Saha Equation

The Saha equation for ionization equilibrium may be written

$$\frac{N_+}{N_0} = A (kT)^{3/2} N_e^{-1} \exp(-\chi_0/kT)$$

where

• $N_+$ is the number of ions per unit volume
• $N_0$ is the number of neutral atoms per unit volume
• $N_e$ is the number of electrons per unit volume

• $T$ is the temperature in Kelvin
• $k$ is the Boltzmann constant
• $\chi_0$ is the ionization energy for the neutral atom
• $A$ is a constant given by the following

$$A = \frac{Q_+}{Q_0} \;\; \frac{2\,(2\pi m_e )^{3/2}}{h^3}$$

where $Q$ is a partition function which normalizes a distribution among all the states of the neutral or ion. For hydrogen

$$\frac{Q_+}{Q_0} = \frac{1}{2}$$

Substituting for $m_e$ and $h$ in SI units we have

$$A = \frac{(2\pi \; 9.1096\times10^{-31})^{3/2}}{(6.6262\times10^{-34})^3}$$

$$A = 4.70679 \times 10^{+55}$$

$$\log_{10} A = +55.6727$$