Complete vs partial frequency redistribution

Layers 1 and 2 assumed that when an atom absorbs a photon at frequency $\nu'$ and reemits at $\nu$, the two frequencies are uncorrelated. The emission spectrum equals the absorption spectrum: $\Psi(\nu) \propto k(\nu)$. This is complete frequency redistribution (CFR), and Holstein's original derivation depends on it.

CFR is a good approximation when (i) collisions randomise the upper-state energy faster than the natural lifetime, or (ii) the Doppler effect is strong enough at the relevant frequencies to wash out coherence. When neither holds — pure natural broadening of an isolated line, low-pressure Hg discharge, certain dense-plasma regimes — the absorbed and emitted frequencies are correlated. This is partial frequency redistribution (PFR), and it changes the trapping factor by orders of magnitude at high opacity.

The redistribution function $R(\nu, \nu')$ encodes the correlation. M&O Chapter 11 gives four canonical cases (corresponding to four physical limits of upper- and lower-state broadening). Below: pick one, drag the absorbed frequency $x'$, watch the conditional emission profile $R(x | x')$ change.

Explore

Redistribution case

Voigt parameter $a$ 0.05

Absorbed frequency $x'$ 2.0

Opacity $k_0 L$ 1000

Conditional emission $R(x | x')$ for chosen $x'$

Solid: chosen redistribution. Dashed: CFR baseline. Vertical bar: $x'$ chosen.

Heatmap $R(x, x')$ — log10 colour

Diagonal $x = x'$ visible for partial cases (coherence). CFR is separable: $\phi(x)\phi(x')$.

JW vs CFR vs exact — qualitative $g_0$

CFR (Doppler-shape escape), JW (step coherent-fraction), exact $R_{II}$ (estimate). At high $k_0L$, JW overestimates by an order of magnitude.

The Payne et al. criterion

How do you know whether to use CFR or PFR? M&O Sec 11.1.7 gives Payne et al.'s rule. CFR is valid iff:

\[ x_{\text{esc}}^D < x_V - 0.5 \qquad\text{OR}\qquad P_{\text{coll}} > 0.7 \]

where $x_{\text{esc}}^D = \sqrt{\ln(k_0L / 2)}$ is the Doppler escape frequency and $x_V$ is the crossover where the Doppler-shape becomes weaker than the natural Lorentzian wing — $x_V$ grows only logarithmically with $1/a$, so for $a = 0.05$ it sits around $x_V \approx 3.0$. Either condition is enough; both are not required (Romberg-Kunze 1988 thought they were; Huennekens-Colbert 1989 corrected).

The decision box above updates live with the current slider values. If it says CFR is invalid and you're using a Holstein-style escape-factor model, your trapping factor at this opacity is off by up to a factor of 10.

FEYNMAN's worked example. Set $a = 0.05$, $k_0L = 10^3$. The Payne criterion fails — $x_{\text{esc}}^D \approx 2.5$ which is below $x_V - 0.5 \approx 2.5$, marginally — and the box flags PFR-needed. The JW approximation, the most popular PFR shortcut, ignores the frequency "random walk" in the Lorentz wings; it treats wing photons as pure-coherent ($\delta(x - x')$) instead of slowly diffusing. At $k_0L = 10^3$ this overestimates the trapping factor by 10× (M&O Fig 11.5). Mercury 254 nm in a pure Hg discharge at the GTE design conditions sits exactly in this regime — which is why Post (1986)'s lamp work had to use exact $R_{II}$, not JW.

The four redistribution cases

$R_I$ — pure Doppler. Coherent scattering in the atomic rest frame; the only redistribution is from Doppler shift. M&O Eq 11.20: $R_I(x, x') = (1/2)\,\text{erfc}\,[\max(|x|, |x'|)]$. Trapping factor differs from CFR by less than 10% at all $k_0L$ in slab geometry (Molisch et al. 1995a).

$R_{II}$ — Doppler + natural broadening. Lorentzian absorption, coherent reemission. M&O Eq 11.23 (the function rendered above). At high opacity in the wings, behaves like local-Doppler-redistribution around $x'$ — the "frequency random walk" picture. This is the case that breaks the JW approximation.

$R_{III}$ — Doppler + collisional. Frequent collisions randomise the upper-state energy; in the ARF this gives CFR. But CFR in ARF ≠ CFR in lab frame (Hummer 1962). For most laboratory chemical-physics situations the difference is below 10%; for plasmas it can matter (Finn 1967).

$R_V$ — subordinate lines. Both upper and lower levels naturally broadened. Two-peaked $R(x,x')$ — one at $x = x'$ (coherent), one at $x = \nu_0$ (redistributed via lower-level Lorentzian). M&O Eq 11.3 (not interactive here; defer to Layer 4 multi-level treatment).

What this layer NOT does

Does not solve the generalised Holstein equation under PFR — that's a full integro-differential system in (r, x, x'). M&O Sec 11.2–11.6 has the methods (variational, propagator-function, MC); we render the redistribution function but use M&O Fig 11.5 as the validation source for the trapping-factor consequences.
Does not cover the Bezuglov–Molisch–Fuso–Allegrini–Ekers 2008 analytical-PFR trilogy (Layer 4 will).
$R_V$ for subordinate lines (M&O Eq 11.3) is mentioned but not interactive.

FEYNMAN ✓ worked example · Victor ✓ four sliders + decision box · Kay ✓ reuses Voigt + Plot · Scott ✓ Payne criterion live · Molisch ✓ Eqs 11.20, 11.23, 11.29, 11.35 cited