= Recombination cooling of ionized hydrogen
In an <optically thin> <plasma>, <radiative recombination> removes the <kinetic energy> of captured <electrons>. A <radiative recombination> <photon> can also carry the <binding energy> released when a free <electron> becomes bound; that part is stored <ionization energy>, rather than newly removed <thermal energy>. The kinetic contribution per event is the <recombination-weighted electron energy>, which need not be $3k_BT/2$.
Equality with mean <photoelectron> heating requires two balances. If <photoionization> is the only ionization and heating process, and <radiative recombination> the only recombination and cooling process, <photoionization equilibrium> equates their event rates. Separately imposing <thermal equilibrium> then equates the mean heating energy per ionization to the mean kinetic cooling energy per recombination. <Photoionization equilibrium> alone does not do this: at the same balanced event rate, unequal energies can still heat or cool the gas. Additional <atomic line cooling>, <collisional ionization> or trapping of emitted <photons> changes the energy accounting.
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