Relativistic theory of resonant X-ray scattering

The task of finding the single particle-like wavefunctions is now in principle equivalent to that within non-relativistic SIC-LSD theory. The four-component nature of the wavefunctions and the fact that neither spin nor orbital angular momentum are conserved separately presents some added technical difficulty, but this can be overcome using well-known techniques (Strange et al., 1984). The formal first-principles theory of MXRS, for materials with translational periodicity, is based on the fully relativistic spin-polarized SIC-LSD method in conjunction with second-order time-dependent perturbation theory (Arola et al., 2004). 

The theory of X-ray scattering is based on the second-order golden rule for the transition probability per unit time  

In relativistic quantum theory, it is the second term in Eq. (51) that is entirely responsible for scattering as it is second order in the vector potential. It is convenient to divide this term into four components, and the X-ray scattering amplitude in the case of elastic scattering may be written as (Arola et al., 1997 Arola and Strange, 2001) 

Equation (54) represents scattering with no photons and positive energy electrons only in the intermediate state. The corresponding theory is only applicable around resonance that makes it complementary to the work of Blume (1985), which is only valid well away from resonance. 

The final expression for the resonant part of the scattering amplitude in Bragg diffraction is 

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