A framework for modeling the cathode fall illustrated with a single beam model

Abstract

A framework for a model of the cathode fall region of a dc glow discharge is presented, and a simple model is solved as an illustration. An extremum condition independent of the model is placed on the electric field behavior to produce a unique solution that agrees with experiment. The zeroth and second moments of the Boltzmann equation are solved for the electrons with a self-consistent electric field. A single-beam model with only two parameters (number density and beam velocity) is assumed for the electron distribution function. Ion motion is modeled with a parametric fit to known ion mobilities. The model is solved for conditions corresponding to the experimental results and to Monte Carlo simulations of Doughty, Den Hartog, and Lawler [Phys. Rev. Lett. 58, 2668 (1987)]. The results are in good qualitative and ‘‘factor-of-two’’ quantitative agreement with the published results.