Saha Equilibrium in Dense Sodium Vapor

Saha equilibrium for partial ionization in dense sodium vapor

The load-bearing claim: at the temperatures where Na vapor goes cooperative — 700 to 1200 K — the thermal ionization fraction is exponentially small, yet a working high-pressure Na lamp runs at $x_e \sim 10^{-3}$. The discrepancy is not a Saha failure; Saha is fine. The discrepancy means the discharge is not in LTE with the neutral vapor. This plate explains why the equilibrium answer is the wrong question, and what the right question is.

The equation, stripped to its bones

For a one-stage ionization $\mathrm{Na} \rightleftharpoons \mathrm{Na}^+ + e^-$ at temperature $T$ in thermal equilibrium, the Saha equation reads

$$ \frac{n_e\, n_i}{n_0} \;=\; \frac{2\, g_i}{g_0}\left(\frac{m_e k_B T}{2\pi \hbar^2}\right)^{3/2} e^{-\chi/k_B T}, \tag{1} $$

where $\chi = 5.139\ \mathrm{eV}$ is the Na first-ionization potential, $g_0 = 2$ (Na ${}^2S_{1/2}$ ground state) and $g_i = 1$ (Na$^+$ ${}^1S_0$ closed shell). The factor of 2 in front is the electron spin degeneracy; the $(m_e k_B T / 2\pi \hbar^2)^{3/2}$ is the inverse thermal de Broglie volume — the same $1/\lambda_{\rm th}^3$ that controls quantum degeneracy and, when inverted on the photon side at the resonance line, controls the cooperative threshold of 01-nlambda3-cooperative-threshold.

[!aside] Why the $\lambda^3$ factor returns The atomic Saha factor and the photonic $N\lambda^3$ factor of the cooperative-emission threshold are not analogies — they are the same quantum phase-space accounting on two different particles. Saha asks: how many translational electron modes fit in a thermal volume? Plate 01 asks: how many photon modes fit in a resonance-line cube? Both answers determine when classical occupation-number reasoning breaks.

Define the ionization fraction $x_e \equiv n_e/(n_e + n_0)$ at total nuclear density $n_{\rm tot}$, with $n_e = n_i$ from charge neutrality. Then (1) becomes a quadratic in $x_e$,

$$ \frac{x_e^2}{1 - x_e} \;=\; \frac{1}{n_{\rm tot}}\left(\frac{m_e k_B T}{2\pi \hbar^2}\right)^{3/2} e^{-\chi/k_B T}. \tag{2} $$

The temperature dependence is dominated by the exponential. Plug in $T = 1000\ \mathrm{K}$: $k_B T \approx 0.0862\ \mathrm{eV}$, so $\chi/k_B T \approx 59.6$, and $e^{-\chi/k_B T} \approx 1.4 \times 10^{-26}$. The thermal prefactor is $(m_e k_B T / 2\pi\hbar^2)^{3/2} \approx 1.0 \times 10^{21}\ \mathrm{cm}^{-3}$.

A worked anchor

To anchor the discussion in a regime where the lamp engineers actually work, take saturated Na vapor at $T = 1000\ \mathrm{K}$. The Nesmeyanov / Honig vapor curve, exposed as the na_saturated_density solver, gives $n_{\rm tot} \approx 3 \times 10^{16}\ \mathrm{cm}^{-3}$ at this temperature. (See the explorable below; this number is computed live.)

Substituting into (2), in the dilute limit $x_e \ll 1$,

$$ x_e \;\approx\; \sqrt{\frac{1}{n_{\rm tot}}\left(\frac{m_e k_B T}{2\pi\hbar^2}\right)^{3/2}}\; e^{-\chi/2k_B T}. $$

For $T = 1000\ \mathrm{K}$ and $n_{\rm tot} = 3\times 10^{16}\ \mathrm{cm}^{-3}$ this gives $x_e \sim 7 \times 10^{-11}$ [unverified — no Saha solver is bound in this build; the figure follows from (2) by hand]. Eleven orders of magnitude below the $10^{-3}$ ionization a working HPS lamp sustains.

The conclusion is not that Saha is wrong; the conclusion is that the discharge plasma sits at a different electron temperature than the neutral vapor. In a high-pressure Na lamp, $T_e \approx 4000\ \mathrm{K}$ while $T_{\rm gas} \approx 1500\ \mathrm{K}$; the electron Maxwellian is the population that Saha needs to balance, and at $T_e = 4000\ \mathrm{K}$ the exponential factor $e^{-\chi/2 k_B T_e}$ leaps by twenty orders of magnitude. Two-temperature plasmas are the rule, not the exception, in this regime.

Where Saha is exactly right, and where it isn't

Saha (1) is derived from detailed balance under three assumptions:

1. Maxwell–Boltzmann electrons. A non-thermal tail in $f_e(\epsilon)$ above $\chi$ — exactly what an applied $E$-field produces — directly violates this.
2. Population equilibrium of all bound states. If radiative cascades drain population faster than collisions can refill it (the corona regime), the ground state is over-populated relative to Saha.
3. No radiation escape. Saha is the equilibrium statement of detailed balance; an optically thin medium leaks recombination radiation, and the ionization balance shifts toward higher $x_e$ at the same $T$.

In a dense Na vapor at the cooperative threshold of plate 01 ($n \sim 3\times 10^{14}\ \mathrm{cm}^{-3}$), assumption (3) breaks the opposite way: the resonance line is so trapped that radiative recombination photons are reabsorbed many times before escape. This pushes the system back toward the Saha prediction, in tension with the radiative escape that pushed it away. The two effects do not cancel cleanly; this is one of the corners where 02-holstein-radiation-trapping is doing more work than the equation suggests.

Two-level partition function — a sidebar on the degeneracy ratio

The $2 g_i / g_0 = 2 \cdot 1 / 2 = 1$ for Na looks accidental. It isn't. Na$^+$ is isoelectronic with neon; its ground state is closed-shell singlet ${}^1S_0$, $g_i = 1$. Na is ${}^2S_{1/2}$, $g_0 = 2$. The electron spin contributes the leading 2. The clean cancellation is a feature of every alkali, and it's why alkali Saha calculations are pedagogically the cleanest — there are no rotational, vibrational, or fine-structure traps to manage. Compare to molecular Saha for $\mathrm{H}_2$, where the rotational partition function alone is several pages of bookkeeping.

For Na, the next-electronic-state correction to $g_0$ is the ${}^2P_{3/2,1/2}$ pair at 2.10 eV — populated to $e^{-2.10/0.0862} \approx 3\times 10^{-11}$ at 1000 K, negligible for ionization balance but not negligible for the resonance-line opacity that 04-voigt-profile cares about.

Where this breaks

Saha breaks (or needs corrections) when:

• Two-temperature plasma: $T_e \neq T_{\rm gas}$, as in every real discharge.
• Non-Maxwellian
  T0
  : $E$-field drift tails, ionizing beams, photoionization by trapped resonance radiation.
• High density: above $n \sim 10^{18}\ \mathrm{cm}^{-3}$ the continuum lowering correction $\Delta\chi \sim (e^2/4\pi\epsilon_0) (4\pi n_e / 3)^{1/3}$ becomes a few percent of $\chi$ and shifts (1) materially. For Na at $n_e = 10^{18}\ \mathrm{cm}^{-3}$, $\Delta\chi \approx 0.08\ \mathrm{eV}$, a 1.5% correction — small, but it goes in the exponent.
• Radiation field present: the photoionization rate from trapped Na resonance photons can compete with collisional ionization in the cooperative regime of 03-dicke-superradiance, breaking detailed balance entirely.

Numerical anchor (solver-bound)

For Na D₂ at $\lambda = 5.89 \times 10^{-5}$ cm, the cooperative threshold of 01-nlambda3-cooperative-threshold sets $n_{\rm crit} = 64/\lambda^3 \approx 3.13 \times 10^{14}\ \mathrm{cm}^{-3}$. The explorable's cooperative_threshold(5.89e-5) binding recomputes this live and overlays it on the saturated-vapor curve from na_saturated_density. For convenient comparison with the canonical Na D₂ case, the live solver gives: - λ = 5.89 × 10⁻⁵ cm - λ³ = 2.04 × 10⁻¹³ cm³ - n_crit = 64 / λ³ ≈ 3.13 × 10¹⁴ cm⁻³

These are the same numbers tabulated in 01-nlambda3-cooperative-threshold; the explorable below recomputes them from cooperative_threshold(5.89e-5).

Saha's exponential suppression is what guarantees that even when $n_{\rm tot} \gg n_{\rm crit}$, the electron density remains far below it — so the cooperative regime of plate 01 is a cooperative regime of neutrals, not of free electrons.

Explorable

The explorable below sweeps temperature, computes the saturated Na density from the Nesmeyanov curve, and overlays it against the cooperative threshold of plate 01. The ionization fraction at the chosen $T$ is computed in the panel by hand from (2); it is not solver-bound (no Saha solver exists yet in /solvers) and is therefore labeled [unverified] in the readout.

## See also - 01-nlambda3-cooperative-threshold — the photonic counterpart of the Saha de Broglie volume - 02-holstein-radiation-trapping — what trapping does to the radiation-field assumption - 04-voigt-profile — the lineshape that controls how trapped recombination photons distribute - 03-dicke-superradiance — the regime where photoionization by the line itself competes with collisions

## Citations - M. N. Saha, "Ionization in the solar chromosphere," Phil. Mag. 40, 472 (1920). - D. R. Bates, A. E. Kingston, R. W. P. McWhirter, "Recombination between electrons and atomic ions," Proc. Roy. Soc. A 267, 297 (1962) — collisional-radiative corrections to Saha. - J. J. de Groot & J. A. J. M. van Vliet, The High-Pressure Sodium Lamp (Philips Tech. Library, 1986). - Y. B. Zel'dovich & Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, ch. III §6 (Dover, 2002 reprint).