Neutron scattering by rough surfaces at grazing incidence

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Abstract
There are a number of calculations in the literature of the effect of statistical surface roughness on the specular and diffuse scattering of neutrons and x rays which impinge on surfaces at grazing incidence. These calculations do not agree with one another and in some cases lack internal consistency. In this paper I show that these discrepancies can be resolved within the distorted-wave Born approximation. The result I obtain for the specular reflectivity is the widely used version of Nevot and Croce, while that for the diffuse scattering is the same as a recent calculation by Sinha, Sirota, Garoff, and Stanley. The approximations made by Nevot and Croce are clearly revealed by the distorted-wave Born approximation. A simple numerical algorithm is proposed for the calculation of diffuse scattering from a surface whose roughness is self-affine, and the form of the scattering is calculated for a range of parameters within this model. The calculation of diffusion scattering is extended in this paper to the case of films with rough surfaces. Correlation between roughness at the two film surfaces is considered and shown to be important for the description of neutron reflectivity data obtained with a thin film of titanium deposited on a sapphire substrate.