Chemical physics 56 (1981) 249-260 0 North-Rolland PublishingCompany : zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC EIGENVALUES OF TFIE BOLT&ANN COLLISION OPERATOR FOR BINARY GASES: MASS DEPENDENCE B. SHIZGAL, M .J. LINDENFELD and R. REEVES Depmmzent of CTmnist~, UniversTty ofBrifish Columbia, Vancouver, Brim Columbia, V62 I YS Gmaaia Received 29 August 1980 Eigenvatues of the Boltzmann collision operator are calcuIatcd versus mass ratio with two different methods. One method involves the expansion of the eigenfunctions in speed polynomi&, whereas with the second method the eigen- functions are evaluated at discrete points based on a particular gaussian quadrature rule. The dkrete ordinate method proved to bc superior provided the mass ratio was neither too large nor too small. The approach of the eigenvaluesto the continuum boundary was also studied for several mass ratios. 1. Introduction mination of the range of vahres of T for which these limiting forms are approximately valid. The approach to equilibrium of a gas, dilutely dis- persed in a second gas at equilibrium is a classic non- equilibrium kinetic theory problem [l] _ It is of parti- cular importance in neutron thermahzation [2] and hot atom chemistry 133 as well as in other related problems [4]. If the relaxing gas is composed of par- ticles without internal degrees of freedom, the linear Boltzmanu equation (BE) governs the approach of the velocity distribution function to equilibrium. The time dependence of the velocity diaabution can be given in terms of the-set of eigenvzdues of the Boltz- mann collision opera<or. In general, rhis eigenvaiue spectrum is composed of both discrete and continu- ous portions. Information with regard to the contin- The calculation of the mass dependence of these eigenvahtesfor a hard sphere cross section was pre- viously considered by Shapiro and Comgold [S] and Hoare and Kaplinsky [6]. The latter express the eigenfunctions in terms of Laguerre polynomials and calculate discrete eigenvahmswith the Rayleigh-Ritz variational method. This method is particularly useful for small $<l), since in the Rayleigh limit the Laguerre poIyuoxnii& are the eigenfunctions of the collision operator. However; for 7 3 1 the method suffers from slow convergence and roundoff errors in the compu- tation of the matrix elements of the Boltzmaun colli- sion operator. The former work [5] employs several alternate schemes including the introduction.of poly- nomials in the reduced speed which give much faster convergence; However, in that work, only the lowest eigemalues were evaluated. , The present paper is au extension of these previous ~&ulations and of an earl& paper b$ one of us [7]. Two methods, one based on a expansion.in speed poly- non&Is and the other on a &rete ordinate &th~d,