LightCell — sodium-vapour 819 nm extraction
This is the terminal layer. Everything that came before — Holstein eigenmodes, multi-mode expansions, PFR taxonomy, isotope-doping, saturation, NLTE iteration, 3D Monte Carlo — converges here, on a single engineering question:
Can a sodium-vapour combustion cell deliver a usable fraction of its input enthalpy as 819 nm photons matched to a GaAs photovoltaic cell?
The 819 nm channel is the 3d → 3p transition in atomic Na. Unlike the 589 nm D-lines (3p → 3s), whose lower state is the ground state and is therefore catastrophically optically thick at design pressures, the 819 nm line's lower state (3p) is Boltzmann-suppressed — at 3000 K its population is $\sim 10^{-2}$ of the ground state. This makes the 819 channel optically thin or moderately trapped where the 589 channel is deeply trapped, and that asymmetry is what makes the LightCell architecture viable in principle.
The framing — LTE-bath, not first-collision-branching
An earlier engineering ledger (ChatGPT deep-research) framed this as a first-collision branching problem: $\eta_{819}$ defined as the fraction of 3d-state excitations that radiate at 819 nm before being collisionally quenched. That framing is wrong as a steady-state observable. The operator corrected the math: "you can set the rate to infinite if the leaking out happens slowly downstream. Ultimately it's a cascade of energy states and there's gradients."
The correct framing: at high enough atom density, fast collisions enforce local thermodynamic equilibrium (LTE) on the entire Na manifold. The 3p and 3d populations follow Boltzmann at the local bath temperature $T_h$. The energy that arrives at the 3d level — whether by combustion-driven electronic excitation or by collisional re-promotion after a quench — leaves the system through one of several drains:
- γ819 (wanted): 3d → 3p radiation, escaping through optical depth $t_{819}$, partially reflected by the 819-window mirror, ultimately absorbed by the PV cell.
- γ589 (drained but recycled): 3p → 3s radiation, escaping through optical depth $t_{589}$, mostly reflected by the hot-side mirror back into the cell. Cavity-Q determines recycling efficiency.
- γIR (drained, partially recycled): thermal continuum radiation from the boundary at $T_b$, with H₂O and CO₂ band emission. Partly reflected by IR mirror.
- γwall (irreversible loss): conduction through ceramic boundary. Recuperator effectiveness ε determines how much is recovered.
- γexhaust (irreversible loss): hot combustion products leaving the cell carrying enthalpy.
\[ \eta_{819} = \frac{\gamma_{819}}{\gamma_{819} + \gamma_{589}^\text{net} + \gamma_\text{IR}^\text{net} + \gamma_\text{wall} + \gamma_\text{exhaust}} \]
The calculator
Na level populations (LTE at $T_h$)
$\eta_{819}$ vs $T_h$ at current settings
Loss-channel breakdown (Sankey-like)
Open question — Na(3d) molecular quenching
The LTE-bath framing assumes the Na(3d) state is in collisional equilibrium with the 3p state — meaning the Na(3d) collisional quenching rate $Q_{3d}$ by combustion-product molecules (N₂, H₂O, CO₂, CO, H₂, O₂) is fast enough to enforce LTE. M&O §3.3 gives a 4-decade range for molecular quenching cross-sections: 10−14 to 10−18 cm². Where Na(3d) sits in that range is not known from the open literature.
If $Q_{3d}$ is near 10−14 cm² (upper bound), $Q_{3d} \sim 10^9$ s⁻¹ — well above the radiative rate $A_{819}/t_{819} \sim 10^6$ s⁻¹ at the design point. LTE holds; the calculator above is valid. If $Q_{3d}$ is near 10−18 cm², $Q_{3d} \sim 10^5$ s⁻¹ — below the radiative rate. LTE breaks down; the 3d population becomes "transparent" to bath thermalisation and the calculator becomes optimistic.
→ The single experiment that gates LightCell design confidence: measure Na(3d) molecular quenching by N₂, H₂O, CO₂, CO, H₂, O₂ at 2500–4500 K combustion conditions. The Hooymayers-Alkemade tradition (1970s, Na 3p) provides the methodology; the 3d state has not been mapped.
How this layer composes from the seven below
- Layer 1 (Holstein eigenmodes) supplies the trapping factors $g_{589}, g_{819}$ as functions of the cell-geometry × lineshape × opacity. The "t_589" and "t_819" sliders here are the user-friendly inputs that Layer 1 derives from physical parameters.
- Layer 2 (modes) tells us we cannot use the escape-factor approximation in the gradient region — the excitation profile (combustion-driven, wall-localized at startup) doesn't project cleanly onto $\psi_0$.
- Layer 3 (PFR) sets the validity of CFR. At 1 atm with combustion-product buffer, $P_\text{coll} > 0.7$ — Payne's criterion is satisfied. CFR holds. ✓
- Layer 4 (multi-level + hfs) is where the Boltzmann ladder lives. Na has only one stable isotope so isotope-doping is unavailable; we rely on the Boltzmann-suppression of the 3p state at $T_h$.
- Layer 5 (saturation) governs whether the high-intensity 589-line trapping bleaches the ground state. At LightCell design intensities, only the very brightest spots saturate. Tracked but small.
- Layer 6 (NLTE/CRETIN) supplies the three-phase architecture: continuum (IR loss), line transfer (589 + 819), spectral diagnostic (matched to PV cell). Layer 8 is structured to mirror that architecture.
- Layer 7 (3D solver) tells us the 2D MC photon-tracing of LightCell's actual geometry will give the gradient-region escape rates more precisely than this slider model — but the design surface here is the right starting point.
What this calculator does NOT do
- Does not solve the full NLTE collisional-radiative system self-consistently. Uses Boltzmann + first-principles γ-rate model with operator-supplied couplings.
- Does not model the spatial gradient explicitly — single bath temperature, single set of trapping factors. Real LightCell has T(x) and t(x).
- Does not include molecular IR emission line-by-line (HITEMP). Approximated via $\sigma T^3$ scaling with effective emissivity.
- Does not include Saha-equilibrium ionization. Na has $\chi = 5.14$ eV, so at 4000 K Saha fraction can be %-level; not modelled.
- Cannot replace the actual Na(3d) quench measurement (see open question above).
FEYNMAN ✓ worked example shows operator's compounding-improvements logic · Victor ✓ eight sliders + live η big-number + composing diagram · Kay ✓ composes Layers 1–7 · Scott ✓ CRETIN three-phase architecture preserved in γ partition · Molisch ✓ M&O §3.3 quench-coefficient range cited; §11.1.7 Payne criterion satisfied for combustion-buffer case · Danielle ⏳ awaiting operator inspection — the terminal-authority sign-off