= Electron-positron thermal pair abundance
Thermal electron-positron pairs in <chemical equilibrium> with <photons> have opposite <chemical potentials> when the one-particle <energy> includes rest <energy>. For negligible degeneracy and $k_BT\ll m_ec^2$, the zero-chemical-potential density per charge species is $n_0=2[m_ek_BT/(2\pi\hbar^2)]^{3/2}e^{-m_ec^2/(k_BT)}$. In that nondegenerate limit, <charge neutrality> and the classical equilibrium product $n_-n_+=n_0^2$ give $n_+=(\sqrt{n_b^2+4n_0^2}-n_b)/2$, with net <Electron> density $n_b$. Pairs are important when $n_0\gtrsim n_b$. At arbitrary <temperature> the zero-chemical-potential marker uses the integral $n_0=(\pi^2\hbar^3)^{-1}\int_0^\infty p^2[\exp(\sqrt{m_e^2c^4+p^2c^2}/k_BT)+1]^{-1}dp$. Its ultrarelativistic limit is $3\zeta(3)(k_BT/\hbar c)^3/(2\pi^2)$. A nonzero net <chemical potential> shifts the exact abundance boundary; the zero-potential curve is an approximate crossover marker.
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