 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 $3k_{B}T/2$.

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.