# Complete treatment of single-photon emission in planar channeling

## TL;DR

This paper derives approximate quantum solutions for ultrarelativistic particles in a one-dimensional periodic potential and compares the resulting photon emission spectra with semi-classical predictions, highlighting differences at high energies.

## Contribution

It provides a full quantum treatment of single-photon emission in planar channeling and compares it with semi-classical methods, revealing energy-dependent discrepancies.

## Key findings

- Quantum calculations differ from semi-classical results at high energies.
- Semi-classical method agrees well at lower energies with shifted harmonic peaks.
- Quantum approach captures effects neglected in semi-classical approximation.

## Abstract

Approximate solutions of the Dirac equation are found for ultrarelativistic particles moving in a periodic potential, which depends only on one coordinate, transverse to the largest component of the momentum of the incoming particle. As an example we employ these solutions to calculate the radiation emission of positrons and electrons trapped in the planar potential found between the (110) planes in Silicon. This allows us to compare with the semi-classical method of Baier, Katkov and Strakhovenko, which includes the effect of spin and photon recoil, but neglects the quantization of the transverse motion. For high-energy electrons, the high-energy part of the angularly integrated photon energy spectrum calculated with the found wave functions differs from the corresponding one calculated with the semi-classical method. However, for lower particle energies it is found that the angularly integrated emission energy spectra obtained via the semi-classical method is in fairly good agreement with the full quantum calculation except that the positions of the harmonic peaks in photon energy and the photon emission angles are shifted.