Energy loss of ions in a magnetized plasma: conformity between linear response and binary collision treatments

The energy loss of a heavy ion moving in a magnetized electron plasma is considered within the linear response (LR) and binary collision (BC) treatments with the purpose to look for a connection between these two models. These two complementary approaches yield close results if no magnetic field is present, but there develop discrepancies with growing magnetic field at ion velocities that are lower than, or comparable with, the thermal velocity of the electrons. We show that this is a peculiarity of the Coulomb interaction which requires cutoff procedures to account for its singularity at the origin and its infinite range. The cutoff procedures in the LR and BC treatments are different as the order of integrations in velocity and in ordinary (Fourier) spaces is reversed in both treatments. While BC involves a velocity average of Coulomb logarithms, there appear in LR Coulomb logarithms of velocity averaged cutoffs. The discrepancies between LR and BC vanish, except for small contributions of collective modes, for smoothened potentials that require no cutoffs. This is shown explicitly with the help of an improved BC in which the velocity transfer is treated up to second order in the interaction in Fourier space.