Numerical calculation of dynamical friction in electron cooling systems, including magnetic field perturbations and finite time effects

The orders-of-magnitude higher luminosities required by future electron–ion collider concepts require a dissipative force to counteract the numerous factors acting to gradually increase the phase space volume of relativistic ion beams. High-energy electron cooling systems could provide the necessary dissipation via dynamical friction, but will have to be designed for new parameter regimes. It is expected that magnetic field errors, finite interaction time and other effects will reduce the dynamical friction and hence increase the cooling time, so improved understanding of the underlying dynamics is important. We present a generalized form of the classical field-free friction force equation, which conveniently captures some of these effects. Previous work (Bell et al 2008 J. Comput. Phys. 227 8714) shows both numerical and conceptual subtleties associated with undersampling of strong collisions, and we present a rigorous mathematical treatment of such difficulties, based on the use of a modified Pareto distribution for the electron–ion impact parameters. We also present a very efficient numerical algorithm for calculating the dynamical friction on a single ion in the field free case. For the case of arbitrary magnetic field errors, we present numerical simulation results, showing agreement with our generalized friction force formula.