Uniqueness theorem for the non-local ionization source in glow discharge and hollow cathode

TL;DR

This paper proves the uniqueness of solutions for the non-local ionization source equation in glow discharges and hollow cathodes, clarifying the physical structure of electron avalanches and the hollow cathode effect.

Contribution

It provides a rigorous mathematical proof of solution uniqueness for the non-local ionization source in complex 3D geometries, enhancing understanding of glow discharge phenomena.

Findings

Established the uniqueness theorem for the ionization source equation.

Derived structure formulae with physical significance.

Clarified the physical construction of electron avalanches.

Abstract

The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.