Scattering by rough surfaces

We have used the first Born approximation to derive analytic expressions for the scattering by an oriented solid-on-solid rough surface. The surface roughness is described by a vertical height profile h(r) which is a random function of the positional vector r in the horizontal plane. The mean-square height difference for such surfaces is assumed to diverge either as a power law or a logarithm of r, i.e., $w_r^2$∝$r^{2\alpha }$ or $w_r^2$∝lnr. We derived expressions for both the specular reflection and the diffuse scattering. Effects due to the surface density profile and to atoms absorbed onto the surface, and relevance to the study of the roughening transition are discussed. We also show that the angular average of our result over all directions of the wave vector q gives a 1/$q^{3+\alpha }$ dependence, in agreement with an earlier calculation for inhomogeneous systems with randomly oriented internal surfaces.