Darwin spherical-wave theory of kinematic surface diffraction

In 1912, von Laue first described X-ray diffraction by approximating as plane waves the spherical waves radiated by atoms in a crystal. Darwin recognized that this approximation is valid only in the limit of very small crystals, and published in 1914 the more general spherical-wave theory based on the reflectivity of individual atomic planes. The Darwin theory is extended here to surface Bragg diffraction from a single-crystalline monolayer, including the rederivation of a `surface' Bragg's law and the reflected intensity versus , the angle of incidence. This more general theory demonstrates how the intensity along a reciprocal-lattice rod associated with diffraction from a semi-infinite crystal is modified by the dependence of the length over which atoms in a plane emit spherical waves that constructively interfere at the detector. The diffracted amplitude is not proportional to the Fourier transform of the charge density. The plane- and spherical-wave models yield identical results for the integrated intensity across a finite detector area, even for incident angles far from a Bragg reflection.