Soft-photon emission in extreme-relativistic Compton scattering by K-shell electrons and connection to photoeffect
Abstract We have recently obtained cross-sections for Compton scattering by K-shell electrons at extreme-relativistic (ER) energies of the incoming photon. Our method is essentially analytical, and only at the end did it require a modest numerical computation. The results are valid for the Compton line of the scattered photon spectrum, but do not cover the infrared divergence at the soft-photon end of the spectrum. This case is treated in the present paper. The method we apply here is an adaptation of the one used for the Compton line. This allows us to make use of many previous results. The quadruply and doubly differential Compton cross-sections can be expressed in analytical forms which allow factorization of the ER photoeffect cross-sections (differential or total, respectively). This result is shown to be a manifestation of the soft-photon theorem connecting the Compton matrix element at low emitted photon frequencies to that of the photoeffect. The computation of ER Compton cross-sections with soft-photon emission is thereby reduced to that of the cross-sections for the ER photoeffect. We carry out a computation for the latter case and extend previously existing results.