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

Abstract

Abstract. 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 suécient 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 diãusion coeécients. 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 diãusion coeécients, D 2 3 =D 3 L . The validity of an empirical approximation in time-of-çight experiments that treats p(x) as a Gaussian distribution is indicated theoretically. It is also found that the diãusion coeécient is deåned as the derivative of a quantity called the cumulant which quantiåes the degree of deviation of a statistical distribution from a Gaussian distribution.

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

“…An advantage of the evaluation of distribution shapes with C x;n and C y;n is that the Hermite components have visual shapes; in contrast to this the deånition of the cumulants is abstract. Sugawara et al (1998) demonstrated the convergence of p(x; t) to a Gaussian distribution by means of C x;n in the previous longitudinal analysis. Similar results for p(y; t) can be seen in ågure 8.…”

“…In the previous work (Sugawara et al 1998), two coordinates v and í were necessary to calculate the longitudinal moment distribution m x;n (v; t). Here one may simply integrate m x;n (v; t) with respect to û because electrons with common velocity components (v; í ) are all equivalent in the contribution to the longitudinal moment equations irrespective of û.…”

“…D T;n can be derived from the moments M y;n (t) in the same way as done for D L;n by Tagashira et al (1977) and Sugawara et al (1998):…”

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

“…Using "n,k and "n,total, Dn,k and Dn,total (n ? : 2) are given as 6 ) Dn,k Figure 2 shows the relations between the moments, cumulants and diffusion coefficients of the component and composite electron swarms. What to be derived is Dn,total = Dn,eq for n ?…”

“…2 ) discussed the difference between physical models based on eqs. (1) and (2), and pointed out the relationship between D3 and D4 and the skewness and kurtosis of the shape of f. Blevin and n!Dn is the time derivative of the nth-order cumulant Kn of f. 6 ) The cumulant is a statistical quantity representing the deviation of distribution shape from a Gaussian distribution. This fact indicates a mutual dependence between Dn and the shape of f. Here, a question arises.…”