Collision integral for ionization by electron impact for nonisotropic electron distribution functions

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Abstract
A collision integral is derived that gives the rate of change of an electron distribution function due to ionizing collisions made by electrons with gas atoms. The treatment is valid for nonisotropic electron distribution functions. The development of the collision integral in this paper uses doubly differential cross sections (DDCS’s) for ionization since accurate analytical approximations for different target gases will be more easily available for DDCS’s than for the other cross sections that give a more complete description of the ionizing collisions. The treatment takes advantage of the inherent incompleteness in the description provided by the DDCS’s and reduces (the incompletely specified) three-body ionization problem into a set of two, fictitious, binary collision processes. These binary collisions are valid in mutually exclusive ranges of parameter space and complement each other in their contributions to the collision integral. The price for this simplification is that these binary collisions conserve neither kinetic energy nor orbital angular momentum. To handle this, the phase space of the gas atoms has been expanded by introducing fictitious scalar and vector parameters, which lend extra internal degrees of freedom to the atoms and destroy the isotropy of space. The overall approach is based on the treatment of S. Chapman and T. G. Cowling $[—The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, London, 1952)].