Henyey-Greenstein Model in the Shape Relaxation of Dilute Gas Mixtures

We study the relaxation of Li+ ions dilutely dispersed in He at equilibrium. We employ the Henyey-Greenstein phase function to model the angular dependence of the differential scattering cross section for Li+-He collisions. We solve the spatially homogeneous linear Boltzmann equation for this model cross section with the collision operator explicitly given in terms of the scattering kernel. With the Quadrature Discretization Method based on the speed polynomials, the Boltzmann equation is reduced to a set of ordinary differential equations. This numerical method provides a rapid convergence for the Li+ distribution function. We study the relaxation of the shape of the Li+ distribution function in terms of the Kullback-Leibler information relative to the steady state and local in time Maxwellians. A comparison of the relaxation times of these two functionals show that there is no formation of a local Maxwellian during the relaxation process. This was verified for several values of the g-parameter in the Henyey-Greenstein phase function model and the initial average energies investigated.