Simulating Quantum States of Positively Charged Particles Channeling along the [111] Direction in a Silicon Crystal

The potential well formed by the repulsive continuous potentials of three neighboring [111] chains in a silicon crystal, for a positively charged particle, exhibits the symmetry of an equilateral triangle, described by the C-3v group. In this case, the previously developed procedure for finding the eigenvalues of the transverse motion energy of channeled positively charged particles (positrons or protons) and the corresponding eigen-functions of the Hamiltonian, implemented on a square spatial grid, leads to artifacts in numerical modeling. We present a modification of the modeling algorithm based on a hexagonal grid, which takes into account the symmetry of the problem. The results of both approaches are compared, demonstrating the absence of artifacts when using a hexagonal grid. Using numerical methods, all discrete energy levels of the transverse motion of channeled positrons with a longitudinal motion energy of 2, 2.5, 3, 3.5, and 4 GeV in the discussed potential well are found. The developed procedure can be used in studies of manifestations of dynamic tunneling and quantum chaos in channeling.