Dynamical X-ray diffraction from imperfect crystals: a solution based on the Fokker-Planck equation

A stochastic model of crystal defects is incorporated into a Fokker-Planck equation describing dynamical X-ray diffraction from imperfect extended-face crystals. The Fokker-Planck equation is solved by forming a set of complex moments describing the reflectance fluctuations in the crystal. This leads to an infinite set of coupled differential equations that are solved by neglecting high-order moments and numerically integrating the equations. The numerical solutions of X-ray rocking curves from a set of imperfect silicon films show excellent agreement with a Monte Carlo simulation and with a kinematical calculation away from the Bragg peak. The dynamical equations are suitable for describing Bragg diffraction from extended-face crystals containing defects, strain and composition variations.