Spatially Dependent Energy Distributions for Electrons Drifting Through a Gas in a Uniform Electric Field

Abstract

The steady-state distribution function is obtained for electrons initially emitted from a point source into a neutral gas and which subsequently drift under the influence of a uniform dc electric field while undergoing elastic collisions with the gas atoms. The usual approximations, regarding the distribution function as almost spherical in velocity space, and regarding the fractional energy gain or loss by an electron upon collision as small are retained. However, the terms in the Boltzmann transport equation involving spatial derivatives of the distribution, which are usually assumed small in comparison to the field and collision terms, are treated exactly. The distribution function is given as a sum of energy modes, each of which decay with distance from the source. The lowest of these modes is the far-distant distribution, while the higher ones, which decrease more rapidly with distance describe the decay of the initial source energy distribution. The complete distribution is obtained in terms of known functions in the case of an energy-independent collision frequency, whereas in the energy-independent cross-section case, only the lowest mode is obtained. The far-distant part of the distribution function is compared with the usual approximate expression which is obtained when the gradient terms are considered small and which is expressed as the density times a normalized energy function. It is shown, that when the gradient terms are correctly considered, the far-distant distribution in energy becomes position-dependent. Furthermore, the deviation from the approximate theory becomes larger, the further the electrons are off the geometrical axis. This position dependence is most important when the electron energy is large in comparison to thermal energies. The interpretation of Townsend method for the determination of the ratio of the diffusion coefficient to the mobility, $\frac{D}{\ensuremath{\mu}}$, is re-examined on the basis of this more exact theory. It is shown that the error in $\frac{D}{\ensuremath{\mu}}$ that results from using the conventional interpretation of this method under typical experimental conditions is never more than about 20%.