International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 2.9, p. 128

Up to this point, we have only considered reflection from smooth, flat surfaces. In reality, however, all surfaces have microscopic or mesoscopic imperfections such as steps, facets and rough hills and valleys. In this case, the potential must be represented by a threedimensional function instead of the simple onedimensional example discussed above. In addition, the roughness may not be confined to the outer surface or substrate, but the imperfections may propagate through several layers. This roughness at the interfaces modifies the specularly reflected beam and adds a diffuse component to the scattered beam (i.e. neutrons scattered at angles other than the incident angle). Theories based on the distortedwave, Born approximation have been developed to describe this type of scattering (Nevot & Croce, 1980; Sinha, Sirota, Garroff & Stanley, 1988; Pynn, 1992; Sears, 1993; Holy, Kubena, Ohlidal, Lischka & Plotz, 1993; de Boer, 1994) for a microscopically rough surface. These theories give results consistent with the earlier work (Nevot & Croce, 1980) for the modification to the specular scattering due to a single rough surface. The reader is referred to Sinha, Sirota, Garroff & Stanley (1988) and Pynn (1992) for a more complete discussion of diffuse scattering.
In order to fit the specular scattering from a rough surface, two simple methods have been employed. First, using the matrix method discussed above, the rough interface can be modelled as a smoothly varying scattering density approximated as a series of steps. This has the advantage that complex interfaces that are combinations of rough surfaces and intermixed layers can be approximated. The other method is to extend the results of Nevot to each successive interface while iteratively calculating the reflection amplitude. This method works well for simple interfaces of Gaussian roughness and is faster, in general, than the matrix method, since fewer calculations are needed for each interface. However, this latter technique suffers from frequently yielding unphysical answers (i.e. surface widths greater than adjacent layer thicknesses). Both of these methods are inadequate, in that there is no way to separate the effects of graded interfaces from rough surfaces. This can only be done with a simultaneous examination of both the diffuse and specular scattering.
References
Boer, D. K. G. de (1994). Influence of the roughness profile on the specular reflectivity of Xrays and neutrons. Phys. Rev. B, 49, 5817–5820.Holy, V., Kubena, J., Ohlidal, I., Lischka, K. & Plotz, W. (1993). Xray reflection from rough layered systems. Phys. Rev. B, 47, 15896–15903.
Nevot, L. & Croce, P. (1980). Caracterisation des surfaces par reflexion rasante de rayons X. Application a l'etude du polissage de quelques verres silicates. Rev. Phys. Appl. 15, 761–779.
Pynn, R. (1992). Neutron scattering by rough surfaces at grazing incidence. Phys. Rev. B, 45, 602–612.
Sears, V. F. (1993). Generalized distortedwave Born approximation for neutron reflection. Phys. Rev. B, 48, 17477–17485.
Sinha, S. K., Sirota, E. B., Garroff, S. & Stanley, H. B. (1988). Xray and neutron scattering from rough surfaces. Phys. Rev. B, 38, 2297–2311.