Compton Scattering by $K$-Shell Electrons. I. Nonrelativistic Theory with Retardation

An exact analytic calculation is presented for the cross section of the nonrelativistic Compton-type inelastic scattering of photons by electrons bound in the ground state of a hydrogenlike atom. The matrix element of the process was taken to be of the Kramers-Heisenberg-Waller form. It was integrated by the Coulomb-Green's-function method, developed previously by the author in connection with Rayleigh scattering. This is based on replacing the sum over intermediate states by the Green's function which is expressed in terms of the integral representation in momentum space given by Schwinger. The final continuum-state wave function was also described by an integral representation. Thus the evaluation of the matrix element requires the carrying out of momentum-space integrations, followed by two contour integrations. Its final form was expressed in terms of hypergeometric functions of four variables of Lauricella's type $F_D$. At small photon energies, where the dipole approximation is valid, the result simplifies considerably. The low-energy end of the scattered-photon spectrum is considered in detail in connection with the infrared divergence problem. The shape of the spectrum is discussed and a comparison with previous results is given.