Runaway Electrons in an Ideal Lorentz Plasma

Abstract

The phenomenon of electron runaway under a constant electric field is treated in the simplified model of a Lorentz gas, in which the electrons experience Coulomb collisions with infinitely massive ions only. This model permits a rigorous systematic analysis of the problem, devoid of arbitrary elements, in the limit of weak electric field. The analysis leads to a decomposition of velocity space into three regions. In the first of these, the low velocity domain, the electron distribution function is dominated by collisions and hence almost isotropic; it obeys a diffusion equation driven by the applied field. The second region, one of intermediate velocity, is characterized by quasi-steady flow in velocity space, for which the low velocity region provides the source. Lastly there is a high velocity region, fed by the intermediate region, in which the electrons run away almost freely under the action of the electric field, with only an exceedingly weak diffusion due to collisions. The problem is solved explicitly in the first and third regions and reduced to solving a single ``universal'' partial differential equation in the second region. The runaway rate, number of runaway electrons, and electric current density (average electron velocity) are obtained explicitly.