THE SPATIOTEMPORAL DEVELOPMENT OF ELECTRON SWARMS IN GASES - MOMENT EQUATION ANALYSIS AND HERMITE POLYNOMIAL EXPANSION

Spatio-temporal development of electron swarms in gases is simulated u sing a propagator method based on a series of one-dimensional spatial moment equations. When the moments up to a sufficient order are calcul ated, the spatial distribution function of electrons, p(x), can be con structed by an expansion technique using Hermite polynomials and the w eights of the Hermite components are represented in terms of the elect ron diffusion coefficients. ii 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 coeff icients, D-3(2)/D-L(3). 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 tl-lat the diffusion coe fficient is defined as the derivative of a quantity called the cumulan t which quantifies the degree of deviation of a statistical distributi on from a Gaussian distribution.