A new procedure for determining the diffusion coefficients of electron swarms according to the modern transport theory

Abstract

In this paper we are concerned with the solution of the kinetic equations which are met when expanding the electron velocity distribution function with respect to the gradient of the electron density in the Boltzmann equation. The approximation of the corresponding zero- and first-order coefficients by an expansion in Legendre, or associated Legendre, polynomials up to a given number (2l)_ of terms, which transform the kinetic equations into three coupled hierarchies for the corresponding expansion coefficients, is assumed. The solutions of these hierarchies permit to determine the drift speed, as well as the longitudinal and transverse diffusion coefficients of an electron swarm. Based on an analysis of the mathematical structure of these singular ordinary differential equation systems (with additional difference terms due to the inelastic,i.e. energy loss, collision processes) a new procedure is developed which isolates and constructs (for arbitrary even approximation order 2l) the nonsingular part of the general solutions (NSPGS) of each hierarchy. The NSPGS is firstly obtained in the region of small electron energies (up to an appropriate connection point) and secondly in the region of large energies (down to the connection point) starting from the two singular points of each hierarchy,i.e. from very small and sufficiently large energies, respectively. The continuous connection of the two NSPGSs, obtained in the two different energy regions, together with the normalization condition, yield the physically relevant solution for each one of the three hierarchies and make it possible to calculate the diffusion coefficients we are particularly interested in. Our new technique, even if much more complex, is a logical generalization of the conventional backward prolongation technique used for the well-known two-term approximation of the stationary and spatially homogeneous Boltzmann equation. The present paper also reports a first application of this new procedure to a model plasma. The diffusion coefficients are calculated with increasing approximation order 2l up to their converged values, under different parameter conditions of the model and are compared with corresponding values found by accurate Monte Carlo simulations as well as with results of the conventional transport theory for the same model. The diffusion coefficients obtained for different (even) order of approximation well illustrate the power of our new procedure. In particular, the converged values of the diffusion coefficients are found to be in very good agreement with the corresponding Monte Carlo values.