The Physical Origin of Anomalous Scattering

Abstract

Owing to the availability of synchrotron radiation, anomalous scattering of X-rays plays an increasing role in crystallography. This makes full understanding of the underlying principles necessary. The first part of the article explains anomalous scattering in terms of the classical theory of scattering by a damped oscillator. The damping accounts for the action of the emitted radiation on the oscillator itself. It causes a phase shift with respect to the motion of an undamped oscillator. This phase shift is the characteric of anomalous scattering and is represented by adding an imaginary component to the real scattering factor. The relations to the macroscopic index of refraction and the absorption coefficient are discussed. To bridge the gap between classical and quantum theory, the harmonic oscillator model is explained in terms of atomic wave functions and their deformation under influence of the electric field. The nonrelativistic quantum theoretical treatment of X-ray scattering is based on perturbation theory whereby use is made of Feynman-type diagrams and their mathematical translations. The origin of Thomson and Rayleigh scattering are given. The second one is characterized by virtual transitions to intermediate states. When the incident radiation corresponds with an absorption edge, the virtual transitions become real ones and decay of the intermediate state sets in. This decay leads to a phase shift of the probability amplitude of the path, the quantum mechanical equivalent of the scattering factor. The relation between the position of the target and the phase of the scattered beam is a matter of convention. This convention also dictates the sign of the phase shift mentioned above. Mathematical tools and detailed derivations are given in an appendix.