The radiation amplification effect in the scattering of a quasi-classical electron by an ion in the weak electromagnetic field

Possibility of amplification of electromagnetic radiation is theoretically studied in the scattering of a quasi-classical electron by an ion in the field of weak linearly polarized wave. Using the limiting process of the total cross-section at small values of adiabaticity parameter to Krainov's formula, the symmetrization of Berson's formula was performed. Expression for the total cross-section (the gain coefficient) for the wide interval of values of adiabaticity parameter was obtained. It was shown, that the wave amplification takes place in the range of values of adiabaticity parameter more than two and can be sufficiently large