Saha Equilibrium in Dense Sodium Vapor

Saha Equilibrium in Dense Sodium Vapor

The degree of ionization in a thermalized vapor is determined by the balance between collisional ionization and three-body recombination. For a monatomic gas like sodium, this balance is expressed by the Saha equation:

$$\frac{n_e n_i}{n_n} = \frac{2}{\lambda_e^3} \frac{g_i}{g_n} \exp\left(-\frac{\epsilon_i}{k_B T}\right)$$

where $\lambda_e = \frac{h}{\sqrt{2\pi m_e k_B T}}$ is the thermal de Broglie wavelength of the electron.

Dense Vapor Context

In the context of 01-nlambda3-cooperative-threshold, we often consider the vapor as a collection of neutral emitters. For Na D₂ at 589 nm: - n_crit ≈ 3.13 × 10¹⁴ cm⁻³

However, as the temperature increases to reach this threshold, the population of free electrons $n_e$ grows. For sodium, the ionization energy $\epsilon_i$ is $5.139$ eV.

At $T = 1000$ K, the saturated number density $n_{total} \approx 3.0 \times 10^{16}$ cm⁻³ is well above the cooperative threshold. The presence of free electrons can significantly alter the 04-voigt-profile via Stark broadening and influence the transition from 02-holstein-radiation-trapping to collective states.

Regimes of Ionization

While 03-dicke-superradiance focuses on the coherent internal states of the atoms, the Saha equilibrium reminds us that the "atomic" nature of the medium is itself a function of the local thermodynamic state. In extreme density regimes, pressure ionization may further lower the effective $\epsilon_i$.

## See also - 01-nlambda3-cooperative-threshold - 02-holstein-radiation-trapping - 03-dicke-superradiance - 04-voigt-profile