Calculation of spontaneous radiation during channeling of relativistic positrons in non-chiral nanotubes using a quadratic approximation

The work investigates the conditions for the possibility of using the quadratic approximation U(ρ) = αρ2 for the interaction potentials of channeled positrons with the inner walls of non-chiral carbon nanotubes of types (n, 0) and (n, n). In particular, (8, 0), (10, 0), (12, 0) and (8, 8), (10, 10), (12, 12) nanotubes were selected. In this case, when calculating the single-particle potential of the carbon atom, only the contribution of valence electrons was taken into account. As a result of this approximation, the parameters α were determined for all the nanotubes studied. Using wave functions and the corresponding quantum levels of transverse energy obtained by solving the Schrödinger equation, the probabilities of occupation of these levels were calculated for positron beams with zero angular dispersion moving along the axes of nanotubes. Based on this information, values of the longitudinal energy of positrons for which the quadratic approximation is applicable were determined for all the studied nanotubes. Spectral distributions of spontaneous radiation were calculated in the dipole approximation for non-dispersive relativistic positron beams, both within the framework of quantum-mechanical and classical approaches.