Thermal free-free emissivity ($j_{\nu }\propto Z^2{n}_e{n}_i{T}^{-1/2}{e}^{-h\nu /(k_{B}T)}{\bar{g}}_{\mathrm{ff}}$)

The emission coefficient per volume, frequency and solid angle of a Maxwellian plasma has $j_{\nu }\propto n_e{\sum }_i{Z}_i^2{n}_i{T}^{-1/2}{e}^{-h\nu /(k_{B}T)}{\bar{g}}_{\mathrm{ff}}$. Define the angular convention before supplying a numerical coefficient. The high-frequency cutoff measures temperature, while the normalization measures a density-squared emission integral. The radio spectrum is almost flat when optically thin and rises as ${\nu }^2$ when optically thick at fixed projected area.

For fully ionized hydrogen and helium with hydrogen mass fraction $x_H$, $n_e=(1+x_H)\rho /(2m_p)$ and ${\sum }_i{Z}_i^2{n}_i=\rho /m_p$. With the same unit Gaunt factor for both species, thermal bremsstrahlung at fixed electron number density is multiplied by $2/(1+x_H)$ relative to pure ionized hydrogen. The factor is one for pure hydrogen and two for pure helium.