Imperfect Crystals and Dynamical X-ray Diffraction in the Complex Reflectance Plane

Abstract

A theoretical framework is developed to describe the dynamical diffraction of X-rays in perfect and imperfect crystals. The propagation of the X-ray beam inside the crystal is described by the evolution of a set of trajectories in the complex reflectance plane. The trajectory path is determined from a form of the Takagi-Taupin equations and leads naturally to simple forms for the crystal reflectivity for perfect crystals. A stochastic model for the effects of crystal defects is developed in terms of the Langevin equation which leads to a description of diffraction from imperfect crystals as the evolution of densities in a parameter space, described by a Fokker-Planck equation.