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

Abstract

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.

An analysis of transverse evolution of electron swarms in gases using moment equations and a propagator method

“…Only a few of them mention the higher-order coeécients D T;n (n = 4; 6; 8; Å Å Å ), and fewer still showed their quantitative results (e.g. Kitamori et al 1980). In the present paper, a calculation technique applicable to derivation of higher-order parameters of the transverse electron swarm evolution is developed based on moment equations derived from the Boltzmann equation.…”

“…On the other hand, mathematical descriptions relating D T with other transport parameters were given by Tagashira et al (1977Tagashira et al ( , 1978, Kitamori et al (1980), Kumar et al (1980), Ikuta and Nakajima (1993) and Robson (1995). Values of D T in real gases were calculated using Boltzmann equation analyses, Monte Carlo simulations, and other numerical methods by Lowke et al (1973) for a CO 2 /N 2 /He mixture, Pitchford and Phelps (1982) and Phelps and Pitchford (1985) for N 2 , Makabe and Shimoyama (1986) and Koura (1987) for Ar or an Ar-like gas, Yachi et al (1991) and Date et al (1993) for CH 4 , etc.…”

Equality of Higher-Order Diffusion Coefficients between Component and Composite Electron Swarms in Gas

“…6,19) Moreover, ftotal (y) may be asymmetrical as long as its components fk (y) are all individually symmetrical, because DTn,total = 0 for odd n when DTn,k = 0 for all k's (the equality of DTn). Figure 14 shows D T3 ,k and D T3 ,total calculated by the MC for the same component and composite electron swarms as have been examined.…”

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

“…Moment (or kinetic) equation analysis is another technique by which to derive the TOF parameters (Kitamori et al 1980, Kumar 1981, Skullerud and Kuhn 1983, Skullerud 1984, Penetrante et al 1985, Robson 1991. Numerically, TOF parameters were calculated using direct estimation of moment (Kitamori et al 1980), the path integral method (Skullerud 1984), multi-term approximation of the Boltzmann equation and Monte Carlo simulation (Penetrante et al 1985), the çight-time integral method (Ikuta and Murakami 1987, Ikuta et al 1988, Robson 1995, Kumar 1995, and so forth.…”

“…Moment (or kinetic) equation analysis is another technique by which to derive the TOF parameters (Kitamori et al 1980, Kumar 1981, Skullerud and Kuhn 1983, Skullerud 1984, Penetrante et al 1985, Robson 1991. Numerically, TOF parameters were calculated using direct estimation of moment (Kitamori et al 1980), the path integral method (Skullerud 1984), multi-term approximation of the Boltzmann equation and Monte Carlo simulation (Penetrante et al 1985), the çight-time integral method (Ikuta and Murakami 1987, Ikuta et al 1988, Robson 1995, Kumar 1995, and so forth. On the basis of the TOF method and its modiåed model of arrival time spectra, which was proposed by Kondo and Tagashira (1990) and Date et al (1992) to follow a practical method of experimental measurements, Nakamura (1987Nakamura ( , 1988, Kurachi and Nakamura (1991), Hasegawa et al (1996) and Yoshida et al (1996) measured the electron drift velocities and the diãusion coeécients in N 2 , CO, CO 2 , SiH 4 /Kr, SF 6 and CH 4 .…”