Saha Equation

The Saha equation describes the ionization equilibrium of a gas in thermal equilibrium. It is fundamental to understanding how plasmas form from neutral gases as temperature rises.

Ionization Equilibrium

For hydrogen, the Saha equation relates the number densities of electrons (n_e), protons (n_p), and neutral atoms (n_H):

$\frac{n_e \cdot n_p}{n_H} = \frac{(2\pi m_e k_B T)^{3/2}}{h^3} \cdot 2 \cdot e^{-\chi / (k_B T)}$

where:

• (m_e = 9.109 \times 10^{-31}) kg is the electron mass
• (k_B = 1.381 \times 10^{-23}) J/K is Boltzmann's constant
• (h = 6.626 \times 10^{-34}) J·s is Planck's constant
• (\chi = 13.6) eV (= 2.179 \times 10^{-18}) J is the hydrogen ionization energy

The factor of 2 accounts for the two spin states of the electron.

Ionization Fraction

For a plasma with total number density (n_{\text{total}}), define the ionization fraction (x = n_e / n_{\text{total}}). Since (n_e = n_p = x \cdot n) and (n_H = (1-x) \cdot n):

$\frac{x^2 n}{1 - x} = S(T)$

where (S(T)) is the right-hand side of the Saha equation. This rearranges to the quadratic:

$n x^2 + S x - S = 0$

Solving with the quadratic formula (taking the positive root) gives the ionization fraction at any temperature and density.

Physical Interpretation

• At low temperatures ((T \ll 10{,}000) K for hydrogen), most atoms are neutral: (x \approx 0)
• At temperatures above ~15,000 K, hydrogen becomes nearly fully ionized: (x \approx 1)
• The transition is sharp due to the exponential dependence on temperature