Electronic energy loss of slow protons channeled in metals

The electronic energy loss for low-velocity protons channeled in the $〈$100$〉$ direction of single-crystal Au and Al is calculated. The proton trajectories are determined by solving the equation of motion. In the proton dynamics two forces are included, a repulsive term arising from the nuclei and core electrons and a friction force, depending on proton velocity, arising from the valence electrons. The repulsive force on the proton is evaluated using a superposition of conservative potentials. The friction coefficient is evaluated by using the local density of valence electrons, and a quantum-mechanical transport cross-section approach with a self-consistent model based on Friedel's sum rule. The results allow us to describe the nonlinear behavior of energy loss with ion velocity observed experimentally in Au, as well as the linear behavior observed in Al.