Saha Equilibrium for Partial Ionization in Dense Sodium Vapor

Saha Equilibrium for Partial Ionization in Dense Sodium Vapor

The Saha equation is usually introduced as the bridge between the kinetic theory of gases and the spectroscopy of stars — the statement that, in thermal equilibrium, a gas decides for itself how many of its atoms shall be stripped of an electron. For dense sodium vapor it is a more delicate instrument than the textbook treatment lets on. The neutral density is enormous, the ionization energy is small ($E_i = 5.139\ \mathrm{eV}$, among the lowest of any stable element), and the free-electron population that results is the very thing that decides whether the vapor radiates as a transparent line emitter or as a trapped, quasi-Planckian slab. Get the electron density wrong by a factor of ten and the entire radiative budget moves.

This plate derives the Saha balance for sodium, evaluates it across the densities relevant to the cooperative-emission regime of 01-nlambda3-cooperative-threshold, and shows the two places where the equilibrium form quietly lies: at high density, where the continuum is depressed and degeneracy enters; and in any real discharge, where the electron temperature has detached from the gas temperature and Saha is no longer the operative balance at all.

The equilibrium statement

Saha's relation is the law of mass action applied to the reaction $\mathrm{Na} \rightleftharpoons \mathrm{Na}^+ + e^-$. Counting the translational partition function of the liberated electron — a free particle with $\Lambda_e^{-3}$ states per unit volume, where the thermal de Broglie wavelength is $\Lambda_e = h/\sqrt{2\pi m_e k_B T}$ — and the internal partition functions $g_i$ of the bound species, one obtains

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

The leading factor of $2$ is the spin degeneracy of the free electron; it is not optional, and it is the single most common factor-of-two error in the literature. For sodium the relevant statistical weights are $g_0 = 2$ for the $^2S_{1/2}$ ground state of the neutral and $g_i = 1$ for the closed-shell $\mathrm{Na}^+$ core, so the prefactor $2g_i/g_0 = 1$. Sodium is, in this one respect, the cleanest possible alkali: the degeneracy bookkeeping collapses to unity and Equation (1) reduces to

$$ \frac{n_e^2}{n_0} = \left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-E_i / k_B T}, \tag{2} $$

where charge neutrality $n_e = n_i$ has been imposed and $n_0$ is the neutral density, which in saturated vapor is set by the vapor-pressure curve through the solver na_saturated_density(T).

[!sidebar] The thermal wavelength is the whole story. The combination $(2\pi m_e k_B T / h^2)^{3/2} = \Lambda_e^{-3}$ is the quantum concentration of the electron — the density at which one electron's de Broglie volume holds exactly one electron. Saha is the statement that the ionized fraction is governed by the competition between this quantum concentration and the Boltzmann penalty $e^{-E_i/k_BT}$. At $T = 1000\ \mathrm{K}$, $\Lambda_e \approx 6.4\ \mathrm{nm}$ and the Boltzmann exponent is $E_i/k_BT \approx 59.6$, so $e^{-59.6} \approx 10^{-26}$. The ionization is exponentially, not algebraically, suppressed. [the figures in this sidebar are analytic, not solver-checked; treat as order-of-magnitude.]

What the numbers say for sodium

Take saturated sodium vapor at the low end of the cooperative regime. Plate 01 establishes that the cooperative-emission threshold for the Na D₂ line at $\lambda = 5.89\times10^{-5}\ \mathrm{cm}$ sits at

$$ n_{\rm crit} = \frac{64}{\lambda^3} \approx 3.13 \times 10^{14}\ \mathrm{cm}^{-3}, \tag{3} $$

In the flat notation the explorable consumes:

• λ = 5.89 × 10⁻⁵ cm
• n_crit = 64 / λ³ ≈ 3.13 × 10¹⁴ cm⁻³

which the explorable below recomputes live from cooperative_threshold(5.89e-5) and which saturated Na vapor reaches near 700–800 K (the solver na_saturated_density(700) returns a neutral density of the same order — see 01-nlambda3-cooperative-threshold). That is a neutral density. The free-electron density that Equation (2) returns at the same temperature is smaller by roughly thirteen orders of magnitude: the ionization fraction $x = n_e/n_0$ in saturated vapor near 1000 K is of order $10^{-11}$ [unverified — no Saha solver is exposed in the catalog; computed analytically from Eq. (2)].

This is the central tension of the dense-sodium program. The neutrals are dense enough to cross into the cooperative regime of 03-dicke-superradiance; the electrons are nowhere close. Whatever cooperative physics the vapor exhibits is therefore cooperative emission of neutrals, not a plasma collective effect — the free-electron Debye sphere is essentially empty. Saha's exponential is precisely what guarantees this separation of scales, and it is why the dense-vapor line emitter can be treated radiatively (Holstein, Dicke) without dragging in the full apparatus of plasma kinetics.

Where the equilibrium form breaks

1. Continuum lowering. Equation (1) treats the ionization energy $E_i$ as a fixed atomic constant. In a dense, partially ionized vapor it is not. The microfields of neighboring ions and electrons depress the continuum edge by $\Delta E_i$; the Stewart–Pyatt or simpler Debye–Hückel correction replaces $E_i \to E_i - \Delta E_i$, which raises the ionized fraction at fixed $T$. For sodium at the densities of interest the correction is small but it enters exponentially, and it is the first thing to check before trusting any ionization fraction below $\sim 10^{-9}$.

2. Degeneracy and the de Broglie crowd. The $\Lambda_e^{-3}$ factor assumes the electron gas is non-degenerate — that $n_e \Lambda_e^3 \ll 1$. In saturated sodium this is overwhelmingly satisfied for the electrons (they are far too sparse to be degenerate), but the same algebra applied to the neutrals is not safe: at $n_0 \gtrsim 10^{18}\ \mathrm{cm}^{-3}$ the neutral de Broglie volumes begin to overlap and the ideal-gas partition function used in Saha is no longer the right object. This is the boundary where Saha equilibrium and the cooperative-threshold of 01-nlambda3-cooperative-threshold are describing genuinely different physics.

3. Two temperatures. The deepest failure is not a correction term — it is the premise. Saha is an equilibrium statement, requiring the free electrons and the heavy gas to share a single Maxwellian temperature $T$. A high-pressure sodium lamp at a neutral temperature of order $1500\ \mathrm{K}$ runs an ionization fraction near $10^{-3}$, not the $\sim10^{-9}$ that Saha at $1500\ \mathrm{K}$ would demand. The discrepancy is not a Saha error: it is the signature of a two-temperature plasma, with an electron temperature $T_e$ driven far above the gas temperature $T_g$ by the applied field. Saha holds — but at $T_e$, not $T_g$, and the engineering of every working discharge takes pains to pry the two apart.

[!synthesis] This framing — that the many orders of magnitude between Saha-at-$T_g$ and the measured ionization of a working HPS lamp is diagnostic of
T1
rather than a failure of equilibrium statistics — is the author's combination of the standard Saha derivation (Mihalas) with the two-temperature discharge literature (Lieberman & Lichtenberg). No single cited source states it in this form for sodium specifically.

Why this matters for the radiative budget

The ionization fraction sets the free-electron density, and the free-electron density sets the electron-impact (collisional) broadening and the continuum (free–free, free–bound) contribution to the spectrum. In the cooperative regime where the neutral vapor emits on the D lines as a superradiant ensemble, the free-electron continuum is the principal parasitic channel — the photons that do not come out at 589 nm. Saha tells you exactly how much of that channel is open: in equilibrium at the gas temperature, almost none. The moment a discharge drives $T_e$ up, the continuum lights up and the line-to-continuum partition — the figure of merit for a chemiluminescent or pumped line source — degrades. The Saha equation is therefore not a curiosity here; it is the gate on how clean a line emitter dense sodium can be.

See also

• 01-nlambda3-cooperative-threshold — the cooperative-emission threshold; source of $n_{\rm crit}$ and the na_saturated_density neutral curve.
• 03-dicke-superradiance — the cooperative emission of neutrals that the empty Debye sphere makes possible.
• 02-holstein-radiation-trapping — the trapped-radiation regime the electrons never reach.
• 04-voigt-profile — the line shape whose collisional wing is set, in part, by the very electron density Saha governs.

References

• D. Mihalas, Stellar Atmospheres, 2nd ed., Ch. 5 — the canonical derivation of Equation (1) including the partition-function bookkeeping.
• M. N. Saha, "Ionization in the solar chromosphere," Phil. Mag. 40, 472 (1920) — the original.
• M. A. Lieberman & A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing — the two-temperature discharge physics behind failure mode (3).
• J. C. Stewart & K. D. Pyatt, "Lowering of Ionization Potentials in Plasmas," ApJ 144, 1203 (1966) — continuum lowering, failure mode (1).