Stopping power of fast protons under channeling conditions

For fast channeled protons the electronic stopping power $S_c$ is investigated in an impact-parameter treatment, where, for single inelastic atomic collisions, the proton moves on a classical straight-line trajectory. The electrons of the crystal atoms are described by hydrogenlike wave functions. In the MeV-energy range, the Born approximation can be applied. An average over all impact parameters yields a mean electronic excitation energy per collision that is twice the binding energy. Using this average for each impact parameter separately, we obtain $S_c$, slowly varying and finite for close collisions and with a long-range part corresponding to distant collisions. The results are in good agreement with experimental data in Si and Ge.