Fokker-Planck Calculation of the Electron Swarm Energy Distribution Function

Abstract

An approximate treatment of the Boltzmann collision integral for electrons in a gas, valid for small, fractional average energy loss per energy transfer collision, is presented and studied. It is essentially a Fokker- Planck expansion in energy space, including mean energy loss (dynamical friction) and energy straggling (coefficient of diffusion). When applied to electron swarms in weakly ionized gases, treating angle variables in the two-term Legendre series, there results a useful, physically meaningful, differential equation for the time evolution of the energy spectrum in a time-dependent electric field. Elastic scattering, and inelastic and super- elastic energy transfer collisions are included. It is valid for fields varying slowly compared with the swarm momentum-transfer collision frequency, but on any time scale relative to the energy-transfer collision frequency.