The double Compton emissivity in a mildly relativistic thermal plasma   within the soft photon limit
  We provide simple and accurate analytic approximations for the low frequency double Compton emission coefficient that are applicable in a broad range of physical situations up to mildly relativistic temperatures. These approximations may be useful for checking in which circumstances the double Compton process is important. We perform series expansions of the double Compton emission integral for low energies of the incident photon and electron and compare the derived analytic expressions with the results obtained by numerical integrations of the full double Compton cross section. We explicitly derived analytic approximations to the low frequency double Compton emission coefficient for initial monochromatic photons and Wien spectra. We show that combining the analytic approximations given in this paper, an accuracy of better than 5% over a very broad range of temperatures and under various physical conditions can be achieved. The double Compton emissivity strongly depends on the ratio of the incoming photon and electron energies: for hard photons and cold electrons the emission is strongly suppressed compared to the case of similar photon and electron energy, whereas in the opposite situation, i.e. hot electrons and soft initial photons, the emission is enhanced. For photons and electrons close to thermodynamic equilibrium the double Compton emissivity increases less rapidly with temperature than in the Lightman-Thorne approximation and the corrections exceed ~10% for temperatures above 4 keV. 
J. Chluba, S.Yu. Sazonov and R.A. Sunyaev