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 sample, 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 illuminated by the incident beam and viewed by the detector. Indeed, one must take account of the changes in the scattering direction that occur during the θ scan, and define at every θ the surface fraction A(θ) that scatters into the detector. In a third step, a rod with finite width is considered, and the spilling and travelling of the diffracted spot, arising from the centre of the sample, over the detector window is described. The spots emerging from any other sample position are then considered. By coupling spatial and angular variables, the scan line shape can be simulated quantitatively, by means of the in-plane intensity distribution of the rod. The resulting integrated intensity provides the correction factor to be applied to the raw data for the derivation of the structure-factor amplitude. This correction factor is compared to the usual correction (ALΔl), where A is assumed to be constant, L is the Lorentz factor and Δl is the l range as integrated during the scan (in the context of the Dirac-like rod). Significant differences occur at large l values when using grazing-incidence conditions.