Moment theory of electron thermalization in gases

Abstract

A time-dependent moment method for solving the Boltzmann equation, not restricted to elastic collisions, is applied to the description of the thermalization of a beam of electrons in a gas. An exact solution is also obtained for diffusion and velocity relaxation by elastic collisions at very short times, before appreciable energy is lost. These results are compared with a recent approximate theory of Mozumder for electron thermalization, using two model systems: the Maxwell model (constant collisions frequency), and the rigid-sphere interaction (constant collision cross section). All results are exact for the Maxwell model, but for rigid-sphere interactions the errors in the velocity and energy relaxation times from Mozumder’s method are approximately 25%. Many real systems are therefore probably described satisfactorily by the approximate theory, unless perhaps the cross sections have a peculiar energy dependence or inelastic collisions are important. If more accurate results are needed, the present moment method gives a systematic procedure for the calculation of higher-order approximations. Although the specific examples treated here consider only elastic collisions, the moment method applies to the case of inelastic collisions as well.