Electron fugacity in the Saha equation ($z_e=e^{{\mu }_e/kT}$)

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.