A stochastic model for X-ray diffraction from imperfect crystals

A model of crystal defects is developed to describe the diffraction of X-rays from imperfect crystals containing defect surfaces and crystal grains. The model, which is based on continuum theory for an isotropic homogeneous elastic medium, leads to a stochastic first-order differential equation, known as a Langevin equation. The solution of this equation is used to derive a correlation function for the strain-dependent term in the formula for the crystal reflectance. A consequence of the model is that the kinematic reflectivity of an imperfect crystal is given by the convolution between the perfect-crystal reflectivity and a function that transforms between a Gaussian and a Lorentzian depending on a correlation length in the crystal.