Splitting of the Transverse-Motion Energy Levels of Positrons during Channeling in the [100] Direction of a Silicon Crystal

The motion of charged particles in the crystal can be both regular and chaotic. Within the quantum approach, chaos manifests itself in the statistical properties of the set of energy levels. The systems in which regions of regular motion are separated by that of chaotic motion in phase space are of special interest. The statistics of levels of these systems is greatly influenced by the possibility of tunneling between phase-space regions dynamically isolated from each other. Matrix elements for such tunneling transitions are estimated in the present paper. To do this, all transverse-motion energy levels of 20 GeV positrons moving in the axial-channeling mode along the Si crystal [100] direction, as well as the Hamiltonian eigenfunctions corresponding to these states, are calculated numerically. The superposition of these eigenfunctions that correspond to classical orbits localized in symmetric but dynamically isolated regions of phase space are found. The energy-level splitting makes it possible to estimate the tunneling-transition matrix elements.