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A study of the mass ratio dependence of the mixed species collision integral

  • A. J. M. Garrett (a1)

Abstract

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. A previous paper studied this expression as a function of the mass ratio for the two species of particle when the test particle distribution function was isotropic in velocity space; this work generalizes that analysis to anisotropic distribution functions by expanding the distribution function in tensorial spherical harmonics. First the limit of zero mass ratio is considered: this simplifies the calculation dramatically. There is no contribution to the collision integral from the zeroth-order spherical harmonic in this limit. Then the main calculation shows how to find the terms arising from the existence of a finite mass ratio as an ascending power series in this quantity, and evaluates for each spherical harmonic the next term, linear in mass ratio. This is checked for two special cases: that of an isotropic distribution function, when the expression reduces to Davydov's form, and that arising from a cross-section inversely proportional to the collision velocity, when a comparison with the exact solution of the associated eigen problem can be made. As in the isotropic case, an exact representation of the collision integral as an expansion in mass ratio must include some terms non-analytic in this quantity and vanishing more quickly than any positive power: it is shown how these arise in the present formalism. The formulae derived here have applications to the transport theory of electrons and light ions in a predominantly neutral gas as governed by the Boltzmann equation.

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A study of the mass ratio dependence of the mixed species collision integral

  • A. J. M. Garrett (a1)

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