Theory of solid-state contributions to the x-ray elastic scattering amplitude

We present a real space Green’s function theory of solid-state contributions and polarization dependence of the x-ray elastic scattering amplitude. In this approach the calculation separates naturally into contributions from the central (embedded) absorbing atom and multiple-scattering contributions from the environment. Both real and imaginary parts of the anomalous x-ray scattering amplitude are calculated simultaneously in the complex energy plane, without the necessity of a Kramers-Kronig transform. This approach also takes into account final-state effects, including core-hole lifetime, the finite temperature Fermi distribution and Debye-Waller factors, as well as experimental resolution. The approach is implemented in a generalization of the ab initio, self-consistent code, FEFF8.10, which permits applications to a number of x-ray spectroscopies for general, not necessarily periodic systems. The solid-state effect on the fine structure in the anomalous scattering amplitude near an absorption edge is illustrated for Cu metal. Calculations are also presented of the x-ray anomalous cross scattering amplitude $F_{\pi \sigma }$ for Cd metal, and x-ray natural circular dichroism in ${\mathrm{LiIO}}_3,$ which are both due entirely to solid-state effects.