Extreme relativistic Compton scattering byK-shell electrons

Abstract

We have derived cross sections for Compton scattering of very hard incident photons $(\ensuremath{\Elzxh}{\ensuremath{\omega}}_{1}\ensuremath{\gg}{\mathrm{mc}}^{2})$ from K-shell electrons, exact in the nuclear charge Z. The nuclear potential was taken to be of Coulomb form. The calculation of the extreme relativistic (ER) S-matrix element involved was carried out analytically. In the present case, this is the viable alternative to an impracticable ab initio numerical computation. In order to obtain the dominant behavior of the matrix element in the large ${\ensuremath{\omega}}_{1}$ limit, the momentum transferred to the nucleus needs to be ascribed a constant (albeit arbitrary) value in the limiting process. The result depends critically on the spectral range in which the scattered-photon energy ${\ensuremath{\omega}}_{2}$ $\mathrm{is}\mathrm{situated}.$ We start by considering the ${\ensuremath{\omega}}_{2}$ range covering the Compton line, for which the ratio ${\ensuremath{\omega}}_{2}/{\ensuremath{\omega}}_{1}$ needs to be kept finite. We show that in the ER limit the Dirac electron spinors and Green's operator entering the S-matrix element can be replaced by their relativistically modified Schr\"odinger counterparts. This allows the application of integration methods developed by us earlier for the nonrelativistic matrix element. Remarkably enough, the sixfold integrals of the ER matrix element can eventually be reduced to single integrals, expressible in terms of generalized hypergeometric functions. The doubly differential cross section ${d}^{2}\ensuremath{\sigma}/d{\ensuremath{\omega}}_{2}d{\ensuremath{\Omega}}_{2}$ for the range of Compton line finally results as a twofold integration, requiring a simple numerical computation. This is a rather unique example of a most elaborate Coulomb problem that could be solved analytically, essentially in closed form. We subsequently consider the low- $({\ensuremath{\omega}}_{2}\ensuremath{\rightarrow}0)$ and high-frequency $({\ensuremath{\omega}}_{2}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{2}^{\mathrm{max}})$ ends of the scattered photon spectrum. For ${\ensuremath{\omega}}_{2}\ensuremath{\rightarrow}0$ we find the expected infrared divergence, and verify the soft-photon theorem, which represents an important check on our calculation. Finally, we present our numerical results for ${d}^{2}\ensuremath{\sigma}/d{\ensuremath{\omega}}_{2}d{\ensuremath{\Omega}}_{2},$ analyzed at fixed ${\ensuremath{\omega}}_{2}$ (angular distributions), or fixed photon scattering angle (spectral distributions). We discuss the ``defect'' and the width of the Compton line for both distributions.