A uniform Wentzel–Kramers–Brillouin approach to electron transport in molecular gases

Abstract

The relaxation of a nonequilibrium distribution of electrons in molecular gases, and the transient and steady electron transport properties are studied with the Boltzmann equation, which accurately accounts for elastic and inelastic electron–moderator collisions. The elastic collision operator is a self-adjoint Fokker–Planck operator, whereas the inelastic collision term is a difference operator. We consider a discretization of the inelastic collision operator which is motivated by the discrete energy losses that occur. For small energy losses, a continuous approximation is introduced such that the inelastic collision term is approximated by a Fokker–Planck operator similar to the elastic collision operator. The transient electron transport properties are studied for an initial electron distribution function in terms of the eigenvalue spectrum of both the elastic and inelastic collision operators. The Fokker–Planck eigenvalue problem is transformed into a Schrödinger equation and the nature of the spectrum is studied in terms of the Wentzel–Kramers–Brillouin and supersymmetric Wentzel–Kramers–Brillouin approximations. The methodology is applied to model systems and the transport of electrons in methane.