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

• Published in: Acta crystallographica. Section A, Foundations and advances Foundations and advances, 2015-11, Vol.71 (6), p.612-627
• Main Authors: Chukhovskii, F. N., Roshchin, B. S.
• Format: Article
• Language: English
• Subjects: Crystallography; eigenfunctions method; Formalism; grazing‐incidence small‐angle X‐ray scattering; Iterative methods; Mathematical analysis; Mathematical models; Scattering; Small angle X ray scattering; statistically rough surface; Surface finish; X-rays
• ISSN: 2053-2733; 0108-7673; 2053-2733
• DOI: 10.1107/S2053273315016666

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.