Comptonization of soft X-ray photons in an optically thin hot plasma

Comptonization of soft X-ray photons in a plane-parallel infinite plasma cloud is analytically solved. The escape probability from the plasma cloud and the energy distribution of single-scattered photons are derived analytically and the output Comptonized spectrum is obtained using the Mellin integral transformation. These solutions are compared with results of Monte Carlo simulations and their agreement is confirmed. The solution is particularly important when the optical thickness of the plasma cloud is small (tau of less than 3).