Treatment of grazing-incidence small-angle X-ray scattering data taken above the critical angle

The equations taking into account refraction at the sample surface in grazing-incidence small-angle X-ray scattering (GISAXS) when the angle between the incoming beam and the sample surface is slightly larger than the critical angle are derived and discussed. It is demonstrated that the refraction of both the incoming and the scattered beam at the sample surface affects the GISAXS pattern and that, when a planar bidimensional detector perpendicular to the incoming beam is used, the effect depends on the azimuthal detector angle. The smearing of the pattern depending on the size of the illuminated sample area in grazing incidence is estimated by simulations with Cauchy functions of different widths. The possibility of integrating the recorded intensities over a suitable azimuthal angular range and then of making the correction for refraction is also analysed, employing simulations involving the intensity function of monodisperse interacting hard spheres. As a case study, the refraction correction is applied to the investigation of a Cu-Ni implant on silica glass.