Complete treatment of single-photon emission in planar channeling

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 semiclassical 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 semiclassical method. However, for lower particle energies, it is found that the angularly integrated emission energy spectra obtained via the semiclassical 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.