a Mathematical Theory of Classical Particle Channeling in Perfect Crystals.
The problem is formulated as the motion of positively charged particles in a classical Hamiltonian system with potential representing the lattice of a perfect, static crystal. The phenomenon of channeling is defined as those motions with large total energy which avoid close encounters with nuclei, in other words as those motions which attain no more than a fixed small value of potential energy. For these motions, a nondimensionalization converts the physical Hamiltonian into nearly-integrable form with small parameter equal to the ratio of maximum potential energy to total energy. It is shown that motions near KAM invariant tori are nonchanneling trajectories, in the sense that such trajectories quickly surpass the maximum allowed value of potential energy for channeling. Using techniques from Fourier analysis, functional analysis, and number theory, a purely mathematical result on the rate of ergodization of nonresonant geodesic flow on the flat Euclidean torus is derived and is then used to bound the maximum time until close encounters with nuclei for nonchanneling trajectories. Away from KAM tori and near low-order resonances, channeling motions are shown to exist and to be stable for times exponentially long in the small parameter. This is accomplished with techniques from the proof of Nekhoroshev's theorem on exponential estimates of stability times for nearly integrable Hamiltonian systems. The heuristically-derived continuum models from channeling physics are shown to coincide with the leading order terms in the resonant normal forms appearing in the proof of Nekhoroshev's theorem. The analysis appears to embody the first rigorous derivations of the planar continuum model, and of the approximate conservation of transverse energy in both axial and planar channeling. As a result, the dissertation gives a satisfying mathematical unity to the phenomenon of particle channeling in crystals, showing that its essential features are contained within the confines of a classical theory.
- Pub Date:
- Mathematics; Physics: Elementary Particles and High Energy; Physics: Condensed Matter