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

Published online by Cambridge University Press:  13 March 2009

A. J. M. Garrett
A. J. M. Garrett
Affiliation:
Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE
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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.

Type
Research Article
Information
Journal of Plasma Physics , Volume 28 , Issue 2 , October 1982 , pp. 233 - 254
Copyright
Copyright © Cambridge University Press 1982

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Andersen, K. & Shuler, K. E. 1964 J. Chem. Phys. 40, 633.CrossRefGoogle Scholar
Braglia, G. L. 1980 Riv. Nuovo Cimento. 3, no. 5.Google Scholar
Cavalleri, G. 1981 Aust. J. Phys. 34, 361.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Davydov, B. 1935 Phys. Zeits. Sovj. Union. 8, 59.Google Scholar
Enskog, D. 1917 Doctoral dissertation, Uppsala.Google Scholar
Garrett, A. J. M. 1982 J. Plasma. Phys. 27, 135.CrossRefGoogle Scholar
Jancel, R. & Kahan, T. 1966 Electrodynamics of Plasmas. Wiley.Google Scholar
Johnson, D. E. & Ikenberry, E. 1958/1959 Arch. Rat. Mech. Anal. 2, 41.CrossRefGoogle Scholar
Johnston, T. W. 1966 J. Math. Phys. 7, 1453.CrossRefGoogle Scholar
Kumar, K., Skullerud, H. R. & Robson, R. E. 1980 Aust. J. Phys. 33, 343.CrossRefGoogle Scholar
Lin, S. L., Robson, R. E. & Mason, E. A. 1979 J. Chem. Phys. 71, 3483.CrossRefGoogle Scholar
Pidduck, F. B. 1915 Proc. London Math. Soc. 15, 89.Google Scholar
Shizgal, B., Lindenfeld, N. J. & Reeves, R. 1981 Chem. Phys. 56, 249.CrossRefGoogle Scholar
Wallace, P. R. 1948 Can. J. Res. 26A, 99.CrossRefGoogle Scholar
Wang, Chang C. S. & Uhlenbeck, G. E. 1952 On the Propagation of Sound in Monatomic Gases, reprinted (1970) in Studies in Statistical Mechanics (ed. de Boer, J. and Uhlenbeck, G. E.), vol. 5. North-Holland.Google Scholar
Wang, Chang C. S. & Uhlenbeck, G. E. 1956 The Kinetic Theory of a Gas in Alternating Outside Force Fields, reprinted (1970) in Studies in Statistical Mechanics (ed. de Boer, J. and Uhlenbeck, G. E.), vol. 5. North-Holland.Google Scholar
Wannier, G. H. 1953 Bell System Tech. J. 32, 170.CrossRefGoogle Scholar
Weinert, U. 1980 Arch. Rat. Mech. Anal. 74, 165.CrossRefGoogle Scholar