Weighted-superposition approximation for x-ray and neutron reflectance.

Abstract

An approximate formula is derived for the x-ray and neutron reflectance of a one-dimensional scattering-length-density (SLD) profile based on the principle of superposition of the wave field. The SLD profile is regarded as being composed of an infinite number of histogramlike differential SLD steps which are distributed along the depth direction. A simple Fresnel reflection is assumed to occur at each differential step. The elemental Fresnel reflections from all the differential steps, weighted by their respective propagation effects, add up to the overall reflectance of the one-dimensional SLD profile in the form of an integral. The reflectance obtained this way is shown to reduce to the Born approximation for large-wave-vector transfer Q and to the modified Born approximation for very thin surface structures. The accuracy of the formula is evaluated through comparisons with Parratt's recurrence formula, the Born approximation, and the distorted-wave Born approximation (DWBA) for a few selected SLD profiles imitating actual experimental SLD profiles. It is concluded that the formula is, in general, more accurate than the Born and DWBA approximations and is valid in the entire range of wave-vector transfer Q except slight deviations in the narrow region around the total reflection edge. The formulation also applies to absorptive materials when the SLD profile is taken to be complex. Owing to the high accuracy and simplicity of the formula, a scheme is proposed to use the formula for the reconstruction of the SLD profile from measured reflectance and reflectivity data.