Time ‐ Dependent Multi ‐ Term Treatment of Plasma Electrons Acted upon by RF Electric Fields

Abstract

A b s t r a c tUsing a new method for solving the time-dependent electron Boltzmann equation in higher order accuracy, studies of the temporal behaviour of electrons in weakly ionized, collision-dominated plasmas under the action of rf fields have been performed. The method is based upon a multi-term approximation of the Legendre polynomial expansion of electron velocity distribution function and is applied to investigate the established periodic behaviour of the electron velocity distribution in helium, argon and CO plasmas.The results obtained in various approximation orders are discussed. The analysis has shown that the simplified treatment using only two terms of the velocity distribution expansion can fail in several conditions. In general, the four-term approximation gives already a good representation of the convergent solution of the electron Boltzmann equation at each instant of the rf period. The discrepancies between two-term and convergent results are found to depend sensitively on the specific atomic data, in particular on the magnitude of the various electron collision cross sections involved.Furthermore, the results obtained in the multi-term approximation are compared with corresponding ones obtained by accurate Monte Carlo simulations. Very good agreement between convergent eight-term Boltzmann and Monte Carlo calculations is found.

Microwave Diagnostics of the Electron Density in Molecular Mixture Plasmas

“…Using a sufficiently small HF field that satisfies the condition E 2 hf E 2 , the phase delay ϕ is obtained within the scope of the multiterm approach from the phase shift between the periodic electric field (7) introduced into the kinetic equation (5) and the calculated periodic current density of the electrons using Eq. (8). Certain differences between two-term and converged results were found concerning the periodic distributionsf andf p 1 , in particular at higher kinetic energies.…”

“…Recent theoretical studies of the behaviour of the electron velocity distribution in plasmas acted upon by RF electric fields have shown that this simplified treatment using only two terms of the velocity distribution expansion can fail in several conditions [8]. In order to check the applicability of the conventional approach, additional calculations have been done by solving the time-dependent electron Boltzmann equation in higher order accuracy by means of a multiterm technique.…”

“…In order to check the applicability of the conventional approach, additional calculations have been done by solving the time-dependent electron Boltzmann equation in higher order accuracy by means of a multiterm technique. Therefore, the system of partial differential equations for the expansion coefficientsf n , n = 0, 1, · · · , l − 1, resulting from the substitution of expansion (6) into the kinetic equation (5), is solved at given atomic data, plasma composition and field parameters as initial-boundary value problem up to the establishment of the periodic state, starting from appropriate initial values [8]. Using a sufficiently small HF field that satisfies the condition E 2 hf E 2 , the phase delay ϕ is obtained within the scope of the multiterm approach from the phase shift between the periodic electric field (7) introduced into the kinetic equation (5) and the calculated periodic current density of the electrons using Eq.…”

“…Alternatively, the periodic behaviour of the electron component in DC discharge plasmas additionally acted upon by microwaves can be determined by solving the time-dependent kinetic equation as an initial-boundary value problem up to the establishment of the periodic state. By using, additionally to the conventional approach, this more time-consuming approach for solving the time-dependent Boltzmann equation of the electrons in higher order accuracy by means of a multiterm technique [8] the impact on the phase delay of higher (than two) terms of the expansion of the distribution function in Legendre polynomials has been studied for various molecular gas mixtures. The comparison between experimental and theoretical results for the phase delay makes it possible to critically evaluate the applicability of the simplified version of the improved microwave resonator method.…”

First-principles particle simulation and Boltzmann equation analysis of negative differential conductivity and transient negative mobility effects in xenon

“…Charged particle swarms have been investigated extensively during the past decades both experimentally and theoretically, see e.g. [1,2,3,4,5,6,7,8]. The experiments, in which the transport of a (usually low density) particle cloud is studied, yield transport coefficients, like the drift velocity, diffusion coefficients, as well as reaction (excitation and ionization) rate coefficients.…”