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.
Highlights
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
Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of nonlocal electron avalanche, which builds a source of ionization in glow discharge at low pressures
The problem of creation of the hollow cathode theory— the theory for a glow discharge device in gases, which was invented by Paschen almost yet hundred years ago [1], producing anomalously high currents at the same voltages of discharge compared the glow discharge devices, which have no geometry of hollow cathode—considerably stipulated for non-local ionization
Summary
The problem of creation of the hollow cathode theory— the theory for a glow discharge device in gases, which was invented by Paschen almost yet hundred years ago [1], producing anomalously high currents at the same voltages of discharge compared the glow discharge devices, which have no geometry of hollow cathode—considerably stipulated for non-local ionization. Though there exists a set of theorems for existence and uniqueness of solutions in the theory of the Fredholm equations [1517], one cannot use them because: 1) the kernel is defined here not but as a solution of the linear boltzmann equation with differential and integral operators, the parameters of which have rather general physical properties; 2) the domain of kernel definition is defined implicitly its geometry is varied in wide range of glow discharge devices, a hollow cathode of arbitrary shape is one of possibilities; 3) the integral term of the equation is usually not of low value in comparison with absolute term (secondary electrons usually contribute more to ionization against primary, cathode, ones); 4) the kernel is not hermitian (or symmetric) one.
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