Transversal Motion Quantum States Populating in Planar Channeling Mode and the Resonant Capture of Relativistic Electrons in the Axial Channeling Mode

The passage features of a beam of relativistic charged particles (electrons or positrons) through a single crystal along densely packed crystallographic planes or axes, as well as the characteristics of the resulting electromagnetic radiation, are determined by the possibility of capturing particles in the channeling mode and their distribution by quantum states with different transverse motion energies. An accurate quantum calculation of the populating coefficients for different quantum states of the transverse energy spectrum is mathematically very difficult and analytically possible only for the simplest model potentials. For the more realistic potential models of plane or axial channels in crystals the only available possibility is to make estimations using classical or quasi-classical approximations. The article provides calculations and evaluations of the quantum states population probabilities for planar channeling. It is demonstrated that even for collimated particle beams, when the particle capture probabilities are high enough, the particles distribution by transverse energies in planar channel is close to even. The possibility of resonant capture in the axial channeling mode for electrons entering single crystal with angular momentum values relative to the crystal axes multiples of the Planck constant is also evaluated.