Abstract

An attempt is made to go beyond the distorted-wave Born approximation addressed to the grazing-incidence small-angle X-ray (GISAX) scattering from a random rough surface. The integral wave equation adjusted with the Green function formalism is applied. To find out an asymptotic solution of the non-averaged integral wave equation in terms of the Green function formalism, the theoretical approach based on a self-consistent approximation for the X-ray wavefunction is elaborated. Such an asymptotic solution allows one to describe the reflected X-ray wavefield everywhere in the scattering (, ) angular range, in particular below the critical angle cr for total external reflection ( is the grazing scattering angle with the surface, is the azimuth scattering angle; 0 is the grazing incidence angle). Analytical expressions for the reflected GISAX specular and diffuse scattering waves are obtained using the statistical model of a random Gaussian surface in terms of the r.m.s. roughness and two-point cumulant correlation function. For specular scattering the conventional Fresnel expression multiplied by the Debye-Waller factor is obtained. For the reflected GISAX diffuse scattering the intensity of the Rdif(, ) scan is written in terms of the statistical scattering factor and Fourier transform of the two-point cumulant correlation function. To be specific for isotropic solid surfaces, the statistical scattering factor and Fourier transform of the two-point cumulant correlation function parametrically depend on the root-mean-square roughness = 0 for = 0 and cumulant correlation length , respectively. The reflected Rdif(, ) scans are numerically simulated for the typical-valued {0, , } parameters array.