Radiation trapping, linearized:
the slope is the survival

Every figure below is live. Every measured number came from a solver whose gates prove it departs from equilibrium only when physics permits — receipts at the end.

I · A photon in a sodium fog

Fill a tube with sodium vapor and excite one atom. It emits a yellow photon — which travels, at flame density, a few microns before another sodium atom swallows it. That atom may re-emit it (the photon lives, direction scrambled) or collide with a bystander first and hand the energy to heat (the photon dies). One number governs everything that follows: the survival probability per capture,

Drag the survival. Watch one photon try to leave.

II · The bathtub identity — true by construction

Call the trapping factor g the ratio by which the tube's light output is slowed compared to a transparent tube at the same excitation. Then the escape rate per stored quantum is A/g, and steady state forces, with no further physics,

This identity cannot be argued with, and isn't where the argument lives.

III · The clamp — and the one-line theorem

Hold the pump fixed instead (the fuel), and let the bath compete. Each stored quantum now leaks two ways — escape at A/g, quench at k_Q — so the inventory is N = R/(A/g + k_Q). Differentiate logarithmically and the entire dispute collapses to one line:

Where photons usually survive to escape (β→1): occupancy rides trapping linearly — the bathtub regime. Where the bath usually wins (β→0): occupancy is clamped to local Boltzmann and trapping only recycles energy through heat. The linearization around any operating point g₀ is honest exactly to the extent that β(g₀) is close to 1 — the slope is the survival.

IV · The mirror made of sodium — and what a mirror is

Sodium vapor is perfectly opaque at its own wavelength: a layer of it is an extreme mirror for sodium light. But opacity is not reflectivity. Each capture inside the layer rolls the ω die — so

The solver measured both faces of this. At 1 atm (ω = 0.04), making the column 100× more opaque lowered the wall occupancy — more sodium meant more funerals. At 0.01 atm (ω = 0.81), the sodium mirror is real — and a second truth appeared: a mirror co-located with its source shields the very atoms it should pump.

Measured, gated solver, fuel-driven, temperatures resolved. The largest departure from Boltzmann ever produced in this model — 4.77× — came from the THINNEST column: flooded from behind a good mirror, not buried inside a bad one.

V · Quench engineering — the actual discipline

Since β decides everything, the design axis is not pressure, not even the mirror — it is who is allowed to collide with an excited sodium atom. Water and CO₂ are voracious quenchers; nitrogen drains to vibrations; noble gases barely touch the excitation. The solver, holding full atmospheric density and swapping the bath:

The low-pressure float, recovered at one hundred times the density. Burn where burning is efficient; radiate where every collision is chosen.

One refinement the ledger forced on us: the 819 nm extraction channel is fed by collisions (3p→3d), so indiscriminate quench-killing starves the tap while filling the reservoir. The precision target is state-selective: suppress k(3p→heat), keep k(3p↔3d). The collision partner table is the design table.

VI · The design point — peak electricity at the self-reversal of 819

The 819 nm line is the machine's tap: it escapes the sodium mirror freely and lands on GaAs. Its absorber is the 3p state — the very state the trap fills. So as the reservoir is driven, the cool skin begins to re-absorb 819 at line center; the line saturates its window precisely when

Below this point the window is open and undersold; at it, the line emits at its source-function ceiling with nothing recycled; beyond it, the self-reversal dip hands power back. The laboratory device measures τ_819,skin ≈ 0.004 — two hundred and fifty times below onset — and its spectrometer agrees: the measured 819 line escapes peaked. The design point lives at the metre-class machine. The small machine measures its own scaling law.

Nomenclature — every symbol, defined (LaTeX beneath each)

A

Einstein spontaneous-emission coefficient of Na(3p→3s), A = 6.16×10⁷ s⁻¹ [NIST].A \equiv A_{3p\to3s} = 6.16\times10^{7}\,\mathrm{s^{-1}}

k_Q

Collisional quench rate of Na(3p) — electronic excitation handed to the heat bath; set by the collision-partner mix and density (the engineered quantity). Flame gas at 1 atm: ≈1.4×10⁹ s⁻¹ [flagged, lit pull E09].k_Q\;[\mathrm{s^{-1}}]

ω

Single-capture survival probability (single-scattering albedo): the chance a captured photon is re-emitted rather than quenched. Range (0,1); ω→1 as k_Q→0; no divergence at 1.\omega \equiv \dfrac{A}{A+k_Q}

g

Trapping factor, output-defined: how many times slower the column releases light than an optically thin column at the same excitation. Equivalently the residence-time multiplier of a stored quantum.g \equiv \dfrac{P_{\mathrm{thin}}}{P_{\mathrm{out}}} = \dfrac{\tau_{\mathrm{res}}}{1/A}

g₀

Operating-point trapping factor — the anchor of the linearization; the tangent drawn at g₀ in Plate III.g_0

β

Escape branching ratio per stored quantum: probability the next exit from storage is an escaping photon rather than quench. THE theorem: the log-slope of occupancy with trapping equals β. Note β(g{=}1)=ω; β = ω/(ω + g(1−ω)).\beta \equiv \dfrac{A/g}{A/g+k_Q},\qquad \dfrac{d\ln N}{d\ln g}=\beta

N

Stored excitation inventory of the 3p ⊕ 589-photon reservoir (count). At fixed pump: N = R/(A/g + k_Q).N = \dfrac{R}{A/g + k_Q}

R

Pump rate into the reservoir (excitations s⁻¹): chemical (Padley–Sugden, radical overshoot Γ), thermal (k_Q f₃ₚ n₃ₛ — the bath, which by detailed balance can never float itself), or external drive.R\;[\mathrm{s^{-1}}]

P_out, hν

Escaping radiant power in the band; photon energy (589 nm → hν = 2.104 eV).P_{\mathrm{out}} = N\,\dfrac{A}{g}\,h\nu

float

Occupancy divergence ratio: n₃ₚ/n₃ₛ measured against the LOCAL kinetic-temperature Boltzmann ratio. 1.000 = clamped; the gated solver's headline observable.\mathrm{float} \equiv \dfrac{n_{3p}/n_{3s}}{(g_1/g_0)\,e^{-E_{3p}/k_BT_{\mathrm{kin}}}}

f₃ₚ(T)

Boltzmann factor of the 3p level at temperature T (statistical weights 6/2 = 3).f_{3p}(T) = 3\,e^{-2.104\,\mathrm{eV}/k_BT}

τ, τ·ω

Line-center optical depth of a sodium layer; the mirror law: opacity says light cannot pass, only survival (ω) says it comes back — a mirror is the product.\tau(\nu) = \int \kappa_\nu\,ds; \qquad \mathrm{mirror} = \tau\times\omega

τ₈₁₉,skin

Optical depth of the 819 nm tap line through the cool skin (the self-reversal budget). THE DESIGN POINT: drive every lever until this product reaches one — peak 819 electricity at self-reversal onset.\tau_{819,\mathrm{skin}} = \sigma_{819}(T_w)\;n_{3p}^{\mathrm{skin}}\;\Delta r_{\mathrm{skin}} \;\overset{!}{=}\;1,\quad n_{3p}^{\mathrm{skin}} = \mathrm{float}\cdot f_{3p}(T_w)\cdot n_{\mathrm{Na}}^{\mathrm{skin}}

σ₈₁₉(T_w), Δr_skin, T_w, n_Na

819 nm line-center absorption cross-section at wall temperature; skin thickness; wick-pinned (→ heat-balance-resolved) wall temperature; free sodium density (salt pressure × speciation).\sigma_{819}(T_w),\;\Delta r_{\mathrm{skin}},\;T_w,\;n_{\mathrm{Na}}

Click any def › tag on an equation to land here; each entry carries its LaTeX source for lifting into the paper.

VII · Provenance

Every population in these results comes from a collisional-radiative cascade that must pass four gates before it may speak: it provably collapses to Saha–Boltzmann on an equilibrated bath (R8a, |b−1| = 2×10⁻¹⁰), provably departs at low density (R8b), bridges monotonically (R8c), and provably pumps to a driven reservoir's chemical potential rather than the kinetic temperature (R8d — added the night the model was caught smuggling Boltzmann in, twice, and corrected in public). Trapping is carried by a Λ-iterated radiation field — no escape-factor knob exists. Temperatures are resolved from the energy balance, closure ≤ 0.01%. Corrections are published as dated revisions, never rewrites.