Selecting an Adequate Potential and a Way of Calculating the Quantum States of Axially Channeled Electrons at High Energies

The authors study the motion of relativistic particles (electrons) along close-packed chains of atoms in single crystals (axial channeling). The motion of electrons is considered in the comoving reference frame (CRF), which itself moves with a velocity equal to the longitudinal velocity component of the channeled particles, relative to the axis of channeling. The motion of an axially channeled particle is two-dimensional (planar) in the CRF. As in the hydrogen atom, it is nonrelativistic for electrons with energies up to several GeV. The quantum characteristics of motion are determined by the particle’s energy (it acts as the mass of the electron in a 2D atom) and the parameters of the average potential of the atomic chain, which depend on the crystallographic orientation and chemical composition of the crystal. Axial channeling can be considered a unique model of a relativistic 2D atom with controlled parameters. The main characteristics of quantum states of the transverse motion of particles during axial channeling are shown to be weakly sensitive to the functional dependence of the parameters of the average potential on transverse coordinates. Bohr’s approximate quantization is used to conveniently calculate these characteristics, allowing the result to be obtained analytically. Modified Bohr quantization can be used to calculate the characteristics of transverse motion, going beyond the nonrelativistic approximation even in the CRF. The energy spectra of the permissible states of transverse orbital motion are calculated for several versions of axially symmetric model potentials. It is shown that despite differences in structure, the average distances between energy levels are insensitive to the choice of potential model. Level energies are found for when a nonrelativistic approximation can be used to describe transverse motion in the CRF, and in a situation going beyond the nonrelativistic approximation.