Nonlinear calculation of stopping powers for protons and antiprotons in solids: The Barkas effect

We calculate the electronic energy loss of protons and antiprotons in solids for a wide range of energies using a self-consistent method based on the extension of the Friedel sum rule to finite velocities. The calculations show the importance of nonlinear effects in the interaction of the moving particles with valence electrons in the intermediate range of energies around the stopping power maximum. The calculations give the magnitude and the velocity dependence of these effects, and provide a consistent theoretical explanation of recent experimental results for aluminum and silicon. [S1050-2947(99)04903-3].