Self-contained solution to the spatially inhomogeneous electron Boltzmann equation in a cylindrical plasma positive column

Abstract

In this paper we develop a self-contained formulation to solve the steady-state spatially inhomogeneous electron Boltzmann equation (EBE) in a plasma positive column, taking into account the spatial gradient and the space-charge field terms. The problem is solved in cylindrical geometry using the classical two-term approximation, with appropriate boundary conditions for the electron velocity distribution function, especially at the tube wall. A condition for the microscopic radial flux of electrons at the wall is deduced, and a detailed analysis of some limiting situations is carried out. The present formulation is self-contained in the sense that the electron particle balance equation is exactly satisfied, that is, the ionization rate exactly compensates for the electron loss rate to the wall. This condition yields a relationship between the applied maintaining field and the gas pressure, termed the discharge characteristic, which is obtained as an eigenvalue solution to the problem. By solving the EBE we directly obtain the isotropic and the anisotropic components of the electron distribution function (EDF), from which we deduce the radial distributions of all relevant macroscopic quantities: electron density, electron transport parameters and rate coefficients for excitation and ionization, and electron power transfer. The results show that the values of these quantities across the discharge are lower than those calculated for a homogeneous situation, due to the loss of electrons to the wall. The solutions for the EDF reveal that, for sufficiently low maintaining fields, the radial anisotropy at some radial positions can be negative, that is, directed toward the discharge axis, for energies above a collisional barrier around the inelastic thresholds. However, at the wall, the radial anisotropy always points to the wall, due to the strong electron drain occuring in this region. We further present pertinent comparisons with other formulations recently proposed in the literature to model the present inhomogeneous problem.