Grazing-incidence small-angle X-ray scattering from a random rough surface: a self-consistent wavefunction approximation.

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 (θ, ϕ) angular range, in particular below the critical angle θ(cr) for total external reflection (θ is the grazing scattering angle with the surface, ϕ is the azimuth scattering angle; θ(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 R(dif)(θ, ϕ) scan is written in terms of the statistical scattering factor eta(theta, theta0) and Fourier transform of the two-point cumulant correlation function. To be specific for isotropic solid surfaces, the statistical scattering factor eta(theta, theta0) and Fourier transform of the two-point cumulant correlation function parametrically depend on the root-mean-square roughness σ [eta(theta, theta0) = 0 for σ = 0] and cumulant correlation length ℓ, respectively. The reflected R(dif)(θ, ϕ) scans are numerically simulated for the typical-valued {θ(0), σ, ℓ} parameters array.