Spin-exchange rates in electron–hydrogen collisions

The spin temperature of neutral hydrogen, which determines the 21-cm optical depth and brightness temperature, is set by the competition between radiative and collisional processes. In the high-redshift intergalactic medium, the dominant collisions are typically those between hydrogen atoms. However, collisions with electrons couple much more efficiently to the spin state of hydrogen than do collisions with other hydrogen atoms and thus become important once the ionized fraction exceeds ∼1 per cent. Here, we compute the rate at which electron–hydrogen collisions change the hydrogen spin. Previous calculations included only S-wave scattering and ignored resonances near the n= 2 threshold. We provide accurate results, including all partial wave terms through the F-wave, for the de-excitation rate at temperatures TK≲ 1.5 × 104 K; beyond that point, excitation to n≥ 2 hydrogen levels becomes significant. Accurate electron–hydrogen collision rates at higher temperatures are not necessary, because collisional excitation in this regime inevitably produces Lyα photons, which in turn dominate spin exchange when TK≳ 6200 K even in the absence of radiative sources. Our rates differ from previous calculations by several per cent over the temperature range of interest. We also consider some simple astrophysical examples where our spin de-excitation rates are useful.