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

Abstract

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 \({{C}_{{3{v}}}}\) 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 eigenfunctions 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.