Sm. Durbin, DARWIN SPHERICAL-WAVE THEORY OF KINEMATIC SURFACE DIFFRACTION, Acta crystallographica. Section A, Foundations of crystallography, 51, 1995, pp. 258-268
In 1912, von Laue first described X-ray diffraction by approximating a
s plane waves the spherical waves radiated by atoms in a crystal. Darw
in recognized that this approximation is valid only in the limit of ve
ry small crystals, and published in 1914 the more general spherical-wa
ve theory based on the reflectivity of individual atomic planes. The D
arwin theory is extended here to surface Bragg diffraction from a sing
le-crystalline monolayer, including the rederivation of a 'surface' Br
agg's law and the reflected intensity versus phi, the angle of inciden
ce. This more general theory demonstrates how the intensity along a re
ciprocal-lattice rod associated with diffraction from a semi-infinite
crystal is modified by the phi dependence of the length over which ato
ms in a plane emit spherical waves that constructively interfere at th
e detector. The diffracted amplitude is not proportional to the Fourie
r transform of the charge density. The plane- and spherical-wave model
s yield identical results for the integrated intensity across a finite
detector area, even for incident angles far from a Bragg reflection.