Electron Velocity Distributions in Gases

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Abstract
An analytical expression for the velocity-distribution function for the low-energy region (u < u1) is obtained from the Maxwell-Boltzmann transport equation on the assumption that inelastic collisions occur at u = u1. The exact linear differential equation that defines the distribution function in the high-energy (u > u1) region is also derived. The mathematical difficulties and the usual approximations that are associated with this equation are briefly examined. An approximate analytical solution is given in terms of Hankel functions of the first kind when the cross section for inelastic collisions is assumed to be proportional to (u − u1)γ. The theory, which is developed in an elementary formalism, applies to electrons that are in a uniform dc electric field. Physical interpretations and the effects of inelastic collisions are emphasized.

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