Investigation of the angle dependence of the photon-atom anomalous scattering factors

The angle dependence of the elastic photon-atom scattering amplitude is investigated using full relativistic S-matrix calculations for guidance. A simple quantitative approach is developed based on the full nonrelativistic amplitude for Rayleigh scattering from the ground state of a hydrogenic ion and it is successfully applied to neutral atoms of low and intermediate nuclear charge. This framework is useful for scattering of X-rays with energies above the K-shell ionization threshold but below energies where major relativistic corrections are necessary. The generalized anomalous scattering factors presented here go beyond the usual angle-independent anomalous-scattering-factor approximation, permitting a discussion of the full angular dependence of the anomalous scattering factors. Simple formulas, expressed in terms of elementary functions and a lookup table, are presented for these generalized anomalous scattering factors. Numerical examples illustrate that the angle dependence can be substantial. A fairly good representation of full numerical results is achieved.