doppler line and the trapping factor increases linearly with the opacity. At higher densities, collisions (and, therefore, the fraction of randomized photons, $P_{\text{coll}}$) increase so that photons now have a chance to get into the wings—the line behaves like a genuine Voigt line. We have already seen in Chapter 7 that the trapping factor for a Voigt line is much lower than for a Doppler line. At some intermediate density, there must thus be a maximum of the trapping factor. The constant value of the trapping factor $g$ at high densities can be explained from the Ladenburg relation. At high densities, the width of the pressure-broadened absorption coefficient increases linearly --- VARIATIONAL SOLUTION [FIGURE: FIG. 11.7. Effective radiative lifetime, $g_0\tau$, of the Hg $6^3\text{P}_1$ level as a function of Hg-density in He-Ar discharges; $R = 7.5\text{ mm}$. From van de Weijer and Cremers (1986).] with the atomic density $N$, as does the integral over the absorption coefficient, so that the centre-of-line absorption coefficient does not change. There are more absorbers, but their absorption is also more smeared out in frequency. The predicted maximum in the trapping factor occurs only under special experimental conditions. Huennekens and Colbert (1989) showed experimentally that the effect does not occur in a sodium vapour for opacities $k_0L < 800$ (in a $0.6\text{ cm}$ thick cell), refuting a claim by Romberg and Kunze (1988) that it actually occurs at $k_0L = 50$ (see also (Huennekens and Gallagher 1983a)). Payne *et al.* (1974) computed the time-decay at early times in argon at high opacities and made corresponding experiments. They theoretically predicted the maximum in the trapping factor, but were unable to observe it experimentally. Of course the presence of this maximum can be destroyed by adding a buffer gas to the vapour. At high buffer gas pressures, the partial frequency coherence is destroyed, and pressure broadening is determined not by the vapour atoms but by the buffer gas, see Fig. 11.7. The Post theory is capable of giving the essential features of trapping in a quite dense atomic vapour. The discovery of a maximum in the trapping factor was quite surprising—even though it was quite simple to give a physical picture for this fact *after* the discovery was made. However, some *caveats* for the use of this theory are necessary. In the derivation of the expressions for the trapping factor, several approximations had to be made. One is the use of the JW approximation for the redistribution function, which can cause an estimated error of up to 30%. Furthermore, it is only valid for a --- high-opacity vapour (above, the factors $\varepsilon$ and $\tau \cdot \gamma$ were neglected). This is no serious restriction, since the effects of PFR will only play a role at very high opacities anyway. An alternative method for these computations would be a Monte Carlo simulation; such a simulation by Vermeersch has shown good agreement with Post's analytical results. More importantly, extensive experiments by Post *et al.* (1986) have proven the trends predicted by the theory, and also the quantitative agreement appears to be quite good (about 20-50%). The Post theory can thus be considered a recommended computational method in case CFR is not valid. Another approximate method was developed by Frisch and Bardos (1981). There, the geometry is divided into an inner region and a boundary layer, and for each region the solution is expanded into a series of a suitable small parameter. An evaluation of trapping with PFR described by $R_{II}$ in an infinite medium is given by Basko (1978). ## 11.3 The velocity distribution of excited atoms In the previous section, we have assumed that Doppler broadening leads to CFR. If we do not wish to use this approximation, there are several possibilities. Naturally, we can make a Monte Carlo simulation—this approach will be described in Sec. 11.5. Another possibility is solving the generalized Holstein equation with the redistribution function for Doppler broadening. A third method is based on an iterative computation of the velocity distribution of the excited-state atoms. This approach, developed by Holt (1976), is a different point of view for the problem of partial frequency redistribution and will allow for some new insights. Up to now, we have always stated that CFR is valid when the absorbed and the reemitted frequencies are completely independent. For Doppler broadening, we can make a different, but equivalent statement. CFR is fulfilled when the velocity distribution of the excited-state atoms is Maxwellian (Bezuglov *et al.* 1977). Correct incorporation of PFR is thus equivalent to computing the actual velocity distribution of the excited-state atoms. Holt did an iterative computation of this velocity distribution. She starts out by defining $n(\mathbf{r}, \mathbf{v}, t)$, the density of excited particles at position $\mathbf{r}$ with a velocity between $\mathbf{v}$ and $\mathbf{v} + \mathrm{d}\mathbf{v}$. The Holstein equation for these particles is $$ \frac{\partial n(\mathbf{r}, t, \mathbf{v})}{\partial t} = -\frac{1}{\tau} n(\mathbf{r}, t, \mathbf{v}) + \frac{1}{\tau} \iint n(\mathbf{r}', t, \mathbf{v}') G(\mathbf{r}', \mathbf{v}', \mathbf{r}, \mathbf{v}) N(\mathbf{v}) \mathrm{d}\mathbf{r}' \mathrm{d}\mathbf{v}' $$ (11.63) The Kernel function $G(\mathbf{r}, \mathbf{r}', \mathbf{v}, \mathbf{v}')$ is the probability that a photon emitted by an excited atom at $\mathbf{r}'$ with velocity vector $\mathbf{v}'$ is absorbed by a ground-state atom at $\mathbf{r}$ with velocity $\mathbf{v}$. The velocity distribution of the ground-state atoms is assumed to be Maxwellian $$ N(\mathbf{v}) = \frac{N}{\pi^{3/2}} \exp \left[ - \left( \frac{|\mathbf{v}|}{v_{\mathrm{D}}} \right)^2 \right] \frac{1}{v_{\mathrm{D}}^3}. $$ (11.64) The density of ground-state atoms with velocity $\mathbf{v}$ is given by $N(\mathbf{v})$, the (constant) total ground-state density by $N$. Atomic velocity $v$ equals $|\mathbf{v}|$. In the following, all velocities are normalized to $v_{\mathrm{D}}$, the most probable atomic velocity, $v_{\mathrm{D}} = (2k_{\mathrm{B}}T/m_{\mathrm{atom}})^{1/2}$. --- THE VELOCITY DISTRIBUTION OF EXCITED ATOMS To start out, we have to find some probabilities: The probability that an atom with velocity vector $\mathbf{v}'$ emits a photon in direction $\rho$ with normalized frequency $x$ is $$ \delta \left[ v'_\rho \frac{\nu_0}{c} - (\nu - \nu_0) \right] $$ (11.65) where $v'_\rho$ is the component of $\mathbf{v}'$ in the direction $\boldsymbol{\rho} = \mathbf{r} - \mathbf{r}'$, and $\rho = |\boldsymbol{\rho}|$. The probability that this photon is absorbed in the volume element $\mathrm{d}\rho$ by an atom with velocity $\mathbf{v}$ is $$ C_{\mathrm{vel}} \mathrm{d}\rho \, \delta \left[ \frac{v_\rho}{c} \nu_0 - (\nu - \nu_0) \right] $$ (11.66) The total probability that a photon of frequency $x$ is absorbed in the volume element $\mathrm{d}\rho$ is $$ k_0 \mathrm{d}\rho \exp \left[ - \left( \frac{\nu - \nu_0}{\nu_0 v_{\mathrm{D}} / c} \right)^2 \right] $$ (11.67) The constant $C_{\mathrm{vel}}$ can be determined by multiplying Eq. (11.66) by $N(\mathbf{v})\mathrm{d}\mathbf{v}$, integrating over velocity space, and equating the result to Eq. (11.67). We get $$ C_{\mathrm{vel}} = 2\pi \sqrt{\pi} \left( \frac{\nu_0}{c} \frac{v_{\mathrm{D}} k_0}{N} \right) $$ (11.68) The probability that a photon transverses a distance $\rho$ is $$ \exp \left[ -k_0 \rho \exp \left( - \left( \frac{\nu - \nu_0}{\nu_0} \frac{c}{v_{\mathrm{D}}} \right)^2 \right) \right] $$ (11.69) We then get for the Kernel function the expression $$ G(\mathbf{r}', \mathbf{v}', \mathbf{r}, \mathbf{v}) \mathrm{d}\mathbf{r} = C_{\mathrm{vel}} \mathrm{d}\rho \exp \left[ -k_0 \rho \exp \left( - \left( \frac{v_\rho}{v_{\mathrm{D}}} \right)^2 \right) \right] \delta(v'_\rho - v_\rho) $$ (11.70) Since $\mathrm{d}\mathbf{r} = 4\pi \rho^2 \mathrm{d}\rho$, the Kernel function is $$ G(\mathbf{r}', \mathbf{v}', \mathbf{r}, \mathbf{v}) = \frac{C_{\mathrm{vel}}}{4\pi \rho^2} \exp \left[ -k_0 \rho \exp \left( - \left( \frac{v_\rho}{v_{\mathrm{D}}} \right)^2 \right) \right] \delta(v'_\rho - v_\rho) $$ (11.71) This Kernel function, in combination with the generalized Holstein equation, gives an exact description under conditions of pure Doppler broadening. Holt then proceeds to compute the decay factor in a sphere for a sharply peaked initial distribution $\delta(r)$ and a Maxwellian initial velocity distribution of the excited-state atoms. She does that by an iterative procedure. The zeroth order approximation for the excited-state distribution is that the excited atoms are all concentrated at the centre, $r = 0$, and have a Maxwellian velocity distribution. This is then inserted into the --- integral of Eq. (11.63), and from this equation, the next approximation for the excited state density $n(\mathbf{r}, t, \mathbf{v})$ is computed. This procedure could be generalized to different initial distributions quite easily, but it is of rather limited value, since it does not give the eigensolutions of the Holstein equation, but only the time-dependent decay factor specifically for the considered initial distribution. We would thus have to make a new computation for each initial distribution. Holt's general formulation is valuable and offers a new interpretation of the redistribution problem. For the actual computation in a practical case, however, Monte Carlo simulations are a more convenient method of obtaining the temporal and spatial distribution of the excited-state atoms. This technique can also be viewed as a special case of the general approach of Borsenberger *et al.* (1985–87), see also Hubeny *et al.* (1983). These authors set up the kinetic equations of the atoms and the photons, with the correct influence of the atom kinetics on the absorption coefficient. They then solve for the excited-state distribution as a function of position and velocity. This approach automatically includes the correct redistribution function, but obtaining self-consistent solutions is quite difficult. ## 11.4 The propagator function method and the PCA method The propagator function method, PFM, is very suitable for the evaluation of the Holstein equation with PFR. We start again with the generalized Holstein equation, which we write down in slightly modified form, $$ \begin{aligned} \frac{\partial n(\mathbf{r}, x, t)}{\partial t} &= E(\mathbf{r}, x, t) - \frac{1}{\tau}n(\mathbf{r}, x, t) + \\ &\quad + \frac{1}{\tau C_x} \int_{-\infty}^{\infty} \int_V R(x, x') \frac{\exp\left[-k(x')|\mathbf{r} - \mathbf{r}'|\right]}{4\pi|\mathbf{r} - \mathbf{r}'|^2} n(\mathbf{r}', x', t) d\mathbf{r}' dx' \end{aligned} $$ (11.72) We define the spatial propagator $$ \mathrm{PS}(x', |\mathbf{r} - \mathbf{r}'|) = \frac{k(x') \exp[-k(x')|\mathbf{r} - \mathbf{r}'|]}{4\pi|\mathbf{r} - \mathbf{r}'|^2}, $$ (11.73) which we can again write as a divergence $$ \mathrm{PS}(x', |\mathbf{r} - \mathbf{r}'|) = -\nabla_{\mathbf{r}-\mathbf{r}'} \left[ \frac{e_{\mathbf{r}-\mathbf{r}'} \exp\left(-k(x')|\mathbf{r} - \mathbf{r}'|\right)}{4\pi|\mathbf{r} - \mathbf{r}'|^2} \right] + \delta^3(\mathbf{r} - \mathbf{r}') $$ (11.74) and the frequency propagator $$ \mathrm{PF}(x, x') = \frac{R(x, x')}{\Phi(x')} $$ (11.75) With these definitions, Eq. (11.72) becomes --- $$ \begin{aligned} \frac{\partial n(\mathbf{r}, x, t)}{\partial t} &= E(\mathbf{r}, x, t) - \frac{1}{\tau}n(\mathbf{r}, x, t) \\ &\quad + \frac{1}{\tau} \int_{-\infty}^{\infty} \mathrm{PF}(x, x') \int_V \mathrm{PS}(x', |\mathbf{r} - \mathbf{r}'|)n(\mathbf{r}', x', t)\mathrm{d}\mathbf{r}'\mathrm{d}x' \end{aligned} \quad (11.76) $$ We discretize this both in space and frequency, using $N_x$ frequency discretization points and $N_r$ space discretization points, $$ \frac{\mathrm{d}n_{k,i}(t)}{\mathrm{d}t} = E_{k,i} - \frac{1}{\tau}n_{k,i}(t) + \frac{1}{\tau} \sum_{l=0}^{N_x-1} \mathrm{PF}_{i,l} \sum_{m=0}^{N_r-1} \mathrm{PS}_{l,k,m} n_{m,l}(t) $$ $$ \mathrm{PF}_{i,l} = \mathrm{PF}(x_i, x'_l), \quad \mathrm{PS}_{l,k,m} = \mathrm{PS}(x_l, |\mathbf{r}_k - \mathbf{r}_m|) \quad (11.77) $$ Equation (11.77) can be solved by the Euler scheme presented in Sec. 5.4. The normalization conditions are $$ \sum_{i=0}^{N_x-1} \mathrm{PF}_{i,l} = 1 \quad \text{and} \quad \sum_{k=0}^{N_r} \mathrm{PS}_{l,k,m} = 1 \quad (11.78) $$ Note that the summation in Eq. (11.78) goes over $N_r + 1$ elements, i.e. we also have to integrate over the exterior of the cell to get the normalization condition. Physically this just means that a photon is either reabsorbed in one of the subcells or it escapes from the vapour. These normalization conditions are always fulfilled when we have a computer with infinite precision—we thus have a useful check on numerical inaccuracies. The spatial mesh points are, as usual, chosen equidistantly. In order to enhance efficiency, the frequency mesh points, however, should be non-equidistant. This is especially true for a high-opacity vapour. The largest influence on the escape will come from photons with a frequency close to $x_{\text{esc}}$. When we use the JW approximation, the discretization near line centre can be quite coarse. When we do not use this approximation, we must take care that the distance between neighbouring frequency points does not become larger than about half a Doppler FWHM, so that we can correctly represent the 'diffusion' in frequency space. We now need expressions for the propagators PF and PS for various geometries. As mentioned in Sec. 5.4, these could for any geometry be computed by a Monte Carlo simulation. For slab, cylinder, and sphere, however, analytical computations can be done, which are both faster and more accurate --- $$ \mathrm{PS}_{l,k,m} = \delta_{k,m} + \eta(x_l, \mathbf{r}_m, \mathbf{r}_k - \Delta/2) - \eta(x_l, \mathbf{r}_m, \mathbf{r}_k + \Delta/2) $$ (11.79) The probability of crossing the boundary of the vapour cell is $$ \mathrm{PS}_{l,Nr,m} = \eta(x_l, \mathbf{r}_m, \mathbf{r}_{\mathrm{bound}}) $$ (11.80) **For the plane-parallel slab** $$ \eta_{l,k,m} = \frac{1}{2} \mathrm{sgn}(z_k - z_m) \mathrm{Ei}_2 (k(x_l)|z_k - z_m|) $$ (11.81) **For the cylinder** $$ \eta_{l,k,m} = \frac{1}{4\pi} \int_{-\infty}^{\infty} \int_{0}^{2\pi} \frac{(r_k - r_m \cos \varphi)r_k}{(z^2 + q^2)^{3/2}} \exp \left( -k(x_l)\sqrt{z^2 + q^2} \right) \mathrm{d}\varphi \mathrm{d}z $$ (11.82) $$ q^2 = r_k^2 + r_m^2 - 2r_k r_m \cos \varphi $$ Performing the $z$-integration, we get $$ \eta_{l,k,m} = \frac{r_k k(x_l)}{\pi} \int_{0}^{\pi} \frac{(r_k - r_m \cos \varphi)}{q} \left[ K_1^B (k(x_l)q) - \mathrm{Ki}_1 (k(x_l)q) \right] \mathrm{d}\varphi $$ (11.83) where $K_1^B$ is a modified Bessel function, and $\mathrm{Ki}_1$ is the first repeated integral of $K_1^B$ (see Appendix B). The integration over $\varphi$ must be performed numerically. **For the sphere, we get** $$ \eta_{l,k,m} = \frac{r_k^2}{2} \int_{0}^{\pi} \frac{(r_k - r_m \cos \vartheta) \exp \left[ -k(x_l) \sqrt{r_k^2 + r_m^2 - 2r_k r_m \cos \vartheta} \right]}{\left[ r_k^2 + r_m^2 - 2r_k r_m \cos \vartheta \right]^{3/2}} \sin \vartheta \, \mathrm{d}\vartheta $$ (11.84) which can be evaluated analytically, see Parker *et al.* (1993). --- We can also describe the propagators $\mathrm{PS}$ in another way. They are the integrands of the $A_{k,m}$ matrix elements for CFR of the PCA method described in Appendix B, with $x = x_l'$. The only difference must lie in the factor $C_x k(x)$, which describes the reemission and which is ascribed to the propagator $\mathrm{PF}$ in the PFR-PFM method. Comparison of Eqs. (11.81)–(11.84) with the equations in Appendix B shows that this is actually the case; $\eta$ is equal to $C_x k(x_l')$ times the integrands of Appendix B. Still missing is the computation of the frequency propagators $\mathrm{PF}_{i,l}$. These are $$ \mathrm{PF}_{i,l} = \frac{R(x_i, x_l)}{\Phi(x_l)} $$ (11.85) This is an expression we are well acquainted with, since we have discussed it in detail in Sec. 11.1. Parker *et al.* use the JW approximation to compute the $\mathrm{PF}_{i,l}$, but this is not --- THE PROPAGATOR FUNCTION METHOD AND THE PCA METHOD essential for the validity of the method. Any other redistribution function could be used as well. The whole problem can also be tackled by the PCA method (Sec. 5.3). Solutions to the generalized Holstein equation are of the form $$ n(z, x, t) = \sum_{jz, jx} \alpha_{jz, jx} \psi_{jz, jx}(z, x) \exp \left[ -t / (g_{jz, jx} \tau) \right] \qquad (11.86) $$ We approximate the eigenfunctions $\psi_{jz, jx}$ by a set of basis functions $$ b_{k,i} = p(z_k) \cdot \delta(x - x_i) \qquad (11.87) $$ With these approximations, the Holstein equation can again be reduced to an algebraic eigenvalue problem $$ \left( 1 - \frac{1}{g} \right) n_{k,i} = \sum_{m,l} A_{k,i,m,l} n_{m,l} \qquad (11.88) $$ The main difference is that now the eigensolutions depend not only on position, but also on the frequency $x$ that is to be emitted by the atom. The size of the matrix $A_{k,i,m,l}$ is considerable --- of excited atoms that corresponds to the shape of the lowest-order mode in the CFR case. From the above discussion, it follows that the eigenvalue method is more efficient than the PFM when we just want to compute the lowest-order trapping factor and the size of the matrices is not too large. However, when we want to compute the complete temporal development of the excited atoms, possibly including temporally and/or spatially varying excitation and de-excitation terms, then the PFM is superior. Of course, we could compute this temporal development also with the eigenvalue method by means of a modal expansion and Laplace transform techniques. However, for this we need both the eigenvalues and the eigenfunctions, which cost more CPU time and are subject to numerical problems. In addition, the modal expansions also cost CPU time, so that the total effort is larger than for the PFM. Despite that, when we want to compute many scenarios with the same geometry and the same vapour but with different excitations, we have to compute the eigensolutions only once and can then solve the problems by simple eigenfunction expansions. In this case, the eigensolution method can offer advantages with respect to CPU time. ## 11.5 Monte Carlo simulations Yet another possibility for the solution of the PFR problem is the Monte Carlo approach. The difference to the basic algorithm described in Chapter 6 is that now the reemission frequency is computed from the absorbed frequency and from the redistribution function. When we use the Jefferies–White approximation, inclusion of the frequency redistribution is straightforward. We know the absorbed frequency $x'$ and can easily compute the fraction of coherently scattered photons $\mathrm{FC}(x')$. We then generate a random number and look whether it is larger than $\mathrm{FC}(x')$. If yes, then chance decided to have a complete redistribution within a Doppler line, so that the reemission frequency is just a random number taken from a Doppler distribution. If no, the reemitted frequency $x$ is simply the same as the absorbed frequency $x'$. Using such an algorithm, Vermeersch *et al.* (1988) showed almost perfect agreement of their Monte Carlo simulation with the analytical results of Post. In the case of pure Doppler redistribution, the implementation is also quite straightforward. One first computes the velocity component that an atom must have in order to be able to absorb a photon of frequency $x'$. Next, the velocity components perpendicular to the direction of the absorbed photon are chosen from the appropriate random distribution. Finally, one chooses the direction of reemission at random, and computes (deterministically) the Doppler shift of the reemitted photon. The situation is more complicated for Doppler plus natural broadening, since the calculation of the velocity component of the absorbing atom in the propagation direction of the photon is somewhat involved. In the following, we describe a step-by-step 'recipe' given by Anderson *et al.* (1995), which is an extension of the algorithm developed by Lee (1974a, 1977, 1982), (see also (Lee and Meier 1980), (Meier 1981)). It requires a --- generator for random variables according to a Voigt distribution; this can be done either by the methods described in Sec. 6.1, or by an algorithm described by Parker *et al.* (1993). --- **Insert 11.3 Calculation of the velocity components of the absorbing atom for an MC simulation of PFR.** The atom velocity component $\hat{v}_p$ in the propagation direction (normalized by $v_\text{D} = (2kT/M)^{0.5}$) must be taken from the distribution $$ \text{pdf}(\hat{v}_p) = \frac{1}{\Phi^\text{V}(x')} \left[ \frac{a}{\pi^{3/2}} \frac{\exp(-\hat{v}_p^2)}{a^2 + (x' - \hat{v}_p)^2} \right] $$ (11.90) To speed up calculations, we distinguish three cases: **(i)** $|x'| > 5$. The distribution $\text{pdf}(\hat{v}_p)$ is approximated as $$ \text{pdf}(\hat{v}_p) = \frac{1}{\sqrt{\pi}} \exp \left[ - \left( \hat{v}_p - \frac{1}{x'} \right)^2 \right] $$ (11.91) For this pdf, the generation of component $\hat{v}_p$ is quite simple: just generate a normal variable with variance one, divide it by $\sqrt{2}$, and add $1/x'$. **(ii)** $5 \ge |x'| \ge 1.5$. The random numbers are generated by the rejection method. In this method, we look for a 'trial function' $q(u)$ so that $q(u) > \text{pdf}(u)$. In order to be useful, this function must be integrable, and the integral must be invertible. Then we generate a random number $u_q$ according to this distribution $q(u)$, and accept it if $\text{pdf}(u_q)/q(u_q) > \text{random}$, where random is a uniformly distributed variable. Otherwise, it is rejected, and we start afresh. We first compute the function $$ \begin{aligned} q(\hat{v}_p) &= -\text{pdf}(1)\hat{v}_p \exp (1 - \hat{v}_p^2) && \text{for } \hat{v}_p < -1 \\ q(\hat{v}_p) &= \max [1.5\text{pdf}(0), \text{pdf}(|x'| - 0.25)] && \text{for } -1 < \hat{v}_p < |x'| - 0.25 \\ q(\hat{v}_p) &= \text{pdf}(|x'| - 0.25)\frac{1 + (0.25/a)^2}{1 + [(\hat{v}_p - |x'|)/a]^2} && \text{for } |x'| - 0.25 < \hat{v}_p \end{aligned} $$ (11.92) The function $Q$ is defined as the integral over $q$ $$ Q(\hat{v}_p) = \int_{-\infty}^{\hat{v}_p} q(u)du $$ (11.93) We then compute a random number as $$ \hat{v}_p = Q^{-1} [\text{random} Q(\infty)] $$ (11.94) Finally, $\hat{v}_p$ is accepted if $$ \text{random} < \frac{\text{pdf}(\hat{v}_p)}{q(\hat{v}_p)} $$ (11.95) --- PARTIAL FREQUENCY REDISTRIBUTION (iii) $|x'| < 1.5$, the trial function is $$ q(\hat{v}_p) = \frac{1}{\pi \Phi^V(x')} \left[ \frac{a}{a^2 + (x' - \hat{v}_p)^2} \right] \qquad (11.96) $$ This function can also be integrated, so we can proceed just like in case (ii). After the determination of the velocity in the direction of the absorbed photon, the rest of the Monte Carlo simulation proceeds just like in the case of the Doppler redistribution. *** Finally, we want to discuss some methods to increase the efficiency of the MC simulations. The opacity is usually very large, otherwise PFR effects are negligible anyway. This means that a photon undergoes many scatterings until it escapes. This means in turn that many redistributed frequencies have to be computed, which costs a lot of CPU time. An approximate technique that circumvents these problems has been developed by Auer (1968). He divides the line into a core and into the wings. In the core, the mean free path is very small, and while a photon stays in the core, it hardly moves in space. It just undergoes a certain number of absorption/reemission processes until it has left the core. This part of the problem can either be treated analytically, or a Monte Carlo simulation is used to estimate how long it takes a photon to leave the core and how far it can move during that time on the average. This result is computed once and is then used any time a photon is in the core. As soon as it has left the core and entered the wings, we do the usual MC simulations with PFR, correctly computing the frequency and mean free path afresh after each absorption/reemission process. A slightly different acceleration technique was developed by Avery and House (1968) and Bonilha *et al.* (1979). These techniques are special cases of the 'biased sampling' that can be used to accelerate MC simulations, see Press *et al.* (1994). ## 11.6 Transfer equation formulations The PFR problem may also be formulated in the transfer-equation-plus-rate-equation formalism. There are two possibilities to express the problem. The first one is to use the usual equation of statistical equilibrium (written here for the two-level case) $$ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -(A_{21} + Q)n(\mathbf{r}, t) + \frac{B_{12}N}{4\pi} \iint I(\mathbf{r}, \mathbf{\Omega}, x, t) C_x k(x) \mathrm{d}\mathbf{\Omega}\mathrm{d}x \qquad (11.97) $$ The equation of radiative transfer can be considerably more complicated, ((Streater *et al.* 1988a), (Heasley and Kneer 1976), for a strict derivation see (Cooper *et al.* 1982)), --- $$ \begin{aligned} \boldsymbol{\Omega} \cdot \nabla I(\mathbf{r}, \boldsymbol{\Omega}, x, t) = -k(x) \Bigg[ I(\mathbf{r}, \boldsymbol{\Omega}, x, t) - \frac{A_{21}}{B_{12}} \frac{n(\mathbf{r}, t)}{N} - \frac{1}{4\pi \Phi(x)} \frac{A_{21}}{A_{21} + Q + C_{\text{ela}}} \cdot \\ \cdot \iint I(\mathbf{r}, \boldsymbol{\Omega}, x', t) \big( R_{II}(x, x') - \Phi(x)\Phi(x') \big) \\ + \varepsilon \big( R_{III}(x, x') - \Phi(x)\Phi(x') \big) \text{d}\boldsymbol{\Omega} \text{d}x' \Bigg] \end{aligned} $$ (11.98) where $Q$ is the quenching rate (direct collisional depopulation) and $C_{\text{ela}}$ the elastic collision rate. The factor $\varepsilon$ is $(C_{\text{ela}} - C_{\text{vc}})/(C_{\text{inela}} + A_{21} + C_{\text{vc}})$, where $C_{\text{vc}}$ denotes the rate of velocity changing collisions. We see that the second correction term vanishes for complete redistribution, since then $R(x, x') = \Phi(x) \cdot \Phi(x')$. For the case of collisional redistribution in the rest frame of the atom, the correction is proportional to $R_{III}(x, x') - \Phi(x)\Phi(x')$ and is thus very small. This was also noted by Streater *et al.* (1988a), who found that even with $\varepsilon = 7$, the correction term is dominated by $R_{II}(x, x') - \Phi(x)\Phi(x')$. In the reference mentioned, this system of equations was solved by a Hankel transform technique (see Sec. 5.2), but of course other numerical procedures are possible. The second possible formulation is to keep the simple transfer equation, but to write the source function in terms of the number of atoms that can emit a photon of frequency $x$ (see e.g. (Milkey and Mihalas 1973)). Specifically, the transfer equation now reads $$ (\boldsymbol{\Omega} \cdot \nabla) I(\mathbf{r}, \boldsymbol{\Omega}, x, t) = -k(x) \left[ I(\mathbf{r}, \boldsymbol{\Omega}, x, t) - \frac{A_{21}}{B_{12}} \frac{n(\mathbf{r}, x, t)}{N \Phi(x)} \right] $$ (11.99) This formulation is very similar to the formulation in Chapter 9. The central difference is that now the source function is frequency dependent. For the rate equation, we get $$ n(\mathbf{r}, x, t) \cdot A_{21} = N \frac{B_{12}}{4\pi} \iint R(x, x') I(\mathbf{r}, \boldsymbol{\Omega}, x', t) \text{d}\boldsymbol{\Omega} \text{d}x' $$ (11.100) This system of equations can be solved by various steady-state techniques. Hummer (1969) used a discrete-ordinate method. Milkey and Mihalas (1973) and Milkey *et al.* (1975a) used the complete linearization technique (see Chapter 13). A similar technique was also proposed by Milkey *et al.* (1975b) for angle-dependent redistribution functions. Gladstone (1982) and Hubeny and Heinzel (1984) used the Feautrier technique. Hubeny (1985b, c) proposed a modification of the Rybicki method, which works, however, only if the JW approximation is used—in that case, the coupling of the frequencies is in the Doppler core the same as in the CFR case, and is non-existent in the wings, so that it can be described by just two parameters. Basically, these solution methods are the same as for the CFR case; the only difference is that in the CFR case, we had a single equation coupling the (integrated) source function to the intensities at frequencies $x_i$. Now we have $N_i$ equations for $N_i$ source functions at frequencies $x_i$ that are all coupled to the transfer equations at all frequencies $x_i$. An iteration technique that requires just the effort of solving one CFR problem in each step was suggested by Heasley and Kneer (1976). However, convergence properties for strong stimulated emission (see Chapter 13) are not favourable. Crivellari and Simonneau (1995) modify their implicit integral method (Sec. 9.4.3) to include PFR. --- One very useful iteration technique in that context is the quadrature perturbation technique. As we will see in Chapter 13, we define a simple approximate operator $\Lambda^*$ that describes the double integral in the rate equation, and then iterate towards the true solution. The obvious choice for the simplified operator is $R(x, x') = \Phi(x) \cdot \Phi(x')$ (Cannon 1984). Taking the anisotropy of the reemission into account is possible, e.g., with Monte Carlo simulations (see Sec. 11.5), and also with the Feautrier technique (Vardavas 1976a), (Vardavas and Cannon 1976). However, the influence on the results is small. A method of solution for the equation of radiative transfer with PFR based on the expansion of the redistribution function into an infinite series is related to the solution methods of Ivanov (1973), but is not in widespread use (see (Yengibarian and Nicoghossian 1973) and (Haruthyunian and Nicoghossian 1978), and references therein). ## 11.7 Frequency diffusion In most laboratory situations, the natural broadening can be considered as a ‘perturbation’ of the escape by frequency redistribution. There are, however, some cases where it is worthwhile to consider the absorption/reemission process as essentially coherent, and the redistribution by the Doppler effect as the perturbation. This might be the case when the opacity --- equation. From this, we compute the average number of scattering processes that a photon undergoes, and the average frequency shift due to the Doppler redistribution (with some weighing factor). We then solve the Milne equation at this new frequency. This will yield a new (lower) average number of scatterings, and thus a lower frequency shift, which we use as the basis for the next iteration step. The iteration is continued until convergence is reached. An alternative technique for Voigt factors larger than unity was developed by Suvorov (1987) and Bulyshev *et al.* (1988). They compute the lowest-order trapping factor as $$ \frac{1}{g_0} = \frac{\lambda_0 C_x}{3L^3} \int_{-\infty}^{\infty} \frac{f(x)}{k(x)} \mathrm{d}x $$ (11.101) The function $f(x)$ is the solution of the differential equation $$ \frac{1}{3k(x)} f(x)\lambda_0 + \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \Delta \nu_s k(x) \frac{\mathrm{d}}{\mathrm{d}x} f(x) = 0 $$ (11.102) where $\lambda_0$ is the eigenvalue of the Laplace operator in this particular geometry and $f(x)$ is normalized so that --- A generalization of these results to include a continuum, i.e. frequency-independent contributions to the absorption coefficient is given by Grachev (1988) and Neufeld (1990); for expanding plasmas it is given by Grachev (1989) and Rybicki and dell'Antonio (1994). Other approximate computation methods include second-order escape probability methods (compare to Sec. 4.5) (Gayley 1992c), and Eddington–Barbier relations (Gayley 1992d). ## 11.8 Large-scale particle flow We have always assumed that the atoms move in the vapour cell only in a random fashion, in other words, that there are no large-scale directional particle streams. This assumption is very reasonable as long as we really *have* a vapour cell, i.e. in most applications of chemical physics (apart from atomic beams, see Chapter 15). The situation is somewhat different in plasmas. There, the atoms have a tendency to move away from the core of the plasma towards the cool outside regions. This is especially true for plasmas that are created by exploding a target with a strong laser pulse, but also for fusion plasmas. There can be strong velocity gradients, which lead to a basic change in the behaviour of the radiation trapping. Before going into the mathematical details, we try to obtain a physical picture of what happens in a vapour with strong velocity gradients. To simplify the discussion, and to work out the essential points, let us assume a plane-parallel slab geometry with a constant velocity gradient $\partial v/\partial z$ and that the opacity $k_0 L$ that would occur if the vapour were stationary, is infinite. In a stationary plasma, photons would not be able to escape from the vapour because of the infinite opacity. The excited-state density at any point would be influenced by the excited-state distribution at all other points in the vapour—remember the discussion on this point in Chapter 4. Due to the velocity gradients, however, the reabsorption properties change. In Fig. 11.8 we see the absorption coefficient as a function of space, once for a stationary plasma, and once for a plasma with a high velocity gradient. Consider a photon that is emitted at frequency $x = 0$ at a certain position $z$. For a stationary plasma, the photon would not be able to escape from the plasma—since the opacity is infinite, it *must* be absorbed somewhere. For the moving plasma, the centre of the absorption line moves away from the frequency $x = 0$. Thus, when the photon is not absorbed near the point with $x = 0$, it gets so far out into the wings of the local absorption coefficient that it cannot be absorbed at all. The size of the velocity gradient determines the size of the region where the photon can be absorbed. From the above discussion, two essential features of moving plasmas can be worked out. (i) The escape probability is not determined by the usual opacity $k_0 L$ of the slab, but by an opacity $k_0 L_{\mathrm{sob}}$, where $L_{\mathrm{sob}}$ is the so-called Sobolev length, determined solely by the velocity gradient. The Sobolev length is defined as the length that a photon has to move for being in a region that is Doppler shifted by one half-width. It can be computed from (Lightbody and Pert 1994) --- LARGE-SCALE PARTICLE FLOW [FIGURE: Absorption coefficient as a function of frequency and position for a Doppler line] FIG. 11.8. Absorption coefficient as a function of frequency and position for a Doppler line, for a normalized velocity gradient $\partial v/\partial z = 0$ on the left, and for $\partial v/\partial z = 5$ on the right. $$ L_{\text{sob}} = \frac{c \Delta \nu_D}{2 \nu_0 \nabla \cdot \mathbf{v}} \qquad (11.107) $$ Using the Sobolev escape factor, we ignore the escape that occurs simply because the vapour has a finite extent. Strictly speaking, the Sobolev approximation is only valid in an infinite plasma. For practical purposes, it is required that $L_{\text{sob}} << L$, where $L$ is a typical dimension of the plasma. (ii) When the velocity gradient is large, the photons that are reabsorbed at a point were all emitted in the vicinity of this point—all other photons have a frequency shift that is too large. We thus no longer have the long-range coupling of the excited-state distribution by the radiation trapping process, and the influence of the radiation can be described in terms of local parameters. This also means that the escape factor method works very well under these circumstances, much better than for static media (Rybicki 1984). The escape factor method will even give a correct prediction of the excited-state distribution. We thus have the astonishing result that the presence of the velocity gradient leads to a *simplification* of the trapping problem. The escape factor method furthermore is *exact* when the excited-state density is constant, a second condition being that either the absorption coefficient is symmetrical with respect to frequency, or that the velocity gradient is constant (Irons 1990c, 1991). For the mathematics, let us assume a constant, positive velocity gradient $v'_s$ along a considered ray. The probability of escape along such a ray with emission frequency $x$ is --- $$ \eta_{\mathrm{ray}}(x) = \exp \left[ - \int_0^\infty k \left( x - \frac{s}{L_{\mathrm{sob}}} \right) \mathrm{d}s \right] $$ (11.108) Introducing the Sobolev optical thickness $\xi_{\mathrm{sob}}$, $$ \xi_{\mathrm{sob}} = \frac{k_0}{\Delta\nu/2} L_{\mathrm{sob}} = \frac{1}{|v'_s|} \frac{hc}{4\pi} N B_{12} $$ (11.109) this can be computed as $$ \eta_{\mathrm{ray}}(x) = \exp \left[ - \xi_{\mathrm{sob}} \int_{-\infty}^x \Phi(x') \mathrm{d}x' \right] $$ (11.110) Averaging the escape probability over all emission frequencies, we get $$ \bar{\eta}_{\mathrm{ray}} = \frac{1 - \exp(-\xi_{\mathrm{sob}})}{\xi_{\mathrm{sob}}} $$ (11.111) Finally, for the mean escape probability, this must also be averaged over the spatial angle by integration over $\mu$. For the plane-parallel slab, $$ \bar{\eta} = \frac{1}{2} \int_{-1}^1 \mathrm{d}\mu \bar{\eta}_{\mathrm{ray}}(\mu) $$ (11.112) where the dependence on $\mu$ is given by $$ v'_s = \mu^2 v'(z) $$ (11.113) where $v'(z)$ is the velocity normal to the boundaries. One factor of $\mu$ stems from differentiation, $\mathrm{d}z = \mu \mathrm{d}s$, and the second from the projection of the velocity along the ray. The escape factor computations for the case of a large velocity gradient were introduced by Sobolev (1957) and are generally known as the ‘Sobolev approximation’, see also Magnan (1974a). Irons (1990a, b) criticized the original derivation and gave an alternative—however, there is no discrepancy between the *results* of these derivations. The escape factors have been computed for various geometries. On the axis of an **expanding cylinder**, the escape probability is Eder (1989), (Shestakov and Eder 1989) (see also (Sasaki *et al.* 1994)) $$ \eta = \frac{1}{3\xi_{\mathrm{sob}}} - \frac{2}{3} e^{-\xi_{\mathrm{sob}}} \left[ \xi_{\mathrm{sob}} K_0^{\mathrm{B}}(\xi_{\mathrm{sob}}) + \left( \frac{1}{2} - \xi_{\mathrm{sob}} \right) K_1^{\mathrm{B}}(\xi_{\mathrm{sob}}) \right] $$ (11.114) Off the axis, the escape factor is $$ \begin{gathered} \eta(r) = \frac{1}{3\pi} \int_0^\pi \left[ \frac{1}{u} - 2e^{-u} \left( u K_0^{\mathrm{B}}(u) + \left(\frac{1}{2} - u\right) K_1^{\mathrm{B}}(u) \right) \right] \mathrm{d}w \\ u = \frac{\xi_{\mathrm{sob}}}{1 + \cos(w) \frac{(\partial v/\partial r) - (v/R)}{\nabla \cdot \mathbf{v}}} \end{gathered} $$ (11.115) It is interesting that for $\partial v/\partial r = v/R$ (self-similar expanding plasma), the variable $u$ becomes equal to the Sobolev opacity $\xi_{\mathrm{sob}}$, and the escape factor $\eta(r)$ has the same value for all positions in --- the cylinder. If the condition for self-similarity is not exactly fulfilled, one can make series expansions and thus avoid numerical integration. Equations for this are given by Shestakov and Eder (1989). For an inhomogeneous cylinder, expressions for the escape factor are given by Kainov *et al.* (1993). Evaluation requires, however, quadruple integrals—it is thus not clear whether anything is gained compared to a full-fledged solution of the transfer equation. Computation in a sphere coupled with variable Eddington factors are discussed by Eastman and Pinto (1993). By **comparison with Monte Carlo simulations**, Lightbody and Pert (1994) analysed the applicability of the Sobolev approximation. They varied the opacity, the velocity gradient, and the inhomogeneity of the ground-state distribution. They found mainly three physical effects: (i) When varying the velocity gradient only, the Sobolev approximation works best for large velocity gradients. This is quite reasonable, since the basic assumption of the Sobolev theory is a large velocity gradient. For a pure Doppler lineshape, the relative error of the Sobolev approximation is about 10% for $L/L_{\text{sob}} = 10$ and 50% for $L/L_{sob} = 3$; it exceeds 100% for $L/L_{\text{sob}} = 1$. (ii) When varying the inhomogeneity only, the agreement was best for a homogeneous vapour, and worse for very strong variations in the ground-state density. In a non-uniform plasma, the 'typical' dimension is the dimension over which the ground-state density varies strongly. Since the Sobolev approximation requires the Sobolev length to be much smaller than the 'typical' dimension, the condition is less fulfilled in a non-uniform plasma as compared to a uniform plasma. An edge-to-centre ratio in ground-state density of 0.01 leads very roughly to an increase of the relative error by a factor 1.5 as compared to the homogeneous case. (iii) When analysing the lineshape, it turned out that the best agreement is achieved with a rectangular absorption coefficient. Doppler lineshapes also work quite nicely, but for Voigt lineshapes with a high Voigt parameter $a$, the agreement is rather bad. Again, there is a physical interpretation for this finding. For a Doppler lineshape, a photon needs to be Doppler shifted at most three Doppler widths before it escapes because the Doppler lineshape drops off so sharply in the wings. A Lorentzian shape, on the other hand, has a much slower drop in the wings. In a high-opacity vapour a photon thus has to go through a much larger frequency shift before it can really escape. Even for a homogeneous vapour and $L/L_{\text{sob}} = 10$, the error is more than 100%. Further comparisons between approximate and exact results are given by Eder and Scott (1991). For **two closely neighboured self-reversed lines**, the shifting of the resonance lines and the easier escape for certain wavelengths makes things especially complicated. Consider two lines that are several (static) Doppler widths apart. Normally, trapping would not influence their intensity ratios, and the lineshapes of the two lines would be distorted separately (i.e. not influenced by the other line). Due to the bulk motion, however, the wing of one line may be shifted into the line centre of the other line (at a different spatial position). One line can thus reabsorb radiation from the other line—this will lead to a strong distortion of the intensity ratios and of the lineshapes (Irons (1980a, b) and Pavlakis and Kylafis (1996)). The Sobolev method is just an approximate method of analysis for large velocity gradients. Due to these gradients, radiation emerging from one region cannot be absorbed in another. For monotonic but otherwise arbitrary velocities, a theory was developed by Hummer and Rybicki (1982a). An approximate method that works also for non-monotonic velocities (which in turn --- leads to coupling between remote regions of the vapour), was developed by Rybicki and Hummer (1978), but seems to be of interest mostly for astrophysical problems. Lucy (1971) and Noerdlinger and Scargle (1972) also made computations where they assume complete coherence in the fluid frame, however, such computations can of course give only very low accuracy. Inclusion of continuum absorption was made by Hummer and Rybicki (1985). A Sobolev theory for PFR was derived by Hummer and Rybicki (1992), and for polarization by Jeffery (1989). **For an accurate solution of large-scale movement**, the hydrodynamic equations have to be coupled with the equation of radiative transfer, and the dependence of the absorption coefficient on the velocity distribution must be included. Such a treatment is extremely computer-time expensive, and to our knowledge has been done only in the astrophysical literature, especially in the context of exploding stars and stellar winds (see (Cannon 1985) and (Mihalas 1978) and references therein, also (Kalkofen and Whitney 1971) ). It is noteworthy, however, that the Feautrier technique cannot be used for asymmetrical velocity fields and line profiles. In the analysis of plasmas, almost exclusively the Sobolev method has been used up to now (see Chapter 18). An analysis of trapping with varying Doppler width in a static medium has been given by Strickland (1979), but seems to be of little use for laboratory situations. An alternative computation method is the Monte Carlo simulation, which can be easily modified to include bulk motion of the vapour; it was applied to stellar atmospheres by Magnan (1970). The influence of microturbulences on the redistribution function was investigated by Magnan (1975), however, this subject is mostly of interest for stellar atmospheres. For partial redistribution, one has either to use the angle-dependent redistribution function, or to exercise extreme care in the choice of the frame in which to average; otherwise spurious solutions can occur (Magnan 1974b). **The velocity gradients can also be included formally in the Holstein equation.** In that case, it is just necessary to modify the Kernel function. It is intuitively clear that, e.g., in the plane-parallel slab case, the Kernel function reads (Mihalas 1978) $$ G(|z - z'|) = \frac{C_x}{2} \int_{-\infty}^{\infty} \int_0^1 \frac{1}{\mu} k(x) k\left(x + \frac{\partial v}{\partial z} \mu z\right) \exp\left[ -\frac{1}{\mu} \int_{z'}^z k\left(x + \frac{\partial v}{\partial z} \mu t\right) \mathrm{d}t \right] \mathrm{d}\mu \mathrm{d}x \qquad (11.116) $$ The Holstein equation can then be solved reasonably simply by using the piecewise-constant approximation. One then just has to use such small subvolumes that the absorption coefficient stays approximately constant within the subvolume. The Holstein equation was recently applied to an expanding plasma by Benredjem *et al.* (1995). **Plasma expansion can also lead to an anisotropy of the radiation.** If a plasma is expanding mainly in one direction, then the absorption coefficient in that direction will be different, and this will lead to an anisotropy (Kawachi *et al.* 1995). In the **astrophyiscal literature** the problem of macroscopically moving particles has been treated in far more detail, where the problem of stellar winds is closely related to these problems. More details can be found in Caroff *et al.* (1972), Noerdlinger and Scargle (1972), Bonilha *et al.* (1979), Mihalas *et al.* (1975), Mihalas *et al.* (1976a, b), Hauschildt (1992a, b), Marti and Noerdlinger (1977), Karp *et al.* (1977), Kunasz and Hummer (1974b), Mihalas (1979), Kunasz (1984), Mihalas and Klein (1982), Baade (1990), Eastman and Pinto (1993), Yin and Miller (1995), Papkalla (1995) and especially in the textbooks by Cannon (1985), Mihalas (1978) and Mihalas and Mihalas (1984) and references therein. --- # 12 # POLARIZATION ## 12.1 Introduction In theory, all radiation trapping problems must be treated under consideration of the polarization state of the radiation. Even with an unpolarized excitation source, the geometry of the problem can introduce some degree of polarization to the fluorescence radiation. Of course, when the excitation source itself is polarized, as is usually the case for a laser, polarization can become even more important. One should not forget, however, that incorporation of polarization makes for a very complicated problem. The Holstein equation will become a Fredholm integral equation for a four-component vector instead of the usual scalar function (the excited-state density). Having seen the difficulties in solving the comparatively simple 'usual' trapping problem, we see that polarization should only be included when it is not to be avoided. In a first step, we derive the transfer equation including polarization. The intensity has to be replaced by the 4-components Stokes vector $\mathbf{I}$, which completely describes polarized radiation. The field at any time $t$ and at any position $z$ is described by the electric field vector $\mathbf{E}(t, z)$. Following Rees (1987), we define the Stokes vector for radiation that propagates in the positive $z$-direction in a right-handed coordinate system $(x, y, z)$. The reference frame is fixed, with the $z$-axis towards the observer: $$ \mathbf{E}(t, z) = \begin{pmatrix} E_{x0} \exp \left[ \mathrm{j}(\varphi_x - \omega t + 2\pi z \nu_0/c) \right] \\ E_{y0} \exp \left[ \mathrm{j}(\varphi_y - \omega t + 2\pi z \nu_0/c) \right] \\ 0 \end{pmatrix} \qquad \mathbf{I} = \begin{pmatrix} I \\ Q \\ U \\ V \end{pmatrix} = \begin{pmatrix} E_{x0}^2 + E_{y0}^2 \\ E_{x0}^2 - E_{y0}^2 \\ 2E_{x0}E_{y0} \cos(\varphi_x - \varphi_y) \\ 2E_{x0}E_{y0} \sin(\varphi_x - \varphi_y) \end{pmatrix} $$ (12.1) Ultimate care must be taken concerning what is termed left-handed and right-handed polarization. As pointed out by Rees (1987), resulting sign errors are present in many transfer codes. A thorough discussion of this subject, as well as a superb introduction into polarized transfer in general, can be found in that reference. In the following, we will for polarized radiation set up both the equation of radiative transfer and the Holstein equation. The transfer equation becomes a matrix differential equation, and we will give algorithms for its formal solution. For a true self-consistent solution, it has to be coupled with the rate equations—analogously to the unpolarized case. However, this time we need the rate equations for a vector source function or the density matrix elements. Similarly, the Holstein equation becomes a matrix Fredholm --- integral equation. Actually, for polarized transfer, the problem lies mainly in setting up the appropriate equations. Once that is done, the numerical solution methods discussed before can be applied, most notably the Feautrier and PCA methods. There are mainly two cases of practical interest. Case one is that there is no magnetic field. Then the whole problem can be described by a two-component vector instead of the full 4-component Stokes vector defined above. Luckily, this case is the more usual in chemical physics experiments, where polarization is most often introduced by the excitation source, i.e. by a laser. In case two there is a magnetic field strong enough to cause appreciable Zeeman splitting. Polarization effects stemming from this might be of interest for experiments involving, e.g., the Hanle effect. ## 12.2 Formal solution of the vector transfer equation In a first step, we derive a formal solution of the transfer equation (including polarization), i.e. we assume that the source function is known. To begin with, we will assume a one-dimensional slab geometry and the presence of a magnetic field that introduces Zeeman splitting. The magnetic field shall have an angle $\vartheta$ to the $z$-axis and an angle $\varphi$ to the $x$-axis. The vectorial transfer equation has a form that is very similar to the basic transfer equation, $$ \frac{\mathrm{d}\mathbf{I}}{\mathrm{d}z} = -\mathbf{k}(\mathbf{I} - \mathbf{S}) $$ --- FORMAL SOLUTION OF THE VECTOR TRANSFER EQUATION The primed lineshapes $\Phi'_{\text{p,b,r}}$ are anomalous dispersion profiles (Faraday–Voigt functions) $2\text{FF}(a, -x - \Delta x_{\text{Zeeman}} \cdot \Delta m)$, where $$ \text{FF}(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{(x - u) \exp(-u^2)}{(x - u)^2 + a^2} \text{d}u \qquad (12.4) $$ A quantum-mechanical derivation of the transfer equation is given by House (1970–71), House and Steinitz (1975), and Mathys (1982). The interaction between atoms and polarized radiation is also reviewed by Lamb and ter Harr (1971), Domke and Hubeny (1988), Streater *et al.* (1988b), Stenflo (1996), and Bommier (1996). Note that now there is no choice, we have to use the angle-dependent redistribution functions. For the isotropic radiator and for the dipole radiator, this redistribution function is given by Hummer (1962). For other types of scattering (e.g. with alkali metal atoms), we have to use the general expressions (see Sec. 12.4 for the phase matrix elements). Symmetry properties of the phase matrix are discussed by Domke and Yanovitsku (1981). It is common to factor the polarization redistribution into the frequency and angle redistributions; however, one has to keep in mind that this is only an approximation. We now give a formal solution of the transfer equation for a known source function. We change notation to the optical depth coordinate $\xi$ along the optical path, so that $\text{d}\xi = -k_0 \Phi_I \text{d}z$. The transfer equation may then written in the form $$ \frac{\text{d}}{\text{d}\xi} \mathbf{I} - \mathbf{I} = \mathbf{\hat{K}} \mathbf{I} - \mathbf{\hat{S}}, \quad \text{where} \quad \mathbf{\hat{K}} = \frac{\mathbf{k}}{k_0 \Phi_I} - \mathbf{1}, \quad \mathbf{\hat{S}} = \frac{\mathbf{kS}}{k_0 \Phi_I}, \qquad (12.5) $$ which has the formal solution $$ \mathbf{I}(\xi) = \int_{\xi}^{\xi_{\text{max}}} \exp \left[ -(\xi' - \xi) \right] \left\{ \mathbf{\hat{S}}(\xi') - \mathbf{\hat{K}}(\xi') \mathbf{I}(\xi') \right\} \text{d}\xi' \qquad (12.6) $$ where $\xi_{\text{max}}$ is the (integrated) opacity along the line-of-sight from cell boundary to cell boundary, i.e. for a plane-parallel slab $\xi_{\text{max}} = k_0 \Phi_I L / (2\mu)$. When we assume that the function in brackets is piecewise-linear, it can be integrated analytically to yield the recursive relation $$ \mathbf{I}_k = \mathbf{A1}_k + \mathbf{A2}_k \mathbf{I}_{k+1}, \quad \text{where} $$ $$ \mathbf{A1}_k = \left( \mathbf{1} + (f1_k - f2_k)\mathbf{\hat{K}}_k \right)^{-1} \left[ (f1_k - f2_k)\mathbf{\hat{S}}_k + f2_k\mathbf{\hat{S}}_{k+1} \right] $$ $$ \mathbf{A2}_k = \left( \mathbf{1} + (f1_k - f2_k)\mathbf{\hat{K}}_k \right)^{-1} \left[ f3_k \mathbf{1} - f2_k\mathbf{\hat{K}}_{k+1} \right] \qquad (12.7) $$ $$ f1_k = 1 - f3_k \qquad \qquad \qquad \qquad f3_k = \exp(-\Delta\xi_k) $$ $$ f2_k = [1 - (1 + \Delta\xi_k)f3_k]/\Delta\xi_k \qquad \qquad \Delta\xi_k = \xi_{k+1} - \xi_k $$ We know the Stokes vector at one boundary since the incident radiation must be specified in order to have a fully determined problem. So we can start at this boundary and then work our way through the geometry. --- As always, there are also other methods for the formal solution of the transfer equation. One is a Feautrier solution (Auer *et al.* 1977), (Rees *et al.* 1989). Rees *et al.* (1989) also propose an operator perturbation method that is extremely fast. E. Landi Degli'Innocenti (1987) computes an 'evolution operator' $\mathbf{O}(\xi, \xi')$ so that $\mathbf{I}(\xi) = \int \mathbf{O}(\xi, \xi')\mathbf{K}(\xi')\mathbf{S}(\xi')\mathrm{d}\xi'/(k_0\Phi_1)$. Despite the analytical elegance of this method, it appears to be less suited for practical computations than Auer's or the Rees method described above. Del Toro Iniesta and Ruiz Cobo (1996) use the evolution operator method for 'inverting' a Stokes profile, i.e. to compute the vapour parameters from the observed Stokes vector. Martin and Wickramasinghe (1979) use a Runge–Kutta solution of the transfer equation. ## 12.3 Trapping problems with polarization The computation of the radiation for a known source function, which we did in the previous section, is an important first step for solving problems with polarization. However, in real problems, we do not know the source function, but have to compute it, too. We will do that in the present section. For this purpose, we will first discuss an important approximate method, the so-called 'field-free approximation'. In this method, we *assume* that the polarization does not changes the source function, so that we can compute it with the usual methods of Chapters 5–11. With this source function, we then solve the polarized transfer equation to get the emerging radiation. In Sec. 12.3.2, we *exactly* solve the vector Holstein equation when no magnetic field is present; in this case, the influence of the polarization on the source function is included. Finally, Sec. 12.3.3 discusses the exact solution in the presence of a magnetic field. ### 12.3.1 The polarization-free approximation The underlying assumption of the 'field-free approximation' of Rees (1969) is that the polarization properties of the radiation do not strongly influence the excited-state density in the vapour. This is usually fulfilled when the polarization is only due to the geometry of the problem, and when the excited-state density is 'thermalized' with respect to the polarization, either by collisions or by repeated absorption/reemission processes. Having thus computed the source function, the polarization of the radiation is found by formal solution of the vector transfer equation, as described in the previous section, Sec. 12.2. As Rees has shown, the error introduced by this approach is usually less than 10% in steady-state problems. When a magnetic field is present, it is preferable to use the so-called 'polarization-free approximation' of Trujillo Bueno and E. Landi Degli'Innocenti (1996) and Bruls and Trujillo Bueno (1996). In this approximation, the Zeeman splitting due to a magnetic field is taken into account in the lineshape, but the source function is computed via the scalar transfer equation. A somewhat more general approach must be taken in time-decay problems with a polarized initial distribution since we have to know the distribution $n_i(z, t)$ of excited-state atoms, excited by photons that have already suffered $i$ absorption/reemission processes. This distribution can be computed by the multiple-scattering technique (Chapter 6). The degree of polarization left after $i$ absorption/reemission processes is easily computed; see also Berberan-Santos *et al.* (1995). --- ### 12.3.2 The Holstein equation without magnetic field In this subsection, we include also the effect of the polarization on the source function. In the most general case, this would lead to a $4 \times 4$ matrix equation. However, for the case of no magnetic field and a slab or sphere geometry, the problem becomes much simpler, because the Stokes vector components $U$ and $V$ vanish. Due to the absence of a magnetic field, the absorption coefficient becomes a scalar. For a complete description, we need an expression for the source function, which is related to the intensity by $$ \mathbf{S} = \frac{4\pi}{\Phi(x)} \int_{-\infty}^{\infty} \int_{4\pi} R(x, x', \mathbf{e_n}, \mathbf{e_n'}) \mathbf{I}(x', \mathbf{e_n'}) \mathrm{d}\Omega' \mathrm{d}x' \qquad (12.8) $$ The problem can be simplified further when the redistribution function $R$ can be factored into a part that describes only the frequency redistribution, and a part that describes the angle redistribution (Rees and Saliba 1982), (Saliba 1985).[^15] Such a factorization is not exact, since the Doppler shift of the emitted radiation is inherently related to the direction of emission as we have reasoned in Sec. 11.1 (a more accurate description is derived by Faurobert (1987), (Domke and Hubeny 1988), (Nagendra 1994), (Frisch 1996)).[^16] Still, the approximation is usually quite good. The source function in a plane-parallel slab is then computed from $$ \mathbf{S}(x, \mu) = \frac{1}{\Phi(x)} \int R(x, x') \int_{-1}^{1} \mathbf{\Gamma}(\mu, \mu') I(x', \mu') \mathrm{d}\mu' \mathrm{d}x' $$ $$ \mathbf{\Gamma}(\mu, \mu') = \frac{3}{8} \begin{pmatrix} 2(1 - \mu^2)(1 - \mu'^2) + \mu^2\mu'^2 & \mu^2 \\ \mu'^2 & 1 \end{pmatrix} \qquad (12.9) $$ This expression for the angular redistribution matrix $\mathbf{\Gamma}(\mu, \mu')$ is only valid when the atom acts as an electric dipole, which results in the highest degree of polarization. It can be combined with the rate equation to a vector Holstein equation for the auxiliary vector $\mathbf{SA}$. For this, we assume complete *frequency* redistribution (but not depolarization) $$ \frac{\partial}{\partial t} \begin{pmatrix} SA_1(z) \\ SA_2(z) \end{pmatrix} = -A_{21} \begin{pmatrix} SA_1(z) \\ SA_2(z) \end{pmatrix} + A_{21} \int \begin{pmatrix} G_{11}(z, z') & G_{12}(z, z') \\ G_{21}(z, z') & G_{22}(z, z') \end{pmatrix} \begin{pmatrix} SA_1(z') \\ SA_2(z') \end{pmatrix} \mathrm{d}z' \qquad (12.10) $$ where $SA_1(z)$ is the usual source function that we would get for isotropic reemission, and $SA_2(z)$ is $$ SA_2(z) = \frac{3}{8} \int \mathrm{d}x' \Phi(x') \frac{1}{2} \int_{-1}^{1} \mathrm{d}\mu' \left[ I(z, x', \mu')(1 - 3\mu'^2) + 3(1 - \mu'^2)Q(z, x', \mu') \right] \qquad (12.11) $$ [^15]: For those atoms that suffer collisions during the natural lifetime, the redistribution function is $R_{\mathrm{III}}$, i.e. CFR, which implies of course destruction of polarization, so that all elements of the phase matrix are equal to unity. [^16]: The Domke–Hubeny form of the redistribution matrix has been in extensive use in the astrophysical literature. --- The Kernel function is a $2 \times 2$ matrix with the elements $$ \begin{aligned} G_{11}(z, z') &= \frac{C_x}{2} \int k(x)^2 \mathrm{Ei}_1 \left[ k(x)|z - z'| \right] \mathrm{d}x \\ G_{12}(z, z') &= \frac{C_x}{2} \int k(x)^2 \left\{ \frac{1}{3} \mathrm{Ei}_1 \left[ k(x)|z - z'| \right] - \mathrm{Ei}_3 \left[ k(x)|z - z'| \right] \right\} \mathrm{d}x \\ G_{21}(z, z') &= \frac{9}{8} G_{12}(z, z') \\ G_{22}(z, z') &= \frac{C_x}{2} \int k(x)^2 \left\{ \frac{5}{4} \mathrm{Ei}_1 \left[ k(x)|z - z'| \right] - 3\mathrm{Ei}_3 \left[ k(x)|z - z'| \right] + \right. \\ &\qquad \left. + \frac{9}{4} \mathrm{Ei}_5 \left[ k(x)|z - z'| \right] \right\} \mathrm{d}x \end{aligned} \qquad (12.12) $$ This set of coupled integral equations can of course be treated with the piecewise-constant approximation, PCA, or with any other method for integral equations described in Chapter 5. As before, the integral equation form is especially suitable for time-decay problems. The two elements of the source function vector $\mathbf{S}$ are then $$ S_1(z, \mu) = S A_1(z) + \left( \frac{1}{3} - \mu^2 \right) S A_2(z), \quad \text{and} \quad S_2(z, \mu) = (1 - \mu^2) S A_2(z) \qquad (12.13) $$ Once the source function is known, the Stokes vector can finally be computed by formal integration. The solution of the vector Holstein equation by the discrete-space theory is given e.g. by Nagendra (1988, 1989, 1994, 1995). In an exact formulation, we have to relate the polarization to the density matrix elements of the atoms (see Appendix F for an introduction to the density matrix). The first attempt in that direction was made by Barrat (1959), who neglected the Doppler broadening. D'Yakonov and Perel (1965) and Perel and Rogova (1972, 1974) gave a formulation that included the Doppler effect. The system of equations describes the problem under the assumption that the natural linewidth is negligible. The vector Holstein equation reads $$ \begin{aligned} \frac{\partial \rho_{k,k'}(\mathbf{r}, \mathbf{p}, t)}{\partial t} &= -A_{21} \rho_{k,k'}(\mathbf{r}, \mathbf{p}, t) + \\ &\quad + A_{21} \int \mathrm{d}\mathbf{p}' \int \mathrm{d}\mathbf{r}' \sum_{m,m'} G_{k,k',m,m'}(\mathbf{r} - \mathbf{r}', \mathbf{p}, \mathbf{p}') \rho_{m,m'}(\mathbf{r}', \mathbf{p}', t) + \\ &\quad + E_{k,k'}(\mathbf{r}, \mathbf{p}, t) \end{aligned} \qquad (12.14) $$ where $\mathbf{p}$ is the momentum and the generalized Kernel function $G$ is --- $$ G_{k,k',m,m'}(\mathbf{r}, \mathbf{p}, \mathbf{p}') = \frac{n_p}{(2\pi)^3} \frac{1}{2\pi|\mathbf{r}|^2} f1_{k,k',m,m'}(E_0\mathbf{n})\delta(\mathbf{n}v_p - \mathbf{n}v_{p'}) \exp\left[-f2(\mathbf{n}v_p)|\mathbf{r}|\right] $$ $$ f1_{k,k',m,m'}(\mathbf{q}) = \sum_{i,i',j,j'} (C_{\mathbf{q},j,m,i'})^*C_{\mathbf{q},j,k,i}C_{\mathbf{q},j',m',i'}(C_{\mathbf{q},j',k',i})^* \frac{1}{2j_1 + 1} \frac{E_0}{A_{21}} $$ $$ f2(\mathbf{n}v_p) = \pi^{3/2} \frac{n_0}{E_0^3 v_{\mathrm{D}}} \exp\left[ - \frac{(\mathbf{n}v_p)^2}{v_{\mathrm{D}}^2} \right] \frac{2j_2 + 1}{2j_1 + 1} A_{21} \qquad (12.15) $$ Here, $j_1$ and $j_2$ are the $j$-quantum numbers of the initial and final stages of the transition. The elements $C$ describe the interaction between atoms and photons, $$ C_{\mathbf{q},j,k,i} = \left( \frac{2\pi E_0^2}{q} \right)^{1/2} \mathbf{e}_{\mathbf{q},j} \mathbf{d}_{k,i} \qquad (12.16) $$ where $\mathbf{e}$ is the polarization vector of the photon, $\mathbf{d}$ is the dipole moment of the resonance transition, $\mathbf{q}$ is the wave vector of the photon, and $E_0$ is the excitation energy. The system of equations describes the problem under the assumption that the natural linewidth is negligible. Similar equations were also derived by Deech and Baylis (1971), Omont *et al.* (1972), Williams *et al.* (1992) and, using semiclassical arguments, by Sandle and Williams (1971). From the density matrix elements, the irreducible tensor elements $T_{0,0}$, $T_{1,0}$, and $T_{2,0}$ can be computed (see Appendix F); these can be interpreted as density, circular polarization, and linear polarization (alignment), respectively. In a plane-parallel slab, $T_{1,0}$ vanishes, and only the alignment is of interest. Decay times of the alignment can be evaluated approximately by a variational technique (D'Yakonov and Perel 1965), see also (Perel and Rogova 1972) and (Hishikawa *et al.* 1995). At early times, the alignment decays much faster than with $g_0$. The most important result is, however, that under optically thick conditions and at late times, the alignment decays with the same time constant as the population; i.e. the relative alignment stays the same. This effect is known as self-alignment. Perel and Rogova (1972) derived that at late times, the ratio of alignment to excited state density is $$ \frac{1}{1.2k_0L\sqrt{2\pi \ln(k_0L)}} \qquad (12.17) $$ Hishikawa *et al.* (1995) have shown that this alignment is too large by about a factor of two, but that the basic tendency is correct. Generally, the results that can be obtained by a variational solution of the above system of equations are rather inaccurate, and a Monte Carlo study (see below), as performed by Hishikawa *et al.* (1995), seems preferable. In particular, it appears to be the only way to treat also the early-time (i.e. multimode) decay. ### 12.3.3 The Holstein equation with magnetic field In the most general case, i.e. allowing for the possible presence of a magnetic field, the vector Holstein equation becomes an equation for a 4-component vector, and the influence of the field on the lineshapes must be considered. However, various simplifications --- justified by the physical circumstances are often possible. In any case, the resulting system of equations is then solved, e.g., by the Feautrier method (Auer *et al.* 1977), (Dumont *et al.* 1977), (Rees and Murphy 1987), by the discrete-ordinate technique (Domke and Staude 1973), or by discrete-space methods (Nagendra and Peraiah 1984, 1985), (Mohan Rao and Rangarajan 1993). Faurobert-Scholl (1991) uses a Fourier expansion of the radiation field with respect to the azimuth and, alternatively, a Feautrier solution. McKenna (1985) computes the moments with respect to the angles, and gets a set of coupled integral equations, which are then approximated as an equivalent algebraic problem. The resulting expressions are, however, extremely complex. One can also treat the Stokes vector component $Q$ as a perturbation. The most promising approach seems to be a solution by an operator perturbation method (Rees and Murphy 1987), (Rees *et al.* 1989). An iterative method is used by Stenflo (1976) and Stenflo and Stenholm (1976). Rees and Geers (1996) propose the use of wavelets to solve such an equation. In the astrophysical field, E. Landi Degli'Innocenti and coworkers have derived in a series of papers an extremely general quantum-mechanical formulation of the interaction between polarized radiation and atoms (usually including the presence of magnetic fields (E. Landi Degli'Innocenti *et al.* 1991a, b), (E. Landi Degli'Innocenti and Bommier 1994), (E. Landi Degli'Innocenti 1996), (Bommier 1996), and references therein). Another classical paper in that context is Omont *et al.* (1973). When the influence of a magnetic field is studied, usually two limiting cases are investigated. In the limit of a weak magnetic field, the Zeeman splitting is smaller than the natural linewidth. In this regime, the Hanle effect (rotation of the polarization) is of the greatest interest (Stenflo 1978), (E. Landi Degli'Innocenti *et al.* 1990), (Faurobert-Scholl 1991, 1993, 1996), (Frisch 1996). Van Ballegooijen (1987) makes computations for a strong magnetic field (i.e. the classical Zeeman effect in pure form) but with many assumptions that are not fulfilled in laboratory situations. Fields that do not fall under these limiting cases are discussed by Bommier and E. Landi Degli'Innocenti (1996) and E. Landi Degli'Innocenti *et al.* (1991a, b, 1994). ## 12.4 Monte Carlo simulations As always, an MC simulation—being the most general technique—can of course be set up to solve a polarized trapping problem. The simplest case occurs when the atom can be considered as an electric dipole whose axis is parallel to the incident electric field vector. Such an electric dipole has a simple cosine-shaped emission characteristic. Molisch *et al.* (1992a) did such a simulation where linearly polarized radiation was normally incident on the slab. At low opacity, $k_0L < 2$, polarized radiation slightly (less that 5%) reduced the trapping factor. Because of the anisotropic reemission process, it is less likely that photons are reemitted parallel to the boundaries of the slab. Photons moving parallel to the boundaries cannot escape in this step and thus increase the average number of reabsorptions. Since under polarized excitation these photon paths are unlikely, the trapping factor is reduced. The influence of the process vanishes at higher opacities, because the multiple absorptions --- and reemissions randomize polarization. In addition, polarization effects can only occur at very low pressures, since collisions tend to destroy polarization. A more general MC procedure was introduced by Wiorkowski (1988). He considered the changes in the Stokes vector by an absorption–reemission process. The vector component $U$ can be made to vanish by an appropriate rotation of the coordinate system. A rotation of the coordinate system by the angle $\varphi$ results in the Stokes vector given by the transformation $$ \mathbf{I}_{\text{rot}} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(2\varphi) & \sin(2\varphi) & 0 \\ 0 & -\sin(2\varphi) & \cos(2\varphi) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \mathbf{I} $$ (12.18) The relation between the Stokes vectors before and after the absorption-reemission process is given by the phase matrix $\mathbf{P}$ $$ \mathbf{I}_{\text{em}} = \mathbf{P} \cdot \mathbf{I}_{\text{inc}} $$ (12.19) Subscripts inc and em stand for incident and emitted photon, respectively. The following insert lists the general expressions for the phase matrix $\mathbf{P}$. --- ### Insert 12.1 General expressions for the phase matrix The general form of the phase matrix is $$ \begin{aligned} \mathbf{P} = \mathbf{P0}(\mu_{\text{em}}, \mu_{\text{inc}}) &+ \\ + 2\sqrt{(1 - \mu_{\text{em}}^2)(1 - \mu_{\text{inc}}^2)} &[\mathbf{P1}(\mu_{\text{em}}, \mu_{\text{inc}}) \cos(\varphi_{\text{em}} - \varphi_{\text{inc}}) + \mathbf{P2}(\mu_{\text{em}}, \mu_{\text{inc}}) \sin(\varphi_{\text{inc}} - \varphi_{\text{em}})] \\ + 2\mathbf{P3}(\mu_{\text{em}}, \mu_{\text{inc}}) &\cos(2(\varphi_{\text{em}} - \varphi_{\text{inc}})) + 2\mathbf{P4}(\mu_{\text{em}}, \mu_{\text{inc}}) \sin(2(\varphi_{\text{inc}} - \varphi_{\text{em}})) \end{aligned} $$ (12.20) where $\varphi$ and $\vartheta$ are the angles with the $x$ and $z$ axis (i.e. the angles in a spherical coordinate system), and $\mu = \cos(\vartheta)$. We define the matrices $\mathbf{M}_{i,j}$, $$ \mathbf{M}_{i,j} = \begin{pmatrix} \delta_{1,i}\delta_{1,j} & \delta_{1,i}\delta_{2,j} & \delta_{1,i}\delta_{3,j} & \delta_{1,i}\delta_{4,j} \\ \delta_{2,i}\delta_{1,j} & \delta_{2,i}\delta_{2,j} & \delta_{2,i}\delta_{3,j} & \delta_{2,i}\delta_{4,j} \\ \delta_{3,i}\delta_{1,j} & \delta_{3,i}\delta_{2,j} & \delta_{3,i}\delta_{3,j} & \delta_{3,i}\delta_{4,j} \\ \delta_{4,i}\delta_{1,j} & \delta_{4,i}\delta_{2,j} & \delta_{4,i}\delta_{3,j} & \delta_{4,i}\delta_{4,j} \end{pmatrix} $$ (12.21) With the help of these matrices, the parameters $\mathbf{P0}, \dots, \mathbf{P4}$ are given as --- } - \mu_{\text{inc}} (\mathbf{M}_{3,1} + \mathbf{M}_{3,2}) \right] \end{array} $$ (12.23) Wait, the image has a bit of vertical space between P1 and P2. `\\[1ex]` might be good, but standard `\\` is fine. * Check P3, P4 formatting. $$ \begin{array}{l} \mathbf{P3} = \frac{3}{16} c1 \begin{pmatrix} (1-\mu_{\text{em}}^2)(1-\mu_{\text{inc}}^2) & -(1-\mu_{\text{em}}^2)(1+\mu_{\text{inc}}^2) & 0 & 0 \\ -(1+\mu_{\text{em}}^2)(1-\mu_{\text{inc}}^2) & (1+\mu_{\text{em}}^2)(1+\mu_{\text{inc}}^2) & 0 & 0 \\ 0 & 0 & 4\mu_{\text{em}}\mu_{\text{inc}} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \\[4ex] \mathbf{P4} = \frac{3}{8} c1 \begin{pmatrix} 0 & 0 & -(1-\mu_{\ --- MONTE CARLO SIMULATIONS $$ \begin{aligned} I(\mathbf{r}) = I'_{\text{inc}} P_{1,1} + Q'_{\text{inc}} P_{1,2} &= I'_{\text{inc}} + 0.5 c1 I'_{\text{inc}} - 1.5 c1 Q'_{\text{inc}} + \\ &\quad + \sin(\vartheta_{\text{em}})^2 \cdot 1.5 c1 Q'_{\text{inc}} - \\ &\quad - \sin(\varphi_{\text{em}})^2 \cdot 0.75 c1 (I'_{\text{inc}} - Q'_{\text{inc}}) \end{aligned} $$ (12.26) Equation (12.26) can be grouped into three terms, an isotropic term, a term that has the emission characteristic of a dipole oriented in parallel to the $z'$-axis, and a term where the dipole axis is parallel to the $x'$-axis. We can thus imagine that the radiation comes from three orthogonal, independent dipoles (along the $x'$, $y'$, and $z'$ axes) whose squares of amplitudes are $$ D_x = \frac{1 - c1}{3} \quad D_y = \frac{1 + 0.5 c1}{3} - 0.5 c1 Q'_{\text{inc}} \quad D_z = \frac{1 + 0.5 c1}{3} + 0.5 c1 Q'_{\text{inc}} $$ (12.27) The simulation of an absorption–reemission process now proceeds along the following lines. (i) We have an incident Stokes vector (normalized to $|\mathbf{I}_{\text{inc}}| = 1$) and a direction in the laboratory coordinate system specified by the angles $\varphi_{\text{inc}}$ and $\vartheta_{\text{inc}}$. We then choose a coordinate system ($x'$, $y'$, $z'$) centred at the considered atom chosen so that $U'_{\text{inc}} = 0$, $\varphi'_{\text{inc}} = 0$, and $\vartheta'_{\text{inc}} = 90^\circ$. (ii) We compute the Stokes vector component $Q_{\text{inc}}$ in the primed coordinate system from Eq. (12.18). (iii) We compute the dipole amplitudes $D_x$, $D_y$, and $D_z$ from Eq. (12.27) and choose the actually emitting dipole by a random choice—in the sense: choose dipole 1 if $\text{random} < D_x$, and so on. (iv) We compute the emission angles $\varphi'$ and $\vartheta'$ from the emission characteristics of a dipole $$ \begin{gathered} \varphi_{\text{dip}} = 2\pi \text{random} \\ \cos(\vartheta_{\text{dip}}) = 2 \cos \left[ \frac{1}{3} \text{acos}(2 \, \text{random} - 1) + \frac{\pi}{3} \right] \end{gathered} $$ (12.28) where $\vartheta_{\text{dip}}$ is the emission angle with respect to the axis of the dipole and $\varphi_{\text{dip}}$ the angle with respect to the plane normal to the axis. (v) We transform the emission angles $\varphi'$ and $\vartheta'$ into the laboratory coordinate system. (vi) We compute the new Stokes vector by multiplying the old Stokes vector by the phase matrix. For a $J = 0 \rightarrow J = 1$ transition, this algorithm becomes equivalent to the simple electric-dipole modelling of the atom mentioned above. This case actually gives the largest influence on the trapping factor. This can be seen easily from Eq. (12.27). For $J = 0 \rightarrow J = 1$, we have only one possible orientation of the emitting dipole so that the decrease in polarization by one absorption–reemission process will not be too large. For other transitions, we have reemitting dipoles that can be oriented in any direction, so that more polarization will be destroyed in an absorption–reemission process. In the --- [FIGURE: FIG. 12.1. Arrangement for the observation of polarized radiation.] extreme case that $D_x = D_y = D_z$, we have isotropic scattering, and polarization will be completely destroyed by one absorption–reemission process. We finally note that a modification of the Monte Carlo procedure to increase its efficiency is given by Voshchinnikov and Karjukin (1994). ## 12.5 Physical effects After this 'recipe' for an MC simulation of trapping of polarized radiation, we turn to the physical interpretation and to a discussion of the importance of polarization effects. Let us first assume that we are only interested in the behaviour of the total emergent radiation, especially at late times. As mentioned above, even for a $J = 0 \rightarrow J = 1$ transition, the influence of polarization on the trapping factor is very small. Including polarization into the simulations is thus usually not worth the effort. The trapping factor changes by less than 5% even when the radiation is initially completely polarized. Next, let us consider the case that we observe the emergent radiation in a specific direction, see Fig. 12.1 for the experimental arrangement. The observed radiation will strongly depend on the polarization. For a linearly polarized laser beam, as shown in the figure, the initially excited atoms are polarized preferentially parallel to the electric field vector. The first emission of photons will thus be mainly in the direction of detector $b$, while very few photons are emitted towards detector $a$. After several absorption–reemission processes, the polarization will be randomized, and emission will be equal towards both detectors. The net effect of this process can be that the radiation detected at detector $b$ monotonically decreases with time, while the radiation at detector $a$ goes through a maximum, see Fig. 12.2. This effect can lead to serious problems in the lifetime measurements when the alignment is not taken properly into account (Hishikawa *et al.* 1995). It seems to be the main source of uncertainty (Fujimoto *et al.* 1983) in lifetime measurements using the high-frequency deflection technique, a method pioneered --- [FIGURE: Emergent intensity from a 202Hg vapour after excitation by a polarized laser pulse.] FIG. 12.2. Emergent intensity from a $^{202}\text{Hg}$ vapour (cell temperature $-12\,^{\circ}\text{C}$) after excitation by a polarized laser pulse. Detection is parallel and perpendicular to the $E$-vector of the laser pulse. From Wiorkowski and Hartmann (1988). by Erman, see Erman (1975), Erman and Huldt (1978), Erman *et al.* (1988), Erman and Sundström (1991), and Erman and Hishikawa (1992). The effects of radiation trapping in collision spectroscopy—atoms are excited by collisions with electrons or protons, and, among other effects, the resulting polarization of the atoms is observed—is described, e.g., in Mikosza *et al.* (1994), Hishikawa *et al.* (1992), and Khakoo *et al.* (1996). We can also make approximate computations for the emergent radiation, by assuming that the distribution of excited-state atoms and its temporal behaviour is not significantly influenced by the polarization of the radiation. This is of course not strictly true. In the case described above, there will be fewer excited atoms near the top of the cylinder, close to detector $a$, than near detector $b$. However, at small or intermediate opacities the distribution of excited-state atoms has only a comparatively weak influence on the emergent radiation. (At high opacities, the relative influence of the polarization is small anyway. It takes about three natural lifetimes to destroy the polarization, which is quite short compared to the imprisoned lifetime of a high-opacity vapour). As a first step, we make the usual modal solution of the trapping problem. From this we compute the probabilities $p_i$ for photons to escape after $i$ absorption/reemission processes, using the relation between these probabilities and the eigenmodes. We know the degree of polarization left after $i$ absorptions/reemissions from the atomic properties, see above. Given that, we can finally compute the polarization of the emergent radiation, see also Berberan-Santos *et al.* (1995). --- [FIGURE: Calculated rubidium and hydrogen polarizations as a function of temperature. From Redsun et al. (1990).] One example where trapping of polarized radiation becomes important is optical pumping of the Zeeman levels of alkali atoms. In that pumping, light with a certain polarization (e.g., $\sigma^+$-polarization) moves an atom from the Zeeman $m = -1/2$ level of the ground state to the $m = 1/2$ state of the first resonance state. The atom then has a probability of $1/3$ to spontaneously decay to the $m = 1/2$ state of the ground state. Hence there is a mechanism that moves some ground state atoms from the $m = -1/2$ to the $m = 1/2$ sub-state, but no mechanism that works the other way round. We thus get a steady buildup of the population in the $m = 1/2$ state. Trapping can now strongly decrease the efficiency of the pumping process. When a photon is reemitted, it need no longer have a $\sigma^+$ polarization. We have seen above that the absorption/reemission process tends to destroy the polarization—it tends to give equal probability for $\sigma^+$ and $\sigma^-$ polarization. Only when the photon has $\sigma^-$ polarization, can it transfer an atom from the $m = 1/2$ state to the $m = -1/2$ state. Trapping has thus decreased the efficiency of the optical pumping (Tanaka *et al.* 1990). The whole process is actually described by a set of *non-linear* Holstein equations when the pumping is so strong that the number of atoms in each Zeeman substate is changed—which is, after all, the purpose of the whole procedure (Ankerhold *et al.* 1993). Solution methods for non-linear problems are described in Sec. 13.5. Computation of this effect is important, e.g., for the determination of the cross-section for depolarizing collisions (Anderson and Walker 1992). Another application is the production of spin-polarized hydrogen by spin-exchange optical pumping. Again, alkali atoms are polarized by optical pumping. They then transfer their orientation to hydrogen atoms by spin-exchange collisions. In a rough picture, one would expect that the higher the alkali density, the more hydrogen atoms one can --- polarize. However, radiation trapping limits the polarization. When the alkali density is too high, optical pumping of the alkalis becomes very inefficient due to trapping, and the polarization of the hydrogen atoms decreases again, see Fig. 12.3 (Redsun *et al.* 1990); see also Peterson and Anderson (1991), Anderson and Walker (1992), Brissaud (1995), and Brissaud and Jacquemin (1995). Experiments with trapping of polarized light are reported by Belsley *et al.* (1986). An MC simulation of the influence of a weak magnetic field is given by House and Cohen (1969). When we additionally have a magnetic field, the absorption cross-section becomes dependent on the angle between the photon flight path and the magnetic field. For a strong magnetic field, and thus with well-separated lines, this is treated by Tupa *et al.* (1986a, b), for a weak magnetic field by Tupa and Anderson (1987), see also L. W. Anderson (1987). --- # 13 ## NON-LINEAR RADIATION TRAPPING In this chapter, we will drop the assumption of linearity that we have kept up till now. Previously, we have always assumed that the excitation of the vapour was weak, so that the radiation had no influence on the absorption coefficient. This assumption had drastically simplified the mathematics of the trapping problem. More importantly, the assumption is fulfilled very well in many practical cases. The mathematical simplifications stem from the fact that for weak excitation, the describing equations became *linear*. This in turn allowed the use of the very efficient mathematical techniques for solving linear problems, most of which are based on the eigenmodes. Until about ten to twenty years ago, there had been practically no laboratory problems where non-linearities played a role. The advent of the laser has, however, changed that situation with a vengeance. From a radiation-trapping point of view, experiments with lasers basically can be divided into three categories. (i) The laser is so weak that no noticeable non-linearities occur so that we can use the methods described in the previous chapters. We will discuss in detail when this is actually the case. (ii) The laser intensity is on the order of the saturation intensity $I_s$, roughly between $0.1 \cdot I_s$ and $100 \cdot I_s$. In that case, the change of the absorption coefficient has to be included in the generalized Holstein equation or in the rate and transfer equations. (iii) The laser intensity is very high. The laser field strongly interacts with the atoms and leads to new effects—like Stark splitting of the energy levels—that also have an influence on the radiation trapping process. The linear case (i) has been thoroughly treated in the previous nine chapters. In this chapter, we will focus on case (ii). This reflects both the fact that a great number of experiments fulfils the assumption of 'not-too-strong excitation' and the fact that most of the research in non-linear radiation trapping up to now has been done in this intensity domain. In Section 13.1, we analyse in detail under what circumstances non-linear trapping computations are required. Section 13.2 very briefly considers some effects of the interaction between strong laser radiation and atoms. The more mathematical part starts with Sec. 13.3, which deals with solutions for non-linear steady-state problems in two-level atoms, i.e. with saturation. Most of the methods employed are based on the solution of the equation of radiative transfer but use special procedures to increase efficiency. Section 13.4 then considers time-dependent radiative transfer, answering questions like 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- what happens after one has switched off the laser. Solutions in this area may be achieved either by extensions of the steady-state techniques, or by solving a generalized, non-linear Holstein equation. Finally, Sec. 13.5 is devoted to depletion effects in three-level atoms. While the mathematics are quite similar to the previous subsections, some completely new physical effects occur. Non-linear radiation trapping is probably the area where most of the near-future research will be done—both because its practical relevance is ever increasing, and because this is a field where the understanding of the physical processes is not yet complete. Especially in this area of research, we observe a convergence of the approaches of physical chemistry, which is the main thrust of this book, of plasma physics, and of astrophysics. ## 13.1 When do non-linearities occur? Non-linear computations are more complicated than linear ones, and require much more CPU time. Before going through all the trouble of a non-linear computation, one is well advised to first check whether it is really necessary to incorporate non-linearities. A first rule of thumb was already mentioned in Chapter 4. We can forget about saturation effects when the flux density of the excitation is smaller than $F_s/g_0$, where the saturation flux density $F_s$ is typically about $10\text{ W/cm}^2$ (see Sec. 2.1), and $g_0$ is the lowest-order linear trapping factor. This rule gives, however, only a first impression. There are cases where this crude rule leads to completely wrong conclusions. These cases require individual treatment. An important concept in the treatment of one aspect of non-linearity, stimulated emission, is the ‘effective absorption coefficient’, $\kappa$. It jointly describes absorption and stimulated emission, in other words, everything that can be described by the Einstein $B$ coefficients. The effective absorption coefficient is defined as $$ \kappa(x, \mathbf{r}, t) = k(x)\frac{n_1(\mathbf{r}, t)}{N} \left[ 1 - \frac{n_u(\mathbf{r}, t)}{n_1(\mathbf{r}, t)} \frac{g'_1}{g'_u} \right] $$ (13.1) where $k(x)$ is the linear absorption coefficient without stimulated emission, as defined in Chapter 2. The symbol $N$ denotes the total atomic density. With a completely saturated transition, atoms have equal probabilities to be found in either of the participating levels, so that the upper and lower state densities, $n_u(\mathbf{r})$ and $n_1(\mathbf{r})$, are in the ratio of their statistical weights, $g'_u$ and $g'_1$. From the definition of the effective absorption coefficient, Eq. (13.1), one sees that a completely saturated vapour is transparent, i.e. it can absorb no additional radiation. This is obviously physically reasonable, since the complete saturation is caused by an infinitely strong excitation to begin with. Note that we have assumed that the non-linearities are due to depopulation of the ground state, which changes the absolute value of the absorption coefficient, and are not due to a change in the shape of the spectral line. Hence, the effective absorption coefficient can be separated into an effective line centre absorption coefficient $\kappa_0(\mathbf{r}, t)$ and into the spectral lineshape $\hat{k}(x)$, with $\hat{k}(x = 0) = 1$, --- [FIGURE: FIG. 13.1. Geometry of a typical experimental situation. A laser shines along the axis of a vapour cell.] $$ \kappa(x, \mathbf{r}, t) = \kappa_0(\mathbf{r}, t)\hat{k}(x) \quad \text{where} \quad \hat{k}(x) = \frac{k(x)}{k_0} \qquad (13.2) $$ We now look at four cases of practical interest and develop individual rules to establish when one may safely assume linear conditions. (i) *Two-level atoms, transient decay.* The basic case to consider is a two-level atom where one observes the decay of an initial excitation. When the upper-state density at all points in the vapour cell at time zero is much smaller than the ground-state density, then the problem is certainly linear—and will at late times become even more so, since the upper-state density can only decrease over time. Mathematically, the condition is $$ \frac{n_u(\mathbf{r}, 0)}{g_u^\prime} \ll \frac{n_l(\mathbf{r}, 0)}{g_l^\prime} \qquad (13.3) $$ Actually, Eq. (13.3) is somewhat too stringent for practical purposes. Consider the setup depicted in Fig. 13.1. We have a cylindrical vapour cell where a small region of radius $R_{\text{NL}}$ is excited. The excitation in this region is so strong that Eq. (13.3) is not fulfilled. In the extreme case of complete saturation, the excited region is even transparent. As these excited-state atoms decay, the effective absorption coefficient in the excited region increases and reaches the linear, unsaturated absorption coefficient $k_0$ at late times. When now $k_0 \cdot R_{\text{NL}} \ll 1$, the reabsorption in the 'non-linear' region is negligible at all times, and the actual effective absorption coefficient is irrelevant. The reabsorption is completely determined by the 'linear' region, and radiation trapping can thus be described by the linear Holstein equation. This case is of great practical interest, since it corresponds to the often encountered situation that a strong laser pulse with small beam diameter has been used to excite the vapour. (ii) *Two-level atoms, steady-state.* The second case of prime interest is the steady-state excited density upon continuous excitation of the vapour by a lamp or laser. Without radiation trapping, the --- saturation conditions for a single atom, described in Chapter 2, would determine the linearity of the problem $$I_{exc} \ll I_s = \frac{A_{ul}}{B_{lu}}$$ (13.4) where $I_{exc}$ stands for the (angle-averaged) intensity of the excitation source. In an ensemble of atoms, radiation trapping leads to an increase in the effective lifetime, while it has no influence on the stimulated emission. When the radiation excites the whole cell, the increase in lifetime is approximately by a factor $g_0$, so that for linearity, the excitation source must fulfil $$I_{exc} \ll I_s = \frac{A_{ul}}{g_0(k_0 R) B_{lu}}$$ (13.5) As before, the condition is too stringent when the laser does not excite the whole cell. The trapping factor $g_0(k_0 R)$ describes the reabsorption probability in the whole cell. However, the reabsorption rate in the directly excited region (which is the most critical part of the cell) is small when the region is small. In the limit of a very small excited region ($k_0 R_{NL} \ll 1$), we have no reabsorption of photons released elsewhere in the prime excited region, and for judgement of linearity, Eq. (13.4) is valid. For the intermediate case—a considerable part of the cell is excited—we can sort the trapped photons into two groups. One group consists of those photons that are reabsorbed in the excited region; these have a trapping factor of $g_0(k_0 R_{NL})$. Photons of the second group are those that are reabsorbed somewhere in the outer region and are then reemitted back to the excited region. For these photons, we make the following estimate. All in all, the steady-state distribution of excited atoms is close to the lowest-order mode, which can be approximated (*very* roughly) by a uniform distribution. The intensity in the cell, averaged over the volume, is approximately $I_{exc} \cdot (R_{NL}/R)^2 \cdot g_0(k_0 R)$ uniformly over the cell. When the excited region is quite small, this quantity could be interpreted as being caused mainly by photons that had already escaped from the directly excited region and are now reemitted back into this region. The worst-case estimate is now to add the contributions of the two groups of the photon distribution, so that $$I_s = \frac{A_{ul}}{\left[ g_0(k_0 R_{NL}) + g_0(k_0 R) (R_{NL}/R)^2 \right] B_{lu}}$$ (13.6) This equation will underestimate the saturation intensity by a factor of two at most. This is amply sufficient, since we only need an order-of-magnitude estimate anyway. In addition, the estimate is on the safe side. Note, that these sophistications only come to bear in a high opacity vapour, where the increase in the apparent lifetime by a large trapping factor $g_0$ considerably lowers the external intensity required for the onset of saturation effects in the vapour. --- A relation that is very useful in that context was derived by Bezuglov (1992). He considers the central region of a partially excited cylinder and investigates how the steady-state density of excited atoms (in the *linear* case) is changed by the presence of the secondary-excitation region. He keeps the radius of the primary excited region constant and increases the radius of the vessel, to get two functions, $$ \begin{aligned} f1(R_{\text{NL}}) &\approx \frac{\pi \gamma}{\sin(\pi \gamma)} \left[ 1 - \frac{\gamma \sin(\pi \gamma)}{\pi(1-\gamma)} \left( \frac{R_{\text{NL}}}{R} \right)^{2(1-\gamma)} \right] \\ f2(R_{\text{NL}}) &\approx \frac{\Gamma(1-\gamma)\Gamma(1+2\gamma)}{\Gamma(1+\gamma)} \left[ 1 - \frac{\Gamma(2+\gamma)(\gamma)^2}{\Gamma(2-\gamma)\Gamma(1+2\gamma)} \left( \frac{R_{\text{NL}}}{R} \right)^{(1-\gamma)} \right] \end{aligned} $$ (13.7) where the parameter $\gamma$ equals 0.25 for a Lorentz line and 0.5 for a Doppler line. Function $f1$ gives the ratio of the excited-state densities at the axis of the cylinder for a vessel of the size of the laser beam compared to a vessel of radius $R$. The function $f2$ is the same ratio for the excited-state densities averaged over the primary-excited regions. We can thus make quite accurate computations of the effects of photons from the outer regions for the considered steady-state case. (Note that there are some typing errors in Bezuglov (1992), as pointed out by N. N. Bezuglov (private communication).) (iii) *Three-level atoms, steady-state.* In three level atoms with one long-lived level, the situation is more complicated. Take the level scheme in Fig. 13.2. Level $c$ has a comparatively short natural lifetime, while level $b$ has a long one. Level $b$ thus acts as a kind of storage reservoir. When we excite the vapour on transition $ac$, we indirectly pump photons from level $a$ via level $c$ to level $b$. (A more detailed account on the physics and mathematics will be given in Sec. 13.5.) For the onset of stimulated emission on the $ac$ transition, the conditions derived above are valid. However, we must remember that non-linearities already occur when the absorption coefficient changes. Since we transfer atoms from level $a$ to level $b$, a change in the absorption coefficient occurs when we have removed a noticeable fraction of state-$a$ atoms. To mathematically account for this effect, we have to multiply the saturation intensities derived in (ii) by the factor $\tau_c/\tau_b$. For typical three-level configurations, this factor is on the order of $10^{-3}$--$10^{-6}$, so that often a few mW of excitation power are sufficient to cause non-linearities. A second condition for linearity is that the absorption coefficient of the $bc$ transition does not change noticeably, in other words, transition $bc$ must stay transparent. Let us again make a worst-case estimate. When transition $ac$ is heavily trapped, all photons sent into the vapour will eventually end up transferring an atom from state $a$ to state $b$. With an excitation power $P_{\text{exc}}$, the number of exciting photons is $N_{\text{photon}} = P_{\text{exc}}/(h\nu_{ac})$, thus transferring $N_{\text{photon}}$ atoms (per unit volume $V$) to state $b$. The loss rate of state-$b$ atoms is $N_b/\tau_b$. The steady-state --- [FIGURE: FIG. 13.2. Energy level scheme of a three-level system with one metastable level.] density of state-$b$ atoms is thus $N_b = P_{\text{exc}}\tau_b/(V h \nu_{ac})$. In order that the vapour is not able to noticeably absorb on the $bc$ transition, we require that $$ \frac{P_{\text{exc}}\tau_b}{V h \nu_{ac}} \sigma_{bc} R \ll 1 \qquad (13.8) $$ (iv) *Three-level atoms, transient decay.* Again, we require that the absorption coefficient of both the $ac$ and the $bc$ transitions does not change noticeably during the whole decay process. This is fulfilled if $n_c$, the density of atoms in state $c$, is much smaller than both $n_a$ and $n_b$—to be precise, when $$ \frac{n_c(\mathbf{r}, 0)}{g'_c} \ll \frac{n_a(\mathbf{r}, 0)}{g'_a} \quad \text{and} \quad \frac{n_c(\mathbf{r}, 0)}{g'_c} \ll \frac{n_b(\mathbf{r}, 0)}{g'_b} \qquad (13.9) $$ The latter condition cannot be fulfilled when there are no atoms in state $b$ to begin with. In that case, we again make the worst-case estimate that all state-$c$ atoms decay to long-lived state $b$. Reabsorption at transition $bc$ is not noticeable when $$ \int n_c(\mathbf{r})\mathrm{d}\mathbf{r} \sigma_{bc} \ll 1 \qquad (13.10) $$ ## 13.2 Interaction of strong laser radiation with atoms When we actually have high-intensity radiation, the physics of the absorption–reemission processes itself will change. The quantum-mechanical study of these effects is a very active field of research, and a complete description of all these effects would require a book of its own. We just consider the effects that occur at moderate intensities; a brief overview of high-field effects can be found in Appendix G. --- Essentially, there are two ways to include the effects of the photons on the absorption and emission processes. The first, and simplest way is to define modified absorption and emission coefficients that include the effect of the field. One example for this approach is the ac-Stark splitting. In the Stark effect, a strong field leads to a splitting of the atomic levels, and thus to a change in the absorption and emission coefficients. The usual trapping computations can then be done with these modified coefficients. Of course one has to take into account that when the radiation intensity is noticeably increased by trapped photons, the absorption coefficient has to be adjusted to include also this intensity contribution. In considering these effects, bandwidth effects generally must also be included for application to trapping problems. We have already noticed in the preface that the semiclassical approach we use in this book is only an approximation. This becomes especially true when dealing with high-intensity radiation fields. The more exact approach is to perform quantum-mechanical computations, either with a classical, or with a quantized description of the field, and to include all the radiation trapping effects in this approach. The density matrix formalism appears to be especially suited for this purpose. This formalism essentially consists of a rate equation for the quantum states, so that additional rates, like the reabsorptions, can be incorporated with manageable effort. A brief introduction to density matrix computations can be found in Appendix G. Formulations of some radiation trapping problems by this method have been undertaken e.g. by Burnett and Cooper (1980a, b) and Burnett *et al.* (1980). However, the description is still much more complicated than with our semiclassical approach. We know of only one paper where the density matrix formalism has been used for the solution of actual non-linear radiation trapping problems (Ellinger *et al.* 1994) and we will not consider it further. The rest of this section deals with modifications that can be done in the framework of semiclassical descriptions, and are important for moderate laser intensities. ### 13.2.1 Selection of velocity groups For predominant Doppler broadening, a narrow-band laser excites only those atoms that have the appropriate velocity component in the direction of the laser beam—e.g., zero velocity if the laser is tuned to the line centre. When the excitation is quite weak, this will not matter very much. The absorption coefficient will not depend on the intensity of the excitation, but will solely be determined by the density of atoms and by the detuning of the laser. The situation is different with a strong narrow-band laser. Let us for a moment imagine a vapour with no homogeneous broadening. If we ignored the velocity selection, a strong laser would cause complete saturation of the vapour. This is, however, physically wrong. Only atoms with the correct velocity will absorb photons (Gainer *et al.* 1982), (Beterov *et al.* 1970) and the laser will ‘burn’ a ‘hole’ into the velocity distribution of the atoms; these holes are known as ‘Bennet holes’. With some amount of homogeneous broadening, also atoms that have approximately the correct velocity can absorb photons. ‘Approximately’ means that the atoms are no more Doppler shifted from the correct velocity than a homogeneous linewidth. For low --- [FIGURE: FIG. 13.3. A narrow-band laser burns a hole into an inhomogeneously broadened absorption coefficient.] laser intensity, the appropriate homogeneous linewidth is the sum of the natural and the collisional broadening contributions. When the intensity is on the order of the saturation intensity or larger, one has to use the power-broadened linewidth, see Appendix G. This is so because at high intensities there is a good chance to excite atoms in the far wings of the homogeneous lines—even at quite large mismatches. Figure 13.3 shows an absorption lineshape with a Bennet hole in it. Note that the same hole will be in the velocity distribution. This velocity selection can mean that only a small fraction of all atoms is initially excited. The effect can be very important for the initial conditions of a trapping computation. The trapping process itself is hardly influenced, because CFR destroys the velocity selection. A typical outcome might be that despite a strong large-area excitation, we can nevertheless make linear trapping computations. A related problem is the computation of PFR including stimulated emission. In a first approximation, the rate equation for partial redistribution is (Baschek *et al.* 1981) (note that this is a correction of the equation given by Mihalas (1978) and Milkey and Mihalas (1973)) $$-n_l B_{lu} \int R(x, x')J(x')dx' + n_u\psi(x) \left( A_{ul} + B_{ul} \int \psi(x')J(x')dx' \right) = 0$$ (13.11) Here, we have written the density of atoms that emit photons of frequency $x$ as $n_u\psi(x)$. It must be stressed, however, that this equation is only approximate, and relies on the --- assumption that the expression $$ \iint I(x, \mathbf{e}_n)\psi(x, \mathbf{e}_n, \mathbf{v})dxd\Omega $$ (13.12) is independent of velocity $\mathbf{v}$. The function $\psi$ denotes the emission profile. This assumption must be checked *a posteriori*, since the intensity $I$, i.e. what we want to compute, enters critically. Furthermore, we have used the angle-averaged redistribution function. There is also the question of what the redistribution function $R(x, x')$ looks like. The derivation of the redistribution functions in Chapter 11 has assumed a Maxwellian velocity distribution for the ground-state atoms. In the linear case, this is a perfectly good assumption, because the unexcited vapour is in thermal equilibrium and, hence, has a Maxwellian velocity distribution. In the linear regime there are comparatively few excited-state atoms, so that the velocity distribution is not changed by the excitation. In non-linear trapping, on the other hand, the velocity distribution may have some distinct 'holes' burned in by a narrow-band source. The required averaging would then have to be done over the modified velocity distribution, resulting in a different redistribution function. To make things even more complicated, the velocity distribution itself depends on the spatial coordinate and is influenced by the trapping process. Accurate computations would have to include all these effects, which leads to an enormous complication of the computation (Simmoneau 1985), (Borsenberger *et al.* 1985–87). Further considerations of the effects of population inversion can be found in the rich literature concerning amplified spontaneous emission and lasers. A further complication is caused by the fact that for high intensities, the semiclassical picture of an atom always being in one of its eigenstates breaks down (Cooper *et al.* 1983). This has important consequences for the profiles of stimulated and of spontaneous emission. While the two are identical in the semi-classical picture, they become different at high intensities—at least in the rate equation, while remaining the same in the transfer equation. For a completely exact treatment, the quantum-mechanical results of Cooper *et al.* (1982) must be used. It has not yet been clarified at which intensities these effects set in to have an important influence, especially in case the radiation is comparatively broadband. We know of no trapping computations that actually make use of these sophisticated considerations. ### 13.2.2 *Burn-through and one-dimensionality* We have emphasized in Chapter 4 that in a cylindrical cell we often cannot make use of the infinite-cylinder approximation even if this would be justified from purely geometrical considerations. We have given the example of a weak laser tuned to line centre, with an optically thick vapour in the cell. The radiation is absorbed close to the entrance window and we have to make two-dimensional computations. When now the laser intensity is very high, the laser will 'burn through' the vapour, and the excitation will be more uniform in the $z$-direction. The assumption of an infinite cylinder will thus be more often justified. Let us assume that the laser pulse is short compared to the natural --- lifetime. In that case, we have no radiation trapping effects during the excitation phase, and the differential equations describing the burn through read $$ \begin{aligned} \frac{\mathrm{d} I(z, x, t)}{\mathrm{d} z} &= -k(x) \frac{N - n_u(z, t)}{N} \left[ 1 - \frac{g_l'}{g_u'} \frac{n_u(z, t)}{N - n_u(z, t)} \right] I(z, x, t) \\ \frac{\mathrm{d} n_u(z, t)}{\mathrm{d} t} &= -A_{ul} n_u(z, t) - \frac{B_{ul}}{4\pi} \int I(z, x, t) \Phi(x) \mathrm{d}x n_u(z, t) + \\ &\qquad + \frac{B_{lu}}{4\pi} \int I(z, x, t) \Phi(x) \mathrm{d}x (N - n_u(z, t)) \end{aligned} $$ (13.13) For such a short excitation, and when Doppler broadening predominates over the homogeneous contributions, we can sort the atoms into 'velocity groups' that are individually excited by the frequency components of the laser radiation. For each velocity group, the laser appears approximately monochromatic, and there is no interaction between the velocity groups. We can thus solve the problem for each velocity group separately. It can be shown, e.g. by using Laplace transform techniques, that since the duration of the laser pulse, $\Delta t$, is much smaller than the natural lifetime, the equations describing the burn-through simplify to $$ \frac{\mathrm{d} I}{\mathrm{d} z} = -k_0 I \frac{1}{1 + \frac{\Delta t}{\tau} \cdot \frac{I}{F_S} \cdot \left( 1 + \frac{g_l'}{g_u'} \right)} = -k_0 I \frac{1}{1 + I / I_s'} $$ (13.14) where $F_S$ now denotes the laser saturation flux at the appropriate frequency, as defined in Eq. (2.15). In most cases, the laser will be narrow-band and tuned to the line centre. Figure 13.4 shows the location where the excited-state density has decayed to 1/e of its maximum value as a function of the laser intensity for two different absorption coefficients. We see that when we require the 1/e-point to be at $L = 1 \text{ cm}$ from the entry point, the necessary intensity is roughly $I = I_s' \cdot k_0 L$. From this, and with the considerations of Chapter 4, we can determine whether the infinite-cylinder assumption is justified. ### 13.3 Steady-state solutions For linear radiation trapping, we had a very simple relationship between the transient and the steady-state solutions. The steady-state solution was just the sum of appropriately weighted eigenmodes. In non-linear trapping, such simple relationships do not exist—there are not even eigenmodes—so that the transient and the steady-state problems have to be treated separately. Most of the solution procedures originally come from astrophysics. There are mainly three basic techniques for the treatment of the non-linearities: operator perturbation, complete linearization, and simple iteration. We will describe the principle of all three methods, and also give the equations necessary for the implementation of the basic routines. Still, these descriptions are only a very brief account of the research done in this --- [FIGURE: Log-log plot of 1/e-point (cm) vs Normalized intensity (I / I'_s) for k_0=5cm^-1 and k_0=50cm^-1] FIG. 13.4. Distance within which the upper-state density has decreased to 1/e of its value at the cell entrance. area. There are literally dozens of papers dealing with specific aspects of these techniques and with methods to decrease computation times. All these specializations of the basic techniques are efficient under specific physical conditions (usually to be found in the atmospheres of some stars) and are thus not described here. For the interested reader, we have assembled most of these papers in the bibliography. A lot of similar work has also been done in the context of targets exploded by strong laser pulses (used for X-ray lasers). This work is currently being declassified, and a lot of papers are being published. However, the emphasis in these papers is usually on the physics of the plasma, and radiation trapping is taken into account by the numerical codes as a side-effect by basically the same techniques. ### 13.3.1 Complete linearization The complete linearization technique (Auer and Mihalas 1969) (see also (Mihalas 1978), (Mihalas and Mihalas 1984)) is one of the most important techniques for treating non-linear problems. Basically, it is a straightforward generalization of Newton's root search algorithm taught at every high school. The detailed equations are, however, extremely complicated, and the efficiency and convergence of this algorithm are a subject of very advanced research. Before going into details, we thus first summarize the basic technique. We are dealing with a mathematical problem of the form $$f(y(x), x) = \Lambda y(x) \qquad (13.15)$$ --- where $\Lambda$ is a linear operator and $f$ is a known non-linear function. Variable $y$ is the unknown. We start out with a guess value $y_0$. The true value for $y$ can be thought to consist of the guess value plus a first order correction $\delta y_0$, $$ y(x) = y_0(x) + \delta y_0(x) \qquad (13.16) $$ We henceforth drop the dependence of $y$ on $x$. We then make a truncated Taylor series expansion, $$ f(y_0 + \delta y_0, x) \approx f(y_0, x) + \delta y_0 \left. \frac{\partial f(y, x)}{\partial y} \right|_{y=y_0} \qquad (13.17) $$ $$ \Lambda(y_0 + \delta y_0) = \Lambda y_0 + \Lambda \delta y_0 $$ Inserting this expansion into Eq. (13.15), we get a linear equation for the correction $\delta y_0$ $$ \Lambda \delta y_0 = \delta y_0 \left. \frac{\partial f(y, x)}{\partial y} \right|_{y=y_0} - \Lambda y_0 + f(y_0) \qquad (13.18) $$ Solution of this equation then gives the next guess value $y_1 = y_0 + \delta y_0$, which is then used instead of $y_0$ in Eq. (13.18) for the computation of $\delta y_1$, and so on, until convergence is achieved. After having delineated the basic principles of the technique, we now go into the detailed linearized transfer and rate equations for a two-level atom in a plane parallel slab. Generalizing the formulation of Chapter 4, the basic equations are the equation of radiative transfer $$ \mu \frac{\partial I}{\partial z} = - \frac{k(x)}{N B_{lu}} [n_l B_{lu} I - n_u B_{ul} I - A_{ul} n_u] \qquad (13.19) $$ where $k(x)$ is the absorption coefficient without stimulated emission (and consequently with all atoms in the ground state), and the rate equation for the upper-state atoms $$ 0 = -A_{ul} n_u + \frac{n_l B_{lu}}{2} \iint I C_x k(x) dx d\mu - \frac{n_u B_{ul}}{2} \iint I C_x k(x) dx d\mu. \qquad (13.20) $$ Again we have dropped the functional dependencies, except for $k(x)$, to simplify notation. The last equation we need is the material conservation, $$ n_u + n_l = N \qquad (13.21) $$ The true solutions are now written as $$ n_u \approx n_{u,0} + \delta n_{u,0}, \quad n_l \approx n_{l,0} + \delta n_{l,0}, \quad I \approx I_0 + \delta I_0. \qquad (13.22) $$ The first guess values are usually derived from some analytical estimate or some very robust and fast but inaccurate numerical technique—we will discuss the possibilities below. When these guesses are physically reasonable, then the material conservation --- $n_{u,0} + n_{1,0} = N$, and the boundary conditions for the radiation will also be fulfilled. In that case, we have a zero boundary condition for $\delta I_0$, and the corrections for the particle density are $$ \delta n_{u,0} + \delta n_{1,0} = 0; $$ (13.23) also for all other higher-order corrections. Inserting the Taylor expansions into the rate and transfer equations, we get $$ \begin{aligned} \mu \frac{\partial \delta I_0}{\partial z} = & - \frac{k(x)}{N B_{lu}} \left[ \delta n_{1,0} B_{lu} I_0 - \delta n_{u,0} B_{ul} I_0 - A_{ul} \delta n_{u,0} \right] - \frac{k(x)}{N B_{lu}} \left[ n_1 B_{lu} \delta I_0 - n_u B_{ul} \delta I_0 \right] \\ & - \frac{k(x)}{N B_{lu}} \left[ n_{1,0} B_{lu} I_0 - n_{u,0} B_{ul} I_0 - A_{ul} n_{u,0} \right] - \mu \frac{\partial I_0}{\partial z} --- same methods, like the Feautrier technique, improved perhaps by Ribycki organization, the variable Eddington factor, etc. After having solved Eq. (13.27) for $\delta I_0$, it is trivial to compute $\delta J_0$ from Eq. (13.25). The corrections to the particle densities $\delta n_{u,0}$ and $\delta n_{l,0}$ are then computed from $$ \begin{aligned} \delta n_{l,0} &= \frac{n_{u,0} B_{ul} J_0 + n_{u,0} A_{ul} - n_{l,0} B_{lu} J_0}{(B_{lu} + B_{ul}) J_0 + A_{ul}} + \frac{n_{u,0} B_{ul} \delta J_0 - n_{l,0} B_{lu} \delta J_0}{(B_{lu} + B_{ul}) J_0 + A_{ul}} \\ \delta n_{u,0} &= -\delta n_{l,0} \end{aligned} $$ (13.29) With the complete corrections, $\delta J_0$, $\delta n_{u,0}$ and $\delta n_{l,0}$, we can now calculate the new estimates $J_1$, $n_{u,1}$, and $n_{l,1}$. We then use these instead of $J_0$, $n_{u,0}$, and $n_{l,0}$ in the next solution of Eq. (13.27). The procedure is repeated until some convergence criterion is fulfilled. The complete linearization technique, CLT, can be made more efficient by the Kantorovich method (Hubeny and Lanz 1992). In order to explain this technique, we first write the basic equation in a slightly different form $$ \Lambda^{\mathrm{NL}} y = 0 $$ (13.30) where $\Lambda^{\mathrm{NL}}$ is a non-linear matrix operator that incorporates both the operator $\Lambda$ and the function $f(y)$ of Eq. (13.15). To be more general, we allow $y$ to be a vector, which is relevant for multilevel problems. The solution for the complete linearization method can be written as $$ \delta \mathbf{y}_n = -\mathrm{Jac}(\mathbf{y}_n)^{-1} \Lambda^{\mathrm{NL}} \mathbf{y}_n $$ (13.31) where $n$ denotes the $n$th iteration, and $$ \mathrm{Jac}_{i,j} = \frac{\partial \Lambda^{\mathrm{NL}}_i}{\partial y_j} $$ (13.32) are the elements of the Jacobian of the system and obviously depend on the values $\mathbf{y}_n$. The Kantorovich method is now to compute this Jacobian and its inverse only up to the $m$th iteration step. For all further steps $n > m$, we still use $\mathrm{Jac}(\mathbf{y}_m)^{-1}$ instead of $\mathrm{Jac}(\mathbf{y}_n)^{-1}$. This eliminates the matrix inversions, and thus saves computer time. However, $\mathbf{y}_m$ must already be close to the final solution, or convergence might fail. An alternative formulation of the complete linearization, which can be used for a Rybicki reorganization, is given explicitly by Auer and Heasley (1976). It is furthermore possible to include the rate equation into the transfer equation instead of writing it as a separate equation (Auer 1973), however, the savings are small. A linearization of the integral formulation of radiation trapping in a plane-parallel slab is given by Kalkofen (1974a). The end result of the formulations is always the same: an equation for the correction terms that has the basic form of the linear equations, but with some inhomogeneous 'driving' term that describes the error in the approximation of the previous step. In plasmas, it may be necessary to linearize not only the transfer and rate equations, but also the equations for hydrodynamic equilibrium, electron density, and temperature. This is then --- true 'complete linearization'. When only the transfer and rate equations are linearized, this is sometimes known as 'partial linearization'. If temperature, electron density, etc. strongly depend on the radiation field, partial linearization might fail to converge (Kalkofen 1985b). However, such strong dependencies are rare in laboratory situations, so that the equations given above are usually sufficient to tackle a problem. A somewhat related technique was also proposed by Skumanich and Domenico (1971). It is essentially a Newton root search for the integral-equation formulation. The technique described by Auer and Mihalas (1968b) could also be viewed as a partial linearization with respect to temperature. ### 13.3.2 *Operator perturbation* The operator perturbation technique, OPT, is based on approximating exact operators, whose computation takes a lot of time, by simple approximate operators. The errors introduced with this approximation are then done away with an iterative procedure (Cannon 1984), (Kalkofen 1985a, 1987a, b). The basic form of the equations in each iteration step is the same as for the original equation, but with the simple operator $\Lambda^*$ replacing the true operator $\Lambda$ (the OPT is thus also often called ALI, for accelerated lambda iteration). The 'driving term' (i.e. inhomogeneity) is always some known function. In the zeroth iteration, it is just the usual excitation term. In the subsequent iterations, the driving term is the difference between the approximate and the exact operator as applied to the solution *of the previous iteration*. The technique is thus an extremely general and powerful one. It can not only be applied to radiation trapping, but to a wide class of physical problems. The efficiency of the OPT essentially depends on a clever choice of the approximate operator, otherwise convergence of the correction series may be slow or non-existent. We believe that these perturbation techniques are among the most important advances in radiation trapping theory since Holstein's pioneering work, and that they will become more and more important as the complexity of the considered problems continues to increase. Many approximate methods which by themselves do not yield sufficient accuracy can be used for the construction of the approximate operators. Examples for such methods are the escape factor or effective lifetime technique (Sec. 4.5), the Milne approximation (Chapter 8), and Sobolev's approximation (Sec. 11.8). The OPT is usually applied to *linear* problems, but we present it at this point, because it *can* also be used for *non-linear* problems. **Basic formulation of the OPT** The OPT always follows the following basic steps. (i) Write down the exact mathematical formulation of the problem in the form $$ f(x) = \Lambda y \qquad (13.33) $$ (ii) Find an approximate operator $\Lambda^*$ which gives similar physics but whose inverse is easier to compute. Then compute the first-order solution $y_0$ from $$ f(x) = \Lambda^* y_0 \qquad (13.34) $$ --- STEADY-STATE SOLUTIONS (iii) Compute the difference between the approximate and the exact operators as applied to the first order solution $y_0$ $$ \varepsilon_0 = (\Lambda^* - \Lambda)y_0 \qquad (13.35) $$ (iv) Compute the new value of y as $$ y_{i+1} = (\Lambda^*)^{-1} (\varepsilon_i + f(x)) \qquad (13.36) $$ (v) Compute the new difference as $$ \varepsilon_{i+1} = (\Lambda^* - \Lambda)y_{i+1} \qquad (13.37) $$ (vi) Go back to step (iv), until the corrections are sufficiently small. The speed and the convergence of this method depend on the choice of the approximate operator $\Lambda^*$. In the extreme case that $\Lambda^* = \Lambda$, we need just one iteration step, but the computation of the zeroth order solution $y_0$ is very CPU-time intensive, since it means just the full solution of the original problem. In the other extreme of a too simple approximate operator, convergence can be slow or non-existent. The 'art' of transfer theorists lies now in finding operators that are best suited for given problems. Unfortunately, most of this work has been done with astrophysical applications in mind. Nevertheless, many of the ideas can also be applied to laboratory situations. Some care must be taken in the application of perturbation techniques. As Kalkofen (1987d) has shown, the original derivation by Cannon is flawed, and does not converge to the correct solution. This derivation was based on a set of differential equations, where the solution converges, but to a wrong value. On the other hand, the basic idea applied to an integral formulation works very well. Thus, when the operator $\Lambda$ is, e.g., the integral operator of the Holstein equation, $\iiint \mathrm{d}\mathbf{r}' G(\mathbf{r}, \mathbf{r}')$, we can use the above recipe without any modifications. When we want to apply the OPT to the transfer-plus-rate differential equation formulation, we have to use a modified approach. Let the basic formulation be given by the set of equations $$ \begin{aligned} I &= (1 - \Lambda_{\mathrm{d}})^{-1} S \\ S &= \Omega I + E \end{aligned} \qquad (13.38) $$ where $\Lambda_{\mathrm{d}}$ is the differential operator of the transfer equation, e.g. a second-order differential operator for the Feautrier formulation. Operator $\Omega$ represents the integration over angle and frequency, and $E$ is the external excitation term. By using analogies with the integral-equation approach, Kalkofen showed that this problem can be treated by the following set of differential equations $$ \begin{aligned} \left[ (1 - \Lambda_{\mathrm{d}}^*) - \Omega^* \right] \delta I_{i+1} &= \varepsilon_i \\ \delta S_{i+1} &= \Omega^* \delta I_{i+1} + \varepsilon_i \end{aligned} \qquad (13.39) $$ where $I_i = I_0 + \Sigma \delta I_i$ and $S_i = S_0 + \Sigma S_i$. --- The error term is now computed in two steps—that is why we use the half-integer indices below. We first make a ‘formal solution’ of the problem, i.e. we find the intensity for a given source function, $$I_{i+1/2} = [1 - \Lambda_d]^{-1} S_i$$ (13.40) The ‘new’ source function is then computed as $$S_{i+1/2} = \Omega I_{i+1/2} + E$$ (13.41) and the error term as $$\varepsilon_i = S_{i+1/2} - S_i$$ (13.42) A problem with these formulations for differential operators is that they are rather complicated to derive, i.e. via the analogy with an integral operator formulation, and are not easily generalized to treat PFR, for example. Using the integral operator formulation thus seems preferable for most purposes. This is not a real problem, since most of the important papers on that subject, e.g. the Scharmer method and the OKAB method (see below) are integral formulations. Only Cannon’s original papers were in the differential equation formulation, plus most reviews of the subject that are based on Cannon’s formulation. ### Quadrature Perturbation The most straightforward approximate operator is one where the number of quadrature points for the angle and frequency integration points is small. In the Feautrier technique, we have operators of the form $$\Lambda\Theta(z, x, \mu) = \sum_{i=1}^{N_x} \sum_{j=1}^{N_\mu} p_i q_j \Theta(z, x_i, \mu_j)$$ (13.43) We bluntly define an approximate operator $\Lambda^*$ with fewer quadrature points $$\Lambda^*\Theta(z, x, \mu) = \sum_{i=1}^{N_x^*} \sum_{j=1}^{N_\mu^*} p_i q_j \Theta(z, x_i, \mu_j)$$ (13.44) so that the solution of the Feautrier equation with $\Lambda$ replaced by $\Lambda^*$ works faster by a factor of $[(N_x N_\mu)/(N_x^* N_\mu^*)]^3$. In the most elemental case, the simplified operator $\Lambda^*$ is just a single-point quadrature, so that the solution of the Feautrier equation is extremely fast. In practice, it is quite common to use $N_\mu^* = 1$ for the number of directions (this is the Eddington approximation, see Sec. 8.2), and to choose the number of frequencies $N_x^*$ on the order of five. For the evaluation of the perturbation term, we have to know the values of the Feautrier intensity $\Theta$ for the frequencies and directions $x_i$ and $\mu_j$ at *all quadrature points of the exact operator*. In order to do this, we first compute the source function --- using our approximate solution for $\Theta$ and then solve the equation of radiative transfer in the Feautrier formulation to get the intensity $\Theta$ at any arbitrary angle and frequency. One important point that remains to be discussed is the convergence of the method. When $N_x^*$ is five or larger, convergence is almost guaranteed. Even with only a single-quadrature point not only for the angle, but also for the frequency quadrature, the series converges in most cases—one possibility that will usually give convergence is to choose this single frequency quadrature point $x_1$ in such a way that $k(x_1)L = \bar{k}L$, where $\bar{k}L$ is the equivalent Milne–Samson opacity, see Sec. 8.4. It is finally interesting to compare the numerical effort of this technique to the piecewise-constant integral equation technique. For a fair comparison, we assume also for the integral equation technique that the $A_{k,m}$ elements are computed by summing over $N_\mu N_x$ quadrature points. The main numerical effort lies then in the inversion of the matrix of size $N_z$. With the QPT, we have to solve $N_z$ very small matrices, then solve the transfer equation $N_x N_\mu$ times, and repeat this until convergence is achieved. For large $N_z$, this can be more efficient than the inversion of a matrix of size $N_z$. Using the operator $\Lambda^*$ with a low number of quadrature points is not efficient in the integral technique, since (for the slab case), the numerical effort for the computation of the $A_{k,m}$ elements scales only linearly with the number of quadrature points. ### Scharmer's method An OPT method that is well known in astrophysics is Scharmer's method (Scharmer 1981, 1984), (Scharmer and Carlsson 1985a, b), (Carlsson 1985) (see also (Kalkofen 1984b) for a relation between Scharmer's method and the operator perturbation technique), and (Nordlund 1984). In this method, the approximate operator is based on the core saturation technique of Chapter 9. The approximate *monochromatic* $\Lambda$ operator, which is the formal solution of the transfer equation, is defined (for a semi-infinite atmosphere) as $$ \Lambda_\nu^*(\xi, \xi') = \begin{cases} \delta(\xi_\nu - \xi_\nu') & \xi_\nu > \xi_{\text{esc}} \\ \frac{1}{2}\delta(\xi_\nu' - \xi_{\text{esc}}) & \xi_\nu < \xi_{\text{esc}} \end{cases} $$ (13.45) where $\xi$ is the opacity to the boundary along a ray with angle $\mu = \cos \vartheta$, and $\xi_{\text{esc}}$ is some arbitrary opacity parameter that determines the boundary between the core and the wings of the line. The parameter is on the order of unity since $\exp(-\xi_{\text{esc}})$ describes the escape probability of a wing photon. This operator is physically a combination of the core saturation technique and of the Eddington–Barbier relation, which states that the source function at optical depth $\xi_\nu = \xi_{\text{esc}}$ determines the emergent intensity at frequency $\nu$. The resulting matrix operator has an upper (or lower) triangular structure, so that it can be solved very quickly. However, as pointed out by Olson *et al.* (1986), the method works efficiently only in one-dimensional astrophysical problems, and is thus of limited use for laboratory situations. Still, the method is in widespread use in the astrophysical community, especially since the program of Carlsson (1985) is easily available and clearly structured. Many astrophysicists have written modifications that add specific effects; e.g. Harper (1994) treats large-scale movements in spherical geometries. Such --- ready-made programs might have strong appeal also for laboratory situations, at least for first estimates. The choice of the frequency grid in this algorithm is discussed by Stift and Moser (1993). A related method was proposed by Hamann (1987) and Werner (1987) (see also Kalkofen 1987b), where $\Lambda^*$ is defined as $$ \Lambda_\nu^*(\xi, \xi') = \begin{cases} \delta(\xi_\nu - \xi_\nu') & \quad \xi_\nu > \xi_{\text{esc}} \\ 0 & \quad \xi_\nu < \xi_{\text{esc}} \end{cases} $$ (13.46) so that the operator is strictly diagonal. This is related to the 'core saturation' idea in the following way. In the line core, we have $\Lambda^* S = S$, whereas we have escape in the line wings, so that we actually consider the physics of core and wings separately. This can also be viewed as 'accelerated $\Lambda$ iteration'. In the usual $\Lambda$ iteration, we need as many iteration steps as absorption/reemission processes are suffered by a photon before escape. With Hamann's operator, this diffusion process is speeded up by a factor that equals 1 over the fraction of wing photons. In Eq. (13.45), the number of iterations is smaller, but the operator is no longer strictly diagonal. ### The OKAB method Another, very important technique was developed by Olson, Kunasz, Auer, and Buchler—henceforth referred to as the OKAB method (Olson *et al.* 1986), (Olson and Kunasz 1987), (Kunasz and Auer 1988). These authors do not use the physics of the trapping problem, but use the mathematics of the basic matrix inversion problem. The main numerical effort in the solution of the transfer problem lies in the inversion of a matrix, which is represented by the operator $\Lambda$. The idea of OKAB is now to define the approximate operator $\Lambda^*$ as a diagonal matrix where the non-zero elements are simply the same as in the exact matrix. One can choose either to use only the main diagonal, in which case the inversion of the matrix is trivial, or a three-diagonal matrix (upper and lower diagonal besides the main diagonal). It can be proven that the scheme achieves convergence. In order to achieve a localized description of the transfer problem, we use the so-called 'short-characteristic' method for the formal solution of the equation of radiative transfer. In a plane-parallel slab, this description reads $$ \begin{aligned} I^+(\xi_i) &= I^+(\xi_{i+1}) \exp(-\Delta\xi_i/|\mu|) + \Delta I_i^+ \\ I^-(\xi_i) &= I^-(\xi_{i-1}) \exp(-\Delta\xi_{i-1}/|\mu|) + \Delta I_i^- \end{aligned} $$ (13.47) where $\xi_i$ is an abbreviation for the opacity term $k(x)z_i$. The dependencies of intensity $I$ on $\mu$ and $x$ have been suppressed for notational convenience. This equation just means a reduction of the transfer problem to the 'local' boundary conditions. For positive directions $\mu > 0$, the intensity at point $\xi_{i+1}$ is equal to the intensity at point $\xi_i$ minus the absorption, plus the increase due to the emission from the source function. The intensity increments $\Delta I^\pm$ must thus depend on the source function. In order to interpolate between the grid points, we can use either linear or parabolic interpolation. At the boundaries, only linear interpolation is possible. --- STEADY-STATE SOLUTIONS In both cases, linear and parabolic, the interpolation equation looks like $$ \Delta I_i^\pm = c1_i^\pm S_{i-1} + c2_i^\pm S_i + c3_i^\pm S_{i+1} \qquad (13.48) $$ The coefficients for the linear interpolation are $$ \begin{aligned} c1_i^- &= d0_i - \frac{d1_i |\mu|}{\Delta \xi_{i-1}} & c1_i^+ &= 0 \\ c2_i^- &= \frac{d1_i |\mu|}{\Delta \xi_{i-1}} & c2_i^+ &= \frac{d1_{i+1} |\mu|}{\Delta \xi_i} \\ c3_i^- &= 0 & c3_i^+ &= d0_{i+1} - \frac{d1_{i+1} |\mu|}{\Delta \xi_i} \end{aligned} \qquad (13.49) $$ For the parabolic interpolation, the coefficients are $$ \begin{aligned} c1_i^- &= d0_i + |\mu| \frac{d2_i |\mu| - (\Delta \xi_i + 2\Delta \xi_{i-1}) \, d1_i}{\Delta \xi_{i-1} (\Delta \xi_i + \Delta \xi_{i-1})} \\ c1_i^+ &= |\mu| \frac{d2_{i+1} |\mu| - \Delta \xi_i d1_{i+1}}{\Delta \xi_{i-1} (\Delta \xi_i + \Delta \xi_{i-1})} \\ c2_i^- &= |\mu| \frac{(\Delta \xi_i + \Delta \xi_{i-1}) \, d1_i - d2_i |\mu|}{\Delta \xi_{i-1} \Delta \xi_i} \\ c2_i^+ &= |\mu| \frac{(\Delta \xi_i + \Delta \xi_{i-1}) \, d1_{i+1} - d2_{i+1} |\mu|}{\Delta \xi_{i-1} \Delta \xi_i} \\ c3_i^- &= |\mu| \frac{d2_i |\mu| - \Delta \xi_{i-1} d1_i}{\Delta \xi_i (\Delta \xi_i + \Delta \xi_{i-1})} \\ c3_i^+ &= d0_{i+1} + |\mu| \frac{d2_{i+1} |\mu| - (\Delta \xi_{i-1} + 2\Delta \xi_i) \, d1_{i+1}}{\Delta \xi_i (\Delta \xi_i + \Delta \xi_{i-1})} \end{aligned} \qquad (13.50) $$ For both interpolations, the following parameters are needed: $$ d0_i = 1 - \exp \left( \frac{\Delta \xi_{i-1}}{|\mu|} \right), \quad d1_i = \frac{\Delta \xi_{i-1}}{|\mu|} - d0_i, \quad d2_i = \left( \frac{\Delta \xi_{i-1}}{|\mu|} \right)^2 - 2d1_i. \qquad (13.51) $$ The normalized intensities $\hat{I}_k$ for inward and outward going rays are then: --- $$ \begin{array}{lc} \text{for} & \hat{I}_k \text{ inward} \\ k < i - 1 & 0 \\ k = i - 1 & c3_{i-1}^- \\ k = i & c3_{i-1}^- \exp\left(-\frac{\Delta\xi_{i-1}}{|\mu|}\right) + c2_i^- \\ k = i + 1 & \left[ c3_{i-1}^- \exp\left(-\frac{\Delta\xi_{i-1}}{|\mu|}\right) + c2_i^- \right] \exp\left(\frac{\Delta\xi_i}{|\mu|}\right) + c1_{i+1}^- \\ k > i + 1 & \hat{I}_{k-1}^- \exp\left(-\frac{\Delta\xi_{i-1}}{|\mu|}\right) \\ \\ \text{for} & \hat{I}_k \text{ outward} \\ k < i - 1 & \hat{I}_{k+1}^+ \exp\left(-\frac{\Delta\xi_k}{|\mu|}\right) \\ k = i - 1 & \left[ c1_{i+1}^+ \exp\left(-\frac{\Delta\xi_i}{|\mu|}\right) + c2_i^+ \right] \exp\left(\frac{\Delta\xi_{i-1}}{|\mu|}\right) + c3_{i-1}^+ \\ k = i & c1_{i+1}^+ \exp\left(-\frac{\Delta\xi_i}{|\mu|}\right) + c2_i^+ \\ k = i + 1 & c1_{i+1}^+ \\ k > i + 1 & 0 \end{array} $$ (13.52) Finally, the elements of the $\Lambda$ matrix in the main, the upper, and the lower diagonal are $$ \begin{aligned} \Lambda_{i-1,i} &= \frac{C_x}{2} \int_{-\infty}^{\infty} \int_{0}^{1} k(x)(\hat{I}_{i-1}^- + \hat{I}_{i-1}^+)\mathrm{d}\mu\mathrm{d}x \\ \Lambda_{i,i} &= \frac{C_x}{2} \int_{-\infty}^{\infty} \int_{0}^{1} k(x)(\hat{I}_{i}^- + \hat{I}_{i}^+)\mathrm{d}\mu\mathrm{d}x \\ \Lambda_{i+1,i} &= \frac{C_x}{2} \int_{-\infty}^{\infty} \int_{0}^{1} k(x)(\hat{I}_{i+1}^- + \hat{I}_{i+1}^+)\mathrm{d}\mu\mathrm{d}x \end{aligned} $$ (13.53) As mentioned above, this defines the approximate operator $\Lambda^*$, which can be inverted easily. The diagonal elements of the inverse of a tridiagonal $N \times N$ matrix can be found in order $N$ operations (Rybicki and Hummer 1991). The method can also be extended to multidimensional geometries, especially to the two-dimensional slab (Kunasz and Auer 1988). The only difference is that the interpolation equations become a bit more complicated, because the intersections between the considered ray and the grid lines are usually not on the specified grid points. Computation of the intersections and interpolation is just a tedious exercise in geometry. More important to note is the fact that the parabolic interpolation is useful only on Cartesian grids, but not in curvilinear ones, e.g. for the finite cylinder. It has been shown that a parabolic interpolation is necessary to achieve the correct limiting --- behaviour at very high opacities—however, in most cases this is not really crucial for laboratory problems. In two- or three-dimensional geometries, it is no longer efficient to include also the upper and lower diagonals of all the nearest neighbours, because they are separated by $N_x$ or $N_z$ elements from the main diagonal. A discussion of this and further numerical details for the two-dimensional slab can be found in Kunasz and Olson (1988) and Auer and Paletou (1994). Puls and Herrero (1988) compared the OKAB method to Scharmer's method (in the implementation of Werner and Hamann), and found it to be superior with respect to convergence speed and stability. Hauschildt *et al.* (1994) analysed the convergence in the one-dimensional case when the approximate operator is the main diagonal plus an arbitrary number of off-diagonals; this means that they trade the computation time in a single iteration cycle against the number of iterations. Väth (1994) extended the method to three dimensions and designed a code for massively parallel computers in order to be able to deal with the large CPU-time requirements caused by 3D-geometries. The addition of partial frequency redistribution was done by Auer and Paletou (1994) and Paletou *et al.* (1993). The OKAB method was applied e.g. by Adam (1990) and Eastman and Pinto (1993). It was modified to include PFR by Griffioen *et al.* (1994). ### Other approximate operators Generally, solutions to simple problems can be used as the basis of the operator perturbation method. In the example above, solutions with a low number of quadrature points in angle and frequency have been used as the basis for high-accuracy solutions. Similarly: **The solutions for the CFR problem can be used as the basis for the PFR problem** (Cannon *et al.* 1975), (Cannon 1984), (Scharmer 1983). The former approaches have to be used with care, since they are based on differential equations (see comments above), so that Scharmer's method seems more appropriate. The generalized Holstein equation is in formal notation $$ S(x) = \frac{1}{4\pi C_x k(x)} \int_\Omega \int_{-\infty}^\infty R(x, x') \Lambda S(x') \mathrm{d}\Omega \mathrm{d}x' + E^S $$ (13.54) where the dependence on the spatial coordinates has been dropped. We consider only angle averaged distribution functions; inclusion of angle dependence is trivial. The approximate operator is the CFR operator, and the first step in the solution method is thus the solution of Eq. (13.54) with $R(x, x') = C_x k(x) \cdot C_x k(x')$, to get the zero-order iteration $S_0^c$. Next, we compute the frequency-dependent error term $\varepsilon_i$ from $$ \varepsilon_i(x) = S_i(x) - \frac{1}{4\pi C_x k(x)} \int_\Omega \int_{-\infty}^\infty R(x, x') \Lambda S_i(x') \mathrm{d}\Omega \mathrm{d}x' - E^S $$ (13.55) and average it by $$ \bar{\varepsilon}_i = \frac{1}{4\pi} \int \int C_x k(x) \Lambda \varepsilon_i(x) \mathrm{d}x \mathrm{d}\Omega $$ (13.56) We now imagine that the frequency-dependent error $\varepsilon_i(x)$ is due partly to a frequency-dependent local error in the source function $S_i^c$, and partly to a frequency-independent error in the average intensity $\delta J_i$. The correction $\delta J_i$ can be computed from --- $$ \delta J_i - \frac{1}{4\pi} \int_{\Omega} \int_{-\infty}^{\infty} C_x k(x') \Lambda \delta J_i \mathrm{d}\Omega \mathrm{d}x' = -\bar{\varepsilon}_i \qquad (13.57) $$ This problem is the usual CFR problem, with the corresponding lower numerical effort. The correction $\delta S_i$ is computed from $$ \delta S_i(x) = -\varepsilon_i(x) + \delta J_i \qquad (13.58) $$ and of course the total source function is given by $S_i = S_{i-1}^c + \delta S_i$. With that, we compute a new error term $\varepsilon_i(x)$, and proceed until convergence is achieved. A generalization to the multilevel case is given by Uitenbroek (1989). An alternative method was proposed by Hubeny and Lites (1995). **For particle movement, the solutions to the static case can be the basis** of the solution including particle movement (Cram and Lopert 1976), (Kalkofen 1987c). Other OPT methods for expanding vapours (stellar atmospheres, but of course also applicable to plasmas) are based on operators derived from comoving frame formulations (Hempe and Schönberg 1986, Schönberg and Hempe 1986), or Sobolev-based methods (Hamann 1985, 1986, 1987), (Swartz 1990). A formulation suitable for shock waves (including discontinuities of the opacity) is given by Ulmschneider (1994). **For two-dimensional problems, the one-dimensional case can be used as a basis.** A combination of perturbations in angle, frequency, geometry, and velocity is described by Cannon (1976). The method of choice for two-dimensional problems, is, however, the OKAB method described above. Yet another operator perturbation method was designed by Hauschildt (1992ab) for a spherically symmetric medium that is moving so fast that the relativistic transfer equation must be used. The convergence properties of the OPT are discussed by Cram (1977). ### Convergence Finally, it is noteworthy that the iteration can be accelerated by a kind of extrapolation, taking a number of previous solutions into account (Ng 1975). Typically between one and four old solutions are used. Let $\mathbf{y}$ denote the solution vector. The accelerated estimate is then $$ \mathbf{y} = \left( 1 - \sum_{m=1}^{M} c_m \right) \mathbf{y}_n + \sum_{m=1}^{M} c_m \mathbf{y}_{n-m} \qquad (13.59) $$ The coefficients $c_m$ of the acceleration are computed as solution of the matrix equation $\mathbf{M}^{\text{acc}} \mathbf{c} = \mathbf{b}$, $$ \begin{aligned} M_{i,j}^{\text{acc}} &= \sum_l \frac{1}{|y_{n-1,l}|} \left[ \Delta y_{n,l} - \Delta y_{n-i,l} \right] \left[ \Delta y_{n,l} - \Delta y_{n-j,l} \right] \\ b_i &= \sum_l \frac{1}{|y_{n-1,l}|} \Delta y_{n,l} \left[ \Delta y_{n,l} - \Delta y_{n-i,l} \right] \end{aligned} \qquad (13.60) $$ where $\Delta y_{n,l}$ is the $l$th component of the vector $\mathbf{y}_n - \mathbf{y}_{n-1}$. --- Second-order acceleration is usually sufficient, but the above general equations allow computation of the coefficients for arbitrary order of acceleration. Another acceleration method was devised by Bond (1984b); however, it requires the knowledge of the largest eigenvalue of the Feautrier matrix and thus seems to be of rather little use in practice (Olson *et al.* 1986). Stability and convergence of the OPT are also treated by Bond (1983b). A similar acceleration scheme is the ‘Orthamin’ accelerator described by Auer (1991). ### 13.3.3 Direct iteration Direct iteration is a very straightforward technique to treat non-linear problems. It is much simpler, and more robust, but also much less efficient than e.g. complete linearization. We start out with a first guess value for the excited-state density and solve the transfer equation with this source function. The thus-computed intensity is then used to solve the rate equation for an improved value of the excited-state density. These excited-state densities are employed for the next solution of the transfer equation, and so on, until convergence is achieved. Direct iteration has the big advantage that during the iteration, all results (also the intermediate results) are physically reasonable. A certain prescribed or assumed excited-state distribution must lead to a physically reasonable distribution of the radiation—after all, we do not know how this distribution was prescribed; it might just as well have been by appropriate exciting electrons and quenching molecules as by the writer of the computer program. The thus-computed radiation distribution must lead to a physically reasonable distribution of excited-state atoms (the same argument applies). While this is not an exact mathematical proof of convergence, it at least indicates a good-natured numerical behaviour. The CLT, on the other hand, has extremely rapid convergence, but might not converge at all when the starting value is badly chosen—again, this is in analogy to the usual Newton root search, which has quadratic convergence if it converges, but might diverge with a bad starting value. It thus may often be a good idea to combine CLT and direct iteration. First use direct iteration to get to the vicinity and then use the CLT to converge rapidly to the true solution. If such a combination is used, it is clear that an approximate description of the problem for the direct iteration part is sufficient (e.g. Eddington approximation, Milne approximation, escape factor technique). In astrophysics and plasma physics, such a first step is often not necessary, because the local thermodynamic equilibrium, LTE, values for the excited-state density are good enough as starting values. For a direct iteration scheme, we can also use some acceleration technique. Examples are Cassinelli (1971) and Matis (1971). It can also be combined with the complete linearization technique (Wu 1992). We now give a short description of the appropriately formulated equations in the Eddington approximation when a laser is incident on the cell. Physically, the Eddington approximation means that fast variations of the intensity with the angle are neglected. This is a good approximation for the spontaneously emitted radiation, which is emitted isotropically (under the assumption of CFR). However, it is a very bad approximation --- for the exciting laser beam, which has the fastest possible angular variations. The vital trick is now to split the intensity into two parts, $I(\Omega) = I_{\text{res}}(\Omega) + I_{\text{laser}}(\Omega)$, and to use the angle approximation only for $I_{\text{res}}$. The equation of radiative transfer for $I_{\text{res}}$ then becomes (see Chap. 8.3) $$ \frac{1}{3} \nabla^2 J_{\text{res}} = \frac{1}{3\kappa} \nabla (J_{\text{res}}) \cdot \nabla(\kappa) + (\kappa)^2 \left( J_{\text{res}} - \frac{A_{\text{ul}}}{B_{\text{lu}}} \frac{n_{\text{u}}}{n_{\text{l}} - n_{\text{u}}(g_{\text{l}}'/g_{\text{u}}')} \right) $$ (13.61) The boundary conditions are that there is no diffusely incident external radiation—only the unidirectional laser beam—so that the Cauchy-type boundary condition reads $$ J_{\text{res}}(\mathbf{r}) = - \frac{2}{3\kappa(\mathbf{r})} \frac{1 + \Xi}{1 - \Xi} \frac{\partial J_{\text{res}}(\mathbf{r})}{\partial \mathbf{n}} $$ (13.62) A possible reflection coefficient $\Xi$ of the boundaries, which may also depend on location, $\Xi(\mathbf{r})$, is incorporated. The normal on the cell boundary is denoted as $\mathbf{n}$. We now assume a finite-cylinder geometry, where the laser radiation enters at the bottom of the cylinder. In order to find the intensity of the laser excitation at point $(r_{\text{e}}, z)$, we have to compute the absorption from the point of entry $(r_{\text{e}}, -L/2)$ to the point of interest $(r_{\text{e}}, z)$ according to $$ I_{\text{laser}}(r_{\text{e}}, z, \nu) = I(r_{\text{e}}, -L/2, \nu) \exp \left[ - \int_{-L/2}^{z} \kappa(r_{\text{e}}, z', \nu) \, \mathrm{d}z' \right] $$ (13.63) We furthermore need the rate equations for the excited-state atoms. The rate equation reads $$ \begin{aligned} 0 &= -A_{\text{ul}} n_{\text{u}} + n_{\text{l}} B_{\text{lu}} \int J_{\text{res}} \Phi(x) \, \mathrm{d}x - n_{\text{u}} B_{\text{ul}} \int J_{\text{res}} \Phi(x) \, \mathrm{d}x - C_{\text{ul}} n_{\text{u}} + C_{\text{lu}} n_{\text{l}} \\ &\quad + n_{\text{l}} B_{\text{lu}} \int I_{\text{laser}} \Phi(x) \, \mathrm{d}x - n_{\text{u}} B_{\text{ul}} \int I_{\text{laser}} \Phi(x) \, \mathrm{d}x + D \nabla^2(n_{\text{u}}) \end{aligned} $$ (13.64) where $D$ is a possible diffusion constant of the excited atoms, and $C_{\text{ul}}$ and $C_{\text{lu}}$ are the rate constants for a possible collisional energy transfer between upper and lower state. A discussion of the boundary conditions for $n_{\text{u}}$ is given in Sec. 10.4. Solution of the transfer and rate equations can be done, e.g., by finite differencing (see also Appendix E) and by direct iteration. This method is described in more detail by Molisch *et al.* (1997). A similar approach is to use the exact transfer equation and to integrate for a given source function along the straight-line path of a ray (Nordlund 1985). This method has the big advantage that the exact transfer equation is only one-dimensional even in a three-dimensional geometry. On the other hand, we need quite a lot of rays, and the interpolation can be non-trivial. --- ### 13.3.4 *Approximate techniques* Non-linear radiation trapping is extremely complicated, and its computation puts heavy demands on the computer-time budget of the researchers. It is thus clear that approximate techniques, which allow a quick first impression of the physical effects, would be desirable. First among those techniques is the escape factor (effective lifetime) technique. In the description of linear radiation trapping, we have already given the appropriate warnings: we cannot expect that this technique gives the correct spatial distribution of excited-state atoms. However, it can give a reasonable guess on the *average* number of excited atoms. This technique was not very useful for linear problems, because other simple analytical techniques exist that give better accuracy with similar numerical effort. However, these techniques were usually based on linear mathematical techniques, and cannot work for the non-linear problems treated now. Escape factor methods are thus often the best choice to get estimates for non-linear problems. The Holstein equation for the steady-state reads $$ E(\mathbf{r}) = \frac{n_u(\mathbf{r})}{\tau} - \int G(\mathbf{r}, \mathbf{r}')n_u(\mathbf{r}')\mathrm{d}\mathbf{r}' \qquad (13.65) $$ which is in the escape factor approximation $$ E(\mathbf{r}) = \eta(\mathbf{r})n_u(\mathbf{r}) \qquad (13.66) $$ where $\eta$ is the escape factor. In the non-linear case, Eq. (13.66) is still valid. The only difference is that now the escape factor depends on the excited-state density, $n_u$, via the effective opacity. Again, we have to make an iteration. We first make an estimate for $n_u$, compute the escape factor, use this to compute a new $n_u$, and so on. The big difference to the previous methods is that the escape factor can be written down (at least approximately) as an *analytical function*, so that a new evaluation in a new iteration step means no numerical effort. Another approximate technique, described already in the previous subsection, is the Eddington approximation—perhaps combined with the Milne approximation. This procedure already gives quite a good approximation also to the spatial distribution of the excited-state atoms, but of course costs much more CPU time than the escape factor approximation. Other types of approximations, e.g. second-order escape factor techniques, are reviewed in Kalkofen (1984a). ### 13.3.5 *Monte Carlo simulations* The simple Monte Carlo procedure described in Chapter 6 is intended for describing a linear problem. However, by a small change in the code, it can also be used for the computation of non-linear problems. In order to account for the non-homogeneous absorption coefficient, the whole vapour cell is divided into subcells. Again, each photon is traced on its path through the vapour, and each time that it is absorbed, the (local) effective absorption coefficient is adjusted accordingly. This is continued until the change --- in the population density is below a certain defined limit. In order to keep the CPU time within reasonable limits, we can reduce the number of photons by some reduction factor. Of course, the number of atoms (and the saturation intensity) also has to be reduced. Physically, we can think of this reduction as photon 'bundles', where all photons have the same properties and are absorbed and reemitted in exactly the same way. The magnitude of the reduction factor determines the accuracy and the runtime requirements. ## 13.4 Transient problems In the linear case, we have usually been able to solve the whole transient trapping problem in a single step, by computing the eigensolutions of the Holstein equation. In the non-linear case, this is not possible, and we have to take a different approach to include the time-dependence. The almost universally used approach is some kind of finite differencing in the time domain. Before going into the algorithmic details, we want to again present reasons why there are no eigenmodes in the non-linear case. We have mentioned in Chapter 4 that only the lowest-order mode makes physical sense, all the higher-order modes also take negative values, which by themselves are not physically meaningful—they are only useful when we build up the total distribution as an appropriate *linear* combination of eigenmodes. Since we now have a non-linear describing equation, it is clear that linear combinations do not make much mathematical sense. There is also one further, more physical reason. For the linear case, the modal description allowed us to separate the spatial and the temporal dependencies of the excited-state distribution. In the non-linear case, the current amplitude and spatial distribution of the excited-state atoms determines the absorption coefficient and thus the temporal behaviour via the escape probability. We can still write the excited-state distribution as the product of a spatial distribution with an exponential time constant, but now the time constant changes with time and depends on the spatial distribution. This formulation is thus of no help in the actual computations—it is just used sometimes to allow for easier comparisons with linear results. ### 13.4.1 Numerical solution of the non-linear Holstein equation In this section, the transient problems in two-level atoms are described by the generalized Holstein equation. Formally, there is very little difference between the linear and the non-linear Holstein equation; we just replace the line-centre absorption coefficient $k_0$ with the effective line-centre absorption coefficient $\kappa_0$. Consequently, the Kernel function now also depends on $n_u(\mathbf{r}, t)$, so that the Holstein equation reads $$ \frac{\partial n_u(\mathbf{r}, t)}{\partial t} = -A_{ul}n_u(\mathbf{r}, t) + A_{ul} \int G(\mathbf{r}, \mathbf{r}', t, n_u)n_u(\mathbf{r}', t)d\mathbf{r}' \qquad (13.67) $$ where the Kernel function $G$ is now --- $$ G(\mathbf{r}, \mathbf{r}', n_u, t) = \frac{1}{4\pi\rho^2} \hat{C}_x \int \hat{k}(x)^2 \kappa_0(\mathbf{r}, n_u, t) \exp\left[ - \int_{\mathbf{r}'}^{\mathbf{r}} \kappa_0(\mathbf{l}, n_u, t)\hat{k}(x)\mathrm{d}l \right] \mathrm{d}x $$ (13.68) and, as before, $\rho = |\mathbf{r} - \mathbf{r}'|$. We note here in passing that this description is not the most general one. It is possible that the lineshape also depends in some way on the spatial coordinate, and even on the excited-state distribution. In that case, one can just replace $\kappa_0(\mathbf{r}, n_u, t) \cdot \hat{k}(x)$ by $\kappa_0(\mathbf{r}, n_u, t, x)$ and then proceed in exactly the same way as described below. However, this situation rarely occurs in chemical physics, so it is not described in greater detail. The change in the formal description of the problem introduced by the non-linearity is very small, just that some additional dependencies occur in the absorption coefficient. The change both in the solution methods and in the physics of the problem is, however, dramatic. The most straightforward solution method for this generalized Holstein equation is a time domain explicit Euler finite differencing scheme. For practical importance, we consider the infinite cylinder geometry. We again approximate $n(r, t)$ as a sum of rectangular pulse functions, but now the amplitudes are time dependent $$ n(\mathbf{r}, t) = \sum_{k=0}^{N_r-1} n_k(t)p(\mathbf{r} - \mathbf{r}_k). $$ (13.69) Inserting this into Eq. (13.67), we get a system of integro-differential equations for the densities $n_k(t)$. We then apply a forward finite-differencing scheme to these equations (the explicit Euler scheme, see e.g. (Press *et al.* 1993)), $$ \frac{\partial n_k(t)}{\partial t} \approx \frac{n_k(t + \Delta t) - n_k(t)}{\Delta t} $$ (13.70) The integro-differential equations thus become a set of algebraic equations $$ \frac{n_k(t_1 + \Delta t) - n_k(t_1)}{\Delta t/\tau} = -n_k(t_1) + \sum_m A_{k,m}(t_1)n_m(t_1), $$ (13.71) where the elements $A_{k,m}$ are computed as in Secs. 5.3 and 10.5, but with absorption coefficient $k(\mathbf{r})$ replaced by $\kappa_0(n_u(\mathbf{r}, t_1)) \cdot \hat{k}(x)$. It is important that the absorption coefficient is now inhomogeneous—it depends on the excited-state distribution, which is a function of position. We thus have to compute the $A_{k,m}$ elements with the formulas developed for an inhomogeneous distribution of absorbers. For the slab case, computation with an inhomogeneous absorber distribution is very simple, because a transformation of variables allows the reduction to the homogeneous case. Now for the homogenous slab case, we could use, e.g., the fitting formula solutions to describe the excited-state distribution. It is thus possible to solve the trapping equation *analytically* for each timestep, resulting in considerable CPU time savings. For the cylinder and the sphere, the numerical techniques described in Chapter 10 allow reasonably fast numerical computations. In any case, the $A_{k,m}$ elements have to be evaluated anew for each --- timestep, with the absorption coefficient $\kappa$ determined by the excited-state distribution at the current time $t_1$. This shows that efficient evaluation of the matrix elements is of utmost importance. This explicit Euler scheme is very similar to the propagator function method described in Sec. 5.4. In the original PFM, the time step $\Delta t$ was constant throughout the computations and chosen equal to the natural lifetime $\tau$, which makes sense for the linear problem. In the non-linear case, an adaptive choice of the stepwidth is necessary. For good accuracy, both the reabsorption rate of photons and the number of excited atoms should change during one timestep by no more than a factor of $c_e$, a constant on the order of 0.1. We can trade off accuracy and CPU time by choosing $c_e$ smaller for better accuracy, or larger for faster computation. When the initial condition is a fully saturated vapour, we will have no trapping in the beginning, and the excited-state population will decrease with the natural lifetime. This means that the number of absorbers will increase by the same amount. For a Doppler lineshape, this is an increase in the average reabsorption rate by $k_0$ (for a Lorentzian it is proportional to $k_0^{0.5}$; the equations below have to be modified accordingly). This means, in turn, that the ratio $\Delta t/\tau$ should be chosen as $c_e/k_0$. At high opacities, this can be a quite stringent requirement. However, we do not have to use such a fine discretization for the whole temporal evolution. At late times, the effect of saturation has almost vanished, and the excited-state density will decay like $\exp(-t/g_0\tau)$, where $g_0$ is the lowest-order trapping factor without saturation. Then, it is sufficient to choose the timestep from the ratio $\Delta t/\tau = c_e \cdot g_0$. Note that the trapping factor $g_0$ is approximately proportional to $k_0$ for a Doppler lineshape. Interpolating between these regions, we choose (for a Doppler lineshape) the stepwidth according to $$ \frac{\Delta t}{\tau} = c_e \frac{(1 + \kappa_0(t_1))^2}{1 + k_0} $$ (13.72) This makes sure that the reabsorption rate and the excited-state density increases by no more than $c_e$ during one timestep. When the excited-state distribution has decreased to about 10% of its value in the fully saturated vapour, we can stop the finite-differencing computations, because then we have reached the easy-to-compute linear regime. When this non-equidistant discretization is used, 100 timesteps or less are usually sufficient to cover the non-linear decay even at high opacities. However, great care has to be exercised in the choice of $c_e$ especially if we are interested in the absolute values of $n_u$ at late times. Too rough a temporal grid can cause considerable inaccuracies. This choice of the stepwidth works very well for the decay of an initial distribution. For a problem with transient excitation by a laser pulse, it is either necessary to make a good estimate for the time within which the absorption coefficient changes—that has to be done on a case-to-case basis—or by a truly adaptive scheme. Essentially, the generalized Holstein equation is of the form $$ \frac{\partial n_u}{\partial t} = f(n_u, t) $$ (13.73) --- Computation of the function $f$ is extremely CPU-time intensive, so that higher-order finite-differencing schemes are preferable. A sensible approach is a fourth-order Runge–Kutta Cash–Karp scheme described by Press *et al.* (1993), because there is a publicly available program for the solution, but of course other procedures work just as well. We know of no investigations of the efficiency of various finite-difference approaches, besides the general opinion of Press *et al.* (1993) that for most physical problems, higher-order schemes are more efficient than low-order schemes when the function $f$ is difficult to compute—which here is certainly the case. The issue of efficiency will thus probably be an area of further research in the future. We note finally that Abramov *et al.* (1967) reduced the steady-state non-linear Holstein equation in a plane-parallel slab to a linear integral equation, but on an unknown scale. This method does not seem to offer special advantages. ### 13.4.2 Analytical approximations The analytical approximation described here is based on the escape factor technique. As mentioned in Sec. 13.3.4, the escape factor technique is quite useful to determine the averaged number of excited state atoms also in the non-linear regime. In order to compute the average excited-state density $\bar{n}_u$, we adopt the procedure of Bezuglov *et al.* (1994, 1995, 1997). We first integrate the generalized Holstein equation, Eq. (13.67) over the whole slab. It then becomes $$ \frac{d\bar{n}_u}{dt} = -A_{ul}\bar{n}_u(t)\eta_{\text{eff}} (\kappa_0(\bar{n}_u)R) \qquad (13.74) $$ where $\eta_{\text{eff}}$ is the escape probability for the effective opacity $\kappa_0(\bar{n}_u)R$ and $R$ is a typical dimension of the cell. We have introduced the opacity $\kappa_0(\bar{n}_u)R$ into Eq. (13.74) as an argument of $\eta_{\text{eff}}$ to emphasize its functional dependence. In order to evaluate the escape factor approximation, Eq. (13.66), we have to know or to assume the spatial distribution of the excited-state density. This will be discussed below. Equation (13.74) combined with the definition of the effective absorption coefficient, Eq. (13.1) gives a differential equation for the average density $\bar{n}_u$, which can be rewritten as $$ \frac{d\bar{n}_u}{\bar{n}_u \eta_{\text{eff}} \left[ k_0 \frac{\bar{n}_l}{N} \left( 1 - \frac{\bar{n}_u}{\bar{n}_l} \frac{g'_l}{g'_u} \right) \right]} = A_{ul} dt \qquad (13.75) $$ The maximum possible value for $\bar{n}_u$ is $$ N_s = N \frac{g'_u}{g'_u + g'_l} \qquad (13.76) $$ As initial condition, we assume complete saturation in a volume $V_{\text{A}}$. The volume of the whole cell is denoted as $V$. Formal integration of Eq. (13.75) then gives $$ t A_{ul} = \int_{\bar{n}_u/N_s}^{V_{\text{A}}/V} \frac{du}{u} \frac{1}{\eta_{\text{eff}} [k_0(1 - u)]} \qquad (13.77) $$ --- Equation (13.77) provides the analytical approximation for the temporal behaviour of $\bar{n}_u$. The left-hand side is the time after the exciting pulse in units of the natural lifetime of the excited state. We can thus make an explicit computation of the time $t$ after which a certain density of excited atoms is attained. The remaining problem is the computation of the effective escape factor. For this, we have to assume a certain spatial distribution of excited-state atoms. It is usually a good choice to use the lowest-order linear mode of the given geometry. For the slab, this means—in the high-opacity limit according to van Trigt, $$ n_u(z) = \left[ 1 - \left( \frac{2z}{L} \right)^2 \right]^{0.5} $$ (13.78) The assumption about the spatial distribution of the excited-state atoms is the main source of error in the derivation of the expression. In reality, the spatial distribution, and thus the escape probability $\eta_{\text{eff}}$, will be different from the assumption and will furthermore change with time. For all times and for all reasonable initial distributions, the escape probability $\eta_{\text{eff}}$ must lie between $\eta_{\text{eff}}$ for a uniform distribution and $\eta_{\text{eff}}$ for a delta distribution. As an example, we consider the error for the slab case. When we have a Doppler lineshape, the error introduced by the assumption of a parabolic excited-state distribution is largest when the actual distribution is spatially uniform—compare Sec. 4.5. In that case, the maximum error in the escape factor at any time is a factor $\ln(k_0L)$, which can become quite large at high opacities. A uniform initial distribution with a pure Doppler lineshape at high opacities could thus be expected to be a kind of ‘worst-case’ scenario for this method.[^17] However, the excited-state distribution does not stay uniform during the decay but soon looks like the lowest-order mode. For the non-linear case, the effective opacity of the slab is very small at early times, so that the escape probability depends very little on the distribution of excited-state atoms. At later times the spatial distribution has already become similar to our assumed distribution (which is similar to van Trigt’s lowest-order mode (van Trigt 1969–75)), so that the computed $\eta_{\text{eff}}$ will be quite realistic. Comparison with numerical analyses shows that the error in $\bar{n}_u$ is always smaller than 30% in slab geometries for both Doppler and Lorentz lineshapes. The assumption of a time-invariant shape of the excited-state distribution can lead to either an over- or underestimation of the excited-state density, depending on the initial distribution. The comparatively small error makes this method quite useful for first estimates. However, the error is small only for the average density $\bar{n}_u$, while for the emergent radiation it can be considerably larger. The emergent flux $Y(t)$ is proportional to $d\bar{n}_u(t)/dt$, and the differentiation tends to increase the error. However, the use of various correction factors allows one to drastically reduce the error both for $\bar{n}_u$ and $Y$. To decrease [^17]: Of course one could construct even worse examples, but these are usually not relevant in practice. --- the errors, Bezuglov *et al.* (1997) introduced some correction factors[^18] that reduced the error in $\bar{n}_u$ to less than 10%, and also gave good results for $Y(t)$. This escape factor method can be applied to various geometries, one just has to use the escape factors appropriate for the given geometry. Again, the accuracy hinges on the clever choice of the spatial distribution of excited atoms. Using some kind of parabolic distribution or the lowest order linear mode usually gives quite good results. Computations in a cylindrical geometry showed that the errors are slightly larger than in the slab case, because the difference in the escape factor $\eta_{\text{eff}}$, between the uniform and delta distribution is larger in a cylinder than in a slab. Finally, we note that the whole method only makes sense when the escape factor can be computed analytically. The escape factor must be physically meaningful at all opacities, also when the initial saturation is very strong. In that case, the vapour is transparent at early times and we cannot use the high-opacity approximation to the escape factor even when the unexcited absorption coefficient $k_0$ would be large. An easy way to circumvent this problem is to use e.g. the fitting equations of Chapter 7, which are meaningful at all opacities. ### 13.4.3 *Numerical solution of the transfer equation* The finite differencing scheme described in Sec. 13.4.1 can be used not only on the Holstein equation, but also on the transfer and rate equations. For the explicit Euler scheme, the approach is then the following. We start out with the initial distribution of excited atoms and solve the equation of radiative transfer with that distribution. The flux density $J(\mathbf{r}, t_0)$ is then known. With that, we can solve the rate equation in the Euler approximation, $$ \frac{n_u(\mathbf{r}, t_0 + \Delta t) - n(\mathbf{r}, t)}{\Delta t} = -A_{ul}n_u(\mathbf{r}, t) + B_{lu}J(\mathbf{r}, t_0)n_l(\mathbf{r}, t_0) - B_{ul}J(\mathbf{r}, t_0)n_u(\mathbf{r}, t_0) $$ (13.79) With these values for the excited density $n_u$, we can then again solve the transfer equation, and iterate towards a solution. The matter of choosing the appropriate time step $\Delta t$ is exactly the same as for the Holstein equation. Again, we can either use *a priori* knowledge, or we can use adaptive stepwidth control. Similar schemes can be set up for the solution of the integral equation for the intensity, i.e. for the Feautrier method and the like. For the operator perturbation technique, it can be efficient to use an approximate operator $\Lambda^*$ that is the exact operator $\Lambda$ of a previous timestep. For all these cases, the trapping is *linear* within the timescale $\Delta t$. This makes it possible to use all the methods described in Part II, with all the advantages and disadvantages discussed there. [^18]: Note, however, that the correction factors $\Delta$ and $\Delta_{\text{A}}$ in Table III of Bezuglov *et al.* should read $\Delta = -0.4\rho^2 + 0.43(1 - \rho^2)$, $\Delta_{\text{A}} = \Delta_{\text{A}}^{\text{ex}} = -0.4$ for the cylinder and $\Delta = -(5/9)\rho^3 + 0.57(1 - \rho^3)$, $\Delta_{\text{A}} = \Delta_{\text{A}}^{\text{ex}} = -5/9$ for the sphere—these values can be achieved by a simpler and more intuitive derivation. Furthermore, there was a bug in the program that produced Figs. 5 and 6, so that these figures should look slightly different (although the difference is not large). --- For the treatment of stellar winds, Cannon (1985) proposes to use an implicit finite-differencing scheme, claiming that it gives advantages with respect to numerical stability. Such a scheme is much more complicated, because we do not already have the terms needed to evaluate the right-hand side of Eq. (13.79). On the other hand, he shows that numerical stability of the implicit scheme is better, so that larger timesteps are possible. We know of no appropriate investigations on its stability and on its efficiency for transient laboratory problems—the time dependence in Cannon's problem is introduced only by large-scale material transport and thus cannot be easily related to the problems we are interested in. ### 13.4.4 *Physical effects* The non-linearities not only require different mathematics, but also give rise to some new physical effects. The first computations of the decay behaviour of a strongly saturated vapour have been done by Stacewicz *et al.* (1993). Figure 13.5 shows their results for the averaged excited-state distribution in a plane-parallel slab with a Doppler lineshape. At early times, the excited-state density decreases very fast (*de facto* with the natural lifetime), since there are no absorbers that can trap the emitted photons. The effective absorption coefficient of a completely saturated vapour is zero. As the excited-state density decreases, the opacity of the vapour increases, and the trapping becomes stronger. At late time, we have weak excitation, and the time decay constant becomes the same as in the unsaturated case. One extremely interesting discovery is the fact that in such a slab, the fluorescent signal can decrease even *faster* than with the natural lifetime. This can be explained the following way: at early times, there is a large number of radiating atoms, and practically no absorption. A bit later, not only has the number of emitters decreased (about with the natural lifetime) but also there are now absorbers that keep the radiation away from the detector. In sum, these effects result in a decrease of the fluorescence that is faster than with the natural lifetime $\tau$. These computations were also confirmed experimentally Chorazy (*et al.* 1996). Figure 13.7 compares the experimental results to the theoretical predictions. Similar theoretical predictions can also be obtained for cylindrical geometries (Molisch *et al.* 1995c). The excited-state distribution will tend to get the shape of the lowest-order linear mode as time $t$ increases. Since the degree of saturation decreases with time, the problem will be linear at late times anyway, see Fig. 13.7. The situation is less dramatic when we saturate a thin layer in the middle of the slab. In that case the fluorescence decays no faster than with lifetime $\tau$. Even at early times, the radiation has to fight its way through a layer of absorbers. The fluorescence will actually increase at early times, before it decreases again. The excited-state density will also decay slower than for a fully saturated slab. Let us next discuss the influence of the pressure of an additional noble gas on the decay of the excited atoms and of the emergent radiation, i.e. the influence of the Voigt parameter on the decay. For a completely excited slab, the effective opacity is zero at the beginning of the decay process—complete saturation, absorption equal to stimulated emission. Thus, we have no reabsorption of photons, and the excited-state density --- TRANSIENT PROBLEMS [FIGURE: Decay of Na(3P) population at the centre of an entirely excited slab.] FIG. 13.5. Decay of Na(3P) population at the centre of an entirely excited slab. Insert: semi-logarithmic scale; straight line represents a decay with decay constant $g_0\tau = 233$ ns. From Stacewicz *et al.* (1993). [FIGURE: Spatial and temporal development of Na(3P) excited population in an entirely excited slab.] FIG. 13.6. Spatial and temporal development of Na(3P) excited population in an entirely excited slab. From Stacewicz *et al.* (1993). --- [FIGURE: Decay of the emerging fluorescence: experimental and analytical results for an entirely excited slab. For comparison: natural decay with $\tau = 16.7$ ns (dotted line). From Bezuglov et al. (1997)] FIG. 13.7. Decay of the emerging fluorescence: experimental and analytical results for an entirely excited slab. For comparison: natural decay with $\tau = 16.7$ ns (dotted line). From Bezuglov *et al.* (1997) will decay with the natural lifetime. This is true regardless of the Voigt parameter $a$. At later times, the effective opacity increases, so that the escape probability decreases. On a variation of the gas pressure, the escape probability becomes smaller as the Voigt parameter becomes smaller. As a consequence, the time decay of the excited-state density at later times will become slower for smaller Voigt parameters. This can also be clearly seen in Fig. 13.8. The situation is somewhat different for the emergent flux $Y(t)$, see Fig. 13.9. Shortly after the exciting pulse, we have a decay time that is smaller than the natural lifetime. For a small Voigt parameter, the escape probability decreases much faster as a function of $(1 - \bar{n}_u/\bar{n}_l \cdot g_l/g_u)$ than for a large Voigt parameter. Thus, shortly after the pulse, the emergent radiation decays faster for smaller $a$-values. At late --- TRANSIENT PROBLEMS [FIGURE: Normalized decay curves for the total number of excited atoms in a strongly excited layer for different Voigt parameters a.] FIG. 13.8. Normalized decay curves for the total number of excited atoms in a strongly excited layer for different Voigt parameters $a$. For all cases the Doppler optical thickness is $k_0L = 170$. The dotted curve with diamond points corresponds to a pure Lorentz shape, while circles correspond to a pure Doppler profile (both are numerical results). The dashed curve represents the natural decay. From Bezuglov *et al.* (1995). [FIGURE: Normalized fluorescence signal registered from an entirely excited sodium vapour layer.] FIG. 13.9. Normalized fluorescence signal registered from an entirely excited sodium vapour layer. The symbols are the same as for Fig. 13.8. From Bezuglov *et al.* (1995). --- describing the change in the escape probability. For infinite intensity, the decay parameter $\gamma(0)$ decreases steadily as we increase the particle density, since the escape factor at time zero equals one—due to the complete saturation, the effective absorption coefficient is zero, independent of the particle density. The decrease in the escape probability, on the other hand, is the stronger the higher the particle density is, so that the decay time constant becomes smaller. Another interesting physical effect that occurs also in steady-state is that a vapour irradiated by high-intensity radiation has a kind of *specular* reflectance (Bulyshev *et al.* 1986), in other words, the radiation emergent from the bottom of a slab that was irradiated there with a strong laser beam has a strongly non-uniform angular distribution. The high-intensity radiation ‘burns out’ a hole in the absorption coefficient on its way into the vapour. Radiation can easily escape through this low-opacity ‘channel’, while it is more strongly absorbed in other directions where the effective absorption coefficient is higher. This leads to a non-uniform distribution of the emergent radiation, i.e. ‘specular reflection’. ### 13.4.5 Monte Carlo simulations For the transient case with stimulated emission, Monte Carlo simulations require some more thought than in the simple linear case, or even in the steady-state case described in Sec. 13.3.4. We cannot follow a photon through a vapour anymore, because during the time that we are following it, the parameters of the vapour change (this is in contrast to the bleaching effects in three-level atoms, see later in this chapter). In order to circumvent this problem, we have to linearize the problem. Essentially, we do a finite-differencing in time, again choosing the size of the timesteps so small that the effective absorption coefficient does not change appreciably during the timestep. We cannot follow the photons uninterruptedly until they escape. Instead, we follow them only until time $\Delta t$ has passed. Then we follow the next photon through that time, and so on. After we have followed all initial photons during that time interval, we look how the effective absorption coefficient has changed. After having adjusted that parameter, we take up again the first photon, and continue to follow it until time $2 \Delta t$. This procedure is repeated until we have covered the whole time that we want to simulate. More details and stability considerations can be found in Brooks and Fleck (1986); see also Fleck (1963), Fleck and Cummings (1971), Giorla and Sentis (1971), Fleck and Canfield (1984), Larsen and Mercier (1987) and Bulyshev *et al.* (1985). Since the distribution of absorbers is inhomogeneous, we have to compute the absorption coefficient for each path separately. The vapour cell is divided into many subcells. The opacity encountered by a photon along a certain direction will vary from subcell to subcell. We thus have to compute the intersections of the photon path with the subcell boundaries and to add up the opacities of the path lengths within each subcell. **Other methods.** Quantum mechanical computations using the density-matrix formalism can be found in Melnikov and Polivenko (1986) and Beterov *et al.* (1970). --- ## 13.5 Multilevel systems Now that we have covered both the steady state and the transient problem of two-level atoms, we want to go one step further and to deal with multilevel atoms. Again, this leads to both mathematical complications and to new physical effects. We will first describe two mathematical techniques: the generalization of the complete linearization technique (CLT) to the multilevel case and the equivalent two-level atom (ETLA). We then outline the physical effects and the mathematical treatment of bleaching (steady-state) in three-level atoms. Finally, we treat time-dependent bleaching in three-level atoms. ### 13.5.1 Complete linearization in the multilevel case For a multilevel system, we first write all densities at a certain spatial grid point $j$ into a vector $\mathbf{n}_j$. The densities are then related to the (frequency- and angle-integrated) radiation intensities $J$, and the first-order perturbations are written as $$ \delta \mathbf{n}_k = \sum_{l=1}^L \left( \frac{\partial \mathbf{n}}{\partial J_l} \right) \Bigg|_{z=z_k} \delta J_{k,l} $$ (13.81) where the index $l$ stands for the frequency. The derivatives of the components for $\mathbf{n}$ with respect to the radiation intensities can be computed analytically from the appropriate rate equations. The system of rate equations generally reads $$ \mathbf{A}_{\mathrm{CLT}} \mathbf{n} = \mathbf{B}_{\mathrm{CLT}} $$ (13.82) The partial derivative of $\mathbf{n}$ with respect to $J$ is given by $$ \frac{\partial \mathbf{n}}{\partial J_l} = \mathbf{A}_{\mathrm{CLT}}^{-1} \left[ \frac{\partial \mathbf{B}_{\mathrm{CLT}}}{\partial J_l} - \frac{\partial \mathbf{A}_{\mathrm{CLT}}}{\partial J_l} \mathbf{n} \right] $$ (13.83) We can thus express the density perturbation $\delta \mathbf{n}$ occurring in the transfer equation by the intensity perturbations $\delta J_{j,l}$ and get a linearized transfer equation, e.g. in the Feautrier form. The multilevel CLT is thus exactly the same as the two-level case—we just have more of everything (especially more work in deriving the appropriate relations between $n$ and $J$). Up to now, we have assumed that the radiation intensities and excited-state densities, which sum up to a constant total atomic density $N$, are the only variable parameters. In some cases, especially in plasmas, it is possible that the free electron density $n_e$, the temperature $T$, and the particle density can vary. In that case, the required total differential $\delta \mathbf{n}_k$ is $$ \delta \mathbf{n}_k = \sum_{l=1}^L \left( \frac{\partial \mathbf{n}}{\partial J_l} \right) \Bigg|_{z=z_k} \delta J_{k,l} + \left. \frac{\partial \mathbf{n}}{\partial n_e} \delta n_e \right|_{z=z_k} + \left. \frac{\partial \mathbf{n}}{\partial T} \delta T \right|_{z=z_k} + \left. \frac{\partial \mathbf{n}}{\partial N} \delta N \right|_{z=z_k} $$ (13.84) --- For a complete description of the multivariables problem, we then have to add an equation for the electron density, the equation of thermodynamic equilibrium, and the vapour pressure equation. A similar technique can also be used for the partial frequency redistribution problem for either the two-level or the multilevel case. Basically, we can view the atoms that can emit at a certain frequency $x_i$ as one distinct level, which is strongly coupled to all other levels. We get a system of coupled equations that can be solved by the CLT described above. For the two-level case, explicit equations for the derivatives $\partial n / \partial J_i$ for the two-level case are given by Milkey and Mihalas (1973), for the multilevel case by Milkey *et al.* (1975a), compare also Freire Ferrero (1985). However, these equations are derived under the assumption that stimulated emission photons from atoms that suffer a collision are redistributed in frequency; this is only an approximation (in reality, they are coherent). Combinations with the OPT are exactly as in the two-level case, see Sec. 13.3.2. ### 13.5.2 Operator perturbation techniques for multilevel systems For a multilevel atom, Werner and Husfeld (1985), Werner (1986, 1987, 1988, 1989), Kubat (1994) and Hillier (1990) combined these --- MULTILEVEL SYSTEMS for the correction terms $\delta n$. Note that if we use the approximate operator Eq. (13.46), the rate equations at all spatial points are decoupled, which allows very fast solution. A very detailed implementation of this step can be found in Werner (1987). (iv) With the new values for $n_{\text{l}}$ and $n_{\text{u}}$, compute the new source function, and go back to step (ii), until convergence is achieved. Hubeny and Lanz (1995) combined the OPT with the complete linearization in a way that allowed them to trade off the advantage of OPT (fast computation time for each iteration cycle) with the advantage of CLT (fast convergence). Rybicki and Hummer (1991, 1992, 1994), developed a scheme called MALI (multilevel accelerated lambda iteration) that does not require linearization, but uses preconditioning of the equations in order to accelerate convergence. This technique has also been applied by Auer *et al.* (1994), Heinzel (1995), and Paletou (1995). ### 13.5.3 *The equivalent two-level atom* The 'equivalent two-level atom' (ETLA) is a technique that bears strong similarities to the direct iteration described in Sec. 13.3.3. In the equivalent two-level approach, we consider a certain transition $ij$. --- the Feautrier equation by the usual techniques. The procedure is repeated for all considered transitions. With the thus-computed intensities, we can update the excited-state densities. These densities result in new values for the opacities. We thus have the new values for updating the coefficients in the equation for the equivalent source function and can start over again. Skumanich and Lites (1985) proposed a procedure to extract the ETLA parameters of each transition from this solution; this might come in handy for the interpretation of the results. Convergence of the ETLA is very fast when the intensities of the lines are not strongly interrelated. In the extreme case that we have only two relevant transitions and they share no common level, the iteration needs only one step. The stronger the interdependence between the lines becomes, the slower the convergence will be. At high opacities, convergence can be accelerated by using ‘effective decay rates’ as proposed by Castor *et al.* (1992), see Appendix C.4, computer program ALTAIR. Convergence can also strongly depend on the actual formulation in each case (Avrett and Loeser 1987). Note, however, when each of the transitions is non-linear, each solution of the equivalent two-level problem requires some iterative procedure to account for the non-linearity. For very strongly coupled lines, convergence can be quite slow. It can be advantageous to simultaneously perform the iteration required to deal with the non-linearities and the basic iteration procedure of the ETLA algorithm. Again, the method of direct iteration is strongly related to ETLA. The procedure is completely analogous to the two-level case. We start out with a guess value for the excited-state densities. With these values, we solve the equation of radiative transfer, which gives the averaged intensities with which we can solve the rate equations and thus update the excited-state densities. It appears to be difficult to decide whether ETLA or complete linearization is preferable for the treatment of multilevel atoms. Different authors have different opinions on that subject; not surprisingly, authors writing about ETLA demonstrate higher efficiency of this algorithm, and vice versa. One thing that seems quite clear is that the standard ETLA technique is not well suited for dealing with population inversion (especially important for lasers). Since the absorption coefficient can become zero or negative, the dependence of the population on the radiation field is extremely strong, and the simple iteration scheme can lead to wild oscillations of the solution as the iteration progresses. The ETLA method works only when the coupling between the lines is not too strong, and when a small change in the excited-state density does not produce a dramatic change in the absorption coefficient. The latter condition is not fulfilled when laser transitions are computed. Alley *et al.* (1982) used a Feautrier method combined with ETLA to compute the population in a plasma where inversion occurs and reported significant numerical problems. Bond (1984a) used a method that is essentially a combination of partial linearization and ETLA. Changes in the absorption and emission coefficient are included to first order. Let us denote the coefficients for absorption and stimulated emission as $\chi^{\mathrm{a}} =$ --- $k(x)n_l/N$, and $\chi^e = B_{ul}\chi^a/B_{lu}$, respectively, and for spontaneous emission as $\chi^s = h\nu n_u A_{ul}C_\nu k(\nu)/4\pi$. We then arrive at the following linearized equation $$ \begin{aligned} \mu \frac{(I_{k+1} - I_k)}{\Delta z} = & - \chi^a I_{k+1} - I_{k+1}^p \frac{\partial \chi^a}{\partial J} (J_{k+1} - J_{k+1}^p) + I_k \chi^e + \\ & + I_k^p \frac{\partial \chi^e}{\partial J} (J_{k+1} - J_{k+1}^p) + \chi^s + \frac{\partial \chi^s}{\partial J} (J_{k+1} - J_{k+1}^p) \end{aligned} $$ (13.89) where superscript $p$ denotes quantities computed in the previous timestep and $J$ is the frequency- and angle-averaged intensity. This linearized equation, combined with the ETLA approach, gives good convergence. Of course, when one has to take the trouble of linearizing the transfer equation one could also decide to go all the way and do a complete linearization. ### 13.5.4 Steady-state in three-level atoms We have already mentioned in Sec. 13.1 that a new effect, 'bleaching', can occur in a three-level atom where one state is a metastable one. Typical examples of such atoms are thallium, barium, mercury, and cesium—in cesium, the hfs splitting of the ground state is so large that the two hyperfine levels have to be considered as separate levels. In the following, we consider thallium as the typical example. Its level scheme is shown in Fig. 13.2, repeated in Fig. 13.10. Since we now have three levels, the absorption coefficient of the ground state can change even when the density of the excited state, 7s, is very small. This happens when many atoms are transferred to the long-lived 6p$_{3/2}$ state, where they rest in peace and can no longer absorb on the 378 nm (uv) resonance transition. This makes the uv transition more transparent, it 'bleaches' this transition. It is very important to distinguish this effect from saturation. The physical effects arising from saturation, i.e. stimulated emission, and from bleaching are quite different. Below, we describe the effects occurring in such a system, and, as an example, present evaluations done by the direct iteration technique, combined with some additional simplifications. This does not mean that direct iteration is preferable for computations of this type. As usual, we have a tradeoff between simplicity, speed, and accuracy of the computations, and all of the presented methods have their advantages and disadvantages. Atoms are excited to the upper state (7s) by external radiation. An upper-state atom can decay either to the ground state (6p$_{1/2}$) or to the metastable state (6p$_{3/2}$), emitting a 378 nm uv or 535 nm green photon, respectively. The reabsorption rate will now depend on how many atoms are in the ground state and how many are in the metastable state. When all atoms are in the ground state, the uv transition will be trapped, while the green transition will not be trapped and vice versa. Suppose now that we have a vapour cell where initially all Tl atoms are in the ground state and we illuminate this cell with external uv 'pump' radiation. In the beginning, the uv transition is trapped, while the green transition is not. This implies that the absorption of one pump photon will lead to the creation of a metastable atom with a probability that is higher than the green-to-uv branching ratio of 50%, because the 'effective decay rate' (Einstein $A$ coefficient --- [FIGURE: FIG. 13.10. Partial energy level scheme of atomic thallium.] times the probability of escape) is higher in the green than in the uv. The metastables are very long lived—they act as a kind of storage reservoir—so that even quite low pump light power will deplete the ground level. This in turn leads to a decrease in the absorption of pump photons (since there are fewer absorbers), to decreased trapping of the uv transition, and, due to the creation of the metastables, to increased trapping of the green transition. These effects result in a decreased production rate of metastables. The whole process is thus highly non-linear. Due to the large lifetime of the metastables, these ‘bleaching’ non-linearities (as opposed to ‘saturation’, which implies the presence of stimulated emission) set in at an intensity that is orders of magnitude lower than what is required for saturation. When we wish to make computations in such a system, the first step is always to set up the rate equations. This is not only a formal mathematical step, but also has an important physical background. We are forced to consider which levels and which processes can contribute to the effects that we are interested in. The example of a thallium vapour is rather simple, because only few levels are involved. Such a situation is typical for chemical physics. In plasma research, the number of involved levels is often huge—a hundred or more levels are by no means uncommon. Our specific example is a vapour cell filled with thallium vapour and argon as a buffer gas. The cell is excited by radiation from a spectral lamp incident on the side walls of the cell. We first set up the steady-state rate equations for the three levels involved. For the metastable state, the density is increased by 7s-atoms that decay to the metastable state. It is decreased (i) by the reverse process–metastable atoms absorbing green photons, (ii) by radiative decay of the metastables to the ground state, (iii) by self-quenching due to Tl-Tl collisions, and (iv) by foreign-gas quenching. Furthermore, the --- spatial distribution of metastables is changed by particle diffusion, and all metastables that hit the cell walls are quenched. The resulting rate equation for the metastables is $$ \begin{aligned} A_g n_{7s}(\mathbf{r}) &= n_m(\mathbf{r})\frac{B^g}{4\pi} \iint I^g(\mathbf{r}, \mathbf{\Omega}, \nu)C_\nu^g k^g(\nu)\text{d}\nu\text{d}\mathbf{\Omega} + \\ &\quad + n_m(\mathbf{r}) \left( \frac{1}{\tau_{\text{nat}}} + \frac{1}{\tau_{\text{q,Tl}}} + \frac{1}{\tau_{\text{q,Bgas}}} \right) - D\nabla^2 (n_m(\mathbf{r})) \end{aligned} $$ (13.90) where superscripts g and uv stand for the green and uv transitions. The variables $n_m$ and $n_{7s}$ are the densities of atoms that are in the metastable and 7s states, respectively. The density of upper-state (7s) atoms is decreased by spontaneous emission, and is increased by absorption of green and uv photons. The rate equation for the 7s atoms is thus $$ \begin{aligned} (A^g + A^{uv})n_{7s}(\ --- [FIGURE: FIG. 13.11. Percentage of Tl atoms that are in the metastable state, averaged over the cell, $\bar{n}_{\text{m}}/N$. The vapour cell is filled with single-isotope Tl and 80 mbar Ar. Pump lamp power is 40 mW.] [FIGURE: FIG. 13.12. Percentage of Tl atoms that are in the metastable state, averaged over the cell, $\bar{n}_{\text{m}}/N$. Tl atomic density is $10^{12}\text{ cm}^{-3}$. Pump lamp power is 40 mW.] broader, so that more pump photons can be absorbed). Figure 13.13 finally shows $\bar{n}_{\text{m}}$ as a function of the pump lamp power. We see that population inversion (because of the degeneracies of the levels, this means $\bar{n}_{\text{m}}/N > 2/3$) can be achieved with only 60 mW pump lamp power. Baranov (1979) has proposed the construction of a laser at $1.28\,\mu\text{m}$ by using the population inversion in a thallium vapour and also experimentally demonstrated the feasibility of such a population inversion. Related problems can occur when the lower state of a considered transition is itself an excited state that is populated by a trapped transition. Astrophysical problems of this --- [FIGURE: Percentage of Tl atoms that are in the metastable state, averaged over the cell] FIG. 13.13. Percentage of Tl atoms that are in the metastable state, averaged over the cell, $\bar{n}_{\text{m}}/N$. Tl density in the cell is $0.3 \cdot 10^{12} \text{ cm}^{-3}$, argon pressure is 250 mbar. kind are discussed by Finn and Jefferies (1969) Quite generally, radiation trapping reduces the optical pumping. Since at each absorption/reemission process a 'pump' photon may change its frequency, there is a higher probability that we get no net increase in the population of the pumped substate (Gornyi *et al.* 1984). On the other hand, there is also an effect called 'optical self-pumping' which is related to the three-level problem. It was observed, e.g. for the resonance line of Thallium, that the intensities of the hfs components of the emerging radiation was not in the theoretical ratio (Aleksandrov and Bezuglov 1978), (Aleksandrov *et al.* 1979). This is due to the fact that the hfs components have different absorption cross-sections, and thus different trapping factors; the effect is very similar to the self-reversal of the lineshape (where the maximum in the emergent line is not at the frequency of maximum emission, i.e. the line centre). ### 13.5.5 *Time-dependent depletion* In the previous sections, we have considered the case of bleaching without saturation. In this section, we also lift this last restriction, and consider a vapour of three-level atoms that is illuminated by a very strong laser beam. We thus get both bleaching (optical pumping) and saturation. A very simple example, which is also of practical importance, is a cesium vapour illuminated by a laser that is tuned to one hyperfine transition. Since we assume a strong laser, the burn-through will create a situation that is very similar to an ideal infinite cylinder. We are thus able to tackle the problem with the generalized Holstein equation. (It is practically impossible to treat two-dimensional non-linear problems with the Holstein equation.) We assume a constant laser intensity within the beam radius $r_{\text{laser}}$ and intensity zero outside. The laser is strong enough to completely saturate the beam region $0 < r < r_{\text{laser}}$. Strong saturation at the Cs resonant wavelength --- requires pulse energies of just a few mJ. We now set up the Holstein equations for the involved atomic levels, the two split ground state levels $6s_{1/2}$, $F = 3$ (level $a$) and $6s_{1/2}$, $F = 4$ (level $b$), and the excited $6p_{1/2}$ level (level $c$). State-$c$ atoms are created (i) by the absorption of laser radiation by state-$a$ atoms, (ii) by reabsorption of $ca$ fluorescence radiation, and (iii) by the reabsorption of $cb$ fluorescence radiation. Atoms leave state $c$ via natural decay or via stimulated emission. The latter is taken into account by using the effective absorption coefficient that describes the difference between absorption and stimulated emission. By balancing between creating and destroying processes, we get the rate equation for state-$c$ atoms $$ \begin{aligned} \frac{\partial n_c(r, t)}{\partial t} &= -\frac{1}{\tau} n_c(r, t) + \frac{\beta}{\tau} \int_0^R n_c(r', t) G^{ac}(n_a, n_c, r, r', t) r' dr' \\ &\quad + \frac{1-\beta}{\tau} \int_0^R n_c(r', t) G^{bc}(n_b, n_c, r, r', t) r' dr' + E(r, t), \end{aligned} $$ (13.92) where $E(r, t)$ is the external excitation by the laser beam. The branching ratio $\beta$ is the ratio of the natural decay probabilities $ca : cb$, that is $\beta = A_{ca} / (A_{ca} + A_{cb})$. Atoms in state $b$ are created by decaying state-$c$ atoms, and are destroyed by reabsorption of $cb$ photons. Their rate equation is then $$ \frac{\partial n_b(r, t)}{\partial t} = \frac{1-\beta}{\tau} n_c(r, t) - \frac{1-\beta}{\tau} \int_0^R n_c(r', t) G^{bc}(n_b, n_c, r, r', t) r' dr' $$ (13.93) Since the Cs density, $N$, must be constant throughout the cell, the rate equation for state-$a$ atoms is simply given by the other two, $$ n_a(r, t) = N - n_b(r, t) - n_c(r, t). $$ (13.94) Solution of this system of equations is by the same procedure as for the two-level case—approximation of the distribution by a set of orthonormal base functions with time-dependent unknown amplitudes, then finite differencing in the time domain. With the above method, we can analyse the temporal behaviour of the excited atoms and of the emergent radiation. It turns out that the combination of trapping, optical pumping, and saturation leads to some interesting effects, which we will show and explain in this section. Most notably, the strength and the temporal dependence of the emergent radiation depends on the duration of the laser pulse. Let us first consider the pump phase, i.e. the time when the laser pulse is on. Figure 13.14 shows the (normalized) densities $\bar{n}_a, \bar{n}_b$ and $\bar{n}_c$ of atoms in states $a, b$, and $c$, averaged over the cell, at the end of the pulse as a function of the pulse duration. First, let the radius of the laser beam be small compared to the cell radius. We see that for very long pulses, the density $\bar{n}_c$ at the end of the excitation is smaller than for very short pulses (a very short pulse means much shorter than the natural lifetime $\tau$). This effect is a consequence of the optical pumping. --- [FIGURE: Graph of Average density of excited atoms vs Pulse duration for densities n_a, n_b, and n_c] FIG. 13.14. Densities $\bar{n}_a$, $\bar{n}_b$, and $\bar{n}_c$ (normalized to $N = 1$) at the end of a strong laser pulse as a function of the pulse duration. Opacity $k_0R = 10$, laser beam radius $r_{\text{laser}} = 0.1 \cdot R$. However, we also see that $\bar{n}_c$ goes through a maximum, and this requires more explanation of the geometry involved. We distinguish between two regions of the cell: the primary-excitation region ($r < r_{\text{laser}}$), where Cs atoms can be directly excited by the laser beam, and the secondary-excitation region ($r > r_{\text{laser}}$), where atoms can only be excited by fluorescence radiation from the primary region. In these two regions, there are competing processes. (i) In the primary-excitation region, optical pumping leads to a decrease of ground state density $n_a$ with time. Density $n_c$ is tightly coupled to $n_a$ because of the strong excitation, so that $n_c$ will also decrease with time. (ii) In the secondary-excitation region, $n_c$ is determined by radiation trapping, i.e. by the reabsorption of fluorescence photons from both excitation regions. If we just had a two-level problem, the absorption of fluorescence from the primary-excitation region would lead to a continuous increase in upper-state atoms until steady-state is achieved ('accumulation' of excitation). In the Cs-problem, however, the primary fluorescence becomes weaker with time, because $n_c$ in the primary-excitation region decreases due to the optical pumping. Thus, $n_c$ in the secondary-excitation region will decrease after some time.[^19] The two competing effects, the accumulation of ex- [^19]: This effect is intensified by the fact that we also have (albeit weaker) optical pumping in the secondary-excitation region, so that the number of absorbers in this region becomes smaller with time. This of course leads to a decrease in secondary excitation. --- [FIGURE: 3D surface plot of n_c(r,t) over Time t and spatial radius R] FIG. 13.15. Spatial and temporal evolution of the Cs excited state density $n_c(r, t)$ during a strong laser pulse. Parameters as in Fig. 13.14. citation, and the decrease of primary fluorescence, cause the density $n_c$ to go through a maximum in the secondary-excitation region. This can also be clearly seen in Fig. 13.15, which depicts the spatial distribution of the excited state density $n_c$ as a function of time. Summarizing, we have two regions, the core with decreasing density $n_c$, and the outer region where $n_c$ first increases and then decreases. It is obvious that depending on the relative importance of these two processes, the average density $\bar{n}_c$ either decreases monotonically, or goes through a maximum. The relative importance of the two processes depends on the opacity, on the size of the excited region, and on the duration of the pulse. In our example, the parameters are such that the two processes cause $\bar{n}_c$ to go through a maximum, see Fig. 13.14. Figure 13.16 shows the average densities $\bar{n}_a(t)$ and $\bar{n}_c(t)$ when the whole vapour cell is excited. In that case, we have no secondary-excitation region that can reabsorb fluorescence, and thus no effects that tend to increase density $n_c$ with time. As we increase the duration of the pulse, $\bar{n}_c$ decreases monotonically. Generally, the maximum --- [FIGURE: Graph of Density of excited atoms vs Pulse duration for average densities $\bar{n}_a$, $\bar{n}_b$, and $\bar{n}_c$] FIG. 13.16. Average densities $\bar{n}_a$, $\bar{n}_b$, and $\bar{n}_c$ at the end of a strong laser pulse as a function of the pulse duration. Parameters: $k_0R = 10$, $r_{\text{laser}} = R$. in the average density $\bar{n}_c$ appears only when the secondary region is both geometrically large enough (as compared to the primary excitation region) and optically thick enough to absorb an appreciable number of photons reemitted in the primary region. The duration of the exciting pulse also influences the shape of the time decay of the excited atoms and of the emergent fluorescence radiation after the excitation is switched off. Let us consider the case where the laser excites the whole vessel. We have seen above that for a long exciting pulse, and thus much optical pumping, both $n_a$ and $n_c$ at the start of the decay phase will be small, while $n_b$ will be large. At this time, transition $cb$ will be strongly trapped, while transition $ca$ will be completely transparent—the effective absorption coefficient being zero because of the saturation. Since both $n_a$ and $n_c$ are small, the opacity of transition $ca$ will stay small even if all state-$c$ atoms decay to state $a$.[^20] Thus, the opacities of both transition $ca$ and transition $cb$ will stay constant throughout the decay process; the former will be very small, while the latter will be very large. We thus have a classical three-level trapping problem with a branching ratio of about 50%, with one rather weakly and one strongly trapped transition, see Sec. 10.2. Hence, density $n_c$ will decay with a time constant that is approximately the decay time constant of the transparent transition. Since the opacities do not change appreciably during the decay, this is also true for the emergent radiation. The situation is completely different for a short exciting pulse. At the beginning of the decay phase, transition $ca$ is completely transparent because of the strong saturation. [^20]: Actually, most atoms in state $c$ will decay to state $a$, since the transition $b$ is strongly trapped and thus has a low effective decay rate. --- However, there is little optical pumping during the pump phase (since it is so short). The number of atoms in state $b$ is thus not affected by the excitation pulse, while the ratio of atoms in states $a$ and $c$ is determined by the ratio of their statistical weights. With increasing time, many of the state-$c$ atoms will decay to state $a$, so that the effective opacity of transition $ca$ increases to considerable values. The opacity of transition $b$ will increase only slightly from its already high value. In the beginning, $n_c$ will thus decay with a time constant that is almost equal to the time constant of the $ca$ transition (as in the above case). At later times, both transitions $ca$ and $cb$ are rather opaque, so that the decay time constant is almost $g_0\tau$ at late times, where $g_0$ is the trapping factor for the non-saturated case. There is thus a strong change in the decay time constant. Since the number of excited atoms decreases and the number of absorbers increases, the decay of the emergent radiation can in the beginning be even faster than with the natural lifetime $\tau$. This situation can only occur when the excitation pulse is short. However, the above effects will be much less pronounced when only a small part of the vapour cell is excited. --- # 14 # COMBINATION OF TECHNIQUES We are now in possession of a wealth of mathematical techniques. It is time to review them, to point out which technique is specifically suited for what physical situation and to discuss how we can combine techniques to get even more efficient algorithms. This chapter is thus both a synopsis of Parts II and III of this book (trapping theory) and an introduction to part IV (applications). In this chapter, we give some concrete recommendations on which techniques should be used for what situations. These comments are based on our personal experience with various techniques, and also on general mathematical and computational considerations. However, they cannot be considered as the 'absolute truth'. There are many ways to deal with a problem, and the reader might find that he personally prefers a different procedure for his given problem than the one we recommend. ## 14.1 The four basic questions Before beginning computations on optical effects in vapours, one has to ponder four questions with regard to radiation trapping. (i) *Is there radiation trapping at all?* This is easily answered by considering the opacity of the vapour. From opacities of about 0.1 onwards, radiation trapping should usually be included in the computations. (ii) *Is the radiation trapping non-linear?* This can be answered by applying the formulas of Sec. 13.1. Nonlinear trapping is usually quite difficult to deal with, so if we have the possibility of tailoring an experiment to what we can easily compute, a configuration that operates in the linear regime is preferable. Unfortunately, the experimental setup is usually dictated by other requirements—quite often it is just these computationally nasty effects that are to be observed. (iii) *Is the incident radiation so strong that non-classical effects occur?* Whether non-classical effects become influential not only depends on the absolute strength of the excitation, but also on other physical effects. When we have a large Doppler width, for example, then the Stark splitting by strong laser radiation can usually be neglected. As mentioned in Sec. 13.2, it is neither within the scope nor within the possibilities of this book to describe all the possible effects of strong radiation (or of all other effects that might come into play). 3BEJBUJPO5SBQQJOHJO"UPNJDTBQPVSESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- (iv) *Is complete frequency redistribution a good assumption?* The conditions for CFR are described in Sec. 11.1, and are usually fulfilled. There are mainly two situations where CFR might not be valid, (a) trapping in a high-opacity vapour where the upper state is very short lived and where no additional gas is present, and (b) radiation trapping in a laser-cooling arrangement. These situations may require quite different treatments than CFR conditions. ## 14.2 Linear trapping with complete frequency redistribution ### 14.2.1 *Solution methods for the classical Holstein equation* Situations with linear trapping under CFR constitute the classical radiation trapping problem, and a huge number of techniques has been developed to treat it. Let us first turn to the methods for the solution of the classical Holstein equation, i.e. with the nine restrictive assumptions listed in Chapter 4. These methods basically can be sorted into four groups: Group (i) *Methods for the computation of the eigensolutions* Group (ii) *Number-of-scattering techniques* Group (iii) *The propagator function method* Group (iv) *Steady-state techniques* (i) **Methods for the computation of the eigensolutions.** * The fitting equations * The variational technique * The piecewise constant approximation, PCA * The integral transform technique * The discrete-ordinate solution These techniques give the eigenvalues and the eigenfunctions of the Holstein equation—or, for the discrete-ordinate technique, of the equivalent transfer equation problem. Among these methods, the fitting equation approach is certainly the principal choice *if it is available*. Its advantage is that the user just has to look up tabulated solutions, and need not invest any effort himself. The drawback is that the solutions exist only for unsplit lines (Doppler, Voigt, and Lorentz) and only in the one-dimensional geometries (slab, cylinder, and sphere). For all other cases, a user has to do his own computations. The variational technique and the piecewise-constant approximation are essentially equivalent; both are based on the solution of a matrix eigenvalue problem. Historically, the variational technique has been used for lower-accuracy computations, and only for the first few lowest-order modes, but for more generally posed problems like the finite cylinder geometry. The piecewise constant approximation is more frequently used for high-accuracy computations—for one-dimensional problems, there is even a publicly available computer program. --- The integral transform techniques provide closed-form solutions, see Chapter 5.2. However, these techniques are mathematically so complicated that it is practically impossible for anyone who just wants to *apply* trapping theory to generalize these results. On the other hand, these solutions can be a good basis for generalizations by the modal expansion technique, see below. The big advantage of the eigensolution techniques is that they solve, in just one step, the whole problem for all times and for all initial conditions, even for time-variant excitation functions. In the case that we have various initial conditions in the same surrounding, we thus should certainly choose an eigensolution technique. In a one-dimensional geometry, the size of the matrix for which we have to find the eigensolutions is quite small (typically 30–80 rows), so that the numerical effort is insignificant. (ii) **Number-of-scattering techniques.** Under this name, we subsume computational methods to solve problems in the multiple-scattering representation: * Monte Carlo simulations, and * the analytical solution of the multiple-scattering representation, AMS. The Monte Carlo technique is easier to implement, and much easier to generalize than the AMS. The main advantage of the AMS is that it gives a certain physical insight into the solution of the problem. Since it is rather awkward numerically, applications are rather restricted. In both cases, the initial result is a statistic on how many photons leave the vapour after being reabsorbed $i$ times. Using the equations of Chapter 4.3, this result can be used to compute the average density of atoms and the emergent radiation. For such computations, typically $10^4$–$10^5$ photons are required to reach 1% accuracy in a one-dimensional geometry. Since computations take longer than with the eigenvalue technique, and have to be done afresh for every new initial distribution, eigenvalue techniques are preferable for the classical trapping problem. MC simulations are preferable for all situations where the solution of the Holstein equation becomes very difficult, e.g. in a three-dimensional geometry. Computation of the excited-state *distribution* requires much longer simulation times than for the average excited-state density, typically by a factor of 100 or more. This can be very tiring when many simulation rounds have to be run. For these cases, it may be preferable to combine the MC simulation with an eigensolution technique. Physically, the $A_{k,m}$ matrix elements of the eigensolution techniques are the probabilities that a photon emitted in subvolume $m$ is reabsorbed in subvolume $k$. These probabilities can also be computed by Monte Carlo simulations. The eigensolutions of the problem $A \cdot \psi = \lambda \psi$ are then computed in just the same way as before. (iii) **The propagator function method, PFM** Just like in a Monte Carlo simulation, also with the PFM we have to start a new run for each initial distribution, and we also have to go through the problem of computing the $A_{k,m}$ matrix elements, as in the eigensolution techniques. One avoids the matrix inversion, but at the price of having to multiply the $A_{k,m}$ matrix with a vector $n_k$ at --- each timestep, which usually costs more computer time. The PFM is better applied to problems with partial frequency redistribution, and to transient non-linear problems than to basic CFR situations. Again, we can compute the propagators, or matrix elements, by a Monte Carlo simulation and then make the time-dependent computations with these propagators. At high opacities, this is much faster than a normal Monte Carlo simulation. (iv) **Steady-state techniques** * Combination of eigenmodes * Piecewise-constant solution of the Holstein equation * The Feautrier technique * $\Lambda$-iteration The combination of eigenmodes is undoubtedly the fastest method when the eigenmodes are already available, e.g. from the fitting equations. It is furthermore the best technique when one wants to analyse the same situation for many different excitation functions. All one needs to do is to make a new eigenmode expansion for each new excitation function—this is in contrast to the other techniques where a complete new run of the problem is required. In a slab geometry, the piecewise-constant solution of the eigenmodes is the second-fastest approach. The largest effort lies in the computation of the $A_{k,m}$ matrix elements. The original Feautrier technique is basically slower, because it requires inversion of quite large matrices. However, using the Rybicki reorganization, it can be made to run as fast as the piecewise-constant solution. In the cylindrical geometry, the computation of the matrix elements for the piecewise constant solution takes a very long time. It could well be that the Feautrier technique is faster in this case, though we know of no systematic investigation on the subject. The $\Lambda$-iteration displays rather slow convergence, especially at high opacities. Its main advantage is ease of implementation. Convergence can be dramatically speeded up by using a Newton acceleration—essentially a combination with the linearization technique—but this goes at the price of potential instabilities. The methods of all four groups of techniques can be combined with the operator perturbation technique, OPT. The efficiency of this technique depends on a good choice of an approximate operator, and on the basic technique with which the OPT is combined. If we take, for example, the variational technique or the piecewise-constant approximation for the computation of the eigenvalues, it is not clear that anything can be gained by applying the OPT. While we would be able to reduce the number of frequency and angle quadrature points by using approximate operators, the evaluation of the perturbation term would cost as much CPU time as was gained in the previous step. From that point of view, it appears that the OPT is especially useful for iterative procedures, or for those cases where large matrices occur—in contrast to the piecewise-constant technique, where the effort lies *in* the computation of each matrix element. --- For a good choice of computation technique, one can roughly proceed along the following lines: a) Are lineshape, geometry, and accuracy given in such a way that there are fitting equations or other ready-made solutions, like the closed-form equations of Chapters 5 and 7? If yes, use these. Otherwise, you have to make computations of your own. b) Do you want to have just a rough estimate on the trapping effects, and require no information about the spatial distribution of the excited-state atoms? If yes, make escape factor computations. c) Are there publicly available computer programs that can be used to solve the problem? If yes, use these. See Appendix C for an overview. d) Are you prepared to write some simple code that can be easily debugged and modified, and do you have extensive computer time? If yes, write a Monte Carlo simulation. e) Is the problem transient? If yes, solution by an eigenvalue technique is recommended—if required, very good eigenvalue solvers (EISPACK) are available free of charge on the Internet. If no, i.e. for steady-state problems, expansion of the excitation function into the eigenmodes is the best way when the eigensolutions are already available, see a), or when solutions for many different excitation functions are required. f) Is the problem two-dimensional? For high opacities and with low accuracy requirements, the geometrical quantization technique solutions, or perhaps the variational solutions, can be used. Otherwise, compute the $A_{k,m}$ elements by Monte Carlo simulations or by Apruzese's method and solve the resulting algebraic system either by an eigensolution technique (when many different initial distributions are to be analysed) or by the propagator function method. We again stress at this point that many of the techniques that were presented in Chapters 5 and 6 are strongly interrelated. All methods proceed essentially in two steps. In a first step, a set of 'characteristic' parameters is computed—the $A_{k,m}$ matrix elements, the probabilities $p_i$, or the matrix elements for basis functions extending over the whole geometry. In the second step, these parameters can then be evaluated by various methods, like eigenvalue analysis. The specific interrelations are depicted in Fig. 14.1. ### 14.2.2 Inclusion of other physical effects Quite often one is confronted not with the pure classical Holstein trapping problem, but with some generalization. Methods for dealing with these generalizations were given in Chapter 10. One has to distinguish between two classes of effects, those that influence the reabsorption integral $\int G(\mathbf{r}, \mathbf{r}') \cdot n(\mathbf{r}') \mathrm{d}\mathbf{r}'$, and those that don't. Among the former effects, wall reflections are the most common. For this problem, there is no other way but setting up a new basic equation, or new boundary conditions for the differential equation formulation, and doing a solution from scratch. --- [FIGURE: Interrelation between computation methods for transient linear problems.] 1 ... Piecewise constant approximation, Sec. 5.3 2 ... Propagator function method, Sec. 5.4 3 ... Apruzese method, Sec. 5.4 4 ... Suggested by various authors 5 ... Classical Monte Carlo computation, Sec. 6.1 6 ... Multiple-Scattering representation, Sec. 6.2 7 ... Extraction of the lowest order mode (Lai), Sec. 6.2 8 ... Variational technique, Sec. 5.1 9 ... van Trigt's approach, Sec. 5.2 FIG. 14.1. Interrelation between computation methods for transient linear problems. --- Among the class of physical effects that do not show up in the reabsorption integral, there are quenching, branching, trapping to another level, and particle diffusion. We have a wide assortment of methods for the inclusion of these effects, with the modal combination technique being probably the most useful one. If we already have the solution to the pure trapping problem in terms of the eigensolutions, we can combine the trapping modes with the modal solutions for the additional effect. For particle diffusion, for example, analytical solutions are available for most geometries. A practically important example is trapping and diffusion in a two-dimensional cylinder. The most efficient solution of the pure trapping problem is given by the geometrical quantization technique (GQT). Full solution e.g. by piecewise-constant approximation, which can be easily generalized, would be extremely CPU-time intensive. On the other hand, generalization of the GQT to include additional effects is extremely difficult. Yet we can take the trapping solutions from the GQT and combine them via the modal combination technique with the modal solutions of the diffusion equation in the finite cylinder (ready for use in Chapter 3) to get the complete solution. The modal combination technique encounters its limits in multilevel atoms with many interconnected trapped transitions. We need the expansion of each eigenmode of each transition into the eigenmodes of each interconnected transition, and the expansion of this expansion for the next transition, and so forth. Under these conditions, the expansions can become more complicated than solution of the complete trapping problem. It is then advisable to change to some other technique. For another choice, the propagator function method, one just has to compute additional propagators $B_{k,m}$ that describe the influence of the added physical effect on the population transfer from subvolume $m$ to subvolume $k$. We then replace the basic $A_{k,m}$ elements by the elements $A_{k,m} + B_{k,m}$, and make the PFM computations exactly as described in Chapter 5. The additional propagators are obtained by solving the appropriate differential equation by Monte Carlo simulation, or by any other technique. Care must be taken whether the new propagator enforces some new boundary condition, like zero excited-state density at the cell walls for the effect of diffusion with wall quenching. When using Monte Carlo simulations, we have to make a new run any time that we include a new physical effect or that we just change a parameter—there is no possibility of efficient combination with other techniques. ## 14.3 Nonlinear trapping with CFR For non-linear radiation trapping, the first and most important step is setting up the basic equations, already including all secondary effects. Due to the non-linearity, one is not able to in some way combine solutions to reduced problems, but has to make a completely new simulation every time a parameter changes. Even for the analytical techniques (using the escape factor), we have to incorporate all secondary effects (like particle diffusion, branching, etc.) into the escape factor and adjust it every time a secondary parameter changes. --- ### 14.3.1 Taking the non-linearity into account For transient computations, solutions are done by a finite-differencing, FD, scheme in the time domain. This means that within one time step, the equations are *linear*. We can thus combine this finite differencing with *any* technique that was developed for the solution of the linear case. We stress that this is *not* only true for the examples mentioned in Part II, but also for techniques that will come up in the future. Considering the fact that in the last decade at least four completely new techniques—the geometrical quantization technique, the multiple-scattering representation, the fitting equations approach, and the propagator function method—for the solution of the classical Holstein equation have been developed, it seems reasonable to anticipate further such developments. There is, however, one important difference between the solution of the linear problem and the step-by-step solution of a non-linear problem: the efficiency of the techniques. The big advantage of the eigenfunction techniques for truly linear problems is that they solve the whole problem in one step. This advantage is lost in the FD solution of non-linear problems. As an example, let us compare the eigensolution method to the propagator function method. For the computation of the $A_{k,m}$ elements, we need the same amount of CPU time, since the matrix elements are the same in both techniques. In the eigensolution technique, we have to solve an $N \times N$ eigenvalue problem only once. The effort required is roughly proportional to $N^3$. For the propagator function technique, we have to calculate a matrix-vector multiplication in every timestep. The effort for this is $N_{\text{timestep}} \cdot N^2$. It is thus usually more efficient to use the eigensolution approach for truly linear problems. In the non-linear case, the matrix elements change with each timestep. In the eigensolution approach, we have to find the eigensolutions of the $A_{k,m}$ matrix for *each* timestep, i.e. $N_{\text{timestep}} \cdot N^3$ operations. For the propagator function method, we still require one matrix-vector multiplication for each timestep, i.e. $N_{\text{timestep}} \cdot N^2$ operations. We see that now the eigensolution technique shows considerable disadvantages for the FD solution of non-linear problems. For steady-state problems, we have mainly the choice between complete linearization and direct iteration. The complete linearization technique requires larger analytical effort for the derivation of the linearized equations, but rewards this effort with faster convergence. As mentioned in Chapter 13, the linearized equations have the form of the truly linear ones, plus some additional ‘driving’ term that represents the correction from the previous iteration step. This means again that we can use essentially all the methods described in Chapters 5–12 for the solution of these equations. The problem is linear within each iteration step also for direct iteration. ### 14.3.2 Multilevel atoms When dealing with multilevel atoms, we essentially have the choice between the equivalent two-level atom technique, ETLA, and the complete linearization. ETLA has the advantage of being easier to implement. It is especially suited when the relevant transitions are not strongly coupled. On the other hand, convergence is not guaranteed (except --- if some special tricks are used) or can be extremely slow (for strongly coupled transitions). The ETLA in the classical sense is somewhat restricted in its choice of method. By definition, it computes the source function of the equivalent two-level atoms, using the Feautrier technique or some related approach. However, we can also define ETLA somewhat more loosely as a direct iteration. We just assume in each iteration step that the solution for each transition is not influenced by the other transitions. We then have the full range of methods for the solution of a linear two-level problem at our disposal. Complete linearization, on the other hand, gives very fast convergence also in the multilevel case. It is thus preferable for high-efficiency programs. However, most programs for plasma problems (hydrocodes), where lots of levels are involved, use an ETLA or direct iteration approach. We have found no justification in terms of increased efficiency as compared to the CLT for this choice—we thus suspect that ease of implementation led to this choice. ### 14.3.3 Required accuracy In any radiation trapping problem, we have a tradeoff between accuracy and speed of solution. While for linear problems there are many tricks that allow an accurate solution with small numerical effort, there is usually a clear-cut tradeoff between accuracy and CPU time in the non-linear case. In the linear regime, the closed-form approximation (based on the escape factor technique) is a very fast approach, but gives low accuracy for the spatial distribution of the excited-state atoms. The Eddington approximation gives about 10% accuracy for the steady-state and reduces the computational effort typically by one order of magnitude. The Milne approximation gives an accuracy of about 20% up to opacities of about one hundred, and results in savings of another one to two orders of magnitude. ## 14.4 Trapping with partial frequency redistribution Complete frequency redistribution, CFR, makes life much easier, and is fortunately fulfilled in most cases. When PFR has to be considered in a linear trapping problem, we can distinguish between two basic cases. a) A high-opacity vapour, where the wings are determined by natural broadening. When the Voigt parameter is very small, and the accuracy requirements are modest, Post's variational technique can be used. The accuracy restriction is not given by the variational ansatz, but rather by the use of the Jefferies–White approximation to the redistribution function. For higher accuracy, the true $A_{k,m}$ matrix elements for PFR should be computed. Solution of the algebraic system of equations is then usually best done with the propagator function method. In case that only the lowest-order mode is required, and that the number of spatial and frequency discretization points is very small, solution of the eigenvalue problem is more efficient. b) Trapping in cold atoms, where the Doppler width is much smaller than the natural width. For a first estimate on the lowest-order mode, the analytical approx- --- imations described in Sec. 11.7 and 15.7 can be used. For very high accuracy, again the PFM should be used. The applicability of the PFM hinges on the available computer memory. In a two- or three-dimensional geometry, the $A_{k,m}$ matrix becomes so large that efficient computations are no longer possible. In all cases, Monte Carlo simulations are an attractive alternative. However, efficient implementation of the exact redistribution function is not trivial. Writing the code is considerably more difficult than in the CFR cases. Monte Carlo simulations are the easiest way to incorporate the influence of polarization on the trapping process. When, however, only the degree of polarization of the emergent radiation is required, one can make normal trapping computations and convert the results with the formulas of Chapter 6 into the statistics of the number of photons that suffer $I$ absorption/reemission processes. From this, it is relatively easy to compute the depolarization that occurred. Alternatively, we can solve the vector equation of transfer with a prescribed source function, as in Sec. 12.2. As long as the PFR problem is linear, the options for combinations with other methods, like e.g. the modal combination technique, are the same as for linear CFR problems. There is only the slight problem that the excited-state distribution now depends on position and frequency, while almost all other physical effects (like particle diffusion) depend on position only. For non-linear trapping problems with PFR, the procedures are essentially the same as for non-linear CFR problems, only that the source function becomes frequency dependent. The selection criteria for efficient methods and the combination opportunities remain the same. --- ## 15 ### MEASUREMENTS IN CHEMICAL PHYSICS In the present chapter we describe some aspects of the interpretation of measurements in chemical physics. The examples will show the difficulties that can occur in the incorporation of radiation trapping. We can also deduce some general guidelines for such experiments that should help to reduce some of the occurring errors. One of the most important points is choosing an appropriate geometry for the experimental vessel. We know that the incorporation of a genuinely three-dimensional geometry is exceedingly difficult. Appropriate mirroring of the cell walls allows one to find an ‘equivalent’ cell that can often be treated more easily than the actual cell. The simplest example is a cylindrical cell with mirrored side walls, which is a good approximation for the plane-parallel slab.[^21] In steady-state experiments, it is often helpful to use an external excitation that is shaped like the lowest-order mode of the Holstein equation. In that case, we can use ‘effective-lifetimes’ in the rate equations and may neglect all higher-order modes. Such a measure is also recommended for pulsed-excitation problems, since then we do not have to wait for ‘late times’, i.e. when the higher-order modes have died out, but when also the signal-to-noise ratio has deteriorated. When shaping of the excitation function is not possible—which is usually the case—incorporation of the higher-order modes is vital for good accuracy. It is very common to use the effective-lifetime approach even if the conditions for its validity are not strictly fulfilled. We will cite in the following many papers that use this approach. In our study of the existing literature, we have discovered essentially three types of these papers: (i) Experiments that were explicitly designed in such a way that the effective-lifetime approximation is justified strictly (or at least in very good approximation). (ii) Experiments where the effective-lifetime approximation is not strictly justified, but where it results in acceptable accuracy *for those specific experiments*. By ‘acceptable accuracy’, we mean that the errors introduced by the effective-lifetime approximation are smaller than other experimental uncertainties. (iii) Papers where no error analysis was performed, and authors just neglected higher-order modes without further consideration. It is often practically impossible for the reader of a paper to distinguish between these categories (especially between (ii) and (iii)). In citing the references, we will thus [^21]: This is of course not true for wall quenching effects. Yet when the trapping problem is linear, we can combine the solutions of the one-dimensional trapping problem and of the two-dimensional diffusion problem by the modal expansion technique of Sec. 10.4. --- not make any further such distinction. In any case, we warn that the use of the effective-lifetime approach in existing papers should *not* be taken as a warrant to use it in new experiments without proper analysis. Another important point is that hyperfine splitting should often be taken into account. Ignoring hfs can easily lead to errors of a factor of two or more. ## 15.1 Atomic lifetimes The measurement of atomic lifetimes seems to be the most direct application of radiation trapping theory. It is evident why radiation trapping leads to an apparent increase in the atomic lifetimes. Nonetheless there are many pitfalls, most of which are associated with higher-order modes. The most straightforward way to measure an atomic lifetime is to excite the vapour with a short pulse of resonance radiation and to observe the response, i.e. the spontaneously emitted radiation. If the vapour has very low opacity, there is a single-exponential decay, and the measured decay time constant gives the desired Einstein $A$ coefficient of the transition. When the opacity is larger, and the shape of the excitation function, e.g. the spatial shape of the laser beam, is very similar to the lowest-order trapping mode, the temporal decay of the excited-state distribution (and of the total emergent radiation) still follows a single exponential law, but now with a time constant $A_{21}/g_0$. All this assumes that the basic assumptions listed in Chapter 4 are fulfilled. However, such ideal conditions are not always realized experimentally. First, one rarely actually observes the excited-state distribution or the total emergent radiation. It is much more common to observe only the radiation emerging into a certain spatial angle. When one uses a monochromator, one observes only radiation that emerges within a certain spatial angle from the vapour cell. Depending on the arrangement of the collection optics, this spatial angle can vary from small (i.e. we observe only the radiation that emerges at a right angle from the vessel surface) to very large (i.e. we observe all radiation emerging from the cell). As we have seen in Sec. 7.6, radiation emerging normally from the cell has a decay time constant that is different from the decay constant of the total emergent radiation. Good design of the collection geometry and its proper modelling is thus of utmost importance. Second, the excitation function usually does not resemble the lowest-order trapping mode. This is especially true for experiments where a spectral lamp illuminates the cell from broadside. When the excitation source is a laser, it is sometimes possible to shape the beam and make it is similar to the lowest-order mode. In many cases such a shaping cannot be realized because of power constraints (e.g. in energy-pooling experiments, where we need a high *density*, not just a high number of excited-state atoms) or because of other experimental problems (e.g. we have to avoid illuminating the side walls of the cell in order to minimize scattering of the laser light). It is often difficult to judge whether the approximation was justified due to an appropriate arrangement of the experimental conditions, or whether the higher-order modes were just simply neglected. --- The interpretation of most time-decay experiments is done with the lowest-order mode approximation. Papers in that context are Dubreuil and Catherinot (1978), Michael and Yeh (1970), Kibble *et al.* (1967), Leveau and Valignat (1981), Klose (1975a, b), Yanson *et al.* (1989), Heron *et al.* (1956), Huennekens and Gallagher (1983a), Huennekens *et al.* (1987), Halstead and Reeves (1982), Husain and Roberts (1988) and Simon *et al.* (1994). There are also many lifetime experiments that confirm trapping theories under various experimental conditions. The low-opacity Milne theory was confirmed by Michael and Yeh (1970), Kibble *et al.* (1967), and Colbert and Huennekens (1990). The hfs-splitting theory described in Chapter 7 was confirmed by Alpert *et al.* (1949) and Huennekens and Colbert (1989). The Voigt-line theory of Walsh (1959) was confirmed by himself and by Hammond and Gallo (1972). The fact that the effective lifetime of self-broadened radiation becomes independent of the pressure at high densities is confirmed by Wieme and Vanmarcke (1979), Millet *et al.* (1982), Turner (1965), and Thonnard and Hurst (1972). The Holstein theory in foreign-gas broadening was confirmed experimentally by Huennekens *et al.* (1987); experiments in the far wings of lines were done by Colbert and Huennnekens (1991). Lifetimes in three-level atoms were investigated by Bicchi *et al.* (1994). Lifetime measurements are especially important in laser media, where a small error in the Einstein $A$ coefficient can make the difference between good and bad laser media. Radiation trapping in these media was explicitly considered by Sumida and Fan (1994). Figure 15.1 shows an overview of measured effective lifetimes for the mercury $6^3\mathrm{P}_1$-state as a function of mercury density (van de Weijer and Cremers 1985). Comparison is done with the theory by Phelps, i.e. essentially with the accurate solution of the Holstein equation by numerical methods. We see that even for such a well-investigated transition, there is considerable uncertainty. This gives us an impression on how much experimental uncertainty (determination of mercury density, higher-order modes, insufficient SNR, etc.) can affect the results. The problem of the higher-order modes can be circumvented in time-decay experiments simply by waiting long enough if the SNR is sufficiently high at late times. In steady-state setups, the situation is even more prone to errors. For atomic lifetime measurements, steady-state normally implies the use of phase fluorometry. The vapour is excited by light that is sinusoidally-modulated with angular frequency $\omega$. As long as radiation trapping is linear, the fluorescence is also sinusoidally modulated, and has a phase shift $\varphi^{\mathrm{ph}}$ with respect to the input radiation that is given by $\varphi^{\mathrm{ph}} = \mathrm{atan}(\omega \cdot \tau_{\mathrm{eff}})$, see Mochizuki and Fowler (1965). However, the effective lifetime $\tau_{\mathrm{eff}}$ obtained in this way will be determined by all modes (Umemoto *et al.* 1980), and it is impossible to separate them solely from the measurement data. Unfortunately, it has become common in the literature to postulate the existence of just one trapping mode, and to do all the computations and interpretations under this assumption. ## 15.2 Collision cross-sections In order to assess the basic influence of radiation trapping on the measurement of collision cross-sections, let us consider the classical problem of measuring the cross-section --- COLLISION CROSS-SECTIONS [FIGURE: Graph of Effective radiative lifetime vs Opacity for Hg 6³P₁] FIG. 15.1. Effective radiative lifetime ($g_0\tau$ in our notation) of the Hg$6^3\text{P}_1$ level as a function of opacity. Full squares (Thomas and Gwinn 1948), full circles (Alpert *et al.* 1949), crosses (Phelps and McCoubry 1960), full triangles (Yang 1966), triangles (Nussbaum and Pipkin 1967), circles (Michael and Yeh 1970), pluses (Halstead and Reeves 1982), squares (van de Weijer and Cremers 1985), trace 1 (Phelps 1958), trace 2 (Walsh 1959). From (van de Weijer and Cremers 1986). for collisions between two excited sodium 3p-atoms, leaving a 4d-excited atom and a ground-state 3s-atom. We populate the 3p-state by exciting a sodium vapour with D1 (or D2) resonance radiation. These 3p-atoms collide, and lead to a population in the 4d-state. We then observe the intensity of the fluorescence radiation from the 4d-state, which is a measure for the number of atoms that have reached the 4d-state through this so-called ‘energy pooling’ process. This measurement can be done either in steady-state, which is the more usual approach, or with a pulsed excitation. The 4d-density of course strongly depends on the 3p-density. For a steady-state experiment, radiation trapping leads to an increase in the 3p-density (often, this is necessary to obtain a sufficiently large excited-state density), and to a change in the spatial distribution of the 3p-population. For a transient experiment, the trapping leads to an increase in the effective decay time, and we have a spatial distribution that changes with time. --- The situation is simplest when we have a steady-state experiment with weak excitation, i.e. linear trapping, and when the spatial shape of the laser beam is similar to the lowest-order trapping mode. In that case, the *shape* of the excited-state distribution remains as it would be without radiation trapping, only the excited-state density is scaled by a factor $g_0$. In that case, we can use the 'effective-lifetime' concept very easily. The collisional excitation rate to the 4d-state can be determined from the measured intensities $I$ of the 3p-3s resonance fluorescence and of the 4d-3p 'higher' fluorescence (Moi 1986), $$ C_{\text{coll}} = c_{\text{cal}} \frac{1}{\beta} \frac{h\nu_{3\text{p}\to 3\text{s}}}{h\nu_{4\text{d}\to 3\text{p}}} \frac{I_{4\text{d}\to 3\text{p}}}{I_{3\text{p}\to 3\text{s}}} \frac{1}{\tau_{\text{eff}}} \frac{\int n_{3\text{p}}(\mathbf{r})\text{d}\mathbf{r}}{\int [n_{3\text{p}}(\mathbf{r})]^2\text{d}\mathbf{r}} \qquad (15.1) $$ where $c_{\text{cal}}$ describes the different sensitivities of the detector at the two observed wavelengths, $\beta$ is the branching ratio for 4d-atoms towards the 3p state; $\tau_{\text{eff}}$ is the effective lifetime $g_0/A$, where $A$ is the Einstein coefficient. An example for this approach is given by Allegrini *et al.* (1983). Similarly, this approach is also justified for pulsed excitation if the spatial shape of the laser pulse matches the lowest-order mode. Compared to steady-state, this approach has the additional advantage that the effective lifetime can be measured rather than calculated. In the pulsed experiment, it is also possible to wait until the higher-order modes have died out (see e.g. (Huennekens and Gallagher 1983b), (Werij *et al.* 1991)). Other experiments that use this effective-lifetime approach are due to Lott *et al.* (1975), Wine and Glick (1976), Huennekens *et al.* (1985), Kowalczyk *et al.* (1985) and Milosevic *et al.* (1995). When we have to determine the (lower, 3p) excited-state distribution, we have basically two possibilities, computation and measurement. Computations have the advantage that they can be done in a very reproducible way, and are often much easier (De Filippo *et al.* 1996). Problems occur when the trapping process (or any other process influencing the excited-state distribution) cannot be modelled completely accurately. A particularly vexing problem is the modelling of the detection geometry, which has a considerable influence on the actual decay time. Measurement of the excited-state distribution requires a more complex measurement setup. Some probe source—a laser or a spectral lamp—is needed to measure the absorption of the probe as it passes through the excited-state distribution. A complication for the measurement arises since we do not need the integral absorption as the probe light passes through the vapour, but we need the excited-state distribution at any point. The reason for this is that the square of the easily measured integrated density is different from the integral of the square of the local density, which determines the rate of the energy-pooling collisions. We thus either have to rely on computations to tell us at least the shape of the excited-state distribution, while we take the absolute value from the measurement, or we need a very elaborate measurement setup, in which we can shift the position of the probe beam and measure the absorption at many points (Molander *et* --- *al.* 1984), (Neuman *et al.* 1994); some simplifications of such a setup can be realized by the approach of Jabbour *et al.* (1996). These considerations have assumed that the transitions from the energy-pooled 4d-state are not trapped. There are, however, many cases where also this higher radiation is quite efficiently reabsorbed. This is especially true when the observed 4d higher-level fluorescence radiation has a larger Einstein $A$ coefficient than the excited transition. Now the observed higher-level radiation always has a branching ratio $\beta$ smaller than unity, thus photons that are reabsorbed have a good chance of being reemitted at a 'useless' wavelength. This effect can lead to a dramatic change in the intensity of the higher-level radiation. An example for accurate computation of this effect is given by Jabbour *et al.* (1996). An alternative is to observe an untrapped transition originating from the higher state. Though such a transition is still affected by trapping of other transitions originating from the same state (the effective branching ratio changes), the sensitivity to computation and modelling inaccuracies is greatly reduced. In any case, the opacity for the higher-level radiation must be computed a posteriori, in order to check whether any trapping did occur (Keramati *et al.* 1988). ## 15.3 Ionization cross-sections The problems occurring in the measurement of ionization cross-sections are very similar to those occurring in energy pooling collisions. A major difference is that one usually does not detect some kind of higher-level radiation in order to determine the actual higher excited state density (in our case now the ion or electron density), but one usually collects the freed electrons. Since now the detection mechanisms are different, measurement parameters like the absolute detector sensitivity, the aperture angle of the detection optics, etc., no longer cancel out. Accurate modelling of the measurement setup thus becomes more important. Radiation trapping can have a very important influence on the interpretation of collisional-ionization experiments. In the 1970s, there were several experiments that baffled researchers at that time. Salter (1979) suggested a qualitative explanation in terms of radiation trapping. Later experiments included the trapping in a more or less careful way. In order to give an idea of the problems occurring in the determination of ionization cross-sections, we describe one (very carefully designed) experiment in more detail. Huennekens and Gallagher (1983d) measured the effective cross-section for associative ionization in collisions between two sodium 3P atoms. The 3P energy level of sodium, Na, is at $16\,968 \text{ cm}^{-1}$. When two such excited atoms collide, they can pool their energy. At about twice the 3P energy, there is a wealth of other atomic levels, and also the 'bottom' of the $\text{Na}_2^+$ energy curve lies below that energy. Table 15.1 lists some of the relevant energy levels. The rate at which $\text{Na}_2^+$ ions are created is easily measured. One just has to 'suck' the ions and electrons to some electrodes. The current thus created is --- MEASUREMENTS IN CHEMICAL PHYSICS **Table 15.1** *Some energy levels of sodium, Na.* | Level | Energy / cm$^{-1}$ | | :---: | :---: | | 3 P | 16968 | | 4 S | 25740 | | 3 D | 29173 | | 4 P | 30271 | | 5 S | 33201 | | 4 D | 34549 | | 4 F | 34587 | | 5 P | 35042 | | Na$_2^+$ | ~33530 | [FIGURE: FIG. 15.2. Layout of the experiment by Huennekens and Gallagher.] $$ \frac{1}{2} e_{-} k_{AI} \int_V [n_{3\text{p}}(\mathbf{r})]^2 \text{d}\mathbf{r} \qquad (15.2) $$ where $V$ is the volume from which the charged species are collected and $k_{AI}$ is the ionization rate coefficient. In order to determine the rate coefficient and the ionization cross-section, we need an accurate determination of the Na 3P densities, and this can cause major problems. The geometry of the experimental setup is sketched in Fig. 15.2. A circular laser beam was sent through a rectangular cell filled with sodium vapour to excite the 3P first resonance level. The 3S ground state vapour density was determined by absorption measurements. We now assume that the excited state density $n_{3\text{p}}(\mathbf{r})$ can be written as $$ n_{3\text{p}}(0, 0, 0) \cdot X(x) \cdot Y(y) \cdot Z(z), \qquad (15.3) $$ i.e. we assume that we can separate the spatial distribution into products of three one-dimensional distributions that depend only on one spatial coordinate $x, y, z$. The spatial --- functions $X$, $Y$, and $Z$ are now determined in different ways. The function $Z$ is determined from the known laser pump power attenuation in the $z$-direction. The function $Y$ is determined by spatially resolved measurements of the 3P$\rightarrow$3S fluorescence emerging through the observation window. Finally, the lateral distribution $X$ is computed from radiation trapping theory for a plane-parallel slab, $-L/2 < x < L/2$. The use of the slab geometry for the given situation is necessary to keep the computational complexity within reasonable bounds, but it can still be a source of errors. The slab geometry implies that in the chosen coordinate system there are no variations in the $y$ and $z$ directions, $\partial/\partial y = \partial/\partial z = 0$. For the $z$-direction this is true when the variations in this direction are on a scale that is much larger than the width of the slab in the $x$-direction, which was 6 mm. This small excited state density variation along the exciting beam was achieved by slightly detuning the laser from the line centre. However, the condition $\partial/\partial y = 0$ cannot be fulfilled, since the exciting laser is circular. Measurement of the spatial distribution $Y(y)$ indeed showed that the population in $Y$ drops to very small values within 6 mm from the laser beam centre. Also problematic is the use of van Trigt's high-opacity eigenfunctions and eigenvalues. In the experimental situation, we have an opacity on the order of 10. In this regime, the eigenvalues and especially the eigenfunctions of van Trigt deviate considerably from the true values (see also Sec. 5.2). We think that the error thus introduced is on the order of 10-20%. On the other hand, there are also many other possible error sources which give a total estimated error of $\pm 40\%$, and comparison with other published results shows differences of up to an order of magnitude. Thus the errors introduced by using the simple slab theory seem tolerable. The steady-state distribution of the excited-state atoms is then computed by the method of Chapter 7: expand the laser beam into the eigenmodes of the slab and get the expansion coefficients from $$ \alpha_j = \int \psi_j(x)n(x, t = 0)\mathrm{d}x $$ (15.4) where the initial excited-state distribution is assumed to follow the laser spatial distribution. Then compute the steady-state distribution as $$ X(x) = \frac{\sum_j g_j \alpha_j \psi_j(x)}{\sum_j g_j \alpha_j \psi_j(0)} $$ (15.5) The inclusion of two to four modes is usually sufficient. A second important point is the determination of the scaling density $n_{3\mathrm{p}}(0, 0, 0)$. Huennekens and Gallagher did this by three different methods, and we will describe two of them here. One method was to compute it from the laser power that is absorbed by the vapour, --- MEASUREMENTS IN CHEMICAL PHYSICS $$ n_{3\text{p}}(0, 0, 0) = \frac{P_{\text{abs}}}{h\nu} \left[ A_{21} \int Z(z)\text{d}z \int Y(y)\text{d}y \int X(x)\eta(x)\text{d}x \right]^{-1} $$ (15.6) where $\eta(x)$ is the angle-averaged escape factor. Now we know that for each mode, $\eta(x) = 1/g_j$, so that $$ \int X(x)\eta(x)\text{d}x = \frac{\sum_j \alpha_j \int \psi_j(x)\text{d}x}{\sum_j g_j \alpha_j \psi_j(0)} $$ (15.7) If only the lowest-order mode would be used, the result for $n_{3\text{p}}(0, 0, 0)$ would be 20% off, even though the incident laser beam and the lowest-order mode have quite similar shape—again demonstrating the importance of considering higher-order modes. Another possibility is to determine $n_{3\text{p}}(0, 0, 0)$ from the 3P+3P$\rightarrow$3S+5S excitation transfer, since the rate coefficient of this process is known. For this method, we observe the intensity of the 5S$\rightarrow$3P fluorescence and compare it to the 3P$\rightarrow$3S fluorescence. We observe only light from a thin stripe near $z = 0$. The intensity of the 3P$\rightarrow$3S fluorescence emerging into the spatial angle $\text{d}\Omega$ is $$ I_{3\text{p}} = h\nu \frac{\text{d}\Omega}{4\pi} \iint \sum_j n_{3\text{p}_j}(x, y, 0) A_{3\text{p}_j\rightarrow 3\text{s}} \eta_{3\text{p}_j\rightarrow 3\text{s}}(x, y, 0) \text{d}x\text{d}y $$ (15.8) where we sum over the fine-structure levels, that is, $j$ takes the values 1/2 and 3/2. Since the vapour has a rather high opacity, we can use the Holstein approximation (for a Lorentz lineshape) for the escape factor $$ \eta_{3\text{p}_j\rightarrow 3\text{s}} = \frac{1}{\sqrt{\pi k_j \left( \frac{L}{2} - x \right)}} $$ (15.9) where $k_j$ is the line-centre absorption coefficient for the radiation from the $3\text{p}_j$ level. From theory and from measurements of the self-broadening rates, one knows that $$ \frac{\sqrt{\pi k_{3/2}}}{1.11} = \sqrt{\pi k_{1/2}} = 609\,\text{cm}^{-1/2}, \quad \text{and} \quad \frac{I_{3\text{p}_{1/2}}}{I_{3\text{p}_{3/2}}} = \sqrt{\frac{k_{3/2}}{k_{1/2}}} \frac{n_{3\text{p}_{1/2}}}{n_{3\text{p}_{3/2}}} $$ (15.10) so that $$ \sum_j \frac{n_{3\text{p}_j}(0, 0, 0)}{\sqrt{\pi k_j}} = \frac{n_{3\text{p}}(0, 0, 0)}{609} \frac{I_{3\text{p}_{1/2}} + I_{3\text{p}_{3/2}}}{I_{3\text{p}_{1/2}} + 1.11 I_{3\text{p}_{3/2}}} = n_{3\text{p}}(0, 0, 0) \cdot c1 $$ (15.11) where $c1$ is just an abbreviation. Inserting Eqs. (15.9)–(15.11) into the expression for the 3p$\rightarrow$3s fluorescence intensity, Eq. (15.8), we get --- $$ I_{3p} = h\nu_{3p} \frac{d\Omega}{4\pi} A_{3p \to 3s} n_{3p}(0, 0, 0) c1 \int Y(y)dy \int X(x) \frac{1}{\sqrt{L/2 - x}} dx $$ (15.12) The equation for the $5s \to 3p$ florescence is similar, but since this transition is not trapped, no escape factor occurs: $$ I_{5s} = h\nu_{5s} \frac{d\Omega}{4\pi} A_{5s \to 3p} \iint n_{5s}(x, y, 0) dx dy $$ (15.13) We further need the rate equation for the $5s$ state, $$ \frac{n_{5s}(x, y, z)}{\tau_{5s}} = \frac{1}{2} C_{3p+3p \to 5s} n_{3p}(x, y, z)^2 $$ (15.14) From the ratio of the fluorescence intensities, Eqs. (15.13) and (15.12), and with the rate equation (15.14), we finally get $$ n_{3p}(0, 0, 0) = 2 \frac{I_{5s}}{I_{3p}} \frac{c_{\text{cal},3p}}{c_{\text{cal},5s}} \frac{h\nu_{3p}}{h\nu_{5s}} \frac{A_{3p \to 3s}}{A_{5s \to 3p}} \frac{c1}{\tau_{5s} C_{3p+3p \to 5s}} \frac{\int Y(y)dy \int \frac{X(x)}{\sqrt{L/2 - x}} dx}{\int (Y(y))^2 dy \int (X(x))^2 dx} $$ (15.15) where the $c_{\text{cal}}$ are the detector efficiencies at the two transition wavelengths. The absolute density $n_{3p}(0, 0, 0)$ can thus be determined from the measured fluorescence. The experiment described includes a very careful consideration of radiation trapping effects. An alternative method is of course to measure the excited-state density by some probe beam (Bonanno *et al.* 1983). For computations of the excited-state density it is quite usual to just modify the Einstein $A$ coefficient, i.e. to use the effective lifetime approach (Measures *et al.* 1981), (Measures and Cardinal 1981), (Bachor and Kock 1981), (Cannon *et al.* 1993). Remember the cautionary remarks that we have made at the beginning of the chapter: the approach is often a good approximation, but the fact that some authors used it justifiably for *their* experimental setup does *not* mean that you can use it for *your* experiments. Another point in case is a method used e.g. by Jahreiss and Huber (1983). There the authors define an 'equivalent' cylinder that has the radius of the exciting laser beam, and trapping is then computed by computing the trapping within this 'equivalent' cylinder. Such an approach is approximately valid in the very specific case that photons leaving the directly excited region are effectively lost for the process of interest (e.g. energy pooling) and the reabsorption from photons originating in 'outer' regions in the directly excited region is negligible. However, the method cannot be generally valid, because the strictly correct way to treat the problem is of course an expansion of the excitation function into the eigenmodes of the real vessel. ## 15.4 Quenching and intermixing cross-sections The experimental situations for the determination of quenching and intermixing cross-sections are not too different from what we have discussed with the energy pooling --- [FIGURE: FIG. 15.3. Partial energy level scheme of sodium, Na.] cross-section. Again, there are some papers that use the lowest-order mode approach, either in the interpretation of time-decay measurements (Matland 1953), (Copley and Krause 1969), (Waddell and Hurst 1970), or of steady-state experiments (Yang 1966). For a steady-state investigation of quenching cross-sections, the assumption of only the lowest order mode leads to a modified Stern–Volmer equation $$ \frac{Y}{Y_0} = \frac{1}{1 + Q g_0 \tau} $$ (15.16) where $Q$ is the quenching rate, and $Y$ and $Y_0$ are the emerging intensities of the resonance radiation with and without quenching. We see that an increase in the trapping factor $g_0$ leads to a decrease in the intensity $Y$. This is intuitively clear, since the longer effective lifetime gives the quenchers more time to destroy the excitation, see also Saenger (1989). We also note that the presence of a foreign gas (quencher) broadens the line and thus leads to a reduction of $g_0$. As an example for the determination of an intermixing cross-section, we describe an experiment by Huennekens and Gallagher (1983a). It was performed in order to determine the cross-section for the sodium $3P_{1/2} \leftrightarrow 3P_{3/2}$ intermixing in collisions with ground-state sodium atoms. The experimental setup is similar to the one described in the previous section. Pulsed excitation is used, and only the ‘late-time’ regime is evaluated. All conditions for the validity of the Holstein approximation are fulfilled, the opacity is rather high, and CFR is valid. Figure 15.3 shows the relevant part of the Na energy level scheme. At time $t = 0$, a laser pulse excites Na ground state atoms to the $3P_{3/2}$ state. Due to the comparatively low vapour densities, energy pooling can be neglected. The rate equations are then --- $$ \frac{\partial n_2}{\partial t} = - \left( \frac{A_{20}}{g_{2\to0}} + C_{21} \right) n_2 + C_{12}n_1 \qquad \frac{\partial n_1}{\partial t} = - \left( \frac{A_{10}}{g_{1\to0}} + C_{12} \right) n_1 + C_{21}n_2 $$ (15.17) where $C_{21}$ and $C_{12}$ are the wanted intermixing rates. Equation (15.17) is a system of two ordinary linear differential equations with constant coefficients that can be solved by standard means. Solutions are of the form $$ n_2 = c1 \exp(-\gamma_- t) + c2 \exp(-\gamma_+ t) \qquad n_1 = c3 \exp(-\gamma_- t) + c4 \exp(-\gamma_+ t) $$ (15.18) with the decay constants $$ \begin{aligned} \gamma_+ &= \frac{1}{2} \left[ \frac{A_{10}}{g_{1\to0}} + \frac{A_{20}}{g_{2\to0}} + C_{21} + C_{12} \right] \\ &\quad + \frac{1}{2} \left[ \left( \frac{A_{10}}{g_{1\to0}} - \frac{A_{20}}{g_{2\to0}} \right)^2 + 2(C_{12} - C_{21}) \left( \frac{A_{10}}{g_{1\to0}} - \frac{A_{20}}{g_{2\to0}} \right) + (C_{12} + C_{21})^2 \right]^{1/2} \\ \gamma_- &= \frac{1}{2} \left[ \frac{A_{10}}{g_{1\to0}} + \frac{A_{20}}{g_{2\to0}} + C_{21} + C_{12} \right] \\ &\quad - \frac{1}{2} \left[ \left( \frac{A_{10}}{g_{1\to0}} - \frac{A_{20}}{g_{2\to0}} \right)^2 + 2(C_{12} - C_{21}) \left( \frac{A_{10}}{g_{1\to0}} - \frac{A_{20}}{g_{2\to0}} \right) + (C_{12} + C_{21})^2 \right]^{1/2} \end{aligned} $$ (15.19) The integration constants $c1$ to $c4$ are determined from the boundary conditions. The excited-state densities are then (for excitation of the $3p_{3/2}$ state) $$ \begin{aligned} n_2(t) &= \frac{n_2(0)}{\gamma_+ - \gamma_-} \left[ \left( \frac{A_{10}}{g_{1\to0}} + C_{12} - \gamma_- \right) e^{-\gamma_- t} + \left( \gamma_+ - \frac{A_{10}}{g_{1\to0}} - C_{12} \right) e^{-\gamma_+ t} \right] \\ n_1(t) &= \frac{n_2(0)}{\gamma_+ - \gamma_-} \left[ e^{-\gamma_- t} - e^{-\gamma_+ t} \right] C_{21} \end{aligned} $$ (15.20) We have thus a double-exponential decay of the excited-state densities. Such double-exponential curves are fitted to the measured output for a range of sodium densities. We thus obtain the decay constants $\gamma_+$ and $\gamma_-$ as a function of the Na density. The ratio of the trapping factors $g_{1\to0}$ and $g_{2\to0}$ is known very well and is fixed at a value that is close to two for a Doppler-broadened line, since the opacities for these transitions are in the ratio 2: 1. The ratio of the wanted intermixing coefficients $C_{12}/C_{21}$ is known to be 1.92 from the principle of detailed balance. We see from Eq. (15.19) that at low densities the decay constant is $$ \gamma_- = \frac{A_{20}}{g_{2\to0}} $$ (15.21) while at high densities $$ \gamma_- \approx 0.34 \frac{A_{10}}{g_{1\to0}} + 0.66 \frac{A_{20}}{g_{2\to0}} $$ (15.22) --- We can thus determine the intermixing rates by inserting this into the expression for the other decay constant, $\gamma_+$. Specifically, in the high-density limit ($C_{12}, C_{21} \gg (A_{10}/g_{1\to0}), (A_{20}/g_{2\to0})$), we find $\gamma_+ \approx C_{12} + C_{21}$. ## 15.5 Atomic beams Some measurements in chemical physics are performed with advantage in an atomic beam. The main benefit is that all atoms have very similar velocity, and the direction of flight is also the same. Hence, Doppler broadening is strongly reduced, and this allows some measurements to be more accurate. The reduction of the Doppler broadening also has important consequences for the radiation trapping. We have seen previously that when there is only Doppler and natural broadening, the absorption is coherent in the rest frame of the atom, and the quasi-CFR is obtained only because the reemission can happen in any direction, and thus with a shift of the emission frequency in the laboratory rest frame. Let us deal with the limiting case that the atomic beam is exactly collimated, i.e. that there is only a single velocity component $v_z$, We start out with an ensemble of excited atoms that all have this velocity in the $z$ direction. An atom emits a photon in a certain direction with angle $\vartheta$ with the $z$-axis. The photon can then be reabsorbed by any atom, --- When the atomic beam comes from an effusive source, i.e. the atoms evaporate from a heated reservoir, then the atoms will not have a single velocity, but velocities according to a Doppler distribution. However, the velocity *vector* of every atom still is only in the $z$-direction. The basic physical processes do not change. Photons can only be reabsorbed by atoms that have the same velocity as the emitting atom. We thus only have to average the excited-state distribution over the frequency distribution, which in turn is determined by the velocity distribution of the initial ensemble of excited atoms. If these initial atoms all have a certain velocity $v_s$ corresponding to a certain frequency $x_s$, then the average number of absorption/reemission processes for a photon emitted by an initially excited atom at point $r$ is $$p_{\text{abs}}(r) = 1 + \frac{3}{2}(\pi - 1)k(x_s)R + \frac{15}{4}k^2(x_s)R^2\left(1 - \frac{r^2}{R^2}\right)$$ (15.25) For an actual atomic beam, the velocities are not completely in the $z$-direction, but there is also a certain divergence of the beam, i.e. the beam has a conical shape. This means, however, that we also have velocity components in the $x$ and $y$ directions. The larger these components are, the more the situation resembles a normal vapour cell, i.e. frequency redistribution sets in. For an opacity of $k_0 R = 3$, an opening angle of the cone equal to 0.04 (i.e. 2.3°) reduces the trapping factor from 17 to 15; an opening angle of 0.2 (i.e. 12°) reduces it to 10. Tables for various opacities and opening angles can be found in Bezuglov and Gorshkov (1984). Bezuglov *et al.* (1982) also made measurements with beams of various elements and found satisfactory greement with the theory. Calculations for the case of two intersecting atomic beams can be found in Bezuglov (1985). Despite the fundamental differences between trapping in beams and in vapour cells (Bezuglov *et al.* 1984), the usual Holstein theory was sometimes used for the computation of trapping in an atomic beam (Selter and Kunze 1978). ## 15.6 Atomic densities The indispensable prerequisite for all trapping computations is the knowledge of the opacity, i.e. the absorption cross-section and the atomic density. While we have always assumed up till now that this information is available, this is not so trivial to obtain. The usual method for determining atomic densities is to measure temperature, and to infer the atomic density from vapour pressure curves. This method has, however, several drawbacks: 1) The vapour pressure curves are roughly exponential functions of temperature, so that a small error in the temperature measurement translates into a large error in the atomic density. Accurate temperature measurements are not very difficult near room temperature, but are very difficult at high temperatures. Since most elements have appreciable vapour pressure only at temperatures exceeding 500 K, this poses serious problems. --- 2) The vapour cell is often not heated uniformly. Frequently, the temperature of a reservoir or of a cold-point is controlled, while the cell itself is at a higher temperature. This avoids condensation on cell windows. After some time, all material has collected at the cold point, and the vapour pressure is solely determined by the temperature of the cold point. However, it may take several days for this equilibrium to set in, especially when a buffer gas is present. When one is not patient enough, the 'effective' temperature is some unspecified mean between the cold point and the cell temperature. 3) Published vapour pressure curves are not completely accurate. For some elements, the densities computed by vapour pressure curves of different authors differ by as much as 100%. In addition, it is often not clear whether the pressure is solely due to single atoms, or whether there are also molecular contributions to the pressure, especially from metallic dimers. A second method for making density measurements is to send a probe beam of resonance radiation through the vapour and to observe the absorption. The method is excellent at comparatively low opacities. However, when the absorption becomes very strong, inaccuracies set in—the difference between 99.99% and 99.995% absorption is very difficult to measure, but translates into a considerable difference in the atomic density. In that case, measurement of absorption in the wings can get around this problem (Huennekens and Gallagher 1983a), (Vadla *et al.* 1996). A third method for the density measurement is to employ radiation trapping itself. When we already know the upper-state lifetime, then the effective lifetime tells us the opacity, and thus the excited-state density. Of course, for such a determination, the setup should be as simple as possible (weak excitation, single-mode, one-dimensional geometry, etc.) because in that case, the effective lifetime can be computed very accurately. Of course, this method is of limited use when we need the opacity for some experiment with radiation trapping. If, e.g. we want to investigate the trapping factor for PFR as a function of the atomic density, we cannot determine the opacity from the trapping factor, since this is exactly the interrelation that we want to measure—in other words, we can't have our cake and eat it. On the other hand, we can use additional trapping experiments under well-defined conditions to determine the opacity, and then go over to the experiment we are really interested in. Trapping has been used for density measurements, e.g., by Garver *et al.* (1982) and Bonanno *et al.* (1983). ## 15.7 Radiation trapping in cold atoms In all laboratory experiments performed near or above room temperature, the Doppler width is larger than the natural linewidth. Doppler widths are typically several hundreds of MHz, while even the extremely short-lived mercury resonance state has a transition with a natural linewidth of only about 100 MHz. However, in recent years, laser cooling of atoms[^22] has become a 'hot' research topic. It has been realized that radiation [^22]: Actually, the field is usually called 'laser cooling and trapping', where trapping means fixing the atom to a certain position by applying radiation pressure. In order to avoid confusion with 'radiation trapping', we summarize all position-fixing mechanisms as 'cooling'. --- trapping is one of the key mechanisms that determine the achievable density of cold atoms. The cooling is very strong; temperatures in the range of microkelvins are habitually achieved, nanokelvins are the current (1997) state of the art, and the race is on for picokelvins. At these low temperatures, the Doppler width is much smaller than the natural linewidth. In a magneto-optical trap, we have three cooling lasers—one for each spatial dimension. Each of them is circularly polarized, and the beams are reflected back on to themselves, with the polarization reversed by the reflection. The beams intersect at the minimum of a magnetic field. We thus get a harmonic potential so that to first order the 'restoring force' (i.e. what keeps the atoms in the trap) is $\mathbf{F} = -k_{\text{spring}} \cdot \mathbf{r}$, where $k_{\text{spring}}$, the 'spring constant' of the system, depends on the magnetic field and on the detuning of the cooling lasers from exact resonance. More details can be found in Arimondo *et al.* (1992). The first investigations of laser cooling safely neglected both the attenuation of the cooling lasers and the radiation trapping, since the total number of atoms was so small, and the equilibrium of the trap was determined mainly by the balance between thermal motion and the cooling laser power. In the meantime, however, the temperatures are so low that thermal losses are almost zero, while the radiation trapping limits the density of atoms (Walker *et al.* 1990), (Hoffmann *et al.* 1992). The existing theory for radiation trapping in cold atoms (Sesko *et al.* 1991) makes a number of simplifications: (i) The opacity of the ensemble of cold atoms is so small that only one reabsorption happens. This condition was fulfilled well at the time that this theory was set up. Today, however, opacities on the order of 5–15 can be achieved, so that this assumption is no longer fulfilled. (ii) The atoms are modelled as ideal two-level atoms. Since most experiments of laser cooling are done with cesium or rubidium, this assumption is not fulfilled. The hyperfine structure of these atoms has also to be taken into account in the experiments. If the cooling lasers were tuned to a transition originating in the F = 3 state, they would transfer (pump) all atoms to the F = 4 state; of course these atoms cannot be cooled any more; this problem is usually avoided by adding a 'repumping laser' which pumps the atoms back into the F = 3 state. The hyperfine structure also has a strong effect on the radiation trapping, and must be included. (iii) The convective motion of the atoms is neglected. (iv) There are no temperature gradients in the trap. (v) The Doppler redistribution is neglected. (vi) Polarization effects are ignored. (vii) Effects of the magnetic field (other than for the restoring force) are ignored. Some of these assumptions were fulfilled at the time when this theory was created (1991), but are fulfilled less and less as experiments become better, and higher opacities --- are reached. There will thus be a need for a more general theory. We will in the following outline the existing theory. The number of absorbed cooling photons per second is given by $\sigma_{\text{pump}}I/h\nu$, where $\sigma_{\text{pump}}$ is the absorption cross-section at the pump frequency. When this radiation is reemitted by an atom, the radiation intensity at distance $d$ is given by $\sigma_{\text{pump}}I/(4 \cdot \pi \cdot d^2)$. If we denote the absorption cross-section for the reemitted radiation as $\sigma_{\text{reem}}$, the repulsive force between two atoms at distance $d$ is $$ F_{\text{repuls}} = \frac{\sigma_{\text{pump}}\sigma_{\text{reem}}I}{4\pi d^2 c} $$ (15.26) Next, we need an expression for the absorption cross-section for the pump radiation. At a point where the intensity is $I$, it is $$ \sigma_{\text{pump}} = \frac{\sigma_0}{1 + \frac{I}{I_s} + \left(2\frac{\Delta\nu}{\Delta\nu^n}\right)^2} $$ (15.27) where $\sigma_0$ is the unsaturated absorption cross-section, $I_s$ is the saturation intensity, and $\Delta\nu$ is the detuning (in Hz) of the cooling laser from the line centre. Now the direct and the reflected beams cause a standing-wave pattern of the intensity $$ I = 6I_{\text{onelas}} \cos^2 \left( \frac{2\pi\nu x}{c} + \frac{2\pi\nu y}{c} + \frac{2\pi\nu z}{c} + \varphi^{\text{ph}} \right) $$ (15.28) where $I_{\text{onelas}}$ is the intensity of one pumping laser, $x, y, z$ is the position in space, and $\varphi^{\text{ph}}$ is an arbitrary phase between $0$ and $2\pi$. Attenuation of the pump laser beams is neglected here. Averaging over one wavelength, the pump-laser absorption cross-section becomes $$ \sigma_{\text{pump}} = \frac{\sigma_0}{\left( 1 + \frac{6I_{\text{onelas}}}{I_s} + \left( 2\frac{\Delta\nu}{\Delta\nu^n} \right)^2 \right)^{1/2} \left( 1 + \left( 2\frac{\Delta\nu}{\Delta\nu^n} \right)^2 \right)^{1/2}} $$ (15.29) Next, we need the absorption cross-section for the reemitted radiation. Since the intensity of this radiation is small, we have no saturation effects, and essentially just have to integrate over the emission spectrum. Computation of the emission spectrum requires, however, knowledge of the redistribution function for the high-intensity cooling laser radiation. The ac-Stark splitting (see Chapter 13 and Appendix G) has a strong influence. Figure 15.4 shows the absorption and emission spectra under typical experimental parameters. The 'overlap' between these spectra gives the average absorption cross-section $\sigma_{\text{reem}}$; see also Bali *et al.* (1996). We thus now have the repulsive force exerted by one atom, which has absorbed a cooling photon, on another atom through the radiation-trapping mechanism. This is a kind of Kernel function. For an arbitrary distribution of such atoms, we get --- [FIGURE: Normalized frequency vs Absorption and Emission lineshapes] FIG. 15.4. Absorption and emission lineshapes for a two-level atom in a one-dimensional standing-wave field of intensity $12\text{ mW/cm}^2$. Laser detuning $-1.5\Delta\nu^n$. The emission profile is in arbitrary units; the zero-width elastic peak (representing 50% of the total intensity) is represented by a spike at $-1.5\Delta\nu^n$. From Sesko *et al.* (1991). $$ \mathbf{F}_{\text{repuls}}(\mathbf{r}) = \frac{\sigma_{\text{pump}}\sigma_{\text{reem}}6I_{\text{onelas}}}{4\pi c} \int n(\mathbf{r}')\frac{\mathbf{e}_{\mathbf{r}-\mathbf{r}'}}{|\mathbf{r}-\mathbf{r}'|^2}\text{d}\mathbf{r}' \qquad (15.30) $$ where $\mathbf{e}_{\mathbf{r}-\mathbf{r}'}$ is the unit vector from $\mathbf{r}'$ to $\mathbf{r}$. This equation looks very much like the integral equation of electrostatics (Gauss' law), and of course also has a differential analogue: $$ \nabla \cdot \mathbf{F}_{\text{repuls}}(\mathbf{r}) = \frac{\sigma_{\text{pump}}\sigma_{\text{reem}}6I_{\text{onelas}}}{c}n(\mathbf{r}) \qquad (15.31) $$ corresponding to Maxwell's third law. Let us next investigate the influence of the attenuation of the cooling lasers, an effect that leads to an attractive force between the atoms. When the attenuation of the laser beam is small, the intensity becomes an approximately linear function of position, and the attractive force between the atoms obeys $$ \nabla \cdot \mathbf{F}_{\text{a}} = -6\sigma_{\text{pump}}^2 I_{\text{onelas}} \frac{n(\mathbf{r})}{c} \qquad (15.32) $$ A third force working on the atoms is the pressure gradient $\nabla \cdot p = T\nabla n + n\nabla T$. Since we have assumed that there are no temperature gradients in the trap, the density gradient determines the pressure gradient. The fourth force working on the atoms is of --- course the 'restoring' force, i.e. the cooling force of the pump laser. The magnetic field gradient causes a harmonic potential so that $\mathbf{F} = -k_{\text{spring}}\mathbf{r}$. Balancing all these forces allows the computation of the excited-state distribution $$ T \nabla^2 \ln (n(\mathbf{r})) = 3k_{\text{spring}} \left( \frac{n(\mathbf{r})}{n_{\text{max}}} - 1 \right) $$ (15.33) where the maximum excited-state density $n_{\text{max}}$ is $$ n_{\text{max}} = c \, k_{\text{spring}} \frac{1}{2\sigma_{\text{pump}} (\sigma_{\text{reem}} - \sigma_{\text{pump}}) I_{\text{onelas}}} $$ (15.34) Numerical solution of the differential equation (15.33) gives the excited-state distribution. We see that radiation trapping limits the achievable density only if $\sigma_{\text{reem}} > \sigma_{\text{pump}}$ (only in this case is Eq. (15.34) valid). The correct computation of the redistribution function is thus of vital importance especially for the low densities considered here. Correct incorporation of radiation trapping is essential for the explanation of many phenomena in laser cooling. Trapping is one of the three or four basic processes, and especially under today's experimental conditions, probably the most important. We expect that the theory of radiation trapping in laser cooling will have to be developed beyond the pioneering but rather simplified theory of Sesko *et al.* An important step in that direction was done by Ellinger *et al.* (1994), who derived quantum-mechanical equations including effects like Raman coupling, and by Hillenbrand *et al.* (1995). ## 15.8 Other measurements A further example for the influence of radiation trapping is the **measurement of broadening coefficients**. On one hand, radiation trapping influences the density distribution of the excited-state atoms. More importantly, radiation trapping leads to a change of the spectral shape of the emergent radiation. An example of such measurements is one by Huennekens and Gallagher (1983c). Kamke *et al.* (1983) used a very simplified theory for low opacities that considers only photons that are reabsorbed no more than once. Radiation trapping can also lead to a **thermalization of velocity-selected excited-state atoms**. For velocity selection, we excite just a certain velocity subgroup of atoms with a very narrow laser linewidth. The reemitted photons are, however, redistributed in frequency, and if they are reabsorbed, also atoms with the 'wrong' velocity can become excited. This effect is described by Huennekens *et al.* (1995). For the determination of **electron excitation cross-sections**, it is also common to set up the rate equations, with the Einstein $A$ coefficient scaled with the escape factor (Gabriel and Heddle 1960). Radiation trapping effects occur also in **cascade experiments** (Nussbaum and Pipkin 1967) and in **stepwise-excitation experiments** (Murray *et al.* (1991), Masters *et al.* (1996)). In these experiments, the polarization effects (see Chapter 12) play an important role. An experimental study of **radiation trapping with spatial resolution of the excited-state density** was done by Molander *et al.* (1984). There, a second beam, which was offset in both time --- and space, excited the excited-state atoms to an even higher state, and the resulting fluorescence was observed. Very few papers treat radiation **trapping in a molecular vapour**. Martinho *et al.* (1989), Berberan-Santos *et al.* (1995), and Nunes Pereira *et al.* (1996) give more or less an independent derivation of the multiple-scattering technique. More importantly, the basic formulation shows that under certain assumptions, the radiation trapping theory of atomic vapours, which is presented in this book, can also be used for molecular vapours. Similarly, Ionikh (1989) develops the Holstein equation for molecular transitions, and then proceeds to derive some approximations when closely spaced lines cannot be resolved by the observation apparatus. --- # 16 # SIMULATION OF OPTICALLY PUMPED GAS LASERS ## 16.1 Trapping in gas lasers The invention of the laser in the early 1960s has completely changed not only optics, but also all other sciences where light is required for experiments. This is especially true for atomic and molecular spectroscopy, but also for such diverse fields as remote sensing, material processing, and astronomy. Most of the millions of lasers currently in use around the world are semiconductor lasers (by far the largest group, due to their applications in consumer electronics and data storage) or solid state lasers with a crystalline host. Gas lasers are in use for special applications, where they cannot be replaced by solid-state lasers yet. When one explains the basic principle of a laser, it is common to ponder a Fabry–Perot cavity filled with an amplifying medium; only the lasing transition is considered. This problem was solved in a landmark paper by Kogelnik and Li (1966) and has been reproduced in practically every book and course on modern optics. The question of how to make the medium amplifying is, for the case of an optically pumped gas, far more difficult. We also have to consider how the optical pumping process influences the lasing transitions. Figure 16.1 shows a term scheme for a three-level laser. The gas is pumped on frequency $\nu_{20}$ and lases at frequency $\nu_{21}$. The lower lasing level is depopulated by emission on frequency $\nu_{10}$. The term scheme makes clear that we have to deal with an extremely complicated radiation-trapping process. We have to include stimulated emission (which is the *conditio sine qua non* for a laser), so that the problem is non-linear. We have *at* [FIGURE: FIG. 16.1. Energy level scheme for a three-level laser.] --- [FIGURE: FIG. 16.2. Model geometry for an optically pumped gas laser.] *least* three atomic levels involved, we have to consider incomplete frequency redistribution (i.e. the non-Maxwellian velocity distribution of the excited atoms), and the geometry of the problem is at least two-dimensional. All these rather nasty properties of the problem make clear that only an approximate treatment of the problem is possible. Such a treatment was given by Morse *et al.* in a series of papers (Morse 1964), (Cipolla and Morse 1971, 1979), (Healy and Morse 1972, 1973). This treatment included trapping, but only as a perturbation term that could be computed by iteration. The iteration is convergent only for rather low trapping factors. There are, however, also laser schemes where this condition is not fulfilled; one example is the mercury-nitrogen laser. Computation methods for this scheme follow basically the methods outlined in Chapter 13. We will give a description of the rate equations, and of the basic physical effects. ## 16.2 Low-opacity formulation ### 16.2.1 *Basic formulation* The geometry of the problem is sketched in Fig. 16.2. We have a plane Fabry–Perot resonator of length $L_y$ and height $L_z$. It extends infinitely in the $x$-direction—this is typically not the case for a real Fabry–Perot, but it allows us to treat only a two-dimensional instead of a three-dimensional geometry. The walls at the top and the bottom, at $z = \pm L_z/2$, are completely transparent for all wavelengths. The lasing radiation propagates in the $y$-direction, between the two partially reflecting mirrors. The length of the cavity is assumed to fulfil the resonance condition to be an integer multiple of the wavelength. We assume that only one optical cavity mode is excited. The treatment here follows closely the one by Healy and Morse (1973). The energy level scheme considered is the one for a three-level laser shown in Fig. 16.1. Each energy level is assumed to be broadened with a width $\Delta E$. Specifically, level 2 is collisionally broadened, a fact that will turn out to be important for proper laser operation. The collisional (and natural) broadening is taken into account by a Milne-like approximation—the true Lorentzian shape (with FWHM $\Delta \nu$) is replaced by a rectan- --- gular lineshape of width $\Delta \nu$, i.e. the collisional lineshape $\Phi_{nm}(\nu)$ for the transition between two levels $m$ and $n$ becomes $$ \Phi_{nm}(\nu) = \frac{\mathrm{Hs}\left[ \left( \nu_{nm} + \frac{\Delta \nu_n}{2} + \frac{\Delta \nu_m}{2} \right) - \nu \right] \cdot \mathrm{Hs}\left[ \nu - \left( \nu_{nm} - \frac{\Delta \nu_n}{2} - \frac{\Delta \nu_m}{2} \right) \right]}{\Delta \nu_n + \Delta \nu_m} $$ (16.1) where Hs is the Heaviside step function and $\Phi$ is normalized so that $$ \int \Phi_{nm}(\nu)\mathrm{d}\nu = 1 $$ (16.2) The Doppler width will separately be taken into account, and Doppler broadening will be related to the particle velocities in the rate equation. The approximation for the collisional broadening makes clear that we cannot use this procedure for a high-opacity vapour. In such a vapour, the very fact that the collisional broadening leads to wings extending into optically thin spectral regions is of utmost importance for trapping. For the occurring linewidths, we can make the following estimates. The linewidth of the pump source will be very large—usually, --- The steady-state equation of radiative transfer in a multilevel atom is $$ (\Omega \cdot \nabla) \hat{I}_{mn}(\nu) = \sum_m \sum_{n (i) The transitions on frequencies $\nu_{21}$ and $\nu_{20}$ do not interact with each other. (ii) No stimulated emission occurs in the depopulating transition. (iii) The laser vessel is usually quite narrow as compared to its length, so that $\partial/\partial y$ in the rate equations can be neglected. (iv) We neglect all collisional excitation and de-excitations ($E_0^{\text{net}}$, $E_1^{\text{net}}$, and $E_2^{\text{net}} = 0$). (v) The laser radiation propagates almost exactly in the $y$ direction. Photons with a considerable $z$-direction component are lost. With these simplifications, the system of equations becomes $$ \begin{aligned} \frac{\partial \hat{I}_{20}}{\partial z} &= \frac{L_z}{2} h \nu_{20} B_{20} \left[ n_{20}^{v\Omega} (n_2) - \hat{I}_{20} n_{20}^{v\Omega} (n_0 - n_2) \right] \\ \frac{\partial \hat{I}_{21}}{\partial y} &= \frac{L_y}{2} h \nu_{21} B_{21} \left[ n_{21}^{v\Omega} (n_2) + \hat{I}_{21} n_{21}^{v\Omega} (n_2 - n_1) \right] \end{aligned} $$ (16.9) $$ \begin{aligned} v_z \frac{\partial n_0}{\partial z} &= +4\pi (A_{20}n_2 + A_{10}n_1) - A_{20}\bar{J}_{20} \cdot (n_0) \\ v_z \frac{\partial n_1}{\partial z} &= +4\pi (A_{21}n_2 - A_{10}n_1) + A_{21}\bar{J}_{21} \cdot (n_2 - n_1) \\ v_z \frac{\partial n_2}{\partial z} &= -4\pi (A_{20}n_2 + A_{21}n_2) + A_{20}\bar{J}_{20} \cdot (n_0) - A_{21}\bar{J}_{21} \cdot (n_2 - n_1) \end{aligned} $$ (16.10) The simplified system of equations Eqs. (16.9) and (16.10), together with the boundary conditions, forms the basis of the subsequent analysis. It is interesting to compare the above formulation with other formulations of the trapping problem. The formulation of Morse *et al.* is the most general insofar as it includes diffusion and incomplete frequency redistribution and correctly incorporates the interdependence between the velocity distribution and the atomic motion. The approach is somewhat similar to that of Holt (1976), which we considered in Sec. 11.3. Cipolla's formulation is even broader, since it includes the atomic motion and the collisional broadening. The above assumption of a rectangular lineshape for collisional broadening does not introduce large errors in the considered range of parameters and could easily be relaxed by using a Lorentzian lineshape for $\Phi(\nu)$ instead of the rectangular profile. This would only increase the computational effort, but pose no conceptual difficulties. It is not possible to make a reasonable comparison with the model of Post (1986), because the two models are for completely different ranges of parameters. Post's model is appropriate for high opacities, no particle diffusion, and a large influence of the natural wings; the Doppler effect is assumed to lead to complete redistribution in the core. Cipolla's model is for low opacities, particle movement, and incomplete redistribution due to the Doppler effect. ### 16.2.2 Approximate solution for single-mode operation The above system of equations cannot be solved analytically, so that we take refuge to an iterative solution. When the initial guesses are not too far off, a single iteration step --- will be sufficient. We have already seen in Chapter 11 that the influence of incomplete frequency redistribution for a Doppler line is rather small. It is thus reasonable to assume for the initial guess that CFR is valid; in other words, we assume a Maxwellian velocity distribution of the particles. For the lasing radiation, we make the initial guess that it is a sharp line propagating along the y-axis. The operator $n_{20}^{\nu\Omega}(n_0)$ can be computed from Eq. (16.6), where $$ n_0(\mathbf{v}, y, z) = \frac{1}{\pi} \exp\left(-v_y^2 - v_z^2\right) n_0(y, z) $$ (16.11) $$ n_{20}^{\nu\Omega}(n_0) \approx \frac{Nc}{\sqrt{\pi} \nu_{20} v_{\mathrm{D}}} \exp\left(-x_{20}^2\right) $$ (16.12) We have used the assumption that the Voigt parameter of the pump transition, $a_{20}$, is very small and that $n_0 \approx N$, i.e. that the large majority of the particles is in the ground state. Inserting Eq. (16.12) into Eq. (16.9), we get a differential equation for the pump intensity --- between amplification or attenuation by the gas. The condition for the assumption of negligible trapping is thus that there is *absolutely no trapping of the depopuating transition*. This cannot be fulfilled in a true three-level laser, but it can be fulfilled when atoms in level 1 decay to the ground state only via some intermediate levels. Under the assumption of no trapping, we get an expression for $\bar{J}_{21}$ the same way as we did for $\bar{J}_{20}$. The result is $$ \bar{J}_{21} = \frac{2\pi L^0}{a_{21}} \left[ \mathrm{Hs}\left(a_{21} + \frac{v_y}{v_\mathrm{D}}\right) \cdot \mathrm{Hs}\left(a_{21} - \frac{v_y}{v_\mathrm{D}}\right) \right] \qquad (16.17) $$ where $L^0$ is a quantity related to the energy in the excited cavity mode. The solutions of the rate equations are now approximated by neglecting the derivatives and solving the resulting system of algebraic equations; i.e. particle motion is excluded. The actual particle density is then computed by integrating over velocity space to get $$ n_2(z) - n_1(z) = \frac{16\pi}{e\sqrt{2}} \frac{2 c1 \nu_{20}^3 v_\mathrm{D}}{L_z c^3 \sqrt{\ln(c2)}} \frac{4\pi(A_{10} - A_{21})}{4\pi(A_{20} + A_{21})4\pi A_{10}} \cosh(z) \cdot \left[ \mathrm{erfc}(a_{21}) + \frac{\mathrm{erf}(a_{21})}{1 + I_s} \right] \qquad (16.18) $$ where $$ I_s = \frac{2\pi A_{21} L^0}{a_{21}} \frac{4\pi(A_{10} + A_{20})}{4\pi(A_{20} + A_{21})4\pi A_{10}} \qquad (16.19) $$ Equation (16.18) is the central result of the papers by Healy and Morse. It allows important conclusions about the dependence of the amount of population inversion on various design parameters. ### 16.2.3 Conclusions A first conclusion about the influence of trapping can be drawn from the results, Eq. (16.18), when we replace the natural decay rates $A_{mn}$ by the effective decay rates $A_{mn}^\mathrm{eff} = A_{mn}/g_{0,mn}$, where the $g_{0,mn}$ are the lowest-order trapping factors. We come to the (intuitively clear) conclusion that $A_{10}^\mathrm{eff}$ should be much larger than $A_{20}^\mathrm{eff}$, so that radiation trapping in the depopulating transition should be avoided. This conclusion has found some important applications recently, because radiation trapping of the depopulating transition is one key mechanism that limits the gain of VUV and X-ray lasers (see e.g. Jaegle *et al.* (1993)). Furthermore, the population inversion is proportional to $$ \frac{1}{A_{20}^\mathrm{eff} + A_{21}} \cdot \frac{1}{\sqrt{\ln(A_{20}^\mathrm{eff})}} \qquad (16.20) $$ This implies that trapping of the pump transition is only of somewhat limited use. As mentioned above, complete trapping of the pump transition leads to only a doubling --- of the pump efficiency, for a branching ratio of the pump transition of 50%; however, this can still be useful (Nilsen 1996). We furthermore emphasize that even this more accurate analysis using the 'effective' decay rates (i.e. still neglecting the higher-order modes) can lead to serious errors. The external excitation is largest at the boundary of the slab and drops towards the middle. Its shape thus looks completely different from the shape of the lowest-order trapping eigenfunction—peaked in the middle and dropping to lower values at the boundary. A really accurate computation would thus require the inclusion of higher-order modes. The population is also a non-linear function, depending on the saturation parameter $I_s$ (which is dimensionless). Let us now consider the case that the expression in the square brackets in Eq. (16.18) vanishes. This occurs either when there is low energy in the cavity mode, $L^0 \rightarrow 0$, or when the collisional width of the lasing transition vanishes, or for the case of large detuning. In all three cases, the lasing transition cannot interact properly with the pump radiation, so that we do not get an efficient laser. The next step is the computation of the gain coefficient $Gain$, which is defined as $$Gain(x_{21}, z) = L_y h \nu_{21} B_{21} n_{21}^{\nu\Omega} (n_2 - n_1)$$ (16.21) The normalized gain is then defined as $$\hat{Gain} = \frac{Gain(x, L^0)}{Gain(0, 0)}$$ (16.22) The gain curve, shown in Fig. 16.3, is in the shape of a Gaussian bell with a hole in the middle. The shape of the hole is triangular, due to the assumption of a rectangular collisional profile. The depth of the hole is determined by the saturation parameter $I_s$, and the width by the Voigt parameter $a_{21}$. As the cavity is detuned, the center of the hole will move to higher or lower frequencies, until the gain becomes less than the system losses and the lasing action stops. The lasing intensity $L^0$ for single-mode operation can be determined from the fact that the system losses must equal the steady-state gain $$\hat{Gain} = \alpha^{\mathrm{loss}}$$ (16.23) where $\exp(-\alpha^{\mathrm{loss}})$ is the loss for a single traverse of the cavity. Healy and Morse (1972) also derived results for multimode operation, which we will not reproduce here, since they use the same mathematical techniques. Although the formulation Eq. (16.9)–(16.10) is quite general, the subsequent analysis contains a great deal of additional approximations, mainly neglecting the trapping of the pump transition and of the depopulating transition. A more general analysis would have to use at least the 'effective decay rate' approximation; an even better computation would also have to include the highest-order modes. Furthermore, wall quenching would have to be included, and more iteration steps should be made (Healy's computation used only a zero-order approximation and one iteration step). All these generalizations were not possible at the time of Cipolla's and Healy's original papers. However, given reasonable computer resources, such an undertaking seems feasible today, and would perhaps lead to new insights into the operation of optically pumped gas lasers. --- [FIGURE: FIG. 16.3. Normalized gain curve of a gas laser.] It is also interesting to compare the formulation of Morse *et al.* to the usual radiation trapping formulation. The zeroth order iteration step completely excludes radiation trapping, and the following steps include it as a kind of perturbation. This will lead to satisfactory convergence of the iterations if the trapping factor is rather low. When there is a rather low branching ratio of the pumping transition, this is actually satisfied, and the method will be useful. There are, however, other laser types where this condition is not fulfilled at all; we will see in the following section how to deal with this case. One important effect of radiation trapping is that the different velocity subgroups of the atoms become intermixed. If we have no such intermixing, then the atoms that have the correct velocity for the current lasing frequency are used for the laser action. The gain profile gets a hole ('Bennet hole') in the shape of the homogeneous broadening. Radiation trapping changes the velocity distribution, and thus the shape of the Bennet holes (Pestov 1991). ## 16.3 The mercury-nitrogen laser ### 16.3.1 *The operating principle* As already mentioned in the introduction, there are cases that cannot be computed very effectively with the method of Morse *et al.* One example is the mercury-nitrogen laser depicted in Fig. 16.4. Mercury atoms are excited by resonance radiation (254 nm) from the $6\text{S}_0$ ground state **0** to the $6\text{P}_1$ state, **2**. From there, they are transferred to the lower-lying $6\text{P}_0$ state, **1**, by collisions with nitrogen ($\text{N}_2$) molecules. The $6\text{P}_0$ state is a metastable state, which acts as a kind of storage for the excitation. The metastables are excited by optical pumping at 405 nm to the $7\text{S}_1$ state **4**. From there, they decay to the $6\text{P}_2$ state **3**—this is the yellow-green lasing transition. The lower level of the lasing transition is depleted by collisions with $\text{N}_2$ molecules, which transfer the $6\text{P}_2$ atoms to the $6\text{P}_1$ state. The optical pumping radiation (254 and 405 nm) comes from a mercury discharge lamp. These lamps are very efficient. From mercury discharge lamps a power on the --- THE MERCURY-NITROGEN LASER [FIGURE: Energy level scheme of the mercury-nitrogen laser.] order of 1 W is easily achieved for the 254 nm radiation, and some 10 mW for the 405 nm radiation. There are several free parameters in the design of the mercury-nitrogen laser, like the nitrogen pressure, the mercury density, and the laser tube diameter. An optimization of these parameters must take the radiation trapping into account. Actually, there are two (weakly coupled) radiation trapping processes. The 254 nm transition is strongly trapped, and also the 405 nm transition can be trapped if the uv pumping and the mercury density are high enough. The geometry of the laser can be approximated very well by an infinite cylinder. The tube filled with mercury and nitrogen is rather long, in order to get good amplification—the amount of pump energy that we can get into the system increases linearly with the length of the laser tube, while the losses due to wall quenching, mirror absorption, etc., stay unchanged. In experiments, it has thus been common to use a one-metre long tube. The laser tube is surrounded by Hg discharge lamps. One can also put two discharge lamps into the foci of an elliptical mirror, and the laser tube into the centre of this arrangement. This leads to an efficient use of the radiation emitted by the lamps. In both configurations, there is a rather uniform flux incident on the side-walls of the cylindrical laser tube, in other words, there are no variations in the $\varphi$-direction of a cylindrical coordinate system. ### 16.3.2 Setup of the rate equations A correct description of the whole problem can be done via the generalized Holstein equation or with the transfer plus rate equations. When we consider just the system $6\text{s}_0$, $6\text{p}_1$, $6\text{p}_0$, ($\mathbf{0}$, $\mathbf{2}$, $\mathbf{1}$) we see that this is a three-level system very similar to the thallium system we have considered in Chapter 13. It is thus obvious to use the same mathematical techniques for its treatment, i.e. to set up the rate equations and the transfer --- [FIGURE: FIG. 16.5. Partial energy level scheme of mercury; shown are all levels and transitions relevant for laser operation. Dashed lines indicate non-radiative transitions. From Holmes and Siegman (1978).] equation, and to use a direct iteration procedure for its solution. For the rate equations, we get the following system of equations; see the sketch of the level scheme in Fig. 16.5. $$ \frac{\partial n_0}{\partial t} = (A_{20} + C_{20}^{\text{coll}})n_2 + C_{10}^{\text{coll}}n_1 - \frac{B_{02}n_0}{4\pi} \int \int C_{v(254)}k_{254}(v)I_{254}\text{d}v\text{d}\Omega - D\nabla^2n_1 $$ $$ \frac{\partial n_1}{\partial t} = C_{21}^{\text{coll}}n_2 + (A_{41} + C_{41}^{\text{coll}})n_4 - \frac{B_{14}n_1}{4\pi} \int \int C_{v(405)}k_{405}(v)I_{405}\text{d}v\text{d}\Omega $$ $$ \quad -(C_{10}^{\text{coll}} + C_{12}^{\text{coll}})n_1 + D\nabla^2n_1 $$ $$ \frac{\partial n_2}{\partial t} = \frac{B_{02}n_0}{4\pi} \int \int C_{v(254)}k_{254}(v)I_{254}\text{d}v\text{d}\Omega - (A_{20} + C_{20}^{\text{coll}} + C_{21}^{\text{coll}})n_2 + C_3^{\text{coll}}n_3 $$ $$ \quad + C_{12}^{\text{coll}}n_1 + (A_{42} + C_{42}^{\text{coll}})n_4 $$ (16.24) $$ \frac{\partial n_3}{\partial t} = W_{43} \left( n_4 - \frac{g_4'}{g_3'}n_3 \right) + (A_{43} + C_{43}^{\text{coll}})n_4 - C_3^{\text{coll}}n_3 $$ --- THE MERCURY-NITROGEN LASER $$ \frac{\partial n_4}{\partial t} = -W_{43} \left( n_4 - \frac{g'_4}{g'_3} n_3 \right) - (A_4 + C_4^{\text{coll}}) n_4 + \frac{B_{14} n_1}{4\pi} \int \int C_{\nu(405)} k_{405}(\nu) I_{405} d\nu d\Omega $$ $$ N_{\text{tot}} = n_0 + n_1 + n_2 + n_3 + n_4 $$ with the boundary condition $$ n_1(R) = 0 \qquad (16.25) $$ In that system of equations, we have already introduced several simplifications. Many collisional transfer rates $C^{\text{coll}}$ that are very small (due to the law of detailed balancing) were neglected completely. Diffusion is only taken into account for the metastables. All other atoms have so low natural lifetime that collisional quenching plays no role. Stimulated emission is allowed for only in the lasing transition 4-3. The rate $W_{43}$ denotes the stimulated emission. In all the equations, we have not explicitly written the dependence of the densities $n_i$ on the spatial coordinate $r$. Since we have a steady state, all temporal derivatives can be set to zero. The symbols $g'_3$ and $g'_4$ stand for the statistical weights of states 3 and 4, respectively. The rate equations contain the integrals over the pumping intensities $I_{254}$ and $I_{405}$, which are determined by the transfer equations $$ \begin{aligned} (\Omega\nabla)I_{254} &= -\sigma_{254} \left( n_0 I_{254} - \frac{A_{20}}{B_{02}} n_2 \right) \\ (\Omega\nabla)I_{405} &= -\sigma_{405} \left( n_1 I_{405} - \frac{A_{41}}{B_{14}} n_4 \right) \end{aligned} \qquad (16.26) $$ Since we have diffuse radiation, it is advantageous to switch over to the Eddington theory, so that we need only consider the angle-averaged radiation $J = \int I d\Omega/4\pi$. The transfer equations then become $$ \begin{aligned} \frac{1}{3} \nabla^2 J_{254} &= \frac{1}{3n_0} \nabla J_{254} \nabla n_0 + (\sigma_{254} n_0)^2 \left( J_{254} - \frac{A_{20} n_2}{B_{02} n_0} \right) \\ \frac{1}{3} \nabla^2 J_{405} &= \frac{1}{3n_1} \nabla J_{405} \nabla n_1 + (\sigma_{405} n_1)^2 \left( J_{405} - \frac{A_{20} n_4}{B_{02} n_1} \right) \end{aligned} \qquad (16.27) $$ where the boundary conditions are $$ \begin{aligned} J_{254} &= -\frac{2}{3\sigma_{254} n_0} \frac{\partial J_{254}}{\partial r} + \frac{F_{254}^{\text{lamp}}}{\pi} \\ J_{405} &= -\frac{2}{3\sigma_{405} n_1} \frac{\partial J_{405}}{\partial r} + \frac{F_{405}^{\text{lamp}}}{\pi} \end{aligned} \qquad (16.28) $$ These simplified transfer equations together with the rate equations give a complete description of the problem, and can be solved, e.g., by direct iteration: assume certain intensities $J$, solve the rate equations with these intensities, with the resulting particle densities then solve the transfer equations with the correct boundary conditions, and so forth. --- ### 16.3.3 Conclusions Up to now, the system of equations has been solved only in an approximate way (Holmes and Siegman 1978). Radiation trapping of the upper transition, 405 nm, was completely neglected. Trapping in the 254 nm transition was taken into account simply by replacing the decay rate $A_{20}$ by $A_{20}/g_0$, where the trapping factor $g_0$ was computed from the Holstein approximation. Furthermore, the wall quenching was taken into account by considering just the lowest-order mode. Finally, the spatial dependence of the distribution was neglected. All these simplifications are not really justified. First, using the Holstein approximation for a pure Doppler lineshape is certainly not admissible in this context. We have hyperfine-split (or rather isotope-split) lines with a Voigt lineshape, due to appreciable pressure broadening by nitrogen. Furthermore, we illuminate the laser tube from the outside. Most of the radiation will be absorbed close to the cell walls (at least if the mercury density is high enough), so that many higher-order modes will be excited. The actual average lifetime will thus differ considerably from the lower-order mode lifetime that is computed by the Holstein approximation. The higher-order diffusion modes will aggravate these problems. Finally, for the computation of the trapping factor $g_0$, it was assumed that all atoms are in the ground state, which is not fulfilled due to the considerable bleaching. Most of these problems were already identified by Holmes and Siegman (1978) themselves, but a full solution of the coupled rate and transfer equation problem was not possible for them. One has also to recognize that even the simplified treatment allows one to identify the general trends and to give a physical interpretation for them. A full solution of the set of equations has recently been done by us (Molisch *et al.* 1996b). There are mainly three parameters that **we can optimize: the nitrogen pressure, the mercury density, and the tube diameter.** Let us first consider the effect on the laser power as we increase the *nitrogen pressure*. At very low pressures, there is little collisional transfer from the $6\text{P}_1$ to the $6\text{P}_0$ state. Thus there are only few $6\text{P}_0$ atoms that can be pumped to the upper laser state, and laser power will be very small. Furthermore, the lower laser level is not depleted fast enough (by collisional transfer to the $6\text{P}_1$ state), so that we might not even achieve population inversion. As we increase the pressure, the collisional transfer will become better, until at very high nitrogen pressures direct quenching also of the ‘useful’ states **4** and **1** will set in, and the laser power will again be very low. In addition, the higher nitrogen pressure leads to more broadening of the lines, so that the (useful) radiation trapping is decreased. It follows immediately that there must be some optimum pressure in between. At very low *mercury density*, there is very little absorption of the 254 nm pump radiation, so that few atoms can get into the $6\text{P}_0$ level even for high pump intensities and proper nitrogen pressures. The positions where the radiation is absorbed are rather uniformly distributed over the radius. We also have no radiation trapping, so that the spatial distribution of the excited atoms will be determined solely by the processes of excitation and diffusion. As we increase the Hg density, the pump radiation is absorbed --- THE MERCURY-NITROGEN LASER [FIGURE: FIG. 16.6. Multimode laser output power as a function of nitrogen pressure, for three tube diameters. Pump lamp diameter is 4 mm in all cases. From Holmes and Siegman (1978).] closer to the sidewalls of the laser tube. Of course, this depends also on the lineshape of the pump radiation—the lamp presumably emits self-reversed lines, which in turn depends on the filling and on the operating temperature—and on the lineshape of the atoms in the laser tube (isotope-splitting, pressure-broadening by N$_2$). Radiation trapping increases the pumping efficiency, and also tends to distribute the excitation more towards the middle of the laser tube—it makes the distribution more similar to a lowest-order mode. On very strong pumping, we get bleaching of the vapour, which in turn decreases the radiation trapping, and lets the pump radiation penetrate deeper into the vapour before it is absorbed. Due to the effects of wall quenching and of radiation trapping, the excited-state distribution will be reasonably uniform for a considerable range of parameters. At very high mercury densities, self-quenching will set in, too much of the radiation is absorbed close to the side walls and will be wall-quenched. Thus the laser power will decrease again. There is an optimum also for the Hg density. The **tube radius** also has an important influence. The larger the tube radius, the lower is the wall quenching given the same pump absorption and radiation trapping. On the other hand, an essentially unchanged number of excited-state atoms is spread over a larger cross-section, which decreases the round-trip gain. Finally, the isotopic composition of the tube filling plays an important role. An isotope-split line has lower absorption and lower radiation trapping. One could also say that only one isotope will usually get above the threshold, so that the pump energy for the other isotopes is wasted. The use of a single-isotope filling is thus preferential from a technical point of view—however this is a very expensive means of improvement. The optima for N$_2$ pressure, Hg density, and lamp radius all depend on each other, so that we must optimize all at once. Figures 16.6 and 16.7 show some examples for measured optima. --- [FIGURE: FIG. 16.7. Multimode laser output power for three tube diameters as a function of mercury density, respectively cold point temperature. Single isotope ($^{202}$Hg) except when noted. From Holmes and Siegman (1978).] There are quite a lot of papers where radiation trapping is taken into account for the operation of gas lasers (and also for X-ray lasers that are built by laser-produced plasmas) (Skippon *et al.* 1983), (Nilsen 1990), (Taniguchi and Saito 1989a, b, 1990), (Carman 1990), (Carman *et al.* 1994), (Kushner 1981), (Harstad 1980), (Srigouri and Prasada Rao 1989), (Srigouri *et al.* 1987), (Fetzer *et al.* 1986), (Lawless 1984), (Batenin *et al.* 1981). However, all of these just replace the Einstein coefficients $A$ by $A/g_0$. We have seen in the example of the mercury-nitrogen laser that such a procedure can give a good qualitative impression, but that an exact treatment is necessary if we really want to simulate and to optimize a laser design. There are also some papers that consider the problem of radiation trapping in the lasing transition of the vapour in a quantum-mechanical way (Pestov 1991), (Maslova and Lerner 1991), (Bachurin 1991, 1992). --- # 17 ## ATOMIC LINE FILTERS In free-space optical information transmission and in LIDAR systems, very narrow-band filtering is often needed to eliminate broad-band solar background noise. Ideally, such filters should have a bandwidth on the order of a few hundred MHz to a few GHz. It is very difficult to produce such narrow-band filters by conventional means, but we know that atomic absorption lines have just that bandwidth. An atomic vapour cell actually is an excellent band-block filter. In order to convert it to a bandpass filter, we just have to find a way to determine whether absorption has taken place. In the case of atomic line filters, one observes the fluorescent radiation that is emitted at a different wavelength, i.e. we use wavelength conversion by the vapour. The atomic line filter (ALF) is then completed by two dyed-glass filters, a shortwave-pass at the front of the cell and a longwave-pass at the back. Together, these filters are completely opaque, and only radiation that is wavelength-converted by the vapour can pass through the ALF, see Fig. 17.1. We are dealing here with passive ALFs (where the input radiation excites ground-state atoms to a higher state), since they are the basic examples. The prototype thereof is the passive cesium-ALF, which is mainly used for submarine communications. Figure 17.2 shows the part of the term scheme of Cs that is relevant [FIGURE: FIG. 17.1. Basic operating principle of a passive atomic line filter.] --- [FIGURE: FIG. 17.2. Part of the energy level scheme of atomic cesium. (hfs of the 7s, 5d, and $7\text{p}_{1/2}$ levels not shown.)] for the operation of the ALF. Input radiation of 455 nm excites a $6\text{s}_{1/2}$ ground state atom to the $7\text{p}_{3/2}$ state. This atom then decays either directly back to the ground state, or via the 7s or 5d states to the 6p state. A 6p atom will then decay to the ground state, emitting a red output photon. The Cs vapour thus achieves a wavelength conversion from the blue input (455 nm) to the red output (895 and 852 nm). ALFs are by definition used only for the detection of very low light levels—otherwise there is no need for such narrow-band filtering to eliminate background noise—so that the ground state is the only state that is appreciably populated. We have three transitions that can be trapped, the $6\text{s}_{1/2}$-$7\text{p}_{3/2}$ input transition and the $6\text{s}_{1/2}$-$6\text{p}_{1/2}$ and $6\text{s}_{1/2}$-$6\text{p}_{3/2}$ output transitions. The input transition includes branching, while the output transitions are the first resonance lines and thus cannot branch to some other (intermediate) state. The ground state of Cs has very large hyperfine splitting. Furthermore, there is usually some noble buffer gas in the cell. Atomic line filters are textbook examples of practical situations where trapping plays a vital role. As in most cases, the situation is much too complicated to treat it exactly. Instead, we have to find just the right amount of simplifications to achieve good results with still reasonable computational effort. It is thus indispensable to first consider the geometry and the physical processes, and to find out what can be neglected and what must be included in the computations. (i) The vapour cell has completely mirrored sidewalls, so that we can describe the cell as a plane-parallel slab. The input window is mirrored for the output radi- --- ation, and the output window is mirrored for input radiation. This mirroring is done in order to increase the efficiency of the ALF. Using the well-known ‘image principle’, see also Chapter 4, we can describe the geometry for the input transition by a substitute slab extending from $+L/2 > z > -3L/2$ (the input window is at $+L/2$) and for the output transition by a slab extending from $+3L/2 > z > -L/2$. (ii) Foreign gas quenching and self-quenching is negligible at the considered densities. We will see below that the typical response time of the Cs-ALF is below $10\,\mu\text{s}$. Quenching cross-sections for noble gases are on the order of $10^{-22}\,\text{cm}^2$, so that the quenching decay time $\tau_\text{q} = 1/(\sigma_\text{q} v N)$ is on the order of $1\,\text{s}$ even for $100\,\text{mbar}$ noble gas pressure. The density of the dimer molecule $\text{Cs}_2$, a rather efficient quencher, is a quadratic function of Cs density, which implies that the Cs density must not become too large—if high opacity is required, the cell length has to be increased. As a coarse estimate, we take the corresponding quenching rate of sodium, which is $3 \cdot 10^{-9}\,\text{cm}^3/\text{s}\cdot\text{molecule}$. Assuming a similar value holds for Cs, the quenching decay time will be on the order of $1\,\text{s}$ for $\text{Cs}_2$ densities below $10^9\,\text{cm}^{-3}$, which corresponds to $10^{13}\,\text{cm}^{-3}$ Cs density. (iii) Wall quenching is neglected. This assumption is only valid if buffer gas of some kind is used. Even $1\,\text{mbar}$ buffer gas is sufficient to make wall quenching almost negligible in a $1 --- on the trapping factors, and there is no physical argument why we could neglect these effects. Summarizing, we have a classical radiation trapping problem that fulfils all simplifying assumptions used in the derivation of the Holstein equation, except for the branching of the input transition. The geometry is a slab, the lineshape is a hyperfine-split Voigt profile. We can thus apply the tools of Chapter 5 to this problem. We wish to compute the conversion efficiency $p_{conv}$ of the filter, i.e. the percentage of photons that is wavelength converted. We also want to compute the response time of the vapour, which limits the information carrying signal modulation bandwidth the filter can accommodate—we want to know for a short pulse of input light, over what time interval the output radiation is distributed. Let us first deal with the problem of efficiency. When an input photon is absorbed, a $7\text{p}_{3/2}$ atom is created. The only chance for such a photon *not* to be wavelength converted is when the $7\text{p}_{3/2}$ atom decays back to the ground state *and* the reemitted 455 nm photon escapes from the cell through the input window. When the 7p atom decays via the 7s or 5d states, wavelength conversion will certainly take place. When it chooses to decay to the ground state, the emitted 455 nm photon may be reused, i.e. absorbed by some other ground-state atom, and the whole process starts anew. The probability of escaping from the cell is thus $$ 1 - p_{conv} = \sum_i p_i \beta^i $$ (17.1) where probabilities $p_i$ are for 455 nm photons leaving the cell after exactly $i$ absorption–reemission processes. The symbol $\beta$ is the branching ratio of the $7\text{p}_{3/2}$–$6\text{s}_{1/2}$ transition, and $p_{conv}$ is the conversion efficiency. In Sec. 4 .3, we have shown that $$ p_i = \sum_j \tilde{\alpha}_j \left( 1 - \frac{1}{g_j} \right)^{i-1} \Bigg/ \sum_j \tilde{\alpha}_j g_j $$ (17.2) so that the conversion efficiency is $$ 1 - p_{conv} = \beta \sum_j \frac{\tilde{\alpha}_j g_j}{g_j - \beta(g_j - 1)} \Bigg/ \sum_j \tilde{\alpha}_j g_j $$ (17.3) where the trapping factors $g_i$ are for the unbranched transitions. We thus only have to compute the solution of the equation $$ \frac{\partial n(z, t)}{\partial t} = - \frac{1}{\tau_{7\text{p}_{3/2}}} n(z, t) + \frac{1}{\tau_{7\text{p}_{3/2}}} \int n(z', t) G(z, z') dz', $$ (17.4) Since the opacity of the input transition in a typical ALF is rather low ($2k_0L \approx 4$), this could be done by the Milne theory or, of course, by the PCA method. It is important to also include the higher-order modes, since photons belonging to these modes have a higher chance of escaping from the vapour than the photons belonging to the lowest-order modes. --- Next, we wish to compute the response time of the vapour. As is common in communications engineering, we are more interested in the signal bandwidth SBW than in a response time. The SBW is defined as the modulation frequency of amplitude-modulated input radiation where the modulation depth of the output radiation $Y(t)$ has dropped to $1/\sqrt{2}$ of the input modulation depth. In a first step, we write down the rate equations $$ \begin{aligned} \frac{\partial n_{7\text{p}_{3/2}}}{\partial t} &= -n_{7\text{p}_{3/2}} \left[ \frac{A_{7\text{p}_{3/2} \to 6\text{s}}}{g_{0,\text{in}}} + A_{7\text{p}_{3/2} \to 5\text{d}_{5/2}} + A_{7\text{p}_{3/2} \to 5\text{d}_{3/2}} + A_{7\text{p}_{3/2} \to 7\text{s}} \right] \\ \frac{\partial n_{7\text{s}}}{\partial t} &= -\frac{n_{7\text{s}}}{\tau_{7\text{s}}} + n_{7\text{p}_{3/2}} A_{7\text{p}_{3/2} \to 7\text{s --- In order to solve the rate equations, Eq. (17.5), we make a Laplace transform. This gives an algebraic system of equations in the Laplace variable $s$. We then replace complex frequency $s$ by frequency $j\omega$ and compute the signal bandwidth as $\text{SBW} = \omega_g = 2\pi f_g$ as the frequency where $$ \left| \tilde{Y}(j\omega_g) \right|^2 = \frac{1}{2} \left| \tilde{Y}(0) \right|^2 $$ (17.7) Figure 17.3 shows the efficiency and the signal bandwidth of a Cs-ALF as a function of the input opacity. The efficiency shown here is the conversion efficiency of the vapour cell multiplied with the absorption efficiency, i.e. the fraction of input photons that are actually absorbed by the vapour, $\exp(-2k_0L)$. We see that increasing the buffer gas pressure slightly decreases the quantum efficiency. This is so because the buffer gas leads to pressure-broadened wings of the lineshape, which reduce the trapping. For the same reason, adding more buffer gas increases the signal bandwidth. The increase in SBW is much more pronounced than the decrease in efficiency, because the influence of the wings is more important at high opacities (i.e. at the 6s-6p transition, which determines the SBW) than at low opacities (at the 6s-7p transition, which determines the efficiency). Although, this comes at some cost, the filter optical bandwidth will increase. Increasing the opacity of course increases the efficiency and decreases SBW, because trapping becomes stronger with increasing opacity. Radiation trapping is the most important process influencing the performance of ALFs. The first investigations of ALFs completely neglected trapping, leading to a large overestimation of the SBW (Marling *et al.* 1979) (about 200 kHz). The next investigations used the Holstein approximation for a pure Doppler line; as a consequence, the SBW was estimated to be less than 1 kHz (Jackson *et al.* 1978). Only in the 1980s was the role of buffer gas for reducing trapping recognized (although Walsh's paper on trapping for a Voigt-broadened line was published more than 20 years earlier) (Nikolai *et al.* 1981). It was not until 1993 that the influence of trapping on the efficiency of the filter was analysed and optimization procedures were published (Molisch *et al.* 1993b). --- ATOMIC LINE FILTERS [FIGURE: FIG. 17.3. Quantum efficiency and signal bandwidth of a passive Cs-ALF.] --- # 18 # DISCHARGE LAMPS AND PLASMAS ## 18.1 Introduction Electric discharge lamps are undoubtedly the economically most important devices where radiation trapping occurs (Waymouth 1986). Mercury discharge lamps in the form of fluorescence tubes are the most widespread source of lighting in the world, and also sodium lamps are ubiquitous. Due to this widespread use, even small improvements in lamp efficiency have tremendous effects on world-wide energy consumption. Rough estimates show that if the efficiency of discharge lamps could be increased by only 1%, this would lead to savings of $10^9$ kWh per year! This is in strong contrast to other trapping-related applications, like e.g. atomic line filters, where 1% more or less efficiency does not play a really vital role in the course of the universe. Since discharge lamps are in such widespread use, they have been studied extensively, and one might think that the physical processes are by now well known, and that all methods of optimization have been investigated. This is, however, far from true. Firstly, the physical processes that occur in these lamps are extremely complex, including effects like particle diffusion, non-Maxwellian energy distribution functions, chemical reactions between filling and cathode material, and so on. Secondly, determination of the relevant cross-sections is very difficult, and data in the literature still show considerable discrepancies. Thirdly, and most relevant for this book, the process of radiation trapping is usually taken into account only in a rather approximate way. In most cases, it is treated by the effective-lifetime approximation, without further check on the validity of this method. This neglect of radiation trapping has serious consequences on the lamp optimization. It was only in the 1980s that scientists at General Electrics (Anderson *et al.* 1985) devised a new method to increase the efficiency of mercury lamps by more than 1% by increasing the percentage of the isotope $^{196}$Hg, although the theoretical ideas required for this approach had been available in the literature for more than 30 years at that time. We think that a more detailed application of trapping theory will allow further optimization also in the future. Apart from these most widespread lamps, there are some other devices, like high-frequency discharge lamps, and other effects, like the optogalvanic effect, which have a similar basic theory, though the applications differ at bit. Finally, radiation trapping in plasmas bears some resemblance to the radiation trapping in lamps. In both cases, we have to include ions and electrons into our computations. On the other hand, plasmas are much more complicated, because they involve 3BEJBUJPO5SBQQJOHJO"UPNJD7BQPVSTESFBT'.PMJTDIFUBM 0Y GPSE6OJW FSTJUZ1SFTT•0Y GPSE6OJW FSTJUZ 1SFTT%0*PTP --- more excited levels, and also higher-ionized states. Furthermore, there are physical processes in plasmas that do not occur in lamps. Radiative transfer in plasmas has up to now been treated mainly by very simple approximations. We will thus give only a short survey of relevant papers. ## 18.2 The theory of discharge lamps The theory of discharge lamps is extremely complicated, since many effects have to be taken into account. We cannot give a complete overview of the literature here; we just mention the classic book by Waymouth (1971), and the recent review by Lister (1992), who cites more than 300 references. A brief but very readable tutorial on the collisional and radiative processes in a discharge is also given by Griem (1986). The three most important classes of lamps (low-pressure, high-pressure, and metal-halide lamps) are reviewed by Jack (1986), Wharmby (1986), and Ingold (1986a). What we present here is an extremely simplified theory—after all, we do not want to describe lamps *per se*, but just want to explain the effects of radiation trapping in these devices. We will furthermore concentrate our discussion on the low-pressure mercury discharge, since this is the most widely used lamp. Apart from this type, there is the low-pressure sodium lamp (Denneman 1981), which is often used for street lighting. It is more efficient than the mercury lamp, but gives off a monochromatic light, which is less comfortable to the human eye than the broad-band radiation emitted by the phosphors which coat the walls of the mercury lamp. The basic principle of a discharge lamp is the following. Some starting process creates a few free electrons in the lamp. These electrons gain energy by drifting through the electric field applied to the lamp. When they collide with mercury ground-state atoms, they excite them to some higher state. These upper-state atoms then either decay back to the ground state, emitting a resonance photon, or are ionized by collisions with other electrons or other upper-state atoms. The ionization process gives additional electrons, which are then also accelerated and collide. Loss of electrons occurs by recombination processes. The 'useful' output of the lamp is the resonance radiation at 185 nm and 254 nm that reaches the walls of the lamp. The walls are coated with phosphor material which converts the narrowband uv radiation to broadband visible light, hence the common name fluorescence tube. A description of the physical processes is achieved by a 'collisional-radiative model'. This is essentially a set of rate equations for the involved atomic levels, the ions and electrons. In the description of these rate equations, we will follow the recent exposition of Lister and Coe (1993). In order to keep the computational effort within reasonable bounds, one usually includes only the following levels: * The mercury ground state $6^1\text{S}_0$ denoted by 1 * The mercury metastable states $6^3\text{P}_0$ and $6^3\text{P}_2$ denoted by 4 and 5, respectively * The mercury singulet resonance state $6^1\text{P}_1$ denoted by 2 * The mercury triplet resonance state $6^3\text{P}_1$ denoted by 3 * The higher mercury states $7^3\text{S}_1$ and $6^3\text{D}_{1,2,3}$ denoted by 6 and 7, respectively --- [FIGURE: FIG. 18.1. Partial energy level scheme for a Hg-Ar discharge lamp. From Wani (1990).] These states are not included by most other authors. - Mercury ions Hg$^+$ and molecular mercury ions Hg$_2^+$ denoted by $i$ The rate equations for the neutral mercury states are then given by Dakin (1986) $$ \begin{aligned} \nabla \left[ D_m n_{\text{BGas}} \nabla \left( \frac{n_m}{n_{\text{BGas}}} \right) \right] &+ n_e \sum_{n \neq m} (C_{nm}n_n - C_{mn}n_m) + \\ &+ \sum_n (n_n \eta_{nm} A_{nm} - n_m \eta_{mn} A_{mn}) \qquad (18.1) \\ &+ \sum_{nwz} (Ca_{nzwm}n_n n_z - Ca_{wmnz}n_w n_m) = 0 \end{aligned} $$ where subscript $m, n, w$ and $z$ can denote any species 1–7. Here, $D_m$ is the diffusion coefficient of species $m$ through the buffer gas at unit pressure, $n_{\text{BGas}}$ is the density of buffer gas atoms (typically argon), $n_e$ is the electron density, $C$ are the collisional coefficients, and $Ca$ are the coefficients for associative processes. Radiation trapping is included by using the escape factor (effective lifetime) approximation; this will be discussed in more detail in the next section. Stimulated emission can be neglected, since the occurring intensities are quite small. The electron density is computed from the Schottky equations for electrons and for (atomic plus molecular) mercury ions, --- THE THEORY OF DISCHARGE LAMPS $$ \begin{aligned} &\nabla \left[ D_{\text{BGas}} \nabla n_e - \frac{\mu_e D_i}{\mu_e + \mu_i} \frac{n_e}{n_{\text{BGas}}} \nabla n_{\text{BGas}} \right] + n_e \sum_n C_{ni} n_n + \sum_{nwz} C_{anwiz} n_n n_w \\ &\quad + \frac{1}{2} \sum_{nz} C_{annz} n_n^2 - C_{aie} n_e n_{\text{Hg}_2^+} = 0 \\ &\nabla \left[ D_{\text{BGas}} \nabla n_{\text{Hg}_2^+} - (D_{\text{BGas}} - D_i) n_e \nabla \frac{n_{\text{Hg}_2^+}}{n_e} - \frac{\mu_e D_i}{\mu_e + \mu_i} \frac{n_{\text{Hg}_2^+}}{n_{\text{BGas}}} \nabla n_{\text{BGas}} \right] \\ &\quad + \sum_{nwz} C_{anwiz} n_n n_w - C_{aie} n_e n_{\text{Hg}_2^+} = 0 \end{aligned} $$ (18.2) where $\mu_e$ and $\mu_i$ denote the electron and ion mobility. The ambipolar diffusion coefficient $D_{\text{BGas}}$ is $$ D_{\text{BGas}} = \frac{\mu_e D_i + \mu_i D_e}{\mu_e + \mu_i} $$ (18.3) Quasi neutrality is assumed, i.e. $$ n_{\text{Hg}^+} + n_{\text{Hg}_2^+} = n_e $$ (18.4) The rate coefficients for collisions with electrons of course depend on the electron energy distribution function. The computation of this function is at the centre of much research in the discharge lamp community. For a long time, it was assumed that the electron energy distribution is Maxwellian, so that it was determined by a single parameter $T_e$, the electron temperature. This is, however, not correct. The high-energy 'tails' of the distribution are depleted by ionization processes. In the last 20 years, the Maxwell distribution has been replaced by more refined models, like two-temperature distributions. Nowadays, many widely used models include (approximate) solutions of the Boltzmann equation, which gives accurate solutions. Finally, the parameters of the lamp must be related to the electric characteristics of the lamp. The energy of the electrons are balanced the following way. The energy flow due to elastic and inelastic collisions must be equated to the energy electrons gain by their drift through the longitudinal electric field $E_z$. For a discharge current $i_e$ this energy gain is given by $E_z \cdot i_e$. Finally, we need the electron mobility equation $$ \frac{i_e}{E_z} = e_- \int_{\substack{\text{cross} \\ \text{section}}} \mu_e n_e \, d\sigma_{\text{cross.sec.}} $$ (18.5) As mentioned above, all this is only a very rough description of the operation of a discharge lamp. Numerous points are still open: how many atomic levels should be incorporated, what additional processes (e.g. involving the buffer gas (Wani 1990)) should be included, what is the reaction with the walls of the lamp and with cathode and anode, and so on (Cayless 1980). This is, however, an entirely different subject from what we are interested in. The simple model described above is sufficient to explain the effects of radiation trapping. --- DISCHARGE LAMPS AND PLASMAS [FIGURE: Graph of Intensity profile vs Radial position r] FIG. 18.2. Mercury 254 nm intensity emerging from a certain radial position for various shapes of the spatial resonance-state distribution. From (Nishimura and Fujimoto 1985). $f(r) = 1$ denotes a uniform distribution, $f(r) = \Psi_0$ the lowest-order Holstein mode, $f(r) = J_0^B$ a Bessel function with a zero at the wall. Discharge current is 3.5 A; wall temperature 250 °C, temperature of the Hg reservoir 44 °C. ## 18.3 Radiation trapping in discharge lamps In the previous section, we have treated radiation trapping the way it is treated in most papers on lamps, by using the ‘effective lifetime’ approximation; i.e. by multiplying the Einstein $A$ coefficient by the escape factor, independent of position. As a matter of fact, this approximation works quite well in lamps, certainly better than in the problems of chemical physics, of lasers, or of atomic line filters. The reason for this behaviour is the following: in a cylindrical discharge lamp, the excitation of ground-state atoms to the resonance states has an approximately parabolic shape. This shape agrees very well with the shape of the lowest-order mode, for an example see Fig. 18.2. As we have discussed in Chapter 7, in this case, only the lowest-order mode is excited, and the escape factor (for the lowest-order mode distribution of excited atoms) agrees with the lowest-order trapping factor. In addition, the spatial distribution of excited-state atoms is not changed by the trapping. Of course, the agreement between the excitation function and the lowest-order trapping mode is only approximate. In addition there are effects that tend to decrease that agreement, like particle diffusion, inhomogeneities in the ground-state and in the metastable density, or inhomogeneous electron temperatures. An accurate model that includes all these effects naturally becomes more complicated. In the following, we will discuss various computational methods that are reasonably efficient. --- RADIATION TRAPPING IN DISCHARGE LAMPS [FIGURE: Trapping parameters $m_j^D$ as function of mode order $j$] FIG. 18.3. Trapping parameters $m_j^D$ as function of mode order $j$: exact solution, solution of the diffusion equation, and escape-factor approximation. From van Trigt and van Laren (1973). Examples for investigations that use the 'effective lifetime' approach are | | | | :--- | :--- | | for Hg discharges | (Kreher *et al.* 1990), (Zissis *et al.* 1992), (Post 1984), (Lama *et al.* 1982) | | for alkali discharges | (van Tongeren and Heuvelmans 1974), (van Tongeren 1974), (Norcross and Stone 1968), (Waszink 1973) | | for noble gas discharges | (Alves *et al.* 1992), (Gousset and Boulmer-Leborgne 1984), (Monteil *et al.* 1977), (Kunc and Gundersen 1984), (Akoshile *et al.* 1985), (Vlcek 1989) (Fujimoto 1979), (Ferreira *et al.* 1985), (Papanyan *et al.* 1995), (Riley *et al.* 1994), (Meunier *et al.* 1995), (Cotrino and Gordillo-Vazquez 1995), (Sommerer 1996), | | for special discharge types | (Petway *et al.* 1989), (Beneking and Anderer 1992), (Lawless and Lam 1986), (Takasu *et al.* 1979). | The first method employed in modelling discharge lamps was the 'diffusion' method analogously to the Milne equation (Cayless 1963). Now we have seen that the Milne equation is not a good approximation at high opacities, and in discharge lamps, the trapping factors can become quite high. Even when the lowest-order trapping factor was fitted to its actual value by some procedure, the higher-order trapping factors thus obtained were not correct, see Fig. 18.3. Attempts to describe trapping by this method thus stopped in the 1960s. When the whole problem is linear, the most efficient exact method is to make an eigenfunction expansion, the method that was described in great detail in Chapter 5. This concept was applied to discharge lamps by van Trigt and van Laren (1973), van --- Trigt and Blom (1975) and van Gemert (1975). They treated an extremely simple model for a sodium discharge, where the excited-state density was given by $$ A_{21}n_2 + n_e n_2 (C_{23} + C_{21}) = n_e n_1 C_{12} + A_{21} \int G(\mathbf{r}, \mathbf{r}')n_2(\mathbf{r}')\mathrm{d}\mathbf{r}' $$ (18.6) which they subsequently solved. However, the usual methods of Chapter 5 are only applicable if we know the electron density, and when the ground-state density is both homogeneous and much larger than the excited-state densities. The homogeneity of the absorber distribution can be disturbed by various effects, like radial cataphoresis, where the Hg atoms in a Hg-Ar discharge may be preferentially ionized and pushed to the wall; other possible effects include e.g. bleaching because of population of the metastable state. Eigenfunction expansions thus seem to be of value only for the investigation of some basic effects, but not for detailed lamp models. Similar investigations, but using $A_{k,m}$ elements, i.e. transforming the problem to an algebraic system of linear equations, were done for noble gas discharges by Phelps (1960) and Sahni *et al.* (1978). Another, more general simulation method is solving the equation of radiative transfer in iteration with the solution of the rate equations, as described in Chapter 13. In the diffusion equations, Eq. (18.1), one would just have to replace the terms $A_{nm} \eta_{nm} n_n$ by the terms $A_{nm} n_n + n_m B_{mn} C_{\nu} \int k(\nu) J(\nu) \mathrm{d}\nu$, where the average intensity $J$ is computed from the solution of the radiative transfer equation. In practice, one could proceed by solving the system of equations described in Sec. 18.1 with the escape factor method (the solution of this system will probably be done by some iterative procedure). The resulting excited-state distributions are then used to solve for the radiative transfer, which gives the (spatially dependent) reabsorption terms. These terms are then inserted into the diffusion equation instead of the escape factor, yielding a new system of equations for the excited-state density, which closes the first iteration round. The procedure is continued until convergence is achieved. Since the excited-state density depends only weakly on the reabsorption terms, the zero-order approximation will already be quite good, and convergence is quite rapid. A somewhat similar scheme is based on an exact computation of the reabsorption terms in the Holstein equation (Dakin 1986). The reabsorption term (net radiative bracket), which is $1 - \int n(\mathbf{r}') G(\mathbf{r}, \mathbf{r}') \mathrm{d}\mathbf{r}' / n(\mathbf{r})$, is used instead of the escape factor in the diffusion equation. In order to compute this factor, we already have to know the excited-state density. For the zero-order iteration, we thus assume a constant density. The reabsorption term is then updated after each iteration. Of course, Monte Carlo simulations can also be used to account for the reabsorption term in an exact way (Wamsley *et al.* 1993). The main conclusions from such investigations are, see Fig. 18.4, — The electron temperature is not appreciably changed by the inclusion of higher-order modes. This seems to hold for almost all usual operating conditions. — The excited-state density can differ significantly when higher modes are included. This difference depends on the operating conditions. However, we also stress that --- [FIGURE: Radial distributions of parameters in a Hg discharge lamp] FIG. 18.4. Radial distributions of parameters in a Hg discharge lamp: (a) electrons, (b) $6^1\text{P}_1$ resonance states, and (c) escape function (local emission rate minus local absorption rate) for the 254 nm photons. Diameter of the lamp 3.6 cm; pressure 2.4 torr Ar; wall temperature 42 °C, discharge current 0.42 A. Computations shown are either with constant escape factor (dashed) or true solutions of the Holstein equation (solid). From Dakin (1986). the errors introduced by neglecting higher-order modes are often smaller than the errors due to the uncertainty in the cross-sections of the various collision processes, and might thus not be decisive for the total error of the computation. A further question in lamps concerns the validity of the assumption of complete frequency redistribution, CFR. In most lamps, there is some noble buffer gas in order to get the discharge started, and also to slow the diffusion of the electrons to the lamp walls, --- [FIGURE: FIG. 18.5. Electron density in a cesium discharge as a function of the trapping factor. Electron density $N_e(\text{Saha})$ is the equilibrium value. From Norcross and Stone (1968).] where they can easily recombine. Since even a quite low buffer gas pressure (a few torr) corresponds to particle densities on the order of $10^{17} \text{ cm}^{-3}$, we have a high probability of collisions within one natural lifetime, and thus validity of the CFR model. The situation is different in a pure mercury discharge (185 nm), and also in a pure neon discharge. The lifetimes of the resonance states of these two atoms are among the shortest of all elements, about 1 ns. This means that for the comparatively low particle densities that occur in a pure discharge (about $10^{14} \text{ cm}^{-3}$), the probability for a collision during 1 ns is low. Furthermore, the short lifetime means a large natural broadening. The combination of these two facts means that CFR is not valid, and PFR theories as described in Chapter 11 have to be used. As a matter of fact, the classical study of PFR by Post (1986) was done in such a pure mercury discharge. Finally, one might wonder what effects radiation trapping generally has on the electron density. The trapping increases the electron temperature, and tends to make the distribution more similar to the equilibrium value (Saha distribution). This can be seen from the example in Fig. 18.5, which shows results from a cesium discharge. ## 18.4 Decreasing radiation trapping in lamps Radiation trapping is quite generally considered detrimental to lamp operation. By keeping the excitation longer in the discharge volume, there is a higher probability of quench- --- ing. One way to increase the efficiency of discharge lamps is thus to decrease the radiation trapping. Increasing the buffer gas pressure would be one way to decrease the trapping factor. As we know from our discussion of trapping with a Voigt lineshape (Chapter 7), increasing the buffer gas pressure, and thus the Voigt parameter, decreases the trapping factor. Unfortunately, this is not a viable solution for lamps. The buffer gas pressure influences many other parameters, mainly by changing the ambipolar diffusion constant, but also by other effects. The optimum buffer gas pressure is thus already more or less determined, and cannot be adjusted to suit the requirements of decreased radiation trapping. One approach that does influence the efficiency is modifying the isotopic composition of the mercury filling. The idea, which was proposed in the 1980s by a group at GTE lighting (Lagushenko *et al.* 1985), (Anderson *et al.* 1985), (Maya *et al.* 1984), see also (Ingold 1986a), is amazingly simple and effective. By adding a rare isotope of mercury to the lamp filling, an additional 'output channel' is provided, through which the radiation can escape. Nothing but the radiative terms have changed, since the collisional processes are the same for all isotopes. The isotope structure of Hg is shown in Fig. 18.6. We see that at reasonably high densities, all hfs components are optically thick, except for the $^{196}\text{Hg}$ line. This is due to the fact that the natural abundance of this isotope is only 0.15%. This percentage is so low that, for complete redistribution, a photon gets into the '196-channel' only after $1/0.0015 \approx 700$ absorption/reemission processes. When we artificially increase the abundance of this isotope, the total trapping factor will first *decrease*, since photons can get into the '196-channel' after fewer absorption-reemission processes while the line is still optically thin. Furthermore, it is important that the 196-hfs component has a large enough distance to all the other hfs components, so that photons emitted in this channel cannot be reabsorbed by other isotopes. If we increase the percentage of this isotope too much, then also the '196-channel' will become optically thick, and the trapping factor will *increase* again. There is thus an optimum isotope composition. The increase in efficiency was computed by the GTE group both by Monte Carlo simulations and by analytical approximations. They based their computations on the assumption that the excitation by electrons had a parabolic spatial shape. Comparison between the computed and the measured exit spectra, see Fig. 18.7, showed that this assumption was fulfilled quite well. Figure 18.8 shows the measured uv output intensity as a function of the Hg density (cold-spot temperature) for various $^{196}\text{Hg}$ concentrations. We see that basically, the uv output goes through a maximum. At too low temperatures, there are not enough Hg atoms that can be excited by the few existing electrons. The electron density is low, because there are few excited Hg atoms that can be ionized. At too high temperatures, there is strong radiation trapping, and the excited-state atoms can be quenched more easily. Addition of the $^{196}\text{Hg}$ isotope increases the efficiency—as anticipated, there is an optimum percentage (2.6%). Furthermore, the optimum temperature depends on the --- [FIGURE: FIG. 18.6. Hyperfine and isotope splitting of the 254 nm line of the stable Hg isotopes. From Grossman et al. (1986).] isotopic composition. For a natural mixture, the optimum is at a lower temperature than for an ‘enhanced’ mixture. Radiation trapping is lower with the improved composition so that the detrimental effects of higher Hg density set in only at a higher temperature, while the beneficial effects of the higher Hg density stay the same. The effects described above apply only when the opacity is small enough that the absorption in one line does not ‘spill’ into the absorption of the other lines. As soon as photons emitted by a $^{196}\text{Hg}$ atom can be absorbed by $^{198}\text{Hg}$, $^{199}\text{Hg}$, or $^{201}\text{Hg}$ atoms, the beneficial effects of the added rare isotope vanishes. At very high opacities, the only ‘escape channels’ for the photons are the low-frequency wing of the lowest-frequency component, and the high-frequency wing of the highest-frequency component. The computations of the GTE group were done with Monte Carlo simulations. Similar results were obtained with a variational technique by Richardson and Berman (1992). Similar effects can also be achieved by applying a magnetic field to the lamp (Sommerer 1993). This field leads to a Zeeman splitting of the levels, and thus also supplies additional ‘channels’ for the radiation to escape. The field strengths necessary to achieve a good effect are on the order of 0.1 T. An increase in the escape probability, and thus in the efficiency of the lamp, can only be achieved for low temperatures. For high-temperatures, the escape probability is only determined by the leftmost and rightmost wings of the complete line, see above. The splitting of the lines, however, shifts part of the strong $^{202}\text{Hg}$ line into the low-frequency wing of the total line, leading to an increased absorption coefficient in the wing. The higher the temperature, the more pronounced this effect will be. For extremely high fields, the whole line will be split --- [FIGURE: FIG. 18.7. Calculated and measured spectra of the Hg 254 nm transition. From Grossman et al. (1986).] into two distinct components, which will then increase the escape probability. We can thus assume that for high temperatures, the escape probability as a function of the field strength will go through a minimum. We can see from the Monte Carlo simulation results reproduced in Fig. 18.9 that this is actually the case. ## 18.5 The optogalvanic effect Optogalvanic effects are changes in the electrical properties of a discharge when it is illuminated by resonance radiation (Lawler *et al.* 1986). These effects are used for diagnostic purposes in spectroscopy. The intuitively most clear effect is the 'positive' optogalvanic effect, where the illumination increases the electric conductance. The radiation increases the number of excited-state atoms, which can be ionized more easily than ground-state atoms, and we thus get more ions and electrons in the illuminated vapour. Also the 'negative' effect can occur; then the radiation depletes a metastable level. Since the metastables can be ionized quite easily, the decrease in metastable density thus decreases the electric conductance of the discharge. The theory behind the optogalvanic effect is essentially the same as for the discharge lamps—we just have to add a term describing the excitation by the external radiation to the rate equations. An explicit investigation of the effects of radiation trapping on the optogalvanic effect in a sodium discharge was carried out by Pepper (1978). The neon positive column, with a negative optogalvanic effect, was studied by e.g. Doughty and Lawler (1983), Kane (1984), Stewart *et al.* (1990) (who use a somewhat unconventional model for trapping in a partially collision-broadened line), and Kumar *et al.* (1994). Extensive experimental investigations of optogalvanic effects in various mercury transitions can be found in van de Weijer and Cremers (1986). --- DISCHARGE LAMPS AND PLASMAS [FIGURE: Graph of Relative uv output versus Cold-point temperature [°C] for different 196Hg concentrations] FIG. 18.8. Relative mercury 254 nm emission versus Hg liquid-vapour equilibrium temperature for different $^{196}$Hg concentrations. The data are normalized to 20.5 °C. From Grossman *et al.* (1986). [FIGURE: Graph of Relative escape probability versus Magnetic field strength [ T ] for various temperatures] FIG. 18.9. Relative escape probability of the 254 nm photons as a function of magnetic field strength in the weak-field limit. Values are shown for various Hg cold-point temperatures, and normalized to zero-field value for each temperature. From Sommerer (1993). --- ## 18.6 Plasmas One aspect of radiation trapping that has gained much attention in recent years is trapping in plasmas (for a review of earlier work, see Griem (1974) and Jaegle *et al.* (1985)). Research on that subject is mainly stimulated by research into nuclear fusion and by investigations of X-ray lasers. Actually, investigations have been going on for a long time, but since almost all was for military purposes, results were mostly classified. Only in the last few years, has declassification led to a strong increase in the number of published papers. In this book, we hardly consider results from this research. The reason for this is not that they are unimportant - quite the contrary. However, the behaviour of the plasma is determined mostly by kinetic effects, and trapping is just an additional effect although it might occasionally lead to important changes in the behaviour of the plasma. Giving physical interpretations of measurements or simulations thus requires a deep knowledge of plasma physics (Kruer 1988), (Goldstein *et al.* 1991), (Marmar and Terry 1991). Such a description is beyond the scope of this book. For experts in hydrodynamics it can also be helpful to concentrate mainly on the hydrodynamics of the plasma, and to include the radiative transfer by treating the photons as a kind of additional 'fluid' with distinct hydrodynamics properties (Mihalas and Mihalas 1984). In the language of our book, this means that the equation of radiative transfer is solved together with the rate equations (which are now hydrodynamic equations). Furthermore, for very dense plasmas, it can be useful to consider the limiting case of 'local thermodynamic equilibrium (LTE)': in LTE, all particle densities are described by thermal excitation at the local temperature. LTE considerations were also done in the early days of trapping research in astrophysics (see, e.g. (Jefferies 1968)). Note, however, that for plasmas, LTE is useful mainly as a check on limiting properties; for other laboratory situations, it cannot be applied at all. There are three main conclusions from basically all investigations: * Radiation trapping *must* be included in the simulation of the plasma. * The velocity gradient must be included in the trapping description. * Computations using the escape factor technique typically have errors on the order of 20% in the excited-state density. This looks quite good at first glance. However, it may be too high an uncertainty for application in X-ray lasers—changing the density of a state by that amount can make the difference between gain and loss. For the applications of chemical physics, trapping is often the dominant mechanism that determines the distribution of the excited-state atoms. It was thus possible to find analytical or semi-analytical methods that allowed fast but accurate computation of trapping effects. Due to the importance of kinetic effects, such approximations are no longer possible for plasmas. The techniques to handle trapping in plasmas are thus a subset of the methods used for trapping in atomic vapours. Computations fall mainly into three groups (see also (R. W. Lee 1982b)): (i) Accurate numerical solutions of the transfer equation. Generally, problems involving plasmas are non-linear, so that the techniques described in Chapter 13 --- must be used. The equation of radiative transfer stays unchanged—except that in some papers the time dependence due to the finite propagation time is considered. In the setup of the rate equations, all kinetic processes have to be included. In chemical physics, solution of the rate equations is usually trivial if the radiation is assumed to be known. In plasma research, however, solution of the rate equation is extremely complicated and requires sophisticated computer programs (hydrocodes). The effects due to radiative transfer then have to be added to these codes. Papers that use this accurate technique are usually restricted to one-dimensional geometries. Back *et al.* (1991a, b) make model computations in a one-dimensional geometry and compare them to experiments tailored to what can be computed. They use the ETLA technique to take the multilevel atoms into account, and solve each two-level problem with accelerated $\Lambda$-iteration. Apruzese (1993) and Lee *et al.* (1990), Nilsen and Chandler (1991) (computer program XRASER) use direct iteration, solving the transfer equation with the excited-state population from the previous timestep. The latter also uses ETLA to account for the multilevel atoms. Scott and Mayle (1994) also use accelerated $\Lambda$-iteration for their code GLF. A method that is similar to the propagator function method combined with Monte Carlo simulations is used by Borovskiy *et al.* (1992). The $A_{k,m}$ matrix elements for each timestep are computed with a ray-tracing program. This is also similar to the Apruzese method, see Sec. 5.4, which is used mostly for plasma applications (Apruzese *et al.* 1978, 1986), (Duston *et al.* 1985), (Mankelevich *et al.* 1990). Anderson (1985a, b, 1988) describes a code that is based on the Feautrier solution, but by appropriately grouping transitions with similar optical properties, many lines ($>1000$) can be treated with still acceptable computational effort. Eder *et al.* (1992) and Benredjem *et al.* (1996) combine the rate equation with a Taylor expansion of the transfer equation to arrive at rapidly convergent solutions. Monte Carlo simulations of expanding plasmas were done by Schulz and Koshelev (1995). (ii) Escape factor approximations.[^23] In order to reduce the computational load, the majority of papers uses the escape factor approximation, usually the Sobolev approximation, see Sec. 11.8, since the plasma is expanding and large velocity gradients are present. Examples of these papers are Barnouin *et al.* (1989), Whitten *et al.* (1988), Bousquet *et al.* (1990), Eder (1989), Epstein (1989), Lee *et al.* (1990), Busquet (1993), Sasaki *et al.* (1994), Derzhiev *et al.* (1988), Derzhiev *et al.* (1990), Makhrov *et al.* (1994), Irons (1975), Irons (1976), Fujimoto *et al.* (1981), Nantel *et al.* (1995) and Djaoui *et al.* (1994). Examples for just the usual escape factor are Chenais-Popovic *et al.* (1986), Otsuka *et al.* (1979), Adams (1978), Breton and Schwob (1965), Pert and Rose (1990) and Carman (1990). Very similar to the escape factor technique is the Biberman approximation (Biberman *et al.* 1971, 1987), and a method introduced by Suckewer (1973), Kuszell and Suck- [^23]: In the Russian literature, the escape factor approximation is sometimes called the Biberman–Holstein approximation. We think that this is somewhat unfortunate, because it is likely to get mixed up with the Holstein equation and the (high-opacity) Holstein approximation, which has nothing to do with the escape factor technique. --- ewer (1973), Suckewer and Kuszell (1976), Biaz and Pham (1972), Biberman and Ulyanov (1964), Biberman and Veklenko (1960), van Blerkom and Hummer (1969), Abramov and Kogan (1966), Pham and Hoe (1972), Cairns *et al.* (1996), Riley *et al.* (1996), Lee and Larsen (1996), Sasaki *et al.* (1995) and Storm and Cappelli (1996). Some papers do a comparison between an accurate solution of the transfer equation and the escape factor technique, coming to the conclusion that the escape factor is 'sufficiently' (whatever that is) accurate. Apruzese (1993) used a time-dependent solution of the equation of radiative transfer, including also retardation effects; however, his geometry is just a plane-parallel slab. For the angle quadrature, he used two or four quadrature points (the results hardly differed, confirming that Schwarzschild–Schuster works quite well). Apruzese *et al.* (1984) compare $\Lambda$-iteration plus core saturation with the escape factor technique. Comparison between the Feautrier method and the escape factor method (for an astrophysical problem) can be found in Collin-Souffrin *et al.* (1981). Similar comparisons have also been done in astrophysics e.g. (Hamann 1981), (de Koter *et al.* 1993), where the agreement under typical parameters appears not so good. (iii) Iterative techniques. The basic idea is to solve the hydrodynamic equations, assuming that no radiation trapping occurs. This gives a certain distribution of excited atoms. This distribution is then used as the excitation term for a formal solution of the transfer equation. This solution then gives the reabsorption rate (photo-excitation) at all points, changing the rate equations, and thus resulting in a new distribution of excited-state atoms. We thus have essentially the direct-iteration technique described in Chapter 13 (R. W. Lee 1982a). A similar technique is applied by Tix and Simon (1994). For plasmas lamps, one also has to consider the redistribution of radiation by scattering from electrons. Basically, the Holstein (or the transfer) equation looks exactly the same as for the pure absorption/reemission case (Auer and Mihalas 1968a). Also the redistribution functions by scattering from electrons are the same as for pure Doppler redistribution, and can be approximated well by CFR. For the influence of PFR and polarization, see Hillier (1996). We thus just have to add a term $I \cdot k_{\text{electron}}/k_{\text{total}}$ in the transfer equation (and analogously in the Holstein equation). Solution of this modified equation can then proceed as in the usual case. A further effect that has to be taken into account is that the Doppler width of the electrons is much larger than the Doppler width of the atoms. We thus need a very large number of frequency grid points, which on one hand accounts for the large Doppler width of the electrons and on the other hand is fine enough to give good resolution of the absorption/reemission line. For a Feautrier solution, this causes serious problems, since we have to use a well-chosen non-uniform frequency mesh. Solution of the Holstein equation might be preferable here. ## 18.7 Concluding remarks The four examples for the application of trapping theory—measurements in chemical physics, optical pumping of gas lasers, atomic line filters, and electric discharge --- lamps—are just selected examples of a wide field of applications. Each real-world problem poses its own problems, and each requires careful consideration of which effects can be included in the calculations, and which can be ignored. It is quite interesting to compare the current state of the art in trapping to that in electromagnetic field computations. Twenty years ago, electromagnetics was a fine art, where extremely refined mathematical methods were used to tackle specific problems—function-theoretical methods to deal with microstrips, modal expansions to deal with hollow waveguides—and careful approximations were used, whose validity had to be checked individually in each case. Nowadays, there are purely numerical program packages where we put in the geometry of the problem, and the computer tells us the solution. Still, the mathematical methods developed at that time are valuable even today, because they provide us with physical insight and with analytical approximations to the results. In trapping, we are at the stage where electromagnetics was twenty years ago. Probably, in twenty years, there will be program packages where we can put in an atomic level scheme, a three-dimensional geometry, and the relevant atomic data, and it will put out the solution. Today, however, the computer can just be a tool to evaluate a problem that we have first modeled and simplified by physical reasoning. Perhaps this is nothing to complain about. After all, ‘the purpose of mathematics is not numbers, but insight’. --- ## APPENDIX A ## ATOMIC STRUCTURE This appendix is a tour through atomic structure and the required quantum mechanics. Starting with the simplest atoms, the hydrogen atom and the alkali atoms, we proceed to atoms with a more complicated structure. A discussion of the hyperfine splitting and of the Zeeman and Stark effects conclude the appendix. The approach is quite heuristic; we try to avoid the sophisticated mathematical descriptions developed for quantum mechanics. More detailed theoretical treatments can be found in many excellent textbooks on atomic physics, e.g. Kuhn (1962), Haken and Wolf (1990), and Atkins (1986). ### A.1 Models of the atomic structure #### A.1.1 *The Bohr–Rutherford model* The starting point of the discussion is the very basic facts. Atoms consist of electrically positive protons, negative electrons, and neutrons, which carry no electric charge. Protons and neutrons are much heavier than the electrons (about 1860 times). They form the nucleus, around which the electrons orbit. The simplest atom is the hydrogen atom, consisting of only one proton and one electron. The structure of the hydrogen atom cannot be described by classical physics. In that picture, the proton and electron form an electric dipole, which would radiate energy of constantly changing frequency. The dipole would continuously lose energy. After some time, the atom would thus collapse. In order to account for the stability of atoms and for the discrete lines of atomic spectra observed by Kirchhoff, Niels Bohr made the following *ad hoc* assumptions: * The classical laws of motion are valid for atoms, but only discrete orbits (with energies $E_n$) are allowed. * Atoms radiate energy only when an electron moves from one discrete orbit into another. The frequency of the emitted radiation is $(E_n - E_m)/h$, where $h$ is Planck's constant and $E_n$ and $E_m$ are the energies of the upper and lower energy levels. * For large orbit radii, the quantum mechanical laws asymptotically converge to the classical physical laws (correspondence principle). An energy state $E_n$ can be associated with a certain radius $r_n$ of the orbit of the electron, $$ E_n = \frac{-e^2}{4\pi\varepsilon_0 r_n}. $$ (A.1) --- Furthermore, the orbiting electron has a certain angular momentum $\mathbf{l}$, where $$ |\mathbf{l}| = m_e r_n^2 \omega_n = n\hbar. $$ (A.2) The electron's mass is denoted by $m_e$; its angular orbiting frequency by $\omega_n$. ### A.1.2 The Schrödinger equation The results above are based on *ad hoc* assumptions that work astonishingly well for an intuitive understanding of simple atoms. However, many experimental results cannot be explained on the basis of these assumptions. A correct treatment requires the solution of the Schrödinger equation, the basic equation of quantum mechanics: $$ -\frac{\hbar^2}{2m_e} \nabla^2 \psi(\mathbf{r}) + V_{\mathrm{pot}}(\mathbf{r}) \psi(\mathbf{r}) = E_{\mathrm{tot}} \psi(\mathbf{r}), $$ (A.3) where $V_{\mathrm{pot}}$ is the potential energy and $E_{\mathrm{tot}}$ is the total energy. In quantum mechanics, we cannot determine the position of a particle, but only the probability that the particle can be found at a certain place. The probability of finding a particle at place $\mathbf{r}$ is given by $\psi(\mathbf{r}) \cdot \psi^*(\mathbf{r})$. Equation (A.3) is a second-order partial differential equation, with $\nabla$ denoting the Nabla operator. Solution of the Schrödinger equation gives the exact behaviour of atoms. Unfortunately, an exact solution is feasible only for the simplest atom, the hydrogen atom. For other atoms, approximate methods have to be used, since the simultaneous solution of all the coupled Schrödinger equations for all particles in an atom (protons, neutrons, and electrons) is more than today's computers can achieve. ## A.2 The hydrogen atom For the simplest element, the hydrogen atom, we wish to determine the movement of a single electron in the electrostatic field of a proton.[^24] The potential energy of an electron in the electrostatic field of a point source of unit charge is $$ V_{\mathrm{pot}}(\mathbf{r}) = \frac{-e^2}{4\pi\varepsilon_0} \cdot \frac{1}{|\mathbf{r}|}. $$ (A.4) To simplify notation, we introduce the abbreviations $$ c1 = \frac{-2m_e E}{\hbar^2}, \quad \text{and} \quad c2 = \frac{m_e e^2}{2\pi\varepsilon_0\hbar^2}. $$ (A.5) Since the problem is spherically symmetric, it is advantageous to use a spherical coordinate system, in which the Schrödinger equation becomes [^24]: Strictly speaking, proton and electron both move around their common “centre-of-gravity”. Due to their large difference in mass, the centre of gravity almost coincides with the location of the proton. --- $$ \frac{1}{r} \frac{\partial^2}{\partial r^2} (r \psi) + \frac{1}{r^2} \Lambda^2 \psi + \frac{c2}{r} \psi = c1 \psi, $$ (A.6) where the operator $\Lambda^2$ is $$ \Lambda^2 = \frac{1}{\sin^2 \vartheta} \frac{\partial^2}{\partial \varphi^2} + \frac{1}{\sin \vartheta} \frac{\partial}{\partial \vartheta} \left( \sin \vartheta \frac{\partial}{\partial \vartheta} \right), $$ (A.7) and $r, \varphi$, and $\vartheta$ are the coordinates of the spherical coordinate system. Equation (A.6) is solved by separation of the variables. The function of three variables $\psi(r, \varphi, \vartheta)$ is set equal to the product of two functions, $R^{\text{H}}(r) Y(\varphi, \vartheta)$, one containing only the radial dependence, the other containing the angular dependences. Equation (A.6) becomes $$ \frac{r}{R^{\text{H}}(r)} \frac{\partial^2}{\partial r^2} (r \cdot R^{\text{H}}(r)) + \frac{1}{Y(\varphi, \vartheta)} \Lambda^2 Y(\varphi, \vartheta) + c2 \, r = c1 r^2. $$ (A.8) The second term, $1/Y \Lambda^2 Y$ does not depend on $r$. When $r$ is varied, this term must not change. It follows that $$ \frac{1}{Y(\varphi, \vartheta)} \cdot \Lambda^2 Y(\varphi, \vartheta) = -\text{const} $$ (A.9) and $$ \frac{r}{R^{\text{H}}(r)} \frac{\partial^2}{\partial r^2} (r R^{\text{H}}(r)) + c2 \, r - c1 r^2 = \text{const}. $$ (A.10) The solutions of Eq. (A.9) are the spherical harmonics $Y_{l, m_l}$, giving the shape of the electron orbit. **Spherical harmonics:** $Y_{l, m_l}$ $$ \begin{aligned} Y_{0,0} &= \sqrt{\frac{1}{4\pi}} & \\ Y_{1,0} &= \sqrt{\frac{3}{4\pi}} \cos \vartheta & Y_{2,0} &= \sqrt{\frac{5}{16\pi}} (3 \cos^2 \vartheta - 1) \\ Y_{1,\pm 1} &= \pm \sqrt{\frac{3}{8\pi}} \sin \vartheta \exp(\pm \underline{j} \varphi) & Y_{2,\pm 1} &= \pm \sqrt{\frac{15}{16\pi}} \sin \vartheta \cos \vartheta \exp(\pm \underline{j} \varphi) \\ & & Y_{2,\pm 2} &= \sqrt{\frac{15}{32\pi}} \sin^2 \vartheta \exp(\pm \underline{j} 2\varphi) \end{aligned} $$ (A.11) with $l = 0, 1, 2, 3, ...$ and $m_l = 0, \pm 1, \pm 2, ..., \pm l$. The quantity “const” in Eq. (A.9) is then $l(l + 1)$. The different discrete states the electron can assume are described by quantum numbers signifying these states. In this case, for the angular dependence of the electron probability distribution, two quantum numbers are necessary, namely $l$ and $m_l$. The quantum number $l$ is not a mere index, but is associated with a certain orbital angular momentum of the electron state. The magnitude of the angular momentum, $|\mathbf{l}|$, --- is given as $\hbar(l \cdot (l + 1))^{1/2}$. The projection of the angular momentum on to an arbitrarily chosen direction (usually designated as the $z$-axis) can only assume $2l + 1$ values of magnitude $m_l \cdot \hbar$. The $2l + 1$ values of $m_l$, the magnetic orbital quantum number, are $l, l - 1, l - 2, ..., -l$. The solutions for the radial dependence of the wavefunction, Eq. (A.10), are of the form $$ R_{n,l}^{\mathrm{H}}(\rho) = \rho^l L_{n,l}(\rho) \exp(-\rho/2) \qquad \text{(A.12)} $$ where $l < n$, $\rho = 2 \cdot r \cdot \sqrt{c1}$, and $L_{n,l}$ are the associated Laguerre polynomials. With the appropriate normalizations, the radial wavefunctions $R_{n,l}^{\mathrm{H}}$ are $$ \begin{aligned} \text{1s: } \quad R_{1,0}^{\mathrm{H}} &= 2 \left( \frac{Z}{a_0} \right)^{3/2} \exp(-\rho/2) \\ \text{2s: } \quad R_{2,0}^{\mathrm{H}} &= \frac{1}{2\sqrt{2}} \left( \frac{Z}{a_0} \right)^{3/2} (2 - \rho) \exp(-\rho/2) \\ \text{2p: } \quad R_{2,1}^{\mathrm{H}} &= \frac{1}{2\sqrt{6}} \left( \frac{Z}{a_0} \right)^{3/2} \rho \exp(-\rho/2) \\ \text{3s: } \quad R_{3,0}^{\mathrm{H}} &= \frac{1}{9\sqrt{3}} \left( \frac{Z}{a_0} \right)^{3/2} (6 - 6\rho + \rho^2) \exp(-\rho/2) \\ \text{3p: } \quad R_{3,1}^{\mathrm{H}} &= \frac{1}{9\sqrt{6}} \left( \frac{Z}{a_0} \right)^{3/2} \rho (4 - \rho) \exp(-\rho/2) \\ \text{3d: } \quad R_{3,2}^{\mathrm{H}} &= \frac{1}{9\sqrt{30}} \left( \frac{Z}{a_0} \right)^{3/2} \rho^2 \exp(-\rho/2) \end{aligned} \qquad \text{(A.13)} $$ where $Z$ is the proton number ($Z = 1$ for the hydrogen atom) and $a_0$ is the Bohr radius. Since the energy of a state appears only in the constant $c1$, and this constant appears only in Eq. (A.10), we see that the quantum number $n$ alone determines the energy state, $$ E_n = - \frac{e^4 m_e}{32\pi^2 \varepsilon_0^2 \hbar^2 n^2} \qquad \text{(A.14)} $$ Each level with a primary quantum number $n$ can thus assume $n^2$ combinations of the quantum numbers $l$ and $m_l$. We say that each level is $n^2$ degenerate. In quantum physics, it is well known that a photon has a “spin” (angular momentum) equal to one. When an atom changes its state by emitting a photon, this implies that the old and the new states must differ by one in their angular momentum, since the total angular momentum has to be conserved. We arrive at an important selection rule for radiative transitions (often also called optical transitions) $$ \text{Selection rule for radiative transitions: } \Delta l = \pm 1 \qquad \text{(A.15)} $$ and $\Delta n$ arbitrary. Transitions that fulfil this criterion are called “dipole-allowed”. --- Up to now, we have encountered three quantum numbers: $n$, $l$, and $m_l$, which can be explained both intuitively and by the solution of the Schrödinger equation. The Schrödinger equation is, however, non-relativistic. A relativistic quantum theory leads to a fourth quantum number, $s$, that has only one possible value, $+1/2$. The quantum number $s$ can be interpreted as the “spin” of the electron. Again there are two possible projections on to the $z$-axis, denoted by the quantum number $m_s = \pm 1/2$. ### A.3 The periodic table of elements With the theoretical tools of quantum mechanics, the structure of the elements and their physical and chemical properties can be explained. One of the most striking features of the elements is that their properties are almost periodic with respect to the nuclear charge $Z$. This periodicity can be explained by the Pauli exclusion principle: “An orbit (described by the quantum numbers, $n$, $l$, and $m_l$) can be occupied by no more than two electrons; if it is occupied by two electrons, these electrons have anti-parallel spin”. Alternatively, when we define a state to be uniquely determined by all four quantum numbers ($n$, $l$, $m_l$, and $m_s$), then no two electrons can occupy the same state. With that knowledge, we can build up the table of elements. For orbits with $n = 1$, we get $l = 0$ (since $l < n$; this was shown in the previous section), $m_l = 0$ ($|m_l| \le l$), and $m_s = \pm 1/2$. The $n = 1$ orbit can thus be occupied by at most two electrons. One proton plus one electron is hydrogen, two protons plus two electrons (plus neutrons) is helium. Helium has the maximum number of electrons for $n = 1$; we call that a complete shell (for a more accurate definition, see Sec A.4). The next element has $n = 2$, $l = 0$, $m_l = 0$ electron in addition to the complete $n = 1$ shell. This element is lithium. At this point, we have to introduce the common nomenclature of atomic physics. Shells with $n = 1, 2, 3, 4, 5$ are called K-shell, L-shell, M-shell, N-shell, and O-shell respectively. Electrons with $l = 0, 1, 2, 3, 4$, are called s, p, d, f, and g electrons. The electron configuration, e.g. for argon, is then denoted as 1s$^2$2s$^2$2p$^6$ 3s$^2$3p$^6$. This means that we have two electrons in the 1s-orbit, two electrons in the 2s orbit, 6 electrons in the 2p orbit, and so on. Complete shells have no total angular momentum. In a first approximation we can treat a nucleus (with charge $Z_1$) plus all electrons that together form complete shells (total charge $Z_2$) like a nucleus with charge $Z_1 - Z_2$. It is mainly the electrons of the outermost shell that are important for the physical and chemical properties. This explains why noble gases are chemically inert, and have a very high ionization potential (it is extremely difficult to tear an electron away from a complete shell) while alkali atoms (with only one electron, the valence electron, in the outermost shell) are very reactive and have a low ionization potential. Elements with an s$^2$ configuration are called alkaline earths. Summarizing, elements with the same electron configuration in the outermost shell have very similar physical and chemical properties and their energy level diagrams look very similar. --- [FIGURE: FIG. A.1. Nucleus, inner electron shells, and valence electrons of an alkali atom.] ## A.4 Alkali atoms ### A.4.1 Lifting of the $l$-degeneracy in alkali atoms Alkali atoms are quite similar to hydrogen atoms. They consist of a nucleus (charge $Z$), one or more complete electron shells, and one s-electron (valence electron) in the outermost shell. In a first approximation, nucleus and complete shells can be considered to represent an “equivalent” nucleus with unit charge, around which the valence electron is orbiting (see Fig. A.1). This assumption is only valid when the valence electron is very far from the nucleus and from the inner electrons. When the valence electron gets very close to the nucleus, it “sees” the full charge of the nucleus. Figure A.2 shows the potential energy for the valence electrons. Near the nucleus, the potential is proportional to $Z/\rho$, while for large $\rho$, it is proportional to $1/\rho$. From Fig. A.3 we see that an s-electron has a much higher probability of being near the nucleus (small $\rho$) than a p- or d-electron, which implies that $V_{\mathrm{pot}}(\rho)$ is shifted to more negative values for the s-electron. Figure A.4 shows a simplified energy level scheme, called a Grotrian diagram after Grotrian (1928). We see that the energy levels show a considerable dependence on the angular momentum. This dependence is often described by the quantum defect $\Delta(n, l)$ $$ E_{n,l} = \mathrm{const} \cdot \frac{1}{(n - \Delta(n, l))^2}. \qquad \text{(A.16)} $$ In the hydrogen atom, the quantum defect is zero, i.e. levels with the same $n$ and different $l$ are degenerate. In this case, Eq. (A.16) becomes the Balmer series equation for the energy terms of hydrogen. Table A.1 lists the quantum defects in sodium. We see that the quantum defect is almost independent of $n$ and decreases with increasing $l$. From our simplified picture, it is also obvious that the quantum defect increases with increasing nuclear charge $Z$, i.e. from Li to Na, K, Rb, and it is highest for Cs. --- ALKALI ATOMS [FIGURE: FIG. A.2. Electrostatic potential $V_{\text{pot}}$ at radius $\rho$ in an alkali atom.] [FIGURE: FIG. A.3. Probability that an electron is at a radius $\rho$, for an s-electron ($n = 3, l = 0$), p-electron ($n = 3, l = 1$), and d-electron ($n = 3, l = 2$).] As mentioned above, dipole-allowed atomic transitions must satisfy the condition $\Delta l = \pm 1$. Transitions from s to p electrons (in the upward direction) belong to the principal series, from p to s to the sharp series, from p to d to the diffuse series, and from d to f to the Bergmann or fundamental series. The energy shifts due to the lifting of the $l$-degeneracy have some interesting consequences for the structure of atoms at higher $n$. As mentioned in Sec. A.3, the configura- --- [FIGURE: Simplified Grotrian diagram of sodium, Na.] **Table A.1** *Quantum defects in sodium, Na (after (Haken and Wolf 1990)).* | **Term** | **n = 3** | **n = 4** | **n = 5** | | :--- | :--- | :--- | :--- | | **l = 0** | 1.373 | 1.357 | 1.352 | | **l = 1** | 0.883 | 0.867 | 0.862 | | **l = 2** | 0.010 | 0.011 | 0.013 | | **l = 3** | - | 0.000 | -0.001 | tion of Ar is $1\text{s}^2 2\text{s}^2 2\text{p}^6 3\text{s}^2 3\text{p}^6$. Intuitively, we would assume that the next element should have a $1\text{s}^2 2\text{s}^2 2\text{p}^6 3\text{s}^2 3\text{p}^6 \mathbf{3d^1}$ configuration. However, potassium, K, the next element has a $1\text{s}^2 2\text{s}^2 2\text{p}^6 3\text{s}^2 3\text{p}^6 \mathbf{4s^1}$ configuration. This is due to the fact that the 4s electrons have a lower energy than the 3d electrons. This is caused by the shift due to the “incomplete screening” of the s-electrons as described above. The expression “complete shell” thus always means that the next electron that is added to this “complete shell” is an s-electron; it does *not* imply that all possible states with a quantum number $n$ are populated. ### A.4.2 *Fine structure of alkali atoms* Moving electric charges cause a magnetic field. When the charge moves circularly around an axis, the magnetic field is directed along this axis and is described by the --- [FIGURE: Precession of the magnetic momentum about a z-axis defined by an external magnetic field.] FIG. A.5. Precession of the magnetic momentum about a $z$-axis defined by an external magnetic field. [FIGURE: Grotrian diagram of sodium, Na (including fine structure).] FIG. A.6. Grotrian diagram of sodium, Na (including fine structure). magnetic momentum $\boldsymbol{\mu}$. An electron orbiting around a nucleus has the magnetic momentum $$ \boldsymbol{\mu}_l = -g_l \mu_B \frac{\mathbf{l}}{\hbar}, \qquad \text{(A.17)} $$ where $\mu_B$ is $e_-\hbar/(2m_e) = 9.274 \cdot 10^{-24} \text{ A m}^2$, $\mathbf{l}$ is the angular momentum, and $g_l$ is the ratio of the magnetic momentum (in units of $\mu_B$) to $\mathbf{l}$ (in units of $\hbar$). In our case this ratio equals one. When there is an external magnetic field $\mathbf{B}$, it defines a preferred spatial direction $z$. We have noted in Sec. A.2 that the $z$-component of the angular momentum can assume --- only certain discrete values $m_l \hbar$, with $m_l = l, l - 1, ..., -l$ (we can make no statements about the $x$- and $y$-components). This can also be interpreted as a precession of the magnetic momentum $\boldsymbol{\mu}$ about the $z$-axis (see Fig. A.5). From relativistic quantum theory and also from earlier experiments it follows that an electron has a spin $\mathbf{s}$ with $$|\mathbf{s}| = \sqrt{s(s + 1)}\hbar$$ (A.18) and a magnetic momentum $$\boldsymbol{\mu}_s = -g_s \frac{e_-}{2m_e}\mathbf{s}.$$ (A.19) $s$ is the spin quantum number already mentioned in Sec. A.2, with the single possible value $+1/2$. The factor $g_s$ is equal to 2.0023. (The slight deviation from 2 is a consequence of the interaction between the electron and the electromagnetic field). In an external magnetic field, the $z$-component of $\mathbf{s}$ can assume only the values $\pm 1/2 \hbar$. We now have two angular momenta, the momentum of the electron moving around the nucleus and the electron spin momentum, which can be added vectorially to give the total momentum $\mathbf{j} = \mathbf{l} + \mathbf{s}$. The absolute value of $\mathbf{j}$ can take the values $$|\mathbf{j}| = \sqrt{j(j + 1)}\hbar, \quad \text{with } j = |l \pm s|.$$ (A.20) The $z$-component of $\mathbf{j}$ is again quantized to the values $m_j \hbar$. The quantum number $m_j$ can assume the values $j, j - 1, j - 2, ..., -j$. An s-electron can only have one value for $j$, namely $j = 1/2$, with two projections on to the $z$-axis, corresponding to $m_j = \pm 1/2$. Hence, an s-electron is two-fold degenerate. A p-electron can have the values $j = 1/2$ and $j = 3/2$. The total angular momentum quantum number $j$ is written as a subscript to the electron state, like $3\mathrm{p}_{1/2}$ or $3\mathrm{p}_{3/2}$. The $j = 1/2$ state has a lower angular momentum and a lower energy than the $j = 3/2$ state. An $n\mathrm{p}_{1/2}$ state ($l = 1, j = 1/2$) is two-fold degenerate, while an $n\mathrm{p}_{3/2}$ state ($l = 1, j = 3/2$) is fourfold degenerate ($m_j = 3/2, 1/2, -1/2, -3/2$). The selection rule for the total angular momentum is $$ \begin{matrix} \Delta j = 0 \text{ or } \Delta j = \pm 1 \\ j = 0 \rightarrow j = 0 \text{ is forbidden.} \end{matrix} $$ (A.21) We can now draw more accurate Grotrian diagrams of the alkali atoms that take the different energies of electrons with different $j$ into account. Figure A.6 shows such a diagram for sodium. The fine splitting increases with the fourth power of the nuclear charge, $Z^4$. It is hardly noticeable in lithium, but leads to a large splitting for the 6p state of cesium — the 6s ground state to 6p resonance transition is split in the two lines 852 nm ($6\mathrm{s}_{1/2}$—$6\mathrm{p}_{1/2}$) and 894 nm ($6\mathrm{s}_{1/2}$ —$6\mathrm{p}_{3/2}$). --- The energy difference between the various fine structure terms follows certain interval rules (Lande rules). The energy of the spin-orbit coupling is given as $$ V_{l,s} = \frac{a_{\mathrm{fs}}}{2} [j(j + 1) - l(l + 1) - s(s + 1)], $$ (A.22) where the coupling constant $a_{\mathrm{fs}}$ can either be measured or computed theoretically. ## A.5 The helium atom The helium atom consists of two protons, two neutrons, and two electrons. In hydrogen and in the alkali atoms, only one electron is in the outermost shell, while now we have two electrons. This leads to some interesting new effects, namely the splitting of the Grotrian diagram into two term schemes (the singulet and the triplet scheme) which cannot interact radiatively. The Schrödinger equation for the helium atom cannot be solved in closed form. For our purpose, we take refuge in some rather heuristic considerations. The angular momenta and spins of the two electrons are coupled. There are two extreme cases of momentum coupling, LS coupling (Russel–Saunders coupling) and j-j coupling (see also Sec. A.6). As usual, the general case lies in between. In LS coupling, the following rules specify how to add the angular momenta of the individual electrons (denoted by lower-case letters) to the total momenta (denoted by capital letters). First, the angular momenta of the electrons are added to give the total angular momentum $\mathbf{L} = \mathbf{l}_1 + \mathbf{l}_2$, where $$ |\mathbf{L}| = \sqrt{L(L + 1)}\hbar $$ (A.23) and $L = l_1 + l_2, l_1 + l_2 - 1, ..., l_1 - l_2$, with $l_1 \ge l_2$. The total electron spin is $\mathbf{S} = \mathbf{s}_1 + \mathbf{s}_2$, where $$ |\mathbf{S}| = \sqrt{S(S + 1)}\hbar, $$ (A.24) and $S = 1/2 \pm 1/2 = 0, 1$. The total angular momentum $\mathbf{J}$ is now computed by adding $\mathbf{L}$ and $\mathbf{S}$ vectorially, $$ |\mathbf{J}| = \sqrt{J(J + 1)}\hbar. $$ (A.25) For $S = 0$, $J$ equals $L$, for $S = 1$, $J$ equals $L + 1$, $L$, or $L - 1$. We see that there are terms with the same $J$ that are single ($S = 0$, singulet system) and terms that are triple[^25] ($S = 1$, triplet system). The selection rules for the angular momentum are $\Delta L = 0, \pm 1$ for the total system and $\Delta l = \pm 1$ for one electron, where $\Delta L = 0$ requires a change of state for both electrons ($\Delta l = +1$ for one electron and $\Delta l = -1$ for the other). The selection rules for the spin are $\Delta S = 0$. The selection rules for the total momentum are $\Delta J = 0, \pm 1$, but the transition from $J = 0$ to $J = 0$ is forbidden. [^25]: With the exception of $L = 1$ terms, which are single. --- ATOMIC STRUCTURE [FIGURE: FIG. A.7. Simplified term scheme of helium.] Due to the selection rule $\Delta S = 0$, optical transitions between the singulet and triplet system are forbidden. Figure A.7 shows the Grotrian diagram of helium. The energy level scheme consists of two parts that do not interact. Spectroscopically, i.e. from the absorption and emission lines, the two systems appear as if they stem from two different chemical elements. In fact, for some time it was indeed believed that there are two different kinds of helium. In the ground state (lowest-energy state), the two 1s-electrons are completely equivalent and thus must have opposite values of $m_s$, so that the ground state can only exist in the singulet system. Note that triplet states are energetically lower than the corresponding singulet states. Now let us take a look at the nomenclature for terms (in Sec. A.3, we have treated the nomenclature for configurations). A term symbol like $^3\text{P}_{1/2}$ contains three pieces of information: * The first (raised) number is $2S + 1$. If $L \ge S$, this is equal to the multiplicity of the term (in the example a triplet). * The capital letter describes the total angular momentum $L$, where $L = 0$ is described by S (do not confuse it with the spin quantum number $S$), $L = 1$ is denoted by P, $L = 2$ is D, $L = 3$ is F, $L = 4$ is G. * The last (lowered) number is the total angular momentum quantum number $J$. --- ## A.6 The alkaline earth elements The Grotrian diagrams of the alkaline earth elements are very similar to the Grotrian diagrams of helium. There is a singulet and a triplet scheme. We have seen that in helium, the triplet terms have lower energy than the singulet terms. This is generally true also in alkaline earth elements. Even more generally, we can state that the higher the multiplicity, the lower the energy.[^26] Up to now, we have assumed LS coupling. Pure LS coupling occurs mainly in light atoms. With heavier atoms, $\mathbf{l}$ and $\mathbf{s}$ couple to $\mathbf{j}$ for each electron and the $\mathbf{j}$ of all the electrons couple to give the total $\mathbf{J}$ ($j - j$ coupling). For $j - j$ coupling, different selection rules apply, $$ \begin{array}{ll} \text{for the total system} & \Delta J = 0, \pm 1 \quad \text{but not } J = 0 \rightarrow J = 0 \\ & \Delta m_J = 0, \pm 1 \text{ but not } m_J = 0 \rightarrow m_J = 0 \text{ when } \Delta J = 0 \\ \text{for one electron} & \Delta j = 0, \pm 1. \end{array} $$ (A.26) Spin-orbit interaction increases with $Z^4$, so that pure $j - j$ coupling occurs only with extremely heavy atoms. In most cases, intermediate coupling occurs, so that the selection rules described above no longer strictly apply. ## A.7 Trivalent elements Trivalent elements have three electrons in their outermost shells. $S$ may thus be $1/2$ or $3/2$, so that we get doublet terms ($J = L + 1/2$, $J = L - 1/2$, just like in alkali atoms) and quadruplet terms ($J = L + 3/2$, $J = L + 1/2$, $J = L - 1/2$, $J = L - 3/2$). The doublet term scheme of aluminium is shown in Fig. A.8. We see that this scheme strongly resembles an alkali scheme with the lowest S-term missing. Boron, gallium, indium, and thallium all have similar schemes. Transitions between the doublet and the quadruplet are forbidden (just as transitions between singulet and triplet terms in helium and alkaline earths are forbidden). There is fine splitting of the doublet terms, lifting the $l$-degeneracy as for the alkali atoms. ## A.8 Hyperfine structure and isotope splitting ### A.8.1 Hyperfine structure Experiments have shown that the nucleus has a spin, too. As the nucleus has a much larger mass than the electron, the magnetic dipole moment of the nucleus is much smaller. The nuclear spin is given as $$ |\mathbf{I}| = \sqrt{I(I + 1)}\hbar. $$ (A.27) The nuclear spin quantum number $I$ is integer or half-integer. For all known atoms $I$ lies between $0$ and $15/2$. For atoms with an even mass number, $I$ is integer; for atoms [^26]: In atoms with three electrons in the outermost shell, there are doublets and quartets; with four electrons, there are singulets, triplets and quintets, etc. --- [FIGURE: Simplified doublet term scheme of aluminium. (Fine structure not shown.)] FIG. A.8. Simplified doublet term scheme of aluminium. (Fine structure not shown.) with an odd mass number, $I$ is half-integer. When both the number of protons and the number of neutrons is even, then $I = 0$. In an external magnetic field, the $z$-component of $\mathbf{I}$ is $|\mathbf{I}_z| = m_I \hbar$ with $m_I = I, I - 1, ..., -I$. The nuclear magnetic moment is $$ \boldsymbol{\mu}_{\mathbf{I}} = \frac{g_I \mu_K}{\hbar} \mathbf{I}, $$ (A.28) where $\mu_K = e_- / (2m_0)\hbar$. The g-factor of the nucleus, $g_I$, is determined experimentally. The nuclear spin $\mathbf{I}$ interacts with $\mathbf{J}$ in the same way as $\mathbf{S}$ interacts with $\mathbf{L}$ (for LS coupling) to give the total vector $\mathbf{F} = \mathbf{J} + \mathbf{I}$. The absolute value of the total momentum is $$ |\mathbf{F}| = \sqrt{F(F + 1)}\hbar, $$ (A.29) where $F = J + I, J + I - 1, ..., J - I$. --- where $a_{\text{hfs}}$ is the hyperfine splitting constant. Equation (A.31) is valid for one-electron structures. In two-electron configurations, $a_{\text{hfs}}$ is replaced by $$ a_J = a_{\text{hfs}} \frac{J(J + 1) + S(S + 1) - L(L + 1)}{2J(J + 1)} \cdot \frac{S(S + 1) + s_1(s_1 + 1) - s_2(s_2 + 1)}{2S(S + 1)} \qquad \text{(A.32)} $$ where $s_1$ and $s_2$ are the spins of the two electrons. For perfect $j - j$ coupling, $a_{\text{hfs}}$ is replaced by $$ a_J = a_{\text{hfs}} \frac{J(J + 1) + s_1(s_1 + 1) - j_2(j_2 + 1)}{2J(J + 1)} \qquad \text{(A.33)} $$ where $s_1$ is the electron spin of the first electron and $j_2$ is the spin-orbit coupling of the second electron. Note that the hfs interaction is three orders of magnitude smaller than the fine structure interaction, so that hfs-energy levels will be very close together (separations on the order of a few MHz to a few GHz). Thus the interval rules, and also the strength rules described in Chapter 2, apply very well. ### A.8.2 *Isotope splitting* Different isotopes of an atom have the same nuclear charge but different nuclear masses. The influence of the nuclear mass on the energy levels of the atom is small but measurable. Two effects contribute to the small isotope dependence of the energies of the atomic states. * The volume effect. Different isotopes have different sizes of the nucleus (due to the different number of neutrons) which causes a somewhat different electric field and thus a different interaction between the electrons and the nucleus. * The electrons and the nucleus move around their common centre of gravity. Nuclei with different masses thus move slightly differently. For the hydrogen atom, we can describe the movement of nucleus and electrons around their common centre of gravity by introducing a reduced mass $$ \frac{1}{\bar{m}} = \frac{1}{m_{\text{nucleus}}} + \frac{1}{m_e} \qquad \text{(A.34)} $$ which has to be inserted into all equations of Sec. A.2 instead of $1/m_e$. Since all energy terms thus depend on $m_{\text{nucleus}}$ it is obvious that energies are different for different isotopes. For some atoms, only one isotope occurs in nature. Many atoms, however, naturally occur with a certain mix of isotopes. Due to their different masses, isotopes can be separated with centrifuges in an extremely costly technique (the most famous example is the separation of the uranium isotopes). Pure single isotopes in small quantities can usually be acquired from nuclear research laboratories. ## A.9 Effects of external magnetic and electric fields ### A.9.1 *The Zeeman effect* In an external magnetic field (along the $z$-axis) the $z$-component of the angular momenta and spins can only assume certain discrete values $m$. If there is no external field, we --- cannot distinguish between states with different $m$, and an $L = 3$ state, e.g., is $2L + 1 = 7$-fold degenerate. In an external field, the spin-orbit interaction gives different energies for each $m$, so that the degeneracy is lifted. The split energy terms are separated equidistantly by $$ \Delta E = g_j \mu_B B_0, $$ (A.35) where $g_j$ is the g-factor, $\mu_B$ is Bohr's magneton, and $B_0$ is the external field strength. In order to get appreciable Zeeman splitting in the visible spectrum, very high magnetic fields are necessary. In the infrared, quite small fields can be sufficient to achieve a noticeable shift. ### A.9.2 The Stark effect In an electric field, the energy levels also shift or split up into sublevels of different energy. There are two kinds of this Stark effect. * The quadratic Stark effect. The external field induces an electric dipole moment $\mathbf{p} = \alpha \mathbf{E}$ in the atom, where $\alpha$ is the polarizability and $\mathbf{E}$ is the electric field. This dipole moment interacts with the electric field. The interaction energy is $$ V_{\text{Stark}} = \frac{1}{2} \alpha |\mathbf{E}|^2. $$ (A.36) In contrast to the Zeeman effect, terms with the same absolute value of $m$ are degenerate. For the (yellow) sodium resonance line, the Stark shift is 5 pm at $|\mathbf{E}| = 10^7$ V/m. However, the Stark shift increases strongly with the primary quantum number $n$. Stark shifts for very high-lying transitions, i.e. near the ionization level, can thus be considerable (Gelbwachs *et al.* 1980). * The linear Stark effect. In hydrogen, levels with the same $l$ are degenerate (we saw above that in alkali atoms, the incomplete screening lifts this degeneracy). An external electric field lifts this degeneracy. The shift of the levels is proportional to the external field strength, hence the name linear Stark effect. --- # APPENDIX B ## VALUES OF THE $A_{K,M}$ MATRIX ELEMENTS FOR THE NUMERICAL SOLUTION OF THE HOLSTEIN EQUATION In this appendix, we give the simplified expressions for the matrix elements that are used to compute the numerical solution of the Holstein equation (see Chapter 5). ### B.1 The slab Insertion of Eq. (IV.16) into Eq. (V.53) yields $$ A_{k,m} = C_x \int_{z_m-\Delta/2}^{z_m+\Delta/2} \int_0^1 \int_{-\infty}^\infty \frac{k^2(x)}{2\mu} \exp(-k(x)|z_k - z'|/\mu)\mathrm{d}x \mathrm{d}\mu \mathrm{d}z', $$ (B.1) where $\Delta = L/N_r$ and $z_m$ is the centre of the $m$th substripe. We interchange the order of integration and first evaluate the integral $$ \int_{z_m-\Delta/2}^{z_m+\Delta/2} \exp(-k(x)|z_k - z'|/\mu)\mathrm{d}z', $$ (B.2) which yields $$ \text{for } k = m \qquad \frac{2\mu}{k(x)} \left[ 1 - \exp\left(-k(x)\frac{\Delta}{2\mu}\right) \right] $$ $$ \text{for } k \neq m \qquad \frac{\mu}{k(x)} \left[ \exp\left(-\frac{k(x)\Delta(|k-m|-0.5)}{\mu}\right) - \exp\left(-\frac{k(x)\Delta(|k-m|+0.5)}{\mu}\right) \right] $$ (B.3) The $j$th exponential integral $\mathrm{Ei}_j(z)$ as defined by (Abramowitz and Stegun 1965) is $$ \mathrm{Ei}_j(z) = \int_1^\infty \frac{\exp(-z \cdot u)}{u^j} \mathrm{d}u. $$ (B.4) Making the substitution $u = 1/\mu$, this can be written as $$ \mathrm{Ei}_j(z) = \int_0^1 \mu^{j-2} \exp\left(-\frac{z}{\mu}\right) \mathrm{d}\mu. $$ (B.5) The $A_{k,m}$ for the slab can thus be written as --- $$ \begin{aligned} \text{for } k = m \qquad A_{k,m} &= C_x \int_{-\infty}^{\infty} k(x) \left[ 1 - \mathrm{Ei}_2 \left( k(x)\frac{\Delta}{2} \right) \right] \mathrm{d}x \\ \text{for } k \neq m \qquad A_{k,m} &= \frac{C_x}{2} \int_{-\infty}^{\infty} k(x) \left[ \mathrm{Ei}_2 \left( k(x)\Delta \left( |k - m| - \frac{1}{2} \right) \right) \right. \\ &\qquad \left. - \mathrm{Ei}_2 \left( k(x)\Delta \left( |k - m| + \frac{1}{2} \right) \right) \right] \mathrm{d}x \end{aligned} $$ (B.6) requiring only (numerical) integration over the (arbitrary) lineshape. The exponential integrals can be evaluated by means of the recursion formula $$ \mathrm{Ei}_{j+1}(z) = \frac{1}{j} (\exp(-z) - z \cdot \mathrm{Ei}_j(z)) $$ (B.7) starting with polynomial approximations for $\mathrm{Ei}_1(z)$ (Abramowitz and Stegun 1965). ## B.2 The cylinder—method 1 Insertion of Eq. (IV.17) into Eq. (V.53) yields the basic equation for $A_{k,m}$. We then rewrite $A_{k,m}$ as $$ A_{k,m} = B_{k,m+1} - B_{k,m}, $$ (B.8) where $$ B_{k,m} = \int_0^{r_m-\Delta/2} G(r_k, r')r'\mathrm{d}r', $$ (B.9) so that $$ \begin{aligned} B_{k,m} = \frac{C_x}{4\pi} \int_0^{r_m-\Delta/2} \int_0^{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} &\frac{k^2(x)r'}{z^2 + r_k^2 + r'^2 - 2r_k r' \cos(\varphi)} \\ &\exp \left( -k(x)\sqrt{z^2 + r_k^2 + r'^2 - 2r_k r' \cos(\varphi)} \right) \mathrm{d}x\mathrm{d}z\mathrm{d}\varphi\mathrm{d}r'. \end{aligned} $$ (B.10) The integral over $r'$ and $\varphi$ extends over a circle with radius $r_m - \Delta/2$. Shifting the origin of the coordinates from the centre of this circle to the point $r_k$ (Golubovskii and Lyagushchenko 1976), the expression for the $B_{k,m}$ terms becomes $$ B_{k,m} = \frac{C_x}{2\pi} \int_0^{\varphi_0} \int_{rb(\varphi)}^{ra(\varphi)} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{k^2(x)q}{z^2 + q^2} \exp \left( -k(x)\sqrt{z^2 + q^2} \right) \mathrm{d}x\mathrm{d}z\mathrm{d}q\mathrm{d}\varphi, $$ (B.11) --- THE CYLINDER—METHOD 1 If $\mathbf{r}_k$ lies within the circle, $$ \varphi_0 = \pi, \text{ and } ra(\varphi) = r_k \cos(\varphi) + \sqrt{\left(r_m - \frac{\Delta}{2}\right)^2 - r_k^2 \sin^2(\varphi)} \qquad \text{(B.12)} $$ $$ rb(\varphi) = 0. $$ If $\mathbf{r}_k$ lies outside the circle, $$ \varphi_0 = \arcsin\left[ \frac{r_m - \Delta/2}{r_k} \right], \quad \text{and} \quad ra(\varphi) = r_k \cos(\varphi) + \sqrt{\left(r_m - \frac{\Delta}{2}\right)^2 - r_k^2 \sin^2(\varphi)} $$ $$ rb(\varphi) = r_k \cos(\varphi) - \sqrt{\left(r_m - \frac{\Delta}{2}\right)^2 - r_k^2 \sin^2(\varphi)}. \qquad \text{(B.13)} $$ We next make the substitution $y = z/q$ and change the order of integration. Equation (B.11) then becomes $$ B_{k,m} = \frac{C_x}{\pi} \int_0^{\varphi_0} \int_{-\infty}^{\infty} \int_0^{\infty} \int_{rb(\varphi)}^{ra(\varphi)} k^2(x) \frac{q^2}{(yq)^2 + q^2} \exp\left(-k(x)q\sqrt{y^2 + 1}\right) dq dy dx d\varphi \qquad \text{(B.14)} $$ Performing the integration over $q$, we get $$ B_{k,m} = \frac{C_x}{\pi} \int_0^{\varphi_0} \int_{-\infty}^{\infty} k(x) \int_0^{\infty} \frac{1}{(1 + y^2)^{3/2}} \left[ \exp\left(-k(x)rb(\varphi)\sqrt{1 + y^2}\right) - \right. $$ $$ \left. - \exp\left(-k(x)ra(\varphi)\sqrt{1 + y^2}\right) \right] dy dx d\varphi. \qquad \text{(B.15)} $$ We make the substitution $\sinh(u) = y$, so that the innermost integral becomes $$ \int_0^{\infty} \frac{1}{\cosh^2(u)} \left[ \exp\left( - k(x)rb(\varphi)\cosh(u) \right) - \exp\left( - k(x)ra(\varphi)\cosh(u) \right) \right] du. \qquad \text{(B.16)} $$ Now, it is well known (Abramowitz and Stegun 1965) that $$ \int_0^{\infty} \frac{1}{\cosh^j(u)} \exp[-z \cosh(u)] du = \mathrm{Ki}_j(z), \qquad \text{(B.17)} $$ where $\mathrm{Ki}_j$ is the $j$th repeated integral of $K_0^B(z)$, the modified Bessel function of zero order, second kind. Equation (B.15) can now be written as $$ B_{k,m} = \frac{C_x}{\pi} \int_0^{\varphi_0} \int_{-\infty}^{\infty} k(x) \left[ \mathrm{Ki}_2(k(x)rb(\varphi)) - \mathrm{Ki}_2(k(x)ra(\varphi)) \right] dx d\varphi. \qquad \text{(B.18)} $$ Using $r_m = m\Delta$ and $\Delta = R/N_r$, we finally get --- THE $A_{K,M}$ MATRIX ELEMENTS $$ \text{for } k \le m \qquad B_{k,m} = \frac{C_x}{\pi} \int_{-\infty}^{\infty} \int_{0}^{\pi} k(x) \{1 - \mathrm{Ki}_2 [k(x)ra_{k,m}(\varphi)]\} \, \mathrm{d}\varphi \mathrm{d}x $$ $$ \text{for } k > m \qquad B_{k,m} = \frac{C_x}{\pi} \int_{-\infty}^{\infty} \int_{0}^{\arcsin \frac{m}{k+0.5}} k(x) \{ \mathrm{Ki}_2 [k(x)rb_{k,m}(\varphi)] \qquad \text{(B.19)} $$ $$ - \mathrm{Ki}_2 [k(x)ra_{k,m}(\varphi)] \} \, \mathrm{d}\varphi \mathrm{d}x $$ where $$ ra_{k,m}(\varphi) = \Delta \left[ (k + 0.5) \cos(\varphi) + \sqrt{m^2 - (k + 0.5)^2 \sin^2(\varphi)} \right] $$ $$ rb_{k,m}(\varphi) = \Delta \left --- ## THE CYLINDER—METHOD 2 We then use the relations (Luke 1962) $$ \int_q^\infty \frac{\exp(-\alpha\rho)\mathrm{d}\rho}{\rho\sqrt{\rho^2 - q^2}} = \alpha \int_1^\infty K_0^B(\alpha q y)\mathrm{d}y $$ (B.25) and (Watson 1944) $$ K_0^B \left( y\sqrt{r^2 + r'^2 - 2rr'\cos(\varphi)} \right) = \sum_{m=-\infty}^\infty K_m^B(yr)I_m^B(yr')\cos(m\varphi), $$ (B.26) where $K_m^B$ and $I_m^B$ are the modified Bessel functions of order $m$, first and second kind, respectively. Inserting Eqs. (B.25) and (B.26) into Eq. (B.24), we get $$ \begin{aligned} A_{k,m} &= \frac{C_x}{\pi} \int_{r_m-\Delta/2}^{r_m+\Delta/2} r' \int_{-\infty}^\infty k^3(x) \int_1^\infty \\ &\quad \sum_{m=-\infty}^\infty K_m^B(k(x)yr --- THE $A_{K,M}$ MATRIX ELEMENTS for $k < m$ $$ \begin{aligned} A_{k,m} &= C_x \int_{-\infty}^{\infty} k^3(x) \int_1^{\infty} \int_{r_m-\Delta/2}^{r_m+\Delta/2} r' K_0^B(k(x)yr')I_0^B(k(x)yr_k)\mathrm{d}r'\mathrm{d}y\mathrm{d}x = \\ &= C_x \int_{-\infty}^{\infty} k^2(x)\Big[ (r_m - \Delta/2)H_1(k(x)(r_m - \Delta/2), k(x)r_k) - \\ &\qquad\qquad\qquad\qquad - (r_m + \Delta/2)H_1(k(x)(r_m + \Delta/2), k(x)r_k)\Big]\mathrm{d}x \end{aligned} $$ For $k = m$ $$ \begin{aligned} A_{k,m} = C_x \int_{-\infty}^{\infty} k^2(x)\Big --- In order to obtain the function $H_0$, we have to evaluate an integral containing the product $K_0^{\mathrm{B}}(\alpha y) \cdot I_1^{\mathrm{B}}(\beta y)$, see Eq. (B.30). Depending on the value ranges of $\alpha$ and $\beta$ the integral has to be evaluated in parts when the approximations of Eq. (B.32) are inserted: **case 1)** $\alpha$ and $\beta > 3.75$ ($\alpha$ is *always* larger than $\beta$), one integral only $$ \int_1^\infty \frac{K_0^{\mathrm{B}}(\alpha y) I_1^{\mathrm{B}}(\beta y)}{y} \mathrm{d}y, $$ (B.33) **case 2)** $\alpha > 2$, $\beta < 3.75$, integral in two parts $$ \int_1^{3.75/\beta} \frac{K_0^{\mathrm{B}}(\alpha y) I_1^{\mathrm{B}}(\beta y)}{y} \mathrm{d}y + \int_{3.75/\beta}^\infty \frac{K_0^{\mathrm{B}}(\alpha y) I_1^{\mathrm{B}}(\beta y)}{y} \mathrm{d}y, $$ (B.34) **case 3)** $\alpha < 2$ (and thus $\beta < 3.75$), integral in three parts $$ \int_1^{2/\alpha} \frac{K_0^{\mathrm{B}}(\alpha y) I_1^{\mathrm{B}}(\beta y)}{y} \mathrm{d}y + \int_{2/\alpha}^{3.75/\beta} \frac{K_0^{\mathrm{B}}(\alpha y) I_1^{\mathrm{B}}(\beta y)}{y} \mathrm{d}y + \int_{3.75/\beta}^\infty \frac{K_0^{\mathrm{B}}(\alpha y) I_1^{\mathrm{B}}(\beta y)}{y} \mathrm{d}y. $$ (B.35) Using these approximations to Eq. (B.30), we get integrals that are either analytically integrable or contain only the exponential integrals $\mathrm{Ei}_j$ and the generalized error function $\mathrm{GE}_j$ $$ \mathrm{Ei}_j(z) = \int_1^\infty \frac{1}{u^j} \exp(-uz) \mathrm{d}u \quad \text{and} \quad \mathrm{GE}_j(z) = \int_z^\infty \exp(-u) u^{j-1/2} \mathrm{d}u. $$ (B.36) The recursion relation for $\mathrm{Ei}_j$ is given in App. B.1, Eq. (B.7). For the generalized error function, the recursion relation is $$ \mathrm{GE}_j(z) = \exp(-z) z^{j-1/2} + \left( j - \frac{1}{2} \right) \mathrm{GE}_{j-1}(z) $$ (B.37) Polynomial approximations for $\mathrm{GE}_0$ are given by Abramowitz and Stegun (1965). ### B.4 The sphere The starting point for our derivation is the equation for the $A_{k,m}$ elements from Cuperman *et al.* (1963) $$ \begin{aligned} A_{k,m} = C_x \int_{r_m-\Delta/2}^{r_m+\Delta/2} \int_0^1 \int_{-\infty}^\infty \frac{k^2(x)}{2\mu} \frac{r'}{r_k} \Bigg[ & \exp \left( -\frac{k(x)|r_k - r'|}{\mu} \right) - \\ & - \exp \left( -\frac{k(x)|r_k + r'|}{\mu} \right) \Bigg] \mathrm{d}x \mathrm{d}\mu \mathrm{d}r'. \end{aligned} $$ (B.38) We first deal with the integral **with only the first term in brackets.** --- **For $r_k > r_m$ this is** $$ A_{k,m}^a = \frac{C_x}{2} \int_{-\infty}^{\infty} \frac{k^2(x)}{r_k} \int_1^\infty \frac{\exp(-k(x)r_k u)}{u} \int_{r_m-\Delta/2}^{r_m+\Delta/2} r' \exp(k(x)r'u) dr' du dx. $$ (B.39) The integral over $r'$ can be evaluated analytically (integration by parts), so that Eq. (B.39) becomes $$ \begin{aligned} A_{k,m}^a &= \frac{C_x}{2} \int_{-\infty}^{\infty} \frac{k^2(x)}{r_k} \int_1^\infty \left[ \frac{r_m + \Delta/2}{k(x)u} \exp(k(x)(r_m + \Delta/2)u) - \right. \\ &\quad - \frac{1}{k^2(x)u^2} \exp(k(x)(r_m + \Delta/2)u) - \frac{r_m - \Delta/2}{k(x)u} \exp(k(x)(r_m - \Delta/2)u) + \\ &\quad \left. + \frac{1}{k^2(x)u^2} \exp(k(x)(r_m - \Delta/2)u) \right] \frac{\exp(-k(x)r_k u)}{u} du dx = \\ &= \frac{C_x}{2} \int_{-\infty}^{\infty} k(x) \frac{r_m + \Delta/2}{r_k} \mathrm{Ei}_2[k(x)(r_k - r_m - \Delta/2)] - \\ &\quad - k(x) \frac{r_m - \Delta/2}{r_k} \mathrm{Ei}_2[k(x)(r_k - r_m + \Delta/2)] - \\ &\quad - \left\{ \frac{1}{r_k} \mathrm{Ei}_3[k(x)(r_k - r_m - \Delta/2)] - \frac{ --- THE SPHERE Since $C_x \int k(x)dx = 1$, the term $A_{k,m}^a$ finally becomes $$ A_{k,m}^a = 1 - C_x \int_{-\infty}^{\infty} k(x)\mathrm{Ei}_2[k(x)\Delta/2]dx \quad (\text{for } k = m) \qquad \text{(B.44)} $$ **For $r_k < r_m$**, the term $A_{k,m}^a$ is $$ A_{k,m}^a = \frac{C_x}{2} \int_{-\infty}^{\infty} \frac{k^2(x)}{r_k} \int_{1}^{\infty} \frac{\exp(k(x)r_k u)}{u} \int_{r_m-\Delta/2}^{r_m+\Delta/2} r' \exp(-k(x)r'u) dr'dudx. \qquad \text{(B.45)} $$ Comparing Eq. (B.45) with Eq. (B.39), we see that they become equal when $r' \to -r'$, $r_k \to -r_k$, and $r_m \to -r_m$, so that if we make these substitutions in Eq. (B.41), $$ \begin{aligned} A_{k,m}^a &= \frac{1}{4} \frac{r_m + r_k - \Delta/2}{r_k} \int_{-\infty}^{\infty} C_x k(x)\mathrm{Ei}_2[k(x)(r_m - r_k - \Delta/2)]dx - \\ &\quad - \frac{1}{4} \frac{r_m + r_k + \Delta/2}{r_k} \int_{-\infty}^{\infty} C_x k(x)\mathrm{Ei}_2[k(x)(r_m - r_k + \Delta/2)]dx + \\ &\quad + \frac{1}{2} \frac{1}{r_k} \int_{-\infty}^{\infty} C_x \exp\big[ - k(x)(r_m - r_k) \big] \sinh\left[k(x)\frac{\Delta}{2}\right]dx. \end{aligned} \qquad \text{(B.46)} $$ The integral Eq. (B.38) **with only the second term in brackets** is for all $k$ and $m$ $$ A_{k,m}^b = \frac{C_x}{2} \int_{-\infty}^{\infty} \frac{k^2(x)}{r_k} \int_{1}^{\infty} \frac{\exp( - k(x)r_k u )}{u} \int_{r_m-\Delta/2}^{r_m+\Delta/2} r' \exp( - k(x)r'u )dr'dudx. \qquad \text{(B.47)} $$ Comparison with (B.39) shows that these expressions become equal when $r' \to r'$ and $r_m \to -r_m$, so that it follows from Eq. (B.41), $$ \begin{aligned} A_{k,m}^b &= \frac{1}{4} \frac{r_k - r_m + \Delta/2}{r_k} \int_{-\infty}^{\infty} C_x k(x)\mathrm{Ei}_2[k(x)(r_k + r_m - \Delta/2)]dx - \\ &\quad - \frac{1}{4} \frac{r_k - r_m - \Delta/2}{r_k} \int_{-\infty}^{\infty} C_x k(x)\mathrm{Ei}_2[k(x)(r_k + r_m + \Delta/2)]dx + \\ &\quad + \frac{1}{2} \frac{1}{r_k} \int_{-\infty}^{\infty} C_x \exp\big[ - k(x)(r_k + r_m) \big] \sinh\left[k(x)\frac{\Delta}{2}\right]dx. \end{aligned} \qquad \text{(B.48)} $$ Using the expressions $\Delta = 1/N_r$, $r_m - r_k = (m - k)\Delta$, and $r_m + r_k = (m + 1/2)\Delta + (k + 1/2)\Delta = (m + k + 1)\Delta$, we define the integrals $$ \begin{aligned} \Upsilon_m^a &= C_x \int_{-\infty}^{\infty} k(x)\mathrm{Ei}_2[k(x)(m + 0.5)\Delta]dx \\ \Upsilon_m^b &= C_x \int_{-\infty}^{\infty} \exp\big[ - k(x)m\Delta \big] \sinh[k(x)\Delta/2]dx \end{aligned} \qquad \text{(B.49)} $$ for $m = 0, 1, ..., 2N_r - 1$. --- THE $A_{K,M}$ MATRIX ELEMENTS With these relations, we can finally write the $A_{k,m}$ elements as $$ \begin{aligned} A_{k,m} &= A_{k,m}^a - A_{k,m}^b \\ A_{k,m}^a &= \frac{1}{4} \frac{m+k+1.5}{k+0.5} \Upsilon_{k-m-1}^a - \frac{1}{4} \frac{m+k+0.5}{k+0.5} \Upsilon_{k-m}^a - \frac{1}{2} \frac{1}{(k+0.5)\Delta} \Upsilon_{k-m}^b & \text{for } k > m, \\ A_{k,m}^a &= -\frac{1}{4} \frac{m+k+1.5}{k+0.5} \Upsilon_{m-k}^a + \frac{1}{4} \frac{m+k+0.5}{k+0.5} \Upsilon_{m-k-1}^a + \frac{1}{2} \frac{1}{(k+0.5)\Delta} \Upsilon_{m-k}^b & \text{for } k < m, \\ A_{k,m}^a &= 1 - \Upsilon_0^a & \text{for } k = m, \\ A_{k,m}^b &= -\frac{1}{4} \frac{m-k+0.5}{k+0.5} \Upsilon_{m+k+1}^a + \frac{1}{4} \frac{m-k-0.5}{k+0.5} \Upsilon_{k+m}^a + \frac{1}{2} \frac{1}{(k+0.5)\Delta} \Upsilon_{m+k+1}^b & \text{for all } k, m \end{aligned} $$ (B.50) This formulation has the big advantage that only $4N_r$ single integrals have to be evaluated instead of the $N_r^2$ integrals that would normally be required (the Kernel function in the sphere case is *not* shift-invariant). --- ## APPENDIX C ### PUBLICLY AVAILABLE SOFTWARE FOR THE COMPUTATION OF RADIATION TRAPPING ### C.1 RAD-TRAP In the course of our work on trapping, we have developed the computer program ‘RAD-TRAP’ for the numerical solution of the Holstein equation in one-dimensional geometries according to the equations given in Chapter 5 and App. B. The program is described by Molisch *et al.* (1993a, d), and is publicly available through the CPC program library (catalogue number ACNX). RAD-TRAP is written in Fortran 77 and was tested on a VAX 3100 under the VMS operating system, on a VAX 4000 Alpha under Open-VMS, and on an IBM-compatible PC (486/66MHz) with a Microsoft Fortran 5.0 compiler under the MS-DOS 5.0 operating system. The program requires the NAG workstation library or the full NAG Fortran library (we used Mark 14 and Mark 15 of the full and Mark I of the workstation library). The final version of the program (Rad-Trap 2) has about 7000 lines of source code, excluding the NAG routines, and requires 500 kByte memory for a typical run. The input of the program is self-explanatory insofar as the user has to answer only very specific questions about parameters of the current problem. The program is modular; the main program just calls the sequence of subroutines and repeats the sequence if another run of the program is required. The subroutine INITIA proposes a certain set of input parameters; changing these default parameters requires a change in the source code (and thus new compiling and linking). The subroutine INPUT shows the current set of parameters to the user; interactive changes are possible: **Geometry.** Slab, cylinder, or sphere. **Lineshape.** Single or hfs-split Doppler, Lorentz, or Voigt lines (Voigt parameter $a$ to be entered) or a user-defined lineshape U_SHAP. Changing U_SHAP has to be done in the source code. **Opacity.** **Accuracy.** The two parameters that can be changed by the user are the integration accuracy and the number of subregions. For the cylinder geometry, the user has to state whether to use algorithm 1 (described in App. B.2) or algorithm 2 (described in App. B.3); see also below. **Output format.** Choose which eigenfunctions, steady-state parameters, and spectra to be output. --- **Steady-state parameters.** Whether to compute the steady-state at all, and if yes, which excitation function to use. One may choose between preprogrammed functions (exponential decay for slab and sphere, Gaussian beam for the cylinder) and user-defined functions, which have to be changed in the source code. **Parameters for the computation of the emergent spectra.** Whether to compute them at all, and if yes, for which angles and frequencies (these can be put in via a file). **Hyperfine structure** (when the chosen lineshape is hfs-split). The hfs-structure is either entered interactively or is read from a file. The spacing between the hfs-components can be put in either in absolute frequencies (Hertz) or normalized to Doppler (or Lorentz) FWHMs. Sorting of the hfs-frequencies and normalization of the amplitudes is done automatically. Entering data via a file is convenient when repeated computations are necessary. Information on the names of the files and on the required format is given in the comment lines at the beginning of the source code of the main program. ### Program description In the subroutine INT_LI, the frequency integration limits for the hfs-split lines are determined. Basically, a frequency integration from $-\infty$ to $\infty$ is required. However, the numerical integration routines of the NAGLIB software package cannot deal with sharp spikes separated by regions where the integrand almost vanishes. Thus a subdivision of the frequency axis into regions where the integrand is approximately monotonic is required. Furthermore, regions with insignificant contributions to the integral are ignored (note that this requires $k(\nu)L$ to be much smaller than both one and the centre-line opacity). For hfs-split lines, the subdivision is done automatically in the subroutine INT_LI. For user-defined lineshapes, the integration limits are entered interactively or via a file in the subroutine INPUT. The subroutine F_NORM computes the frequency normalization constant $C_x$. For (single or hfs-split) Doppler and Lorentz lineshapes, this computation is done analytically; for Voigt lineshapes and the user-defined profile, it is done numerically. In the subroutine MAT_EL, which calls the subroutines SLAB_M, CYL_M1, CYL_M2, or SPHE_M, the $A_{k,m}$ matrix elements are computed according to the equations given in App. B. The functions K0I1 and I0K1 compute the integrals of Eq. (B.30). The subroutine MAT_IN computes the eigenvalues and the eigenvectors of Eq. (5.54) by calling the appropriate routines of NAGLIB. The computation of the excitation vector for the steady state is done in the subroutine EX_VEC. The steady-state equation is solved in MATST, by Choleski decomposition for the slab case, and by LU factorization with partial pivoting and iterative refinement for cylinder and sphere. Computation of the spectra and output to the files 'steads.dat' (for the steady-state spectra) and 'eigspe.dat' (for the spectra of the eigenfunction) is done in the subroutine SPECTR. Output of the eigenfunction, the eigenvalues, and the steady-state distribution of excited atoms to the files eigfun.dat, eigval.dat, --- and steadf.dat, with some selected results sent to the screen, is done in the subroutine OUTPUT. The recommended number of subregions $N_r$ and the integration accuracy depend on the geometry, on the required accuracy, and on the available CPU time. For reasonable accuracy of the eigenvalues, $N_r$ should be at least twice the order of the highest mode that is required. For good accuracy of the eigenfunctions, $N_r$ should be at least 8–10 times the highest mode order. Remember that $\psi$ is assumed constant within each subregion. Usually, no more than 10 modes are required, and the higher eigenfunctions can be approximated from the orthonormality condition, so that $N_r = 20-30$ is a useful compromise in many cases. When large $N_r$ is required in a cylinder geometry, algorithm 2 should be used, especially at higher opacity, the reason being that in algorithm 1, the $A_{k,m}$ elements are computed as the difference $B_{k,m+1} - B_{k,m}$, and the higher $N_r$, the closer $B_{k,m+1}$ and $B_{k,m}$ become. Extremely high accuracy in the $B_{k,m}$ elements would be needed for reasonable accuracy in the $A_{k,m}$. For large opacities, $k_0 R > 30$, integration accuracies of $\approx 10^{-5}$ or better are required for 1% accuracy in $g_0$ at $N_r = 30$. Such high accuracies either cost much CPU time or cannot be achieved at all due to roundoff errors (this is especially true for Doppler lines or Voigt lines with small $a$). In that case, it is advantageous to use routine 2. For low opacities (and comparatively low integration accuracies, e.g. $10^{-3}$), routine 1 is faster by about a factor of 2. The runtime of the program is mainly determined by the numerical integrations for the $A_{k,m}$ computations. We can give the following rough estimates (for unsplit Doppler lines): in a slab, run time $t$ in seconds is about $N_r \cdot 0.01$, in a sphere $t = N_r \cdot 0.08$, and in a cylinder $t = N_r^2 \cdot 3$ on a VAX 3100 or on a 486/66 MHz PC. On a DEC Alpha, these times are reduced by about a factor 15. Note that the accuracy entered by the user is the integration accuracy for the $A_{k,m}$ matrix elements and may differ significantly from the accuracy for $g$ or for $\lambda = 1 - 1/g$. As a rule of thumb, $10^{-3}$ integration accuracy and $N_r = 30$ will usually give about 1% accuracy for the trapping factors $g_0, \dots, g_9$. The upper limits for the array sizes are defined in the file rt.inc. For a PC, these limits must be rather small because the Fortran compiler we used is confined to the 640k DOS memory. For the same reason, the program had to be split into 6 modules that have to be compiled separately (rtmain.for and rta,...,rte). All functions that can be changed by the user are in the module rta. ## C.2 McTrap Monte Carlo simulations are a versatile tool for the computation of trapping when we do not want to make many simplifying assumptions. A program that can deal with a wide range of situations is the program McTrap (Molisch *et al.* 1996a). McTrap is written in Pascal, and has been tested both on a DEC-Alpha 4000 workstation under OPEN-VMS and on a PC under DOS. Currently implemented are the atomic constants for thallium (in the constant-block) but of course these are easily --- changed to other atoms. The program can treat three-level atoms with bleaching, but also linear problems and two-level atoms (by setting the branching ratio $\beta$ to one). The assumed geometry is a two-dimensional cylinder with arbitrary reflection coefficients at the walls. By choosing appropriate reflection coefficients, one-dimensional geometries can be simulated. In a Monte Carlo simulation of a trapping problem, we trace each photon on its paths through the vapour. Frequency, direction, and length before reabsorption, etc., are chosen at random from the appropriate statistical distributions (for the computations of random numbers from a certain distribution see Sec. 6.1). The MC simulation of a general three-level situation faces two major problems: (i) the inhomogeneity of the distribution of absorbers and (ii) bleaching, which implies non-linearity. Note, however, that McTrap is not suitable for saturation problems. (i) **Inhomogeneity of absorbers**. In order to deal with the inhomogeneity of the absorbers, the cell is divided into subcells; we assume a constant density of absorbers within each subcell. Since the spatial distribution of absorbers is smooth and non-oscillating, the number of subcells required for good accuracy is quite low ($\approx 10$ cells in each direction). When a photon is created, we first compute the opacity $kL$ a photon can cover before being reabsorbed according to Beer's law (Random=$\exp(-kL)$). We then compute the intersection of the photon path with the subcell boundaries. From the distance between the point of emission to this intersection and from the density of absorbers in the considered subcell, we can compute the opacity $k_s L$ of this part of the photon path. If this opacity is larger than $kL$, the photon is absorbed in the subcell, and the point of absorption is found by interpolation. If this partial opacity is smaller than $kL$, we compute the 'residual' opacity $kL - k_s L$, which is the opacity the photon still has to cover before it is reabsorbed. We are now in a new subcell. We compute the intersection of the photon path with the new subcell, compute the opacity of this length (with the density of absorbers in the new cell), and look whether this opacity is smaller or larger than the 'residual' opacity. This process is repeated until the photon is absorbed. The main computational effort lies in the computation of the intersections of the photon path with the cell boundaries. It is *much* more efficient to compute them by the methods of analytical geometry than by trigonometric relations. (ii) **Bleaching**. Consider the level scheme shown in Fig. 10.2. If a $bc$ photon is absorbed and an $ac$ photon is reemitted, the number of state-b atoms is decreased by one; if an $ac$ photon is absorbed and a $bc$ photon emitted, the number of state-b atoms increases. The number of absorbers can thus be updated at each absorption–reemission process; this takes care of the bleaching non-linearity of the vapour. If we are just interested in the steady-state distribution of atoms, it is advantageous to assume a certain distribution (which should be a good guess of the actual distribution) already at the start of the program in order to save CPU time. If we are interested in the transient behaviour, we have to start out with the actual distribution of atoms. The diffusion of the state-a and state-b atoms through the vapour is an additional difficulty (diffusion of state-c atoms is neglected, since state c has by assumption a short natural lifetime). Diffusion is a --- random process, so that it could be treated by a Monte Carlo simulation. However, the CPU-time requirements for such a simulation are prohibitive if an atom suffers many collisions. We thus chose to compute the diffusion analytically. Before the simulation starts, we compute the distribution of metastables that results when a state-b density $n_b$ of $1$ inside one subcell and $0$ elsewhere diffuses for $t_1$ seconds. This computation is done for all subcells. During the simulation, we let the photons excite the vapour for $t_1$ seconds and compute the distribution of metastables according to the bleaching processes described above; diffusion is neglected during that time. At the end of that time, we compute $n_b(\mathbf{r})$ that results when this distribution of metastables diffuses through the buffer gas for $t_1$ seconds. For sufficiently small time intervals $t_1$, this gives quite accurate results. However, due to numerical problems, $t_1$ must not become *too* small ($t_1 \approx 0.05\tau$ is usually a good compromise). ### Description of the subroutines All parameters of the atom and of the buffer gas are defined in the constant-block. In the current version, they are for a natural mix of thallium isotopes and for argon as buffer gas. The main program just calls the subroutines INPUT1, INPUT2, INITIA, LINEAR, and NONLINEAR. In INPUT1, all parameters of the vapour cell are entered. The input is menu-guided and self-explanatory. A large number of parameters can be defined, namely cell dimensions, number and dimensions of subcells, number of modes for the solution of the diffusion equation, the reduction factor, Tl density, temperature (for the Doppler velocities), buffer gas pressure, timestep $t_1$, the number of timesteps, and the reflection properties of the cell walls (for each subcell that borders a cell wall, the absorption and reflection coefficients for bc- and ac-photons can be defined; specular or diffuse reflection can be chosen). In INPUT2, the parameters of the light source are defined. Three kinds of sources are available: a toroidal light source at the side walls, a flat ring at the side walls, and a circular source at the top of the cell. The probability distribution of the emission frequency is given by the function ALLG_FREQ_SOURCE1,2,3. For sources 1 and 2, the currently implemented function corresponds to a spectral lamp emitting a self-reversed line of ac-radiation. For source 3, we have implemented a narrow-band source at the centre frequency of the bc transition. The emission angles are chosen according to the probability functions ALLG_PHI1,2,3 and ALLG_THETA1,2,3. Theta is the angle between the direction of emission and the normal on the emission area. Currently, we have implemented a Lambertian distribution for source 1 and 2 and a unidirectional distribution for source 3. The emission coordinates $z$ (for source 2) and $r$ (for source 3) are given in the functions ALLG_Z and ALLG_R, respectively. For source 1, emission is independent of angle with the $z = 0$ plane. The procedure INT_PROFILE integrates and normalizes the probability function and stores the results in the vector **pf**. The procedure ALLG_FREQ determines a random number according to a given probability function from a vector **pf**, according to the method described in Chapter 6. Procedure INTERSECTION_WITH_SUBCELL computes the intersection point of a photon path with the boundaries of the subcell. --- POINT_OF_ABSORPTION determines the point of absorption of a photon or whether it has left the cell or was absorbed at the cell walls. The procedures EXPANSION_COEFF and DIFFUSION are required to compute the influence of the particle diffusion according to the method described above. The actual simulations for bleaching or no bleaching are done in NONLINEAR and LINEAR, respectively. Additional information, especially on the format of the required data files, can be found in the comment lines of the source code at the beginning of the program. ## C.3 SLAB3 Kunasz and Kunasz published SLAB3, a computer program available from the CPC program library (Kunasz and Kunasz 1975). SLAB3 allows the computation of trapping including particle transport, where the upper level can be a doublet, with the two fine-split levels collisionally coupled. The program is based on the discrete-ordinate method described in Sec. 10.4.3. SLAB3 is written in FORTRAN IV, and is still based on punch cards. Some modifications might thus be necessary if one wants to run it on a modern machine. However, FORTRAN IV is more or less a subset of today's standard FORTRAN 77, so that the necessary modifications are expected to be minor. One further slight restraint is that the program is not interactive. These drawbacks are due to the fact that the program was written in 1975; it should be easy for anyone familiar with FORTRAN to make appropriate modifications to the source code. ### Description of the subroutines Routine DRIVER calls the subroutines, and can make repeated runs of the main loop. The comment lines specify the necessary input data and the common blocks. The function PROFIL computes the function $\Phi(x) = C_x \cdot k(x)$; choices for the profile are Doppler, Lorentz, and Voigt. The type of profile and the Voigt parameter are handed over in a common block. For a Voigt line, the profile is computed in the function VOIGT according to an algorithm by Armstrong and Nicholls (1972). The subroutine FINDER computes the frequency of a monochromatic input beam when the relative transmission through the slab is given. Routine CHRTS calls the functions of the first overlay (i.e. the functions BASIQ, BMIN, and ZERZ. Note that there are no overlays in FORTRAN 77). The subroutine BASIQ inputs the atomic data and computes the poles of the characteristic equation. Routine PWI computes the quadrature points and weights; an equidistant grid is used. The highest occurring frequency is chosen so that $T \cdot \Phi(x_n) = 0.005$. Subroutine BMIN orders the poles. ZERZ computes the roots $k_\alpha^2$ of the characteristic equation and checks their product with the theoretical root product TEST computed in the subroutine BASIQ. The root search is done between two poles; an artificial pole at 5 times the location of the largest pole is added in BMIN so that also the largest root can be found. If ZERZ finds the wrong number of sign changes in the characteristic equation, there are numerical troubles and variable ISHOW is set to one to inform the user. For the root finding, a positive chop technique in routine BISECT is used. The --- number of grid points is given by NGRID, which is currently preset to 55. If NGRID is too small, zeroes of the characteristic equation might be overlooked; if it is too large, CPU time requirements are high. The function GF(DIL), which also calls the functions SL1, SL2, ..., SZ5, computes the characteristic function at a certain distance to the next pole. Subroutine SOLVE calls the routine BEAM to read in the boundary conditions, and calls EQSLV to obtain the integration constants $L_{\alpha}$ and $M_{\alpha}$. Subroutine RESULT computes and prints source functions, intensities, and densities for the two transitions. Routines DENSIT and SINS are called to compute densities and intensities at the slab walls. SORC computes the source functions and INTENS the emergent intensities at angle $\mu =$CMU. The subroutine TSINK finally computes the thermal losses. A more complete description of the program, as well as several test runs, can be found in Kunasz and Kunasz (1975). The program is quite long (about 3000 lines of source code), but fast (typically 10 s run time on a CDC6400, which we expect to translate into milliseconds on a good workstation today) and can deal directly with the steady-state distribution on a collisionally coupled doublet including quenching, wall reflections, and particle transport. ## C.4 ALTAIR ALTAIR is a very general program for multilevel problems (Klein *et al.* 1989), (Castor *et al.* 1992), (Dykema *et al.* 1996). Although originally intended for astrophysical applications, it can be applied to plasma computations (Back *et al.* 1991b). ALTAIR is based on the equivalent two-level atom ETLA iteration scheme (see Chapter 13) to deal with the interactions of the lines, and also of the continuum. It can deal with both steady-state and transient problems. In the latter case, it uses a simple first-order backwards finite differencing scheme. The stepwidth can be adjusted so that the maximum rate of change in the ionic population does not exceed a certain value. The information on the geometry and the atomic data is provided by an external database. This database can be quite large. Computations were performed by Castor *et al.* (1992) for 4000 radiative lines, 8000 collisional transitions, and a lot of other processes. The type of computation of radiative transfer can be specified for each transition. Choices are no trapping (for optically thin lines), LAMBDA for direct formal solution of the transfer equation, or ETLA. The ETLA iteration is accelerated by the following scheme. Usually, one computes the upwards and the downwards radiative rates for one transition, while all the other transitions remain fixed. These rates are then used for the next transition. In the accelerated scheme, one sets the upwards rate to zero, and the downwards rate is replaced by the net radiative bracket (i.e. the difference between spontaneous decay and reabsorption of photons). When all populations already have the correct value, nothing is changed. However, when the upper and lower levels have inconsistent values, the accelerated scheme gives better convergence than changing both rates. The scheme displays good stability at high opacities, while at low opacities, it is better to use the direct rates. It is --- thus necessary to "blend" from the direct to the effective rates, which is implemented in ALTAIR. The scheme is further enhanced by Chebyshev acceleration, a general technique for iteratively solving linear systems of equations (Manteuffel 1977, 1978). The non-linear transfer equation must first be linearized around the current iterated value. The same scheme is used for the equations of population kinetics. Stimulated emission is included in an iterative way. The stimulated emission terms are based on the level population in the *previous* iteration step. This greatly accelerates the computations, because the source function becomes a linear function of the average radiation intensity, the drawback being that other authors have observed that ETLA in general can have convergence problems when close to population inversion. No statements about convergence specifically for ALTAIR are available; the computation is simply switched off for a population-inverted transition. The radiative transfer equation for a given source function is solved by the discontinuous finite-element method (DFE) as described in Chapter 9. Alternatively, it can be computed by accelerated lambda iteration, which is essentially the operator perturbation method as used by Olson *et al.* (1986), and described in Chapter 13, and double splitting (see Chapter 9). An adaptive frequency mesh assigns frequencies to either 'optically thick' or 'optically thin' groups (actually, the subdivision is a bit finer). ALTAIR appears to be one of the most general programs for multilevel problems, and seems especially suited for plasma problems. We do not know, however, whether it is publicly available. ## C.5 TLUSTY Hubeny (1988) published the program TLUSTY in the CPC program library (catalogue number ABFK). It is a program based on the complete linearization technique (see Sec. 13.3) and handles a large number of chemical elements (about 30). The main restriction is that it is intended for a plane-parallel slab geometry. A large number of options allows one to choose various simplifications (e.g. some atoms contribute not to the opacity, but only to the charge equilibrium equations) and computation methods (e.g. choose between a standard second-order form or Auer's fourth-order Hermitian form of the Feautrier equation). The lineshapes can either be user-supplied or chosen from a number of pre-programmed functions (Doppler or Voigt, where the damping can be natural, Stark, van der Waals, etc.). The frequency integration points have to be chosen by the user; some tips on how to choose them can be found in Sec. 5 of Hubeny (1988). The main program consists of the routines START, which controls the input and the initialization of the parameters, RESOLV, which controls the formal solution step, and SOLVE, which performs the actual linearization. Typical runtimes are on the order of 10 minutes on a VAX 8600, which should translate into a few seconds on a state-of-the-art (1997) workstation or personal computer. The program was written explicitly for astrophysical applications, which is reflected in the choice of the typical parameters, and the chosen geometry. However, the large --- number of allowed chemical elements might make it also suitable for the treatment of plasmas. TLUSTY can also be used for chemical-physics application, where the slab geometry is often fulfilled. It would, however, be necessary to write modifications in order to include the effects of laser radiation. For problems without saturation, the use of TLUSTY would be a bit of an "overkill"; the programs described in Sec. C.1– C.3 are probably preferable in that case. ### C.6 Other programs Carlsson (1985) gives a FORTRAN 77 code for computations according to Scharmer's method, which is essentially based on an operator perturbation technique. While this method is especially efficient for astrophysical problems, the program is worth mentioning since it is publicly available and several authors have added improvements that are also accessible. The solution of the transfer equation for a known source function (in other words, the emerging lineshape) can be done with PROFILE by Tallents (1982). TRANSPHERE by Hummer *et al.* (1973) deals with the formal solution of the transfer equation in spherical geometries. The radiation trapping problem in such a sphere is solved in DUSTCD by Spagna and Leung (1983). Further there are several astrophysical programs that are available from the authors, but not listed in a formal library (like the CPC library). Since authors' addresses and affiliations frequently change, it is not practical to list them here. A search on the Internet, however, will produce many of these sources. --- ## APPENDIX D ### FITTING EQUATIONS FOR THE EIGENVALUES AND EIGENFUNCTIONS OF THE HOLSTEIN EQUATION In this appendix, we summarize the coefficients for the fitting equations to the trapping factors and eigenvalues of the Holstein equation in a one-dimensional geometry (see also Chapter 7). ### D.1 The slab In the slab case, the trapping factors for a Doppler lineshape can be written as $$ g_j^D = 1 + \frac{1}{m_j^D} \cdot k_0 L \sqrt{\ln\left(\frac{k_0 L}{2} + e\right)} - \frac{c_{0j}^D k_0 L \ln(k_0 L) + c_{1j}^D k_0 L + c_{2j}^D (k_0 L)^2}{1 + c_{3j}^D k_0 L + c_{4j}^D (k_0 L)^2}, \quad (D.1) $$ where the coefficients $m_j^D$ and $c_{ij}^D$ are given in Table D.1. In the Lorentz case, the trapping factors are given by (Table D.2) $$ g_j^L = \frac{1}{m_j^L} \sqrt{\pi k_0 L + (m_j^L)^2} - \frac{c_{0j}^L k_0 L \ln(k_0 L) + c_{1j}^L k_0 L + c_{2j}^L (k_0 L)^2}{1 + c_{3j}^L k_0 L + c_{4j}^L (k_0 L)^2}. \quad (D.2) $$ The lowest-order even and odd eigenfunctions are given as (Table D.3) $$ \psi_0(z) = \sqrt{\frac{2/L}{1 + si(\zeta_0 \pi)}} \cos(\zeta_0 \pi \frac{z}{L}), \quad \psi_1(z) = \sqrt{\frac{2/L}{1 - si(\zeta_1 \pi)}} \sin(\zeta_1 \pi \frac{z}{L}), $$ $$ \text{where} \qquad \zeta_0 = d_1 \left( \frac{k_0 L + d_2}{k_0 L + d_2 d_3} \right)^{d_4}, \quad \zeta_1 = 1 + d_5 \left( \frac{k_0 L + d_6}{k_0 L + d_6 d_7} \right)^{d_8}. \quad (D.3) $$ ### D.2 The cylinder The trapping factors for the cylinder case have the same functional form as for the slab, only the fitting coefficients differ; they are given in Tables D.4 for the Doppler case and in D.5 for the Lorentz case. The lowest-order eigenmodes are approximated as (Table D.6) $$ \psi_0(r) = \frac{1}{R} \sqrt{\frac{2}{J_0^2(\zeta_0) + J_1^2(\zeta_0)}} J_0(\zeta_0 \frac{r}{R}), \quad \text{where} \quad \zeta_0 = d_1 \left( \frac{k_0 R + d_2}{k_0 R + d_2 d_3} \right)^{d_4} \quad (D.4) $$ --- ## D.3 The sphere The trapping factors in the spherical case again have the same functional form as for the slab. The fitting coefficients differ and are given in Tables D.7 for the Doppler case and D.8 for the Lorentz case. The lowest-order eigenfunction is given as $$ \psi_0(r) = 2 \sqrt{\frac{\zeta_0}{R[2\zeta_0 - \sin(2\zeta_0)]}} \frac{\sin \left(\zeta_0 \frac{r}{R}\right)}{r}, \quad \text{where } \zeta_0 = d_1 \left( \frac{k_0 R + d_2}{k_0 R + d_2 d_3} \right)^{d_4} $$ (D.5) The coefficients $d_1, ..., d_4$ are listed in table D.9. **Table D.1** *Fitting coefficients for the trapping factor for a Doppler line-shape in a slab.* | $j$ | $m_j^D$ | $c_{0j}^D$ | $c_{1j}^D$ | $c_{2j}^D$ | $c_{3j}^D$ | $c_{4j}^D$ | |---|---|---|---|---|---|---| | 0 | 1.025 | 0.1226 | -0.1169 | 0.1449 | 0.6871 | 0.001007 | | 1 | 2.441 | 0.01129 | 0.1284 | 0.1456 | 0.8262 | 0.001006 | | 2 | 3.826 | 0.01647 | 0.0 --- **Table D.2** *Fitting coefficients for the trapping factor for a Lorentz lineshape in a slab.* | $j$ | $m_j^L$ | $c_{0j}^L$ | $c_{1j}^L$ | $c_{2j}^L$ | $c_{3j}^L$ | $c_{4j}^L$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 1.146 | 0.1288 | 0.4577 | 0.01303 | 1.154 | 0.1189 | | 1 | 1.892 | 0.06225 | 0.3249 | 0.006925 | 0.7969 | 0.04529 | | 2 | 2.397 | $8.684 \cdot 10^{-5}$ | 0.1594 | 0.001451 | 0.2198 | 0.009393 | | 3 | 2.821 | -0.01123 | 0.1065 | 0.00233 | 0.06789 | 0.008303 | | 4 | 3.184 | -0.008319 | 0.08315 | 0.003315 | 0.04629 | 0.009888 | | 5 | 3.52 | 0.001161 | 0.07962 | 0.0001894 | 0.112 | 0.001909 | | 6 | 3.82 | 0.0007058 | 0.06792 | 0.0001364 | 0.09328 | 0.001355 | | 7 | 4.10 | 0.0003931 | 0.05846 | 0.0001427 | 0.07694 | 0.001128 | | 8 | 4.37 | 0.0004455 | 0.05094 | 0.0001412 | 0.06611 | 0.001006 | | 9 | 4.60 | 0.0001114 | 0.04687 | 0.0001015 | 0.05992 | 0.0006963 | | 10 | 4.84 | 0.0004174 | 0.04243 | $6.813 \cdot 10^{-5}$ | 0.05668 | 0.0005476 | | 11 | 5.06 | $4.066 \cdot 10^{-5}$ | 0.03887 | $6.736 \cdot 10^{-5}$ | 0.04898 | 0.0004666 | | 12 | 5.28 | $8.333 \cdot 10^{-5}$ | 0.03576 | $4.791 \cdot 10^{-5}$ | 0.0454 | 0.0003769 | | 13 | 5.47 | 0.0001964 | 0.03356 | $4.474 \cdot 10^{-5}$ | 0.04362 | 0.0003126 | | 14 | 5.66 | $5.481 \cdot 10^{-5}$ | 0.03161 | $4.193 \cdot 10^{-5}$ | 0.04016 | 0.0002664 | | 15 | 5.85 | $9.527 \cdot 10^{-5}$ | 0.02938 | $3.918 \cdot 10^{-5}$ | 0.03703 | 0.0002405 | | 16 | 6.01 | $5.256 \cdot 10^{-5}$ | 0.02805 | $4.322 \cdot 10^{-5}$ | 0.03491 | 0.0002156 | | 17 | 6.20 | 0.0001357 | 0.02637 | $2.559 \cdot 10^{-5}$ | 0.03357 | 0.0001669 | | 18 | 6.33 | $6.354 \cdot 10^{-5}$ | 0.02557 | $3.286 \cdot 10^{-5}$ | 0.03176 | 0.0001523 | | 19 | 6.49 | 0.0001578 | 0.02418 | $3.402 \cdot 10^{-5}$ | 0.03031 | 0.0001466 |
**Table D.3** *Fitting coefficients for the lowest-order eigenmodes in a slab.* | Shape | $d_1$ | $d_2$ | $d_3$ | $d_4$ | $d_5$ | $d_6$ | $d_7$ | $d_8$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Doppler | 0.9029 | 0.9603 | 1.686 | 1.644 | 0.8828 | 2.459 | 1.727 | 1.603 | | Lorentz | 0.805 | 0.7186 | 1.653 | 1.583 | 0.7761 | 2.106 | 1.492 | 1.892 | --- THE SPHERE **Table D.4** *Fitting coefficients for the trapping factors in a cylinder for a Doppler lineshape.* | $j$ | $m_j^{\mathrm{D}}$ | $c_{0j}^{\mathrm{D}}$ | $c_{1j}^{\mathrm{D}}$ | $c_{2j}^{\mathrm{D}}$ | $c_{3j}^{\mathrm{D}}$ | $c_{4j}^{\mathrm{D}}$ | |---|---|---|---|---|---|---| | 0 | 0.8889 | -0.004447 | 0.2464 | -0.0002139 | 0.0165 | $6.57 \cdot 10^{-6}$ | | 1 | 2.2711 | 0.002771 | 0.2015 | 0.001472 | 0.01393 | $2.03 \cdot 10^{-5}$ | | 2 | 3.6601 | 0.01095 | 0.1415 | 0.001667 | 0.01613 | $2.071 \cdot 10^{-5}$ | | 3 | 5.0507 | 0.01196 | 0.107 | 0.002868 | 0.02478 | $4.716 \cdot 10^{-5}$ | | 4 | 6.4419 | 0.008564 | 0.0941 | $-1.07 \cdot 10^{-5}$ | 0.002772 | $3.078 \cdot 10^{-7}$ | | 5 | 7.8334 | 0.009355 | 0.07763 | $-2.9 \cdot 10^{-5}$ | 0.002572 | $2.187 \cdot 10^{-7}$ | | 6 | 9.2251 | 0.01065 | 0.06361 | $-3.584 \cdot 10^{-6}$ | 0.003166 | $4.297 \cdot 10^{-7}$ | | 7 | 10.6168 | 0.009667 | 0.05656 | $-5.587 \cdot 10^{-6}$ | 0.002666 | $1.151 \cdot 10^{-6}$ | | 8 | 12.0087 | 0.01014 | 0.0482 | $1.694 \cdot 10^{-5}$ | 0.003142 | $1.786 \cdot 10^{-6}$ | | 9 | 13.3988 | 0.01004 | 0.04228 | $1.849 \cdot 10^{-5}$ | 0.003174 | $2.317 \cdot 10^{-6}$ | **Table D.5** *Fitting coefficients for the trapping factors in a cylinder for a Lorentz lineshape.* | $j$ | $m_j^{\mathrm{L}}$ | $c_{0j}^{\mathrm{L}}$ | $c_{1j}^{\mathrm{L}}$ | $c_{2j}^{\mathrm{L}}$ | $c_{3j}^{\mathrm{L}}$ | $c_{4j}^{\mathrm{L}}$ | |---|---|---|---|---|---|---| | 0 | 1.1227 | 0.1204 | 0.8738 | -0.003321 | 1.877 | 0.1166 | | 1 | 1.8547 | 0.07711 | 0.3981 | 0.0006754 | 0.9185 | 0.0352 | | 2 | 2.3725 | 0.04259 | 0.2186 | 0.0005424 | 0.5071 | 0.01357 | | 3 | 2.7964 | 0.02177 | 0.1433 | 0.0003337 | 0.3052 | 0.006095 | | 4 | 3.1642 | -0.01347 | 0.09756 | 0.0002741 | 0.0577 | 0.001294 | | 5 | 3.4935 | -0.0141 | 0.08046 | 0.0002714 | 0.02386 | 0.0009119 | | 6 | 3.7943 | -0.01322 | 0.07234 | 0.0001625 | 0.01794 | 0.0004828 | | 7 | 4.0730 | -0.01055 | 0.06298 | 0.0002232 | 0.01528 | 0.0006919 | | 8 | 4.3339 | -0.01059 | 0.05868 | 0.0001853 | 0.009522 | 0.0005132 | | 9 | 4.5796 | $5.53 \cdot 10^{-5}$ | 0.04673 | $3.486 \cdot 10^{-5}$ | 0.05186 | 0.0005444 | **Table D.6** *Fitting coefficients for the lowest-order eigenmodes in the cylinder case.* | Lineshape | $d_1$ | $d_2$ | $d_3$ | $d_4$ | |---|---|---|---|---| | Doppler | 2.202 | 0.9545 | 1.582 | 1.048 | | Lorentz | 2.024 | 0.7057 | 1.504 | 1.007 | --- ## FITTING EQUATIONS **Table D.7** *Fitting coefficients for the trapping factors in a sphere for a Doppler lineshape.* | $j$ | $m_j^{\text{D}}$ | $c_{0j}^{\text{D}}$ | $c_{1j}^{\text{D}}$ | $c_{2j}^{\text{D}}$ | $c_{3j}^{\text{D}}$ | $c_{4j}^{\text{D}}$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 1.2090 | -0.01467 | 0.2704 | 0.000355 | 0.007323 | $4.247 \cdot 10^{-6}$ | | 1 | 2.6209 | 0.00932 | 0.1808 | 0.003258 | 0.02823 | $3.074 \cdot 10^{-5}$ | | 2 | 4.0491 | 0.01416 | 0.1262 | 0.003041 | 0.02947 | $2.955 \cdot 10^{-5}$ | | 3 | 5.4850 | 0.01523 | 0.09422 | 0.002739 | 0.02955 | $2.938 \cdot 10^{-5}$ | | 4 | 6.9283 | 0.01494 | 0.0737 | 0.002119 | 0.02588 | $2.503 \cdot 10^{-5}$ | | 5 | 8.3739 | 0.01341 | 0.06134 | 0.001971 | 0.02542 | $2.707 \cdot 10^{-5}$ | | 6 | 9.8206 | 0.01298 | 0.05091 | 0.001799 | 0.02506 | $2.721 \cdot 10^{-5}$ | | 7 | 11.2668 | 0.01252 | 0.04286 | 0.001682 | 0.02504 | $2.793 \cdot 10^{-5}$ | | 8 | 12.7129 | 0.01179 | 0.03713 | 0.001587 | 0.02493 | $2.915 \cdot 10^{-5}$ | | 9 | 14.1576 | 0.01262 | 0.02971 | 0.00141 | 0.0244 | $2.692 \cdot 10^{-5}$ |
**Table D.8** *Fitting coefficients for the trapping factors in a sphere for a Lorentz lineshape.* | $j$ | $m_0^{\text{L}}$ | $c_{0j}^{\text{L}}$ | $c_{1j}^{\text{L}}$ | $c_{2j}^{\text{L}}$ | $c_{3j}^{\text{L}}$ | $c_{4j}^{\text{L}}$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 1.3368 | -0.04229 | 0.4281 | 0.0004927 | 0.586 | 0.01763 | | 1 | 1.9940 | 0.00122 | 0.2463 | 0.0009306 | 0.3512 | 0.01419 | | 2 | 2.4858 | 0.009816 | 0.1678 | 0.000536 | 0.2701 | 0.008151 | | 3 | 2.8971 | 0.000929 | 0.1173 | 0.0005416 | 0.1529 | 0.004579 | | 4 | 3.2582 | -0.001344 | 0.09097 | 0.0004244 | 0.103 | 0.00294 | | 5 | 3.5839 | 0.0003746 | 0.07239 | 0.001316 | 0.08239 | 0.004801 | | 6 | 3.8829 | -0.0008453 | 0.06027 | 0.0008988 | 0.0578 | 0.003262 | | 7 | 4.1605 | 0.0005394 | 0.05423 | 0.0006895 | 0.0635 | 0.002504 | | 8 | 4.4204 | -0.0007005 | 0.04444 | 0.000257 | 0.03508 | 0.001314 | | 9 | 4.6652 | -0.001313 | 0.04317 | 0.0001924 | 0.0365 | 0.000945 |
**Table D.9** *Fitting coefficients for the lowest-order eigenmode in the sphere case.* | Lineshape | $d_1$ | $d_2$ | $d_3$ | $d_4$ | | :--- | :--- | :--- | :--- | :--- | | Doppler | 2.904 | 1.543 | 1.444 | 0.8827 | | Lorentz | 2.710 | 1.149 | 1.286 | 1.0310 | --- ## APPENDIX E ### FINITE DIFFERENCE SOLUTION OF THE INHOMOGENEOUS EQUATION OF RADIATIVE TRANSFER IN A FINITE CYLINDER The equation of radiative transfer in the Eddington approximation described in Chapter 8 (and applied in Chapter 13) have the general form $$ \nabla^2 u + e \cdot u + f \cdot (\nabla u) \cdot (\nabla g) + s = 0. $$ (E.1) In a cylindrical coordinate system with $\partial/\partial\varphi = 0$, this becomes $$ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^2 u}{\partial z^2} + eu + f \left( \frac{\partial u}{\partial r} \frac{\partial g}{\partial r} + \frac{\partial u}{\partial z} \frac{\partial g}{\partial z} \right) + s = 0 $$ (E.2) with the boundary conditions $$ \begin{array}{lllll} \frac{\partial u}{\partial r} + \alpha u = \beta & \text{at } r = R & \text{and} & \frac{\partial u}{\partial z} + \alpha u = 0 & \text{at } z = L'/2 \\ \frac{\partial u}{\partial r} = 0 & \text{at } r = 0 & \text{and} & \frac{\partial u}{\partial z} = 0 & \text{at } z = 0, \end{array} $$ (E.3) where $e, f, g, h, \alpha$, and $\beta$ are known functions of position $(r, z)$ and $u$ is an unknown function of position. As mentioned in Sec. 13.5, non-equidistant discretization is necessary. The boundary conditions imply that external radiation can be incident on the side walls of the cylinder; this radiation is assumed to be symmetric with respect to the $z = 0$ plane. Other cases of incident radiation can be treated completely analogously. For a FD solution (Press *et al.* 1993), (Schwarz 1986) we lay a (non-equidistant) grid of the points where we wish to compute the solutions in the area of interest ($0 < r < R$, $0 < z < L/2$). At an arbitrary point $\mathbf{r}$ near a grid point $\mathbf{r}_0$, we make a two-term Taylor expansion of the function $u$: $$ \begin{aligned} u(r, z) = u_0 &+ \frac{u_E - u_W}{h_W + h_E}(r - r_0) + \frac{u_N - u_S}{h_N + h_S}(z - z_0) + \\ &+ \frac{\frac{u_E - u_0}{h_E} + \frac{u_W - u_0}{h_W}}{h_E + h_W}(r - r_0)^2 + \frac{\frac{u_N - u_0}{h_N} + \frac{u_S - u_0}{h_S}}{h_N + h_S}(z - z_0)^2 \end{aligned} $$ (E.4) The first and second derivatives can then be approximated as --- FINITE DIFFERENCE SOLUTION [FIGURE: FIG. E.1. Numbering system for the finite difference solution.] [FIGURE: FIG. E.2. Numbering system of the grid for the FD solution.] $$ \begin{aligned} \frac{\partial u}{\partial r} &= \frac{u_E - u_W}{h_E + h_W} \\ \frac{\partial u}{\partial z} &= \frac{u_N - u_S}{h_N + h_S} \\ \frac{\partial^2 u}{\partial r^2} &= \frac{2}{h_E + h_W} \left[ \frac{u_E}{h_E} + \frac{u_W}{h_W} - u_0 \left( \frac{1}{h_E} + \frac{1}{h_W} \right) \right] \\ \frac{\partial^2 u}{\partial z^2} &= \frac{2}{h_N + h_S} \left[ \frac{u_N}{h_N} + \frac{u_S}{h_S} - u_0 \left( \frac{1}{h_N} + \frac{1}{h_S} \right) \right] \end{aligned} $$ (E.5) Figure E.2 shows the definitions for $h_E$, $h_W$, $h_N$, and $h_S$. Figure E.1 shows the numbering system of the grid points, so that if $u(r, z)$ at the grid point $\mathbf{r}_0$ has the index $i$ ($u$ at that point equals $u_i$), then at the neighbouring points, the indices are $$ \begin{matrix} & u_N = u_{i+N_r+1} & \\ u_W = u_{i-1} & & u_E = u_{i+1} \\ & u_S = u_{i-N_r-1} & \end{matrix} $$ (E.6) --- FINITE DIFFERENCE SOLUTION The number of grid points in the $r$ and $z$ directions are $N_r + 1$ and $N_z + 1$, respectively. The index $i$ runs from 0 to Nmax, where Nmax $= (N_r + 1)(N_z + 1) - 1$. For the inner points of the grid, we insert the approximations for the differential operators, Eq. (E.5), into the differential equation (E.2). From this we can compute the matrix elements of the FD matrix $\mathbf{A} \cdot \mathbf{u} = \mathbf{B}$ (Note that the $A$-matrix elements have double indices. The first one is always $i$, and has been omitted for brevity). $$ \begin{aligned} A_i &= -\frac{2}{h_E + h_W} \left( \frac{1}{h_E} + \frac{1}{h_W} \right) - \frac{2}{h_N + h_S} \left( \frac{1}{h_N} + \frac{1}{h_S} \right) + e_0 \\ B_i &= -s_0 \\ A_{i-1} &= \frac{2}{h_W(h_E + h_W)} - \frac{1}{r_0(h_E + h_W)} - f_0 \frac{g_E - g_W}{(h_E + h_W)^2} \\ A_{i-N_r-1} &= \frac{2}{h_S(h_N + h_S)} - f_0 \frac{g_N - g_S}{(h_N + h_S)^2} \\ A_{i+1} &= \frac{2}{h_E(h_E + h_W)} + \frac{1}{r_0(h_E + h_W)} + f_0 \frac{g_E - g_W}{(h_E + h_W)^2} \\ A_{i+N_r+1} &= \frac{2}{h_N(h_N + h_S)} + f_0 \frac{g_N - g_S}{(h_N + h_S)^2} \end{aligned} $$ (E.7) At the points where $z = 0, r \neq 0$ or $R$, we have $u_N = u_S$ (because of the symmetry that is shown mathematically in the Neumann boundary conditions). The matrix elements are then $$ \begin{aligned} A_i &= -\frac{2}{h_E + h_W} \left( \frac{1}{h_E} + \frac{1}{h_W} \right) - \frac{2}{h_N^2} + e_0 & B_i &= -s_0 \\ A_{i-1} &= \frac{2}{h_W(h_E + h_W)} - \frac{1}{r_0(h_E + h_W)} - f_0 \frac{g_E - g_W}{(h_E + h_W)^2} \\ A_{i+1} &= \frac{2}{h_E(h_E + h_W)} + \frac{1}{r_0(h_E + h_W)} + f_0 \frac{g_E - g_W}{(h_E + h_W)^2} & A_{i+N_r+1} &= \frac{2}{h_N^2} \end{aligned} $$ (E.8) At the point $r = 0, z = 0$, we have furthermore $u_W = u_E$, so that the matrix elements are $$ \begin{aligned} A_i &= -\frac{2}{h_E^2} - \frac{2}{h_N^2} + e_0 & B_i &= -s_0 \\ A_{i+1} &= \frac{2}{h_E^2} & A_{i+N_r+1} &= \frac{2}{h_N^2} \end{aligned} $$ (E.9) For the points with $r = 0, z \neq 0$ or $L/2$, we have $u_E = u_W$, so that --- $$ \begin{aligned} A_i &= -\frac{2}{h_E^2} - \frac{2}{h_N + h_S} \left( \frac{1}{h_N} + \frac{1}{h_S} \right) + e_0 \\ & \qquad \qquad \qquad \qquad B_i = -s_0 \\ & \qquad \qquad \qquad \qquad A_{i-Nr-1} = \frac{2}{h_S(h_N + h_S)} - f_0 \frac{g_N - g_S}{(h_N + h_S)^2} \\ A_{i+1} &= \frac{2}{h_E^2} \qquad \qquad \quad A_{i+Nr+1} = \frac{2}{h_N(h_N + h_S)} + f_0 \frac{g_N - g_S}{(h_N + h_S)^2} \end{aligned} $$ (E.10) The situation is considerably more complicated at the boundary points where we have Cauchy boundary conditions. Take, for example, a point $\mathbf{r}_i$ with $r = R$, $z \neq 0$ or $L/2$. We approximate $u(\mathbf{r})$ at any point $\mathbf{r}$ in the vicinity of $\mathbf{r}_i$ by a two-term Taylor expansion. In particular, for the point $\mathbf{r} = \mathbf{r}_{i-1}$ (i.e. the western neighbour), $$ u_{i-1} = u(r_i - h_W, z_i) = u(r_i, z_i) - h_W \frac{\partial u}{\partial r} + \frac{1}{2} h_W^2 \frac{\partial^2 u}{\partial r^2}. $$ (E.11) Inserting the boundary conditions $\partial u/\partial r = \beta - \alpha u$ into this relation, we get an expression for $(\partial^2 u)/(\partial r^2)$ at $r = R$, $$ \frac{\partial^2 u}{\partial r^2} = \frac{2}{h_W^2} [u_W - u_0 + h_W \beta - h_W \alpha u_0]. $$ (E.12) the first derivative, $\partial u/\partial r$, is of course simply $\beta - \alpha u$. At $z = L/2$, we get in a completely analogous way $$ \frac{\partial^2 u}{\partial z^2} = \frac{2}{h_S^2} [u_S - u_0 - h_S \alpha u_0], $$ (E.13) and $\partial u/\partial z = -\alpha u$ ($\beta = 0$ at these points). Inserting the above expressions into the differential equation (E.2), we get for the matrix elements at the points $r = R$, $z \neq 0$ or $L/2$ $$ \begin{aligned} A_i &= -\frac{2}{h_W^2} - \frac{2}{h_N + h_S} \left( \frac{1}{h_N} + \frac{1}{h_S} \right) - \frac{2}{h_W}\alpha^r - \frac{\alpha^r}{r_0} - f_0 g_r \alpha^r + e_0 \\ & \qquad \qquad \qquad \qquad B_i = -\frac{2\beta}{h_W} - \frac{\beta}{r_0} - f_0 \beta g_r - s_0 \\ & \qquad \qquad \qquad \qquad A_{i-Nr-1} = \frac{2}{h_S(h_N + h_S)} - f_0 \frac{g_N - g_S}{(h_N + h_S)^2} \\ A_{i-1} &= \frac{2}{h_W^2} \qquad \qquad \quad A_{i+Nr+1} = \frac{2}{h_N(h_N + h_S)} + f_0 \frac{g_N - g_S}{(h_N + h_S)^2} \end{aligned} $$ (E.14) The symbol $\alpha^r$ denotes $\alpha$ at $r = R$; $\alpha^z$ is $\alpha$ at $z = L/2$. The symbols $g_r$ and $g_z$ are short for $\partial g/\partial r$ and $\partial g/\partial z$, respectively. --- At the points $z = L/2, r \neq 0, R$, we get $$ \begin{aligned} A_i &= -\frac{2}{h_E + h_W} \left( \frac{1}{h_E} + \frac{1}{h_W} \right) - \frac{2\alpha^z}{h_S} + e_0 - f_0 \alpha^z g_z - \frac{2}{h_S^2} & B_i &= -s_0 \\ A_{i-1} &= \frac{2}{h_W(h_E + h_W)} - \frac{1}{r_0(h_E + h_W)} - f_0 \frac{g_E - g_W}{(h_E + h_W)^2} & A_{i-Nr-1} &= \frac{2}{h_S^2} \\ A_{i+1} &= \frac{2}{h_E(h_E + h_W)} + \frac{1}{r_0(h_E + h_W)} + f_0 \frac{g_E - g_W}{(h_E + h_W)^2} \end{aligned} $$ (E.15) At $z = L/2, r = 0$, we have again $u_E = u_ --- otherwise the iteration becomes unstable and gives unphysical results. The fact that finite difference solutions may become unstable for a grid that is too coarse is well known (Press *et al.* 1993). If such instabilities occur, increasing the number of grid points and reducing the difference between neighbouring steps $h$ will resolve the problem. --- ## APPENDIX F ### THE DENSITY MATRIX We have mentioned in the main part of the book that for some problems formulations in terms of the density matrix are preferable. These situations occur mainly when a semiclassical description is no longer appropriate—we will see that the density matrix allows a correct quantum-mechanical description of the considered system. While we have not used such a description very often (mostly because the equations would become too complicated) it might be desirable from time to time. This appendix gives a brief account of the basics of the density matrix formalism. The reader interested in more details may want to consult the textbook (Blum 1996), which this appendix follows closely, or any textbook on quantum mechanics. ### F.1 The density matrix for atomic states We first give a formal derivation for the density matrix of atomic states. After that, we turn to a simple example that allows a more intuitive interpretation. A quantum-mechanical particle is described by its wavefunction $\psi(\mathbf{r}, t)$, see Appendix A. However, we are often either not able or not interested in finding the wavefunction exactly. We only want to know the probability of finding a particle in a certain state. For this purpose, we first expand the wavefunction in terms of an orthonormal system of functions $$ \psi(\mathbf{r}, t) = \sum_j \alpha_j(t) \psi_j(\mathbf{r}) $$ (F.1) Let now $A$ be an operator that corresponds to an observable of the system. The expectation value of $A$ is $$ \langle A \rangle = \sum_{j,k} \alpha_j^*(t) A_{j,k} \alpha_k(t) $$ (F.2) where the $A_{j,k}$ matrix elements are $$ A_{j,k} = \int \psi_j^*(\mathbf{r}) A \psi_k(\mathbf{r}) \mathrm{d}\mathbf{r} $$ (F.3) As mentioned above, we often do not know the wavefunctions, and thus not the expansion coefficients $\alpha$. However, we can find out the ensemble average (denoted by a bar) of the observable $A$, so that $$ \overline{\langle A \rangle} = \sum_{j,k} \overline{\alpha_j^*(t) \alpha_k(t)} A_{j,k} $$ (F.4) --- The density matrix elements $\rho_{j,k}(t)$ are then defined as $$ \rho_{j,k}(t) = \overline{\alpha_j^*(t)\alpha_k(t)} $$ (F.5) By definition, $\rho$ is a Hermitian matrix, i.e. $\rho_{j,k} = \rho_{kj}^*$. Furthermore, $\mathrm{tr}(\rho) = 1$, where $\mathrm{tr}$ denotes the trace of the matrix. Let us turn to a more intuitive interpretation of the density matrix. For that purpose, we consider particles with spin one half (e.g. hydrogen). These particles have only two possible states, namely $s= 1/2$ and $s= -1/2$. They can thus be easily described in terms of the basis states $|+1/2\rangle$ and $|-1/2\rangle$. The states can be easily distinguished by means of a Stern–Gerlach-type experiment, where particles in an atomic beam are deflected in different directions, depending on their spin. We prepare now an ensemble of atoms in such a way that the state vector $|\chi\rangle$ can be expanded in terms of the basis states as $$ |\chi\rangle = \alpha_1 |+1/2\rangle + \alpha_2 |-1/2\rangle = \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} $$ (F.6) The density matrix is now the “outer product” $|\chi\rangle\langle\chi|$, i.e. $$ |\chi\rangle\langle\chi| = \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} (\alpha_1^*, \alpha_2^*) = \begin{pmatrix} |\alpha_1|^2 & \alpha_1\alpha_2^* \\ \alpha_1^*\alpha_2 & |\alpha_2|^2 \end{pmatrix} $$ (F.7) Since the probability that a particle is in state $i$ ($i = \pm 1/2$) is equal to $|\alpha_i|^2$, the diagonal elements of the density matrix can be directly interpreted as the probability of finding the particle in the base state $i$. From this, we can also interpret the mathematical condition $\mathrm{tr}(\rho) = 1$; the sum of the probabilities of finding the particle in a state must be unity, since it must be in *some* state. The off-diagonal elements are not so easily interpreted. They basically describe the coherence between the states. Physically, the density matrix elements describe the spatial orientation of the polarization vector $\mathbf{P}$ with components $P_x$, $P_y$, $P_z$. The system is “completely polarized” if $\mathrm{tr}(\rho^2) = \mathrm{tr}(\rho)^2$, or alternatively if $|\mathbf{P}| = (P_x^2 + P_y^2 + P_z^2)^{0.5} = 1$. It can be shown that the time dependence of the density matrix is given by $$ \frac{\partial \rho}{\partial t} = -\frac{2\pi j}{h} \mathbf{H}^{\mathrm{H}} \rho $$ (F.8) where $\mathbf{H}^{\mathrm{H}}$ is the Hamiltonian of the system. ## F.2 The density matrix for photons We can also describe photons (including their polarization properties) in terms of the density matrix. Let us assume that the radiation is propagating in the $z$-direction. Then the vector of the electric field strength can be written as (see Chapter 12) $$ \mathbf{E}(t, z) = \begin{pmatrix} E_{x0} \exp[\mathrm{j}(\varphi_x - \omega t + 2\pi z \nu_0/c)] \\ E_{y0} \exp[\mathrm{j}(\varphi_y - \omega t + 2\pi z \nu_0/c)] \\ 0 \end{pmatrix} $$ (F.9) --- THE DENSITY MATRIX FOR PHOTONS After normalization, this becomes $$ \mathbf{E}(t, z) = E_{\text{tot}} \left[ \text{j}(\varphi_x - \omega t + 2\pi z \nu_0 / c) \right] \begin{pmatrix} \alpha_1 \\ \alpha_2 \exp \left[ \text{j}\varphi \right] \\ 0 \end{pmatrix} \quad \text{(F.10)} $$ so that the polarization vector is simply given by $$ \mathbf{\hat{E}} = \begin{pmatrix} \alpha_1 \\ \alpha_2 \exp \left[ \text{j}\varphi \right] \\ 0 \end{pmatrix} \quad \text{(F.11)} $$ When we thus take polarization along the $x$- and $y$-axes as "basis states", we can write any polarization state as a combination of the basis states, with expansion coefficients $\alpha_1$ and $\alpha_2 \exp(\text{j}\varphi)$. Equations (F.9–F.11) completely characterize polarized light. For unpolarized or partially polarized light one has to use the density matrix description, or alternatively the Stokes parameters. We have already seen in Chapter 12 that the polarization properties of radiation can be described in terms of the Stokes vector $\mathbf{I}$. Since this vector carries the same information as the --- ## F.3 Interaction of atoms and radiation Next, let us look at the interaction of atoms and photons. For the sake of simplicity, we assume that only two atomic levels (with energies $E_1$ and $E_2$) play a role, and that these states are not hyperfine- or Zeeman-split. The polarization properties of the radiation thus do not enter the computations, and the density matrix for the atoms is just a $2 \times 2$ matrix. The total Hamiltonian of the system is $$ \mathbf{H}^{\mathrm{H}} = \mathbf{H}_0^{\mathrm{H}} + \mathbf{H}^{\mathrm{H,inter}} $$ (F.16) where $\mathbf{H}_0^{\mathrm{H}}$ is the Hamiltonian without any field. The eigenfunctions of this unperturbed Hamiltonian must of course fulfil $\mathbf{H}_0^{\mathrm{H}} \psi_n = E_n \psi_n$. The interaction Hamiltonian is $\mathbf{H}^{\mathrm{H,inter}} = -\mu \mathbf{E}(t)$, where $\mu$ is the component of the atomic dipole operator in the direction of the field $\mathbf{E}$. Since we are considering dipole transitions from state 1 to state 2, the diagonal elements of the dipole operator matrix $\boldsymbol{\mu}$ must equal zero, and we set the phase of the transition elements so that $\mu_{21} = \mu_{12} = \mu$. From the knowledge of the Hamiltonian and with Eq. (F.8), we can set up the differential equation for the density matrix elements $$ \begin{aligned} \frac{\partial \rho_{2,1}}{\partial t} &= -\frac{2\pi \mathrm{j}}{h} \left[ (\mathbf{H}_0^{\mathrm{H}} + \mathbf{H}^{\mathrm{H,inter}})\boldsymbol{\rho} \right]_{2,1} \\ &= -\frac{2\pi \mathrm{j}}{h} \left[ H_{2,1}^{\mathrm{H,inter}} (\rho_{1,1} - \rho_{2,2}) + (E_2 - E_1)\rho_{2,1} \right] \end{aligned} $$ (F.17) Defining the resonance frequency as $\omega_0 = 2\pi(E_2 - E_1)/h$, this can be written as $$ \frac{\partial \rho_{2,1}}{\partial t} = -\mathrm{j}\omega_0\rho_{2,1} + \frac{2\pi \mathrm{j}\mu}{h}E(t)(\rho_{1,1} - \rho_{2,2}) $$ (F.18) and similarly for the other density matrix elements: $$ \begin{aligned} \frac{\partial \rho_{2,2}}{\partial t} &= -\frac{2\pi \mathrm{j}\mu}{h}E(t)(\rho_{2,1} - \rho_{2,1}^*) \\ \frac{\partial (\rho_{1,1} - \rho_{2,2})}{\partial t} &= \frac{4\pi \mathrm{j}\mu}{h}E(t)(\rho_{2,1} - \rho_{2,1}^*) \end{aligned} $$ (F.19) A loss of phase coherence can be included by adding a term $-\rho_{2,1}/\tau_{\mathrm{coherence}}$ on the r.h.s. of Eqs. (F.18) and (F.19). We again stress that the above derivation is valid only under very special simplified circumstances. While it is possible to derive some important properties, like e.g. the Kramers–Kronig relations from them (Yariv 1989), a full-fledged description of the time-decay of excited atoms (including polarization) is much more complicated. Details can be found e.g. in Chapter 5 of Blum (1996). Also the phenomena of Rabi oscillations, of self-induced transparency, and the like, can be analysed in that picture. --- ## F.4 State multipoles We have stated above that the diagonal elements of the density matrix have a clear physical interpretation as the probability of finding a particle in a certain state. The off-diagonal elements are related to the coherence between the considered states, but have no direct physical significance. Their importance can only be worked out by applying certain transformations. In this subsection, we want to describe these transforms and the interpretation of the transformed variables. ### F.4.1 Definition of tensor operators Let us consider two ensembles of particles, the first with the angular momentum $J$, the second with the angular momentum $J'$. When these two states interact, the resulting possible states have a total angular momentum $K$ and a $z$-component $Q$, where of course the transition rules $$|J - J'| \leq K \leq J + J' \quad \text{and} \quad -K \leq Q \leq K$$ (F.20) must be fulfilled. We now define a set of operators $\mathbf{T}(J', J)_{K,Q}$ by $$\mathbf{T}(J', J)_{K,Q} = \sum_{M',M} (-1)^{J-M} (J', M', J - M | K, Q) |J', M'\rangle \langle J, M|$$ (F.21) Note that $K$ and $Q$ are not subscripts denoting matrix elements, but rather parameters for the tensor $\mathbf{T}$. The $(\cdot|\cdot)$ expressions denote the Clebsch–Gordan coefficients. When we use the “$3j$-coefficients”, this reads $$\mathbf{T}(J', J)_{K,Q} = \sum_{M,M'} (-1)^{J'-M'} (2K + 1)^{1/2} \begin{pmatrix} J' & J & K \\ M' & -M & -Q \end{pmatrix} |J', M'\rangle \langle J, M|$$ (F.22) When we apply the “bra” $\langle J', N'|$ and “ket” $|J, N\rangle$ to Eq. (F.21), where $N' = -J', \dots, J'$ and $N = -J, \dots, J$, we get $$\langle J', N' | \mathbf{T}(J', J)_{K,Q} | J, N \rangle = (-1)^{J-N} (J', N', J - N | K, Q)$$ (F.23) which in terms of the $3j$-coefficients reads as $$\langle J', N' | \mathbf{T}(J', J)_{K,Q} | J, N \rangle = (-1)^{J'-N'} (2K + 1)^{1/2} \begin{pmatrix} J' & J & K \\ N' & -N & -Q \end{pmatrix}$$ (F.24) The elements of the operator $\mathbf{T} = \mathbf{T}(J', J)_{KQ}$ can then also be written as $$ \mathbf{T} = \begin{pmatrix} \langle J', J'|\mathbf{T}|J, J\rangle & \langle J', J'|\mathbf{T}|J, J - 1\rangle & \dots & \langle J', J'|\mathbf{T}|J, -J\rangle \\ \langle J', J' - 1|\mathbf{T}|J, J\rangle & \langle J', J' - 1|\mathbf{T}|J, J - 1\rangle & \dots & \langle J', J' - 1|\mathbf{T}|J, -J\rangle \\ \vdots & \vdots & & \vdots \\ \langle J', -J'|\mathbf{T}|J, J\rangle & \langle J', -J'|\mathbf{T}|J, J - 1\rangle & \dots & \langle J', -J'|\mathbf{T}|J, -J\rangle \end{pmatrix} $$ (F.25) --- For the interpretation of these results, we now consider the case of a sharp angular momentum $J' = J$. Then the operator with $K = 0$, $\mathbf{T}(J)_{00}$ is an operator that remains invariant under all rotations $$ \mathbf{T}(J)_{0,0} = \frac{1}{(2J + 1)^{1/2}} \mathbf{1} \quad \text{(F.26)} $$ where $\mathbf{1}$ is the unit matrix of size $(2J + 1)$. For the operators with $K = 1$ (vector operators), it can be shown that $$ \mathbf{T}(J)_{1,Q} = \left[ \frac{3}{(2J + 1)(J + 1)J} \right]^{1/2} J_Q \quad \text{(F.27)} $$ where we define the spherical vector components as $$ \begin{aligned} J_{+1} &= -\frac{1}{\sqrt{2}} (J_x + \mathrm{j} J_y) \\ J_0 &= J_z \\ J_{-1} &= +\frac{1}{\sqrt{2}} (J_x - \mathrm{j} J_y) \end{aligned} \quad \text{(F.28)} $$ and $J_x$, $J_y$, $ --- $$ = \text{tr} \left( \boldsymbol{\rho} \mathbf{T}(J', J)_{K,Q}^* \right) $$ (F.31) The conversion between $\rho$ and the statistical tensors is now given as $$ \boldsymbol{\rho} = \sum_{J,J',K,Q} \langle \mathbf{T}(J', J)_{K,Q}^* \rangle \mathbf{T}(J', J)_{K,Q} $$ (F.32) In the reverse direction, the conversion equation is $$ \langle J', M' | \boldsymbol{\rho} | J, M \rangle = \sum_{K,Q} (-1)^{J'-M'} (2K+1)^{1/2} \begin{pmatrix} J' & J & K \\ M' & -M & -Q \end{pmatrix} \langle \mathbf{T}(J', J)_{K,Q}^* \rangle $$ (F.33) These quantities now have a direct physical interpretation: $\langle \mathbf{T}(J)_{0,0} \rangle$ is a normalization constant. The values $\langle \mathbf{T}(J)_{1,Q} \rangle$ are the components of the orientation vector of the ensemble of atoms. The system is called oriented if at least one of the components of this vector does not vanish. This is so because the orientation vector is proportional to the net magnetic dipole vector of a given system. Similarly, the parameters $\langle \mathbf{T}(J)_{2,Q} \rangle$ are proportional to the spherical components of the electric quadrupole tensor. A system is thus called “aligned” if at least one the components of this quadrupole tensor is non-vanishing. Finally the system is called “polarized” if at least one of the multipoles (apart from $K = 0$) is non-zero. --- ## APPENDIX G ### HIGH-FIELD EFFECTS This appendix describes some of the effects that high-intensity radiation can have on atoms. These effects occur at *very* high intensities, and are not considered in the main part of the book.[^27] We just wish to give the reader a brief impression of the occurring effects. For more details, see e.g. Knight and Milonni (1980), Loudon (1983), Yariv (1989), Demtröder (1982), and references therein. ### G.1 Absorption and emission coefficients of homogeneously broadened lines Let us first deal with atoms with a homogeneously broadened lineshape interacting with a strong narrow-band laser field. The point of interest is the shape of the absorption and reemission coefficients. The most well-known effect in a strong laser field is “power-broadening”. Assume a homogeneously broadened lineshape of the absorption coefficient, and shine with a narrow-band laser with flux density $F$ on the vapour. If we now record the upper-state population as a function of detuning from the line centre, see Fig. G.1, we find that it follows a Lorentzian shape with FWHM $$ (\Delta \nu^{\mathrm{broad}})^2 = (\Delta \nu^{\mathrm{n}})^2 \left( 1 + \frac{2\Omega^2}{(2\pi \Delta \nu^{\mathrm{n}})^2} \right) $$ (G.1) which can also be written as $$ \left(\Delta \nu^{\mathrm{n}}\right)^2 \left( 1 + \frac{F}{F_R} \right) $$ (G.2) The Rabi frequency $\Omega$ will be defined below. It is, however, not necessary to modify absorption or emission coefficients to approximately take power broadening into account. Actually, using the rate equation including absorption, stimulated emission, and spontaneous emission leads exactly to the equations mentioned above, so that a quasi-classical picture is quite sufficient in that context. When we employ a strong pump laser at the line centre to cause the saturation, and then probe the absorption with a weak probe laser at $x'$, the absorption coefficient encountered by the probe laser will be $$ k(x') \frac{1}{1 + (F(0)/F_R)} $$ (G.3) [^27]: The only exception to this is the power broadening, which is, however, included in a different way in the radiation trapping computations. --- AC-STARK SPLITTING AND MOLLOW TRIPLETS 441 [FIGURE: FIG. G.1. Power broadening of a Lorentzian line.] A quantity that is strongly related to power broadening is the so-called Rabi frequency at zero detuning $$ \Omega = \frac{e_- E_0}{\hbar} \cdot \frac{|\mathbf{D}_{12}|}{\sqrt{3}} $$ (G.4) where $\mathbf{D}_{12}$ is the vector of the dipole moment of the considered transition and $E = E_0 \cos(\omega_0 t)$ the electric field of the exciting radiation. This equation assumes that the atoms are oriented randomly in space, which is usually the case in an atomic vapour. The physical interpretation of this quantity is the following. When a laser interacts with an atom, the probability of finding this atom in the upper state is (for an infinite upper-state lifetime) $$ \frac{1}{2} [1 - \cos(\Omega t)] $$ (G.5) When the laser radiation is detuned by a natural frequency offset $\Delta\omega$ from the line centre, the generalized Rabi frequency is $$ \Omega_R = [\Omega^2 + \Delta\omega^2]^{1/2} $$ (G.6) ## G.2 AC-Stark splitting and Mollow triplets Fundamental to power broadening and high-field effects is the ac-Stark effect, also --- [FIGURE: FIG. G.2. Sidebands in the spectrum due to ac-Stark splitting.] A somewhat more detailed picture can be obtained by visualizing the electric dipole moment of the atom as a vector. Due to the electric field of the radiation, the vector will perform a precession around its zero-field position, where the precession rate will of course depend on the field strength. The sidebands occur at frequencies $\pm\Omega_R$, see Fig. G.2. Quantitative results for the intensities and for the half-widths of the components must be computed strictly quantum-mechanically. The results are given below. We have stated in the main part of the book that for pure natural broadening in an atom at rest, the absorbed and reemitted frequencies must be the same. We have reasoned that this must be true because of energy conservation. For a strong field, this is no longer valid. Quantum-mechanical computations show that there is an elastic component, which has this complete frequency coherence, and there are inelastic components, which have Lorentzian lineshapes. The ratio of the intensity of the elastic component to the total intensity is $$ \frac{I_{\mathrm{coh}}}{I_{\mathrm{tot}}} = 2\frac{\kappa^2 + \Delta\nu^2}{\hat{\Omega}^2 + 2(\kappa^2 + \Delta\nu^2)} $$ (G.7) where $\kappa = A_{ul}/4\pi$ and $\hat{\Omega} = \Omega/2\pi$. In other words, when the laser radiation is very strong, most of the light is scattered inelastically. Quite generally, the emission spectrum is given by $$ 2\pi f2 \delta(\nu - \nu_0) + 2\mathrm{Re}\{f1(s)|_{s=-\mathrm{j}(\nu-\nu_0)}\} $$ (G.8) where $$ f2 = \hat{\Omega}^2 \frac{\kappa^2 + \Delta\nu^2}{[\hat{\Omega}^2 + 2(\kappa^2 + \Delta\nu^2)]^2} $$ (G.9) and $$ f1(s) = -\frac{f2}{s} + \frac{ \frac{\hat{\Omega}^2/4}{\Delta\nu^2+\kappa^2+\hat{\Omega}^2/2} [(s+2\kappa)(s+\kappa-\mathrm{j}\Delta\nu)+\hat{\Omega}^2/2] + \frac{\hat{\Omega}^2/4}{\Delta\nu^2+\kappa^2+\hat{\Omega}^2/2} (\kappa+\mathrm{j}\Delta\nu)(1+2\kappa/s)(s+\kappa-\mathrm{j}\Delta\nu) }{ (s+\kappa+\mathrm{j}\Delta\nu)[(s+2\kappa)(s+\kappa-\mathrm{j}\Delta\nu)+\hat{\Omega}^2/2] + (s+\kappa-\mathrm{j}\Delta\nu)\hat{\Omega}^2/2 } $$ (G.10) --- EFFECTS OF AC-STARK SPLITTING ON RADIATION TRAPPING For zero detuning and at high intensity, the emission spectrum simplifies to $$ 2\pi \left(\frac{\kappa}{\hat{\Omega}}\right)^2 \delta(\nu-\nu_0) + \frac{\kappa/2}{(\nu - \nu_0)^2 + \kappa^2} + \frac{3\kappa/8}{(\nu - \nu_0 - \hat{\Omega})^2 + 9\kappa^2/4} + \frac{3\kappa/8}{(\nu - \nu_0 + \hat{\Omega})^2 + 9\kappa^2/4} \quad \text{(G.11)} $$ The redistributed light comes in three components, one at the resonance frequency, plus two sidebands. | The ratios | | | | | | | :--- | :---: | :---: | :---: | :---: | :---: | | of the **peak intensities** are | $1/3$ | : | $1$ | : | $1/3$ | | of the **half-widths** are | $3/4 \cdot \Delta\nu^n$ | : | $1/2 \cdot \Delta\nu^n$ | : | $3/4 \cdot \Delta\nu^n$ | | of the **integrated intensities** are | $1/2$ | : | $1$ | : | $1/2$ | Experiments are in complete agreement with these theoretical predictions. Further discussions of the redistribution function at high intensities are given by Panteleev and Starostin (1992). Note that in typical trapping problems, many frequencies are present, which strongly modifies the nature of the redistribution. Bandwidth and coherence effects become important, so that the resulting redistribution is not simply the sum of three peaked spectra resulting from individual frequency components. At least qualitatively, we can describe the occurring effects also from a semiclassical point of view. The strong laser field leads to a splitting of the atomic states into upper and lower levels that have somewhat different decay time constants. These states are approximately symmetrical around the position of the undisturbed state. Another high-intensity effect, which however plays little role at optical frequencies, is the Bloch-Siegbert shift. This names a shift of the centre of the resonance frequency of a transition in a strong radiation field. However, this shift is only on the order of $\Omega^2/\nu_0$, so that it is typically only a few Hz for optical transitions and can thus be neglected. The high-field effects (modifications of the absorption and emission processes) also play an important role for degenerate four-wave mixing (Mulgan and Ballagh 1993). There are a lot of other phenomena, most of which are caused by coherence effects, like Rabi oscillations, self-induced transparency, photon echoes, and the like. These phenomena cannot be described by semi-classical pictures. Fortunately, in experiments where these effects occur, radiation trapping is usually non-existent, so they are of no further interest for us. ### G.3 Effects of AC-Stark splitting on radiation trapping As we have discussed above, strong radiation leads to a modification of the lineshape. The ac-Stark effect, for example, leads to a splitting of the line. This splitting has important consequences for the radiation trapping. First, the splitting leads to a reduction in the trapping factor. This is analogous to the usual hyperfine splitting – we simply have more frequency channels through which the radiation can escape. Second, the splitting occurs only as long as the laser is on, so that a modulation of the laser leads to a modulation of the trapping factor. For a third consequence, consider the situation sketched in Fig. G.3. The upper state is first populated, e.g. by a laser pulse. After that pulse, the state-$b$ population decays as in a usual radiation trapping problem. When we then apply a time-delayed laser pulse to --- HIGH-FIELD EFFECTS [FIGURE: FIG. G.3. Level scheme for a situation where a strong laser pulse on transition b - c leads to a splitting of state b.] the transition $bc$, states $b$ and $c$ are split into two substates each. We write the split states of level $b$ as $b_l$ and $b_u$. Now, transitions $b_l - a$ and $b_u - a$ are not resonant transitions, and cannot reabsorb, so that $b_l$ and $b_u$ are not trapped as long as the laser on transition $b - c$ is on. When this laser is finally switched off, the transition $b - a$ is trapped again, and the absorption coefficient suddenly increases (Fini and Mazzoni 1995). In Chapter 13, we have assumed that even when there is saturation, the ac-Stark splitting is not noticeable. This is not a contradiction, as might appear at first glance. The lineshape is mostly determined by Doppler broadening, so that the splitting $\Omega$ can be much smaller than the linewidth even if $\Omega > A_{ul}$, which is the requirement for noticeable saturation. The great bandwidth ensures that the simple transfer and rate equation is still valid (Cooper *et al.* 1982), and these equations still include power broadening and saturation as a valid manifestation of the high intensities. ## G.4 The validity of the transfer equation In Chapter 4, we have derived the equation of radiative transfer in a rather *ad hoc* manner, as a rate equation for the photons. This approach is very appealing and leads to intuitively clear results. The transfer equation was also derived quantum-mechanically by Cooper and Zoller (1984), see also Streater *et al.* (1988a) and Veklenko and Tkachuk (1975). These authors point out some limitations that have to be fulfilled if the transfer equation is to be valid. (i) The bandwidth of the radiation field must be much larger than the Einstein $A$ coefficient, which in turn must be larger than the decay rate of the intensity. This condition is usually fulfilled for experiments at room temperature, but might be violated for experiments with cold atoms. --- (ii) Furthermore, it is pointed out that all absorptions must be in the far field of the emitter. This condition can be violated when the particle density is very high. However, since only a small proportion of the photons in the near field can participate in a resonant photon exchange (Cooper and Stacey 1974), the problem is somewhat mitigated. (iii) The atoms must be located at random, so that no grating effects occur. This might be violated for lattices, where the atoms are held — e.g. by laser beams — in fixed positions that form a lattice. (iv) The bandwidth of the radiation must be small compared to the carrier frequency — this is always fulfilled for optical frequencies. The question of when the comparatively simple transfer equation can be used for the description of light and its interaction with atoms is one that is also covered in several books on quantum mechanics. However, it appears that at the moment, complete descriptions of the quantized fields are too complicated for practical computations.