Coulomb Milne problem

The transport properties of an ensemble of charged test particles diffusing from infinity through a second charged species that fills the positive half-space are studied. It is assumed that there are no incoming particles at the boundary between the medium in the positive half-space and the vacuum that fills the negative half-space. This problem has been referred to as the Milne problem and has previously been considered for neutral species [M. J. Lindenfold and B. Shizgal, Phys. Rev. A 27, 1657 (1983)]. The present paper examines the rarefied-gas dynamical effects for the Coulomb interaction between the two species. The distribution function of the background gas is taken to be a Maxwellian and the test-particle distribution is determined from a solution of the Boltzmann equation with its expansion in Burnett functions, products of Laguerre polynomials, and spherical harmonics. The extrapolation length is calculated together with the density and temperature profiles and compared with the Chapman-Enskog results valid far from the boundary. A comparison with the previous results for the hard-sphere cross section describing collisions between two neutral species is also presented.