Grazing-incidence small-angle X-ray scattering in a twofold rough-interface medium: a new theoretical approach using the q-eigenwave formalism

Based on the rigorous Green function formalism to describe the grazing-incidence small-angle X-ray scattering (GISAXS) problem, a system of two linked integral equations is derived with respect to amplitudes of the reflected and transmitted plane q-eigenwaves (eigenstate functions) propagating through two homogeneous media separated from each other by a rough surface interface. To build up the coupled solutions of these basic equations beyond the perturbation theory constraint 2kσθ₀ < 1, a simple iteration procedure is proposed as opposed to the self-consistent wave approach [Chukhovskii (2011). Acta Cryst. A67, 200–209; Chukhovski (2012). Acta Cryst. A68, 505–512]. Using the first-order iteration, analytical expressions for the averaged specular and non-specular scattering intensity distributions have been obtained. These expressions are further analysed in terms of the GISAXS parameters {k, θ, θ₀} and surface finish ones , where θ and θ₀ are the scattering and incidence angles of the X-rays, respectively, σ is the root-mean-square roughness, is the correlation length, h is the fractal surface model index, k = 2π/λ, and λ is the X-ray wavelength. A direct way to determine the surface finish parameters from the experimental specular and diffuse scattering indicatrix scan data is discussed for an example of GISAXS measurements from rough surfaces of α-quartz and CdTe samples.