Nonlinear Eigenvalue Problem of a Self-Sustained Electrical Discharge

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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 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 is numerically equal to a certain function of the dimensionless eigenvalue . For example, it must be (i) for , i.e. when recombination is neglected, α = 0, and (ii) for , 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 versus Z and of the density distribution n(x) of the charged particles are given.

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