Darwinʹs theory for the grazing incidence geometry

Darwinʹs dynamical theory of X-ray diffraction was studied in the case of grazing incidence geometry. It was shown that in such geometry the Darwin theory is not correct. In the two-beam case, the reflectivity of the specular reflection calculated by the original Darwinʹs theory was slightly different from that calculated by the well-known Fresnel formula. Furthermore, in the three-beam case, there were large discrepancies between the calculations by the Darwin theory and those by the extended dynamical theory proposed by Kishino and Kohra. These discrepancies were resolved when multiple scattering in an atomic plane was taken into account.