Coupling between spatial and angular variables in surface X-ray diffraction: effects on the line shapes and integrated intensities

The intensity line shape, as provided by a rocking scan, and the derivation of the structure factor from the integrated intensity are reviewed in the context of surface X-ray diffraction, focusing on the z-axis geometry. In a first step, under the assumption of a Dirac-like rod and a point-like samp le, the effects of the detector slit settings on the scan width and on the integrated rod height are described. In a second step, it is shown that it is incorrect to treat the integrated intensity as being proportional to the active area A of the surface, defined as the sample area that is illuminat ed by the incident beam and viewed by the detector. Indeed, one must take a ccount of the changes in the scattering direction that occur during the the ta scan, and define at every theta the surface fraction A(theta) that scatt ers into the detector. In a third step, a rod with finite width is consider ed, and the spilling and travelling of the diffracted spot, arising from th e centre of the sample, over the detector window is described. The spots em erging from any other sample position are then considered. By coupling spat ial and angular variables, the scan line shape can be simulated quantitativ ely, by means of the inplane intensity distribution of the rod. The resulti ng integrated intensity provides the correction factor to be applied to the raw data for the derivation of the structure-factor amplitude. This correc tion factor is compared to the usual correction (AL Deltal), where A is ass umed to be constant, L is the Lorentz factor and Deltal is the l range as i ntegrated during the scan (in the context of the Dirac-like rod). Significa nt differences occur at large l values when using grazing-incidence conditi ons.