TITLE: Extreme-relativistic Compton Scattering by K-shell electrons

ABSTRACT:

We have obtained cross sections for Compton scattering of very hard incident
photons \left( \hbar \omega _{1}\gg mc^{2}\right)  from K-shell electrons,
exact in the nuclear charge Z. For the calculation of the extreme
relativistic (ER) form of the S-matrix element involved, we have pursued
an analytical method as far as it could go. In the present case, this is the
viable alternative to an impracticable (\it ab initio) numerical
computation. In order to obtain the dominant behavior of the matrix element
in the large \omega _{1} limit, the momentum transferred to the nucleus
need be ascribed a constant value in the limiting process. The result
depends critically on the spectral range in which the scattered-photon
energy \omega _{2} is situated. We start by considering the \omega
_{2}  range covering the Compton line, for which the ratio \omega
_{2}/\omega _{1} need 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 Schroedinger
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}\sigma /d\omega
_{2}d\Omega _{2} describing the range of Compton line introduces an extra
integration. The final twofold integration requires a simple numerical
computation. This is a rather unique example of a most elaborate (Coulomb)
problem that could be solved analitically, essentially in closed form. We
subsequently consider the low-frequency  \left( \omega _{2}\rightarrow 0\right)
end of the scattered photon spectrum. Some of the analytic procedures used
for the range of the Compton line can be applied, leading to the limiting
forms of the matrix elements. For \omega _{2}\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 atomic numbers Z=13, 50 and 82.

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