Grazing-incidence small-angle X-ray scattering in a twofold rough-interface medium: a new theoretical approach using theq-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 planeq-eigenwaves (eigenstatefunctions) 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 \{ \sigma, \ell, h\}, where θ and θ0are the scattering and incidence angles of the X-rays, respectively, σ is the root-mean-square roughness, \ell is the correlation length,his 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.