Abstract
The matrix element for Rayleigh scattering by atomic -shell electrons is evaluated in the limit of high photon energies at finite momentum transfers . The limiting form of the matrix element is derived from its relativistic expression at finite energies in which the one-electron Green's function is replaced by its nonrelativistic approximation with adequately modified parameters. The expression obtained for is exact in and the atomic number , and is equivalent to the one found by Goldberger and Low. The evaluation of the matrix element is carried out in momentum space for the case of a Coulomb atomic field. Exact integral representations are used for and the ground-state eigenspinors. The integration of is carried out analytically as far as possible and at one stage the high-energy limit is taken. For one -shell electron, when electron spin-flip is possible, is expressed in terms of three real amplitudes , , and , whereas the matrix element for the closed shell, , depends only on . The amplitudes are obtained as one-dimensional integrals over Appell functions . They simplify considerably in the case of forward scattering, and the connection of their imaginary parts with absorption cross sections is discussed. Expansions of the forward amplitudes in powers of are also given. Further, a connection between and the nonrelativistic form factor is established, and the asymptotic behavior of with respect to is derived. Then, a description is given of the numerical methods used. is computed for and . Of the electron spin-flip amplitudes only is computed for forward scattering ( vanishes in this case). The numerical results are discussed and comparison is made with other works. The validity of the high-energy result at lower photon energies is considered. Finally, the magnitude of the Rayleigh matrix element is compared with the one for Delbrück scattering.
- Received 29 December 1975
DOI:https://doi.org/10.1103/PhysRevA.14.211
©1976 American Physical Society