Kazutaka Kitamori

Development of electron avalanches in argon-an exact Boltzmann equation analysis

Development of the electron avalanche in argon has been studied at E/N=141, 283 and 566 Td by two exact Boltzmann equation methods, a Fourier expansion (FE) of the distribution function of electrons by Tagashira et al. (1978), and a direct estimation of moments (DEM) of the electron density distribution in the real space by starting from a Boltzmann equation. Elastic momentum transfer, total electronic excitation and ionisation collisions are considered. The electron swarm parameters obtained by FE agree exactly with those by DEM, justifying the expansion used in FE, and the swarm parameters obtained by two-term expansion (TE) are in good agreement with those obtained by the exact FE and DEM except the longitudinal diffusion coefficient DL and a higher order coefficient D3 at E/N=566 Td and the transverse diffusion coefficient DT at all the E/N studied, suggesting that the disagreement with DL and D3 is due simply to breakdown of TE at high E/N but the disagreement with DT is due to more essential insufficiency in TE. The electron velocity distributions are also obtained and discussed.

Relaxation process of the electron velocity distribution in neon

The relaxation process from an initial velocity distribution to the equilibrium distribution for electrons in neon is calculated by a finite difference method for the ratios of electric field to gas number density E/N between 56.6 and 566 Td (E/p0=20 and 200 V cm-1 Torr-1 at 0 degrees C) without using the usual two-term spherical harmonics expansion of the velocity distribution. The pulsed Townsend condition, in which the evolution of all the electrons involved in an avalanche is observed as a function of time only, is assumed. The results suggest that the electron velocity distribution reaches through randomisation the equilibrium distribution which has a structure with a minimum near the origin in the velocity space.

Self-consistent Monte Carlo modelling of RF plasma in a helium-like model gas

A self-consistent modelling and simulation technique of RF nonequilibrium plasmas using a Monte Carlo method is presented. Under the condition that the product of the gas pressure and the gap length is small and/or the local electric field changes rapidly, as is commonly the case with RF plasmas, the electron energy distribution shows a nonequilibrium effect against the local electric field. The kinetics of electrons and ions is calculated by a Monte Carlo method which enables the authors to deal fully with the nonequilibrium effect. The electric field in the plasma is self-consistently determined by solving Poissons equation. With a particle model, the statistical fluctuation tends to be large in the sheath in which the number densities of electrons and ions are low. A scaling technique which enables the authors to diminish the fluctuation, and therefore the instability of the simulation is used. The simulation method is applied to an RF plasma in an He-like model gas and the results suggest that this model can adequately simulate RF plasmas.

A multi-term Boltzmann equation analysis of electron swarms in gases

An accurate and efficient method for solving the Boltzmann equation for electron swarms in gases is proposed. The method utilises the conventional two-term expansion and a Galerkin technique for solution of the Boltzmann equations in an amalgamated form; the first two terms of the Legendre polynomial expansion of the electron energy distribution are obtained by the two-term method while the third and higher-order terms are deduced by the Galerkin method. Although a relaxation procedure has to be used for connecting the solutions and making them self-consistent by the two methods, the present amalgamated method can reduce the size of the matrix for the Galerkin technique, which may serve to make the computational time shorter. The present technique is applied to deduce the electron energy distribution and swarm parameters in methane gas, and good agreement with those calculated by Monte Carlo simulation is obtained. This shows the validity of the proposed method. The computational time of the present method is estimated to be roughly of the order of the geometric mean of those by the two-term expansion and Monte Carlo methods.

A multi-term Boltzmann equation analysis of electron swarms in gases-the time-of-flight parameters

An accurate and efficient method for solving the Boltzmann equation is presented. Using a multi-term expansion technique, the time-of-flight (TOF) electron swarm parameters (such as the centre-of-mass drift velocity Wr, the longitudinal DL and transverse DT diffusion coefficients) can be evaluated. The method utilizes an amalgamation of the conventional two-term expansion and the Galerkin method. The first two terms of the Legendre polynomial expansion of the electron energy distribution are obtained by the two-term method, while the third-order and higher-order terms are deduced by the Galerkin method. The present technique is applied to determine the electron energy distribution using Fourier components to solve the Boltzmann equation and TOF swarm parameters in methane gas. Good agreement with Monte Carlo simulation is obtained and the method is also applied to a previously described model gas.

Development of electron avalanches parallel and perpendicular to the electric field-a Boltzmann equation analysis

An exact Boltzmann equation method for analysing the electron swarm motion parallel and perpendicular to the electric field is proposed and a solution for the distribution function given. Preliminary numerical results for argon are presented. The transverse diffusion coefficient and the continuity equation are discussed.

The spatio-temporal development of electron swarms in gases : moment equation analysis and Hermite polynomial expansion

Spatio-temporal development of electron swarms in gases is simulated using a propagator method based on a series of one-dimensional spatial moment equations. When the moments up to a sufficient order are calculated, the spatial distribution function of electrons, p( x), can be constructed by an expansion technique using Hermite polynomials and the weights of the Hermite components are represented in terms of the electron diffusion coefficients. It is found that the higher order Hermite components tend to zero with time; that is, the normalized form of p( x) tends to a Gaussian distribution. A time constant of the relaxation is obtained as the ratio of the second- and third-order diffusion coefficients, . The validity of an empirical approximation in time-of-flight experiments that treats p( x) as a Gaussian distribution is indicated theoretically. It is also found that the diffusion coefficient is defined as the derivative of a quantity called the cumulant which quantifies the degree of deviation of a statistical distribution from a Gaussian distribution.