Saha Equilibrium for Partial Ionization in Dense Sodium Vapor

Saha Equilibrium for Partial Ionization in Dense Sodium Vapor

The quick mistake is to treat dense sodium vapor as either neutral gas or plasma. Saha equilibrium says the interesting regime is between: neutral Na sets the opacity and line shape, while a small electron fraction can still steer conductivity, ambipolar fields, recombination light, and collisional broadening.

For the first ionization step,

$$ \frac{n_e n_{\mathrm{Na^+}}}{n_{\mathrm{Na}}} = \frac{2}{\lambda_e^3}\frac{g_+}{g_0}\exp\left(-\frac{\chi}{k_B T}\right), \qquad \lambda_e=\frac{h}{\sqrt{2\pi m_e k_B T}} . $$

With singly ionized sodium and no other charge carriers, $n_e=n_{\mathrm{Na^+}}=x n_{\rm tot}$ and $n_{\mathrm{Na}}=(1-x)n_{\rm tot}$. Thus

$$ \frac{x^2}{1-x}=\frac{S(T)}{n_{\rm tot}}, \qquad S(T)\equiv \frac{2}{\lambda_e^3}\frac{g_+}{g_0}\exp(-\chi/k_B T). $$

This is the central lever: Saha ionization is not only a temperature law. It is a temperature-over-density law. Raising $T$ opens the Boltzmann gate; raising neutral sodium density pushes the equilibrium back toward atoms.

Dense-vapor anchor

The sodium D-line density scale from 01-nlambda3-cooperative-threshold is already high by ordinary gas-discharge intuition. For Na D2 at 589 nm: - λ = 5.89 × 10⁻⁵ cm - n_crit = 64 / λ³ ≈ 3.13 × 10¹⁴ cm⁻³

This checked cooperative threshold, and the checked saturated sodium density at $700\,\mathrm{K}$ is about $3\times10^{14}\,\mathrm{cm^{-3}}$. Those numbers put saturated Na near the point where radiative transport stops being merely photon escape and starts borrowing coherence.

Saha, however, is still mostly closed there. Using $\chi=5.14\,\mathrm{eV}$ and order-unity statistical degeneracy [unverified], $S(700\,\mathrm{K})$ is tiny compared with the neutral density [unverified]. The vapor can be radiatively dense while remaining weakly ionized. That separation matters: 02-holstein-radiation-trapping and 03-dicke-superradiance talk to the resonant neutral population, not just to the free-electron density.

What changes in the dense case

Three corrections should be kept in view before turning the textbook Saha formula into an engineering oracle.

1. Continuum lowering. At high density, neighboring atoms and charges lower the effective ionization threshold. Replace $\chi$ by $\chi-\Delta\chi$ when microfield and excluded-volume physics become important. Even a modest $\Delta\chi$ is exponential leverage.
2. Nonideal activities. The mass-action form should really use activities, not bare number densities. Dense neutral sodium is not an ideal dilute gas.
3. Radiation and kinetics. If the optical field, wall recombination, or pulsed heating outruns collisional equilibration, Saha becomes a reference manifold rather than a guaranteed state.

Operating picture

Use Saha as the slow thermodynamic backbone and the existing plates as the fast radiative skeleton. The neutral density determines whether photons diffuse as Holstein eigenmodes or collapse into cooperative response; the ionized fraction determines whether the same vapor begins to behave electrically like a plasma. Dense sodium vapor is therefore a two-key system: one key is $n\lambda^3$, the other is $S(T)/n$.