X-ray propagation and diffraction in distorted, anisotropic, dielectric and magnetic crystals

The interaction of X-rays in distorted anisotropic crystals is derived using a semi-classical approach based on Maxwell's equations. The X-ray wave is represented by a vector potential that is coupled to the charge and the magnetic moments of the electrons in the crystal by scattering factors obtained from non-relativistic quantum theory. Distortions are introduced through a slowly varying displacement field that shifts the positions of each unit cell in the crystal from its perfect-crystal position. The result is a generalization of the Takagi-Taupin equations in the form of a matrix equation for the electric field vector of the X-rays that includes magnetic scattering and the mixing of X-ray polarization states. The solutions of the equations in one-beam and two-beam cases are discussed.