A mathematical theory of planar particle channeling in crystals

We use a simple two-phase averaging method for ODEs to characterize the class of energetic positively charged particle motions in the relativistic perfect crystal model that remain uniformly close to motions in Lindhard's planar continuum model over long time intervals that grow with the particle energy. Our averaging method includes a generalization of a method of Besjes to quasiperiodic functions with truncated diophantine conditions, which allows us to exclude resonances due to low-order axial directions. Although our methods are mathematically rigorous, we believe the results are physically realistic, notably the estimates of the critical energy and of the initial directions for planar channeling.