Nonlinear screening of ions in plasmas: A general phase-shift sum rule

A very basic aspect in the interaction of ions with plasmas is the effect of screening of the ion charge. In the case of classical plasmas this effect is usually described by linear theories. However, for dense quantum plasmas, deviations from the linear models arise and new effects appear. These nonlinear effects have been extensively studied in the case of solid targets where the screening may be very appropriately modeled using the Friedel sum rule for the phase shifts of the scattering waves, as a constraint to adjust the screening potential. This approach considers the screening media as a degenerate electron gas. In the present work the formulation is extended to the general case of plasmas with arbitrary degree of degeneracy. The study covers the whole range of plasma conditions going from dense and degenerate plasmas to nondegenerate ones. A form of the phase-shift sum rule for the case of moving ions in plasmas is obtained. The formulation provides several closed expressions and analytical approximations to characterize the screening effects. Analytical expressions for the linearized regime for the cases of the Fermi-Dirac and Maxwell-Boltzmann distributions are obtained. The calculations show a weakening of the screening and a disappearance of nonlinear effects with increasing plasma temperature, plasma density, or ion speed, and a convergence of the nonlinear results to the linear ones when these conditions are reached. The expressions derived here are of general applicability to a wide range of plasma conditions.