Improved formulae for X-ray Reflectivity

In the conventional X-ray reflectivity (XRR) analysis, the reflectivity is calculated based on the Parratt formalism, accounting for the effect of roughness by the theory of Nevot-Croce. However, the calculated results of the XRR done in this way often showed strange results where the amplitude of the oscillation due to the interference effects would increase for a rougher surface. For the solution to this problem, we have developed an improved formalism in which the effects of the roughness-induced diffuse scattering are included correctly. In this paper, for deriving more accurate formalism of XRR, we introduce the effective roughness depending on the angle of incidence X-ray in XRR measurement. The new improved XRR formalism derives more accurate surface and interface roughness with depending on the size of coherent X-rays probing area, and derives the roughness correlation function and the lateral correlation length.