WEIGHTED-SUPERPOSITION APPROXIMATION FOR X-RAY AND NEUTRON REFLECTANCE

An approximate formula is derived for the x-ray and neutron reflectanc e of a one-dimensional scattering-length-density (SLD) profile based o n the principle of superposition of the wave field. The SLD profile is regarded as being composed of an infinite number of histogramlike dif ferential SLD steps which are distributed along the depth direction. A simple Fresnel reflection is assumed to occur at each differential st ep. The elemental Fresnel reflections from all the differential steps, weighted by their respective propagation effects, add up to the overa ll reflectance of the one-dimensional SLD profile in the form of an in tegral. The reflectance obtained this way is shown to reduce to the Bo rn approximation for large-wave-vector transfer Q and to the modified Born approximation for very thin surface structures. The accuracy of t he formula is evaluated through comparisons with Parratt's recurrence formula, the Born approximation, and the distorted-wave Born approxima tion (DWBA) for a few selected SLD profiles imitating actual experimen tal SLD profiles. It is concluded that the formula is, in general, mor e accurate than the Born and DWBA approximations and is valid in the e ntire range of wave-vector transfer Q except slight deviations in the narrow region around the total reflection edge. The formulation also a pplies to absorptive materials when the SLD profile is taken to be com plex. Owing to the high accuracy and simplicity of the formula, a sche me is proposed to use the formula for the reconstruction of the SLD pr ofile from measured reflectance and reflectivity data.