Nonlinear Eigenvalue Problem of a Self-Sustained Electrical Discharge

Abstract

An analytical solution is presented for the nonlinear boundary-value problem of a one-dimensional, electrical model discharge, in which the production of plasma by ionization is balanced by the plasma losses due to recombination and diffusion. The maximum value [Formula: see text] of the electron–ion density in the center of the discharge is given as the positive-real eigenvalue of a transcendental equation resulting from the boundary conditions. It is shown that a self-sustained, steady-state discharge exists if the dimensionless discharge number [Formula: see text] is numerically equal to a certain function [Formula: see text] of the dimensionless eigenvalue [Formula: see text]. For example, it must be (i) [Formula: see text] for [Formula: see text], i.e. when recombination is neglected, α = 0, and (ii) [Formula: see text] for [Formula: see text], i.e. when recombination is considered, α > 0 (a = half wall distance; ε, D, α = ionization, diffusion, and recombination coefficients respectively). Plots of the eigenvalue [Formula: see text] versus Z and of the density distribution n(x) of the charged particles are given.