Metastable particle eigenmodes of a surface-driven discharge

Abstract

A set of two differential equations has been formulated to model the behaviour of metastable particle density under constant electric field conditions in a weakly ionized plasma in which the metastable particles can 'self' regenerate. In the present case the self-regeneration occurs indirectly through the production of secondary electrons at the cathode surface by metastable particle impact, the secondary electrons then travelling across the gap space, under the influence of the constant electric field, to produce further metastable particles. Previously this problem has been approached by first solving the diffusion equation in the non-regenerative situation in which the eigenmodes are simple sinusoidal functions, and the decay times obtained are those applicable to no regeneration. The decay rates of the metastable particle density in the regenerative case are then obtained by various iterative procedures involving assumptions about the form of the metastable species' decay. No specific form for the eigenmodes of the actual physical system under consideration can be obtained by this method. The present approach takes a more physically significant and fundamental view by obtaining the eigenmodes and decay rates of the actual physical system under consideration directly from the set of coupled differential equations. The set of differential equations is solved by assuming that the transit time of the electrons is very fast compared with the average lifetime of the metastable particles in the gap space. In most cases, the solution for the metastable particle density equation can be written as an infinite sum of terms, each of which consists of the product of an exponentially decaying part and a spatial eigenmode, which remains constant in form during the decay period. Under certain conditions, however, some of the eigenfunctions and their decay rates can become complex. In this case pairs of complex eigenmodes can be combined to form real terms, although the decay of these terms is not a simple exponential one. Forms for the metastable particle density, the individual metastable particle eigenmodes and their corresponding decay rates have been obtained under the various conditions outlined above. The metastable particle density is then used to predict the theoretical form of the decay of current in a typical pre-breakdown nitrogen discharge, which is then compared with the experimentally obtained current.