Saha Hydrogen Ionization Balance

Theoretical Background

The Saha ionization equation relates the ionization state of a gas to its temperature and density. For pure hydrogen in LTE, charge neutrality requires the electron density to equal the proton density, n_e = n_p = x * n_{total}. The statistical factor is simplified to 2U_p / U_H = 1. The equilibrium condition gives the ratio R = (2π * m_e * k_B * T / h²)³ᐟ² * exp(-13.6 eV / (k_B * T)) / n_total.

The ionization fraction x is found by solving the quadratic equation x² / (1 - x) = R, yielding the stable root x = 2√R / (√(R+4) + √R).

Worked Example

Consider a hydrogen gas at T = 10000 K and n_total = 10²² m⁻³.

1. Calculate the term 2π * m_e * k_B * T / h² ≈ 1.80 × 10¹⁷.
2. Raise this to the power of 3/2 and multiply by the exponential factor to find R = 0.03378794663084566.
3. Calculate x = 2√0.03378794663084566 / (√(0.03378794663084566+4) + √0.03378794663084566).
4. Result: Ionized fraction x ≈ 0.16769571481809095.

Interactive Experiment

Explore the temperature and density dependence of the ionization fraction in the interactive module.

1. Set Temperature to 10000 K and Density to 1e+22. Note the partial ionization.
2. Increase Density to 1e+24. The ionization fraction drops significantly as recombination is favored.
3. Increase Temperature to 20000 K. The gas becomes highly ionized despite the high density.

Limitations

This idealized model assumes Local Thermodynamic Equilibrium (LTE) and neglects excited atomic states, yielding a partition function ratio of 1. It also ignores pressure ionization at very high densities and non-ideal plasma effects. The T=0 case is included as a formal mathematical limit, not a physical regime where classical LTE holds.

References

The original formulation of the ionization equilibrium equation was introduced by Meghnad Saha. This module follows modern pedagogical presentations.

Baring, M. G. (2021). ASTR 350 Lecture Notes. Rice University. These lecture notes provide a detailed derivation of the Saha equation for hydrogen and discuss the assumptions involved in Local Thermodynamic Equilibrium.