Determination of macroscopic quantities for a Townsend discharge in helium by both Boltzmann equation and Monte Carlo methods

Abstract

We compare the results obtained in the study of a weakly ionized gas for a Townsend discharge which is developped in a steady uniform field. This study has been carried out in two laboratories: firstly at the Centre de Physique Atomique de Toulouse using Boltzmann equation (1) and secondly in Paris (Ecole Supe'rieure dl~ lectricitk) by Monte Carlo simulation. A lot of papers has been devoted to the microscopic study of weakly ionized gases in steady uniform fields. However, most of them deal with homogeneous states, which are attained far from the walls ie when the electrons are in equilibrium with the electric field. In many cases too, these studies deal with low values of the ratio electric field upon pressure (E/P); we then have a great number of collisions, and the mean free paths are much smaller than the gap length. For high values of E/P, when the mean free paths are almost as great as the gap length, the methods of resolution of the Boltzmann equation normally used are no more valid. For such a case we developped a new technique based on the collision probability method well known in neutron physics. The method consists in a splitting of the phase space in spherical shells. The distribution function is assumed constant in each shell. This generates a linear system of equations in which the elements P. of the