An Algorithm for Solving Multiple-Wave Dynamical X-ray Diffraction Equations

An algorithm is proposed for solving the fundamental equations of the wavefield for multiple-wave dynamical X-ray diffraction with grazing incidence and scattering geometry. The algorithm is developed based on the representation of the electric fields and wavevectors in a single Cartesian coordinate system with one of the axes in the direction normal to the crystal surface. With this representation, the fundamental equations of the wavefield can be solved as an eigenvalue-eigenvector problem involving a 4N × 4N scattering matrix in which the matrix elements are not related to the polarization, N being the number of waves. The polarization factors are absorbed in the vector components of the eigenvectors. This simplifies the process in solving the fundamental equation, avoids unnecessary approximations on polarization in the matrix calculation and makes the algorithm very generic so that it can be applied to multiple diffractions of all kinds, including grazing-angle and wide-angle geometries. The intensity distribution of an Umweganregung of a specularly diffracted wave in a polarization-forbidden state is calculated using this algorithm as a demonstration.