Saha neutral fraction from a measured neutral density ($f_{\mathrm{HI}}=(1+\sqrt{S/n_{\mathrm{HI}}}{)}^{-1}$)

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.