Novel theoretical approach to the GISAXS issue: the Green function formalism using the q-Eigenwaves propagating through a twofold rough-surfaced medium.

To describe the 1D and 2D patterns of the grazing-incidence small-angle X-ray scattering (GISAXS) from a rough fractal surface, the novel integral equations for the amplitudes of reflected and transmitted waves are derived. To be specific, the analytical expression for the 2D total intensity distribution [Formula: see text] is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular [Formula: see text] and diffuse [Formula: see text] patterns, respectively. Hereafter, [Formula: see text] is the scattering meridian angle, [Formula: see text] is the scattering azimuth angle; [Formula: see text] is the angle of incidence. By using the above analytical expressions, the 1D and 2D GISAXS patterns are numerically calculated. Some new experimental measurements of the specular reflectivity curves R([Formula: see text]) related to the fused quartz and crystal Si(111) samples have been carried out. Based on the theoretical approach developed, a direct least-squared procedure in a χ-fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.