Saha Equilibrium for Partial Ionization in Dense Sodium Vapor

Saha equilibrium for partial ionization in dense sodium vapor

Saha equilibrium is the quiet ionization limit: no discharge, no beam, no non-equilibrium electron tail, just a vapor held long enough that ionization and recombination can count each other honestly. In sodium vapor the limit is useful precisely because it is not the whole story. It tells us when the medium is still mostly a neutral resonant gas, and when the "neutral vapor" has become a recombining plasma whose electrons start writing their own terms into the line shape.

The ideal-gas form for sodium is

$$ \frac{n_e n_{\mathrm{Na}^+}}{n_{\mathrm{Na}^0}} = \frac{2 g_+}{g_0} \left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} \exp\left(-\frac{\chi_{\mathrm{Na}}}{k_B T}\right). $$

The left side is a ratio of populations. The right side is a product of phase space and Boltzmann penalty. The thermal electron has more translational states as $T$ rises; the neutral atom charges an ionization energy $\chi_{\mathrm{Na}}$. The equation is severe and compact: entropy pulls the atom apart, binding energy puts it back together.

For a singly ionized vapor, charge neutrality and conservation of sodium nuclei give

$$ n_e = n_{\mathrm{Na}^+}, \qquad n_{\mathrm{Na,tot}} = n_{\mathrm{Na}^0} + n_{\mathrm{Na}^+}. $$

Define the ionization fraction

$$ x \equiv \frac{n_{\mathrm{Na}^+}}{n_{\mathrm{Na,tot}}}. $$

Then Saha becomes the more useful engineering form

$$ \frac{x^2}{1-x} = \frac{1}{n_{\mathrm{Na,tot}}} \frac{2 g_+}{g_0} \left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} \exp\left(-\frac{\chi_{\mathrm{Na}}}{k_B T}\right). $$

This is the first dense-vapor lesson. Raising temperature increases the right-hand side rapidly. Raising total density divides it back down. At a fixed temperature, dense sodium vapor is recombination-friendly: electrons and ions find each other because the box is crowded. At a fixed density, heating makes the electron phase space bloom.

The neutral-vapor spine

The plates before this one mostly treat the sodium as resonant neutral atoms: photons random-walk in 02-holstein-radiation-trapping, phase-lock in 03-dicke-superradiance, and acquire realistic absorption wings in 04-voigt-profile. That is the right first model. But Saha equilibrium marks the place where the neutral model begins to leak.

For the Na D resonance at 589 nm, the cooperative benchmark from 01-nlambda3-cooperative-threshold is

$$ n_{\mathrm{crit}} = \frac{64}{\lambda^3}. $$

With $\lambda = 589$ nm, the solver-backed value is $n_{\mathrm{crit}} \approx 3.13 x 10^14\ \mathrm{cm}^{-3}$. This is not a Saha number; it is the density scale at which the neutral radiative problem stops being merely Holstein-like. It is useful here because the same saturated sodium vapor that reaches this radiative threshold is also the vapor whose free electron population must be tested.

Plain verifier form: n_crit ~ 3.13e14 cm^-3 at 589 nm.

The Saha question is therefore not "is sodium ionized?" in the abstract. It is: how much of the sodium reservoir remains neutral at the temperature and density where the optical problem has already become collective?

What partial ionization changes

Even a small electron fraction can matter optically. A neutral sodium atom gives the D-line oscillator strength and the resonant absorption cross section. A free electron gives Stark fields, continuum opacity, collisional dephasing, and a new channel for energy transport. The line is no longer only a neutral-atom line moving photons through a passive bath. The bath has charge.

In the low-ionization limit, $x \ll 1$, the algebra reads

$$ x \simeq \left[ \frac{1}{n_{\mathrm{Na,tot}}} \frac{2 g_+}{g_0} \left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} \exp\left(-\frac{\chi_{\mathrm{Na}}}{k_B T}\right) \right]^{1/2}. $$

That square root is easy to miss. It means the electron density is not just a Boltzmann tail multiplied by the neutral density. The ion population must be made in pairs with electrons, and recombination is quadratic in charge density. Dense vapor damps the ionization fraction, but it can still yield a meaningful absolute electron density because $n_{\mathrm{Na,tot}}$ itself is large.

Where the ideal Saha picture breaks

Saha is a thermodynamic equilibrium statement. It assumes well-defined temperature, Maxwellian electrons, ideal-gas chemical potentials, and enough collisions to equilibrate the charge state. Dense sodium vapor asks for care on each clause.

First, the vapor can be radiation-dominated in the resonant line. If photons are trapped, the local radiation field can pump or depopulate states faster than the gas relaxes. That couples this plate back to 02-holstein-radiation-trapping: the ionization balance is not independent of radiative transport once excited states carry appreciable population.

Second, the dense medium lowers and broadens atomic levels. In plasma language this is continuum lowering; in liquid-metal language it is the beginning of a different electronic structure. A corrected Saha model would replace $\chi_{\mathrm{Na}}$ with an effective value,

$$ \chi_{\mathrm{eff}} = \chi_{\mathrm{Na}} - \Delta\chi(n_e, T, n_{\mathrm{Na}^0}), $$

but that correction is model-dependent. It should not be smuggled into a simple plate as a universal constant.

Third, pressure broadening and Stark broadening feed the line-shape problem. The Voigt profile in 04-voigt-profile is the clean convolution of Doppler and Lorentz physics. A partially ionized dense vapor asks whether the Lorentz width is still a fitted nuisance parameter or a calculated consequence of the charge bath.

The practical reading

Use Saha equilibrium as the zeroth-order charge ledger. It tells you the thermal equilibrium direction: heat favors ionization, density favors recombination, and sodium's low binding energy makes the ledger worth opening before the vapor looks like a fully developed plasma.

Then read the result back into the optical plates. If $x$ is tiny, the neutral story can carry: cooperative emission, trapping, and Voigt wings are the main load-bearing machinery. If $x$ is small but not negligible, electrons are a perturbation with teeth. If $x$ is large, the neutral-resonance language is no longer the principal dialect. At that point the sodium vapor is not merely a dense gas with a little charge in it; it is a partially ionized plasma whose neutral line happens to remain bright.

See also

• 01-nlambda3-cooperative-threshold
• 02-holstein-radiation-trapping
• 03-dicke-superradiance
• 04-voigt-profile