The influence of the dechannelling process on the photon emission by an ultra-relativistic positron channelling in a periodically bent crystal

We investigate, both analytically and numerically, the influence of the dechannelling process on the parameters of undulator radiation generated by ultra-relativistic positron channelling along a crystal plane, which is periodically bent. The bending might be due either to the propagation of a transverse acoustic wave through the crystal, or due to the static strain as it occurs in superlattices. In either case the periodically bent crystal serves as an undulator which allows the generation of x-ray and γ-radiation. We propose a scheme for the accurate quantitative treatment of the radiation in the presence of dechannelling. The scheme includes: (a) the analytic expression for the spectral-angular distribution which contains, as a parameter, the dechannelling length; (b) the simulation procedure of the dechannelling process of a positron in periodically bent crystals. Using these we calculate the dechannelling lengths of 5 GeV positrons channelling in Si, Ge and W crystals, and the spectral-angular and spectral distributions of the undulator over broad ranges of the photons. The calculations are performed for various parameters of channel bending.