AXIAL CHANNELING IN PERFECT CRYSTALS, THE CONTINUUM MODEL, AND THE METHOD OF AVERAGING

We present a mathematically rigorous treatment of axial channeling motions of energetic, positively charged particles based on the classical relativistic perfect crystal model. More specifically, we reduce the study of motions in a six-dimensional phase space to the study of associated motions in a four-dimensional space. Our main mathematical tool is a recently improved version of the classical method of averaging for ordinary differential equations, which we discuss separately. Applying the method at orders one, two, and three, we extract successively better approximations to perfect crystal model motions (the time of validity is the same for each approximation and scales inversely with the square root of incident particle energy). We call these approximations first-, second-, and third-order continuum model solutions, respectively, and our first-order continuum model solutions are precisely those arising in Lindhard's classical continuum model for axial channeling. The second-order solutions introduce the effects of lattice periodicity in what we believe to be the simplest possible way, e.g., in a computationally simpler way than the standard constant longitudinal momentum approximation. The third-order continuum model solutions introduce the first nontrivial relativistic corrections, and are also, we believe, the most precise approximations which may be useful in applications. After introducing these approximations and discussing their use, we conclude by briefly disussing the present state of the mathematics of channeling. © 1991.