Existence and Stability of Particle Channeling in Crystals on Timescales Beyond all Orders

Abstract

Particle channeling in crystals is an important aspect of particle-solid interactions for which I have developed a mathematical theory using modern techniques in Hamiltonian perturbation theory. This article describes that theory in broadest terms; ample discussion and detailed proofs have been or will be shortly published elsewhere (cf. Dumas 1988, 1989, 1991, and Dumas and Ellison 1991). A number of people helped me develop these ideas, but I am especially grateful to my thesis advisor, Jim Ellison, whose extensive work on mathematical aspects of channeling was the starting point for what is described here. I am also indebted to Pierre Lochak and to other members of the CMA at the École Normale Supérieure (Paris) for their assistance during the year 1986–87. It was there that I learned many of the mathematical tools used to formulate the channeling theory: a collection of results comprising KAM theory and the related but less familiar theory due to Nekhoroshev (1971, 1977, 1979) and, more recently, to Bennetin, Galgani, and Giorgilli (1985) and to Bennetin and Gallavotti (1986). The latter group of mathematical physicists first recognized the advantages—in principle at least—of Nekhoroshev’s approach in applications to physical systems where rigorous results on long-time stability of motion are desired; this approach is closely related to some of the techniques discussed in Professor Marsden’s article (these proceedings) on adiabatic invariance.