Analysis of spherical harmonic expansion approximations for glow discharges

The use of spherical harmonic expansions to solve the Boltzmann equation for glow discharge systems is evaluated with finite element-based simulations of the time-dependent Boltzmann equation in three-phase dimensions (position, axial velocity, and radial speed). The anisotropy of the electron energy distribution function is investigated for two cases: the cathode fall of a direct current (DC) glow discharge, and a collisionless electron beam. For each case, the first ten coefficients of the spherical harmonic expansion are evaluated. The relatively high values of higher order coefficients show that the electrons are anisotropic in the cathode fall, especially close to the boundaries, and it is concluded that for the given conditions, the commonly used two-term approximations are not valid