Saha ionization equation

Chemical equilibrium for $A^{+}+e^{-}\leftrightarrow A^0+\gamma$ givesafter neglecting the ion--atom mass difference in the translational prefactor.

For charge-neutral pure hydrogen in thermal equilibrium, write the Saha equation as $n_e{n}_p/n_{\mathrm{HI}}=S(T)$. A measured neutral number density fixes $n_e=n_p=\sqrt{S(T)n_{\mathrm{HI}}}$, so the neutral fraction is $f_{\mathrm{HI}}=n_{\mathrm{HI}}/(n_p+n_{\mathrm{HI}})=(1+\sqrt{S/n_{\mathrm{HI}}}{)}^{-1}$. At high ionization it scales as $\sqrt{n_{\mathrm{HI}}/S}$, rather than $n_{\mathrm{HI}}/S$. The latter mistake confuses measured neutral number density with total number density. Photoionization equilibrium with an external nonthermal field generally requires a rate equation instead of the Saha equation.

For classical nondegenerate electrons, the free-particle Boltzmann occupancy has the form $z_e\exp [-p^2/(2mkT)]$, with chemical potential measured relative to its rest energy. Counting the two spin states in phase space gives $n_e=2z_e(2\pi mkT/h^2{)}^{3/2}$. A bound level of energy $-I_n$ has relative occupancy $(g_n/g_{+})z_e{e}^{I_n/kT}$ against its parent ion. Eliminating $z_e$ gives the level-specific Saha ionization equation.

Because equilibrium reactions balance chemical potentials and photons have zero chemical potential,The rest-mass difference in the Maxwell--Boltzmann densities then supplies the Boltzmann factor $e^{-I_A/(k_{B}T)}$.

For $p+e^{-}\leftrightarrow H+\gamma$, the hydrogen binding energy is the rest-energy defect