X-ray specular scattering from statistically rough surfaces: a novel theoretical approach based on the Green function...

The Green function formalism is applied to the problem of grazing-incidence small-angle X-ray scattering from statistically rough surfaces. Kirchhoff's integral equation is used to describe the X-ray wavefield propagation through a single rough surface separating vacuum and medium. Taking into account multiple diffuse X-ray scattering effects, the reflection R(coh)(θ) and transmission T(coh)(θ) coefficients of the specular wave are obtained using the Gaussian statistical model of rough surfaces in terms of the two-point height-height correlation function. In the limiting cases when the correlation length xi is equal to zero or infinity, analytical formulae for the reflection R(coh)(θ) and transmission T(coh)(θ) coefficients of the specular wave are obtained. It is important that in the case xi --> infinity they coincide with the corresponding reflection R(DW)(θ) and transmission T(DW)(θ) coefficients related to the conventional Debye-Waller approximation for describing the grazing X-ray scattering from a rough surface. In the case of finite values of correlation length \xi the reflection |R(coh)(θ)|(2) and transmission |T(coh)(θ)|(2) scans are numerically calculated.