Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 10E / Solution

Expand the relativistic energy as $ϵ(p)=m{c}^2+p^2/(2m)+O(p^4/(m^3{c}^2))$. In the dilute nonrelativistic regime the occupation denominator is dominated by its exponential, so the Fermi-Dirac distribution reduces to the Maxwell-Boltzmann distribution. The number density is thenThe assumed low temperature makes the contributing momenta nonrelativistic; the large positive $(m{c}^2-\mu )/(kT)$ makes occupation numbers small.

For chemical equilibrium of hydrogen ionization, ${\mu }_p+{\mu }_e={\mu }_H+{\mu }_{\gamma }$, and thermal photons have ${\mu }_{\gamma }=0$. Apply the same dilute-gas expression to protons, electrons and ground-state hydrogen. Their density ratio isAssume charge neutrality with no other important ionic species, so $n_p=n_e$; neglect the binding-energy mass correction in $m_p/m_H$, so this ratio is approximately one. Including proton and electron spin gives $g_p=g_e=2$ and ground-state $g_H=4$, hence $g_p{g}_e/g_H=1$. Equivalently one may consistently omit nuclear spin from both proton and hydrogen degeneracies. Thus the Saha equation isThe assumptions are thermal and chemical equilibrium at a common temperature, dilute nonrelativistic species, neutral total charge, predominantly ground-state hydrogen, and negligible interaction/partition-function corrections. The Maxwell-Boltzmann density formula applies to dilute hydrogen whether its composite spin makes it a boson or a fermion.