Theory of the electron distribution function in a Lorentz gas at highE/n0

The stationary electron Boltzmann equation is solved for high reduced electric-field strengths E/${\mathit{n}}_0$. We consider both the cases of a homogeneous system and of a Townsend ionization avalanche (exponentially growing electron density). The usual calculation scheme based on the two-term spherical-harmonic expansion fails due to the formation of runaway electrons at high energies. To account for this effect, the velocity space is separated into different energy regions. For energies below the ‘‘runaway threshold’’ (v<${\mathit{v}}_{\mathit{R}}$) Boltzmann’s equation is solved by a modification of the usual calculation scheme. The expansion breaks down near the runaway threshold. For v>${\mathit{v}}_{\mathit{R}}$ we construct an alternative expansion that is suitable to describe the formation of a runaway beam. In the homogeneous case, the runaway flux excludes exactly stationary solutions. Under avalanche conditions, the runaway effect is ‘‘hidden’’ because the runaways follow a Maxwellian distribution. Nevertheless, it has a distinct influence on the ionization coefficient α. This is demonstrated with numerical results for a weakly ionized He plasma, where α decreases strongly with E for high E/${\mathit{n}}_0$ values.