THE TRANSVERSE CORRELATION LENGTH FOR RANDOMLY ROUGH SURFACES

It is shown by numerical simulations for a random, one-dimensional surface defined by the equation x3=ζ(x1), where the surface profile function ζ(x1) is a stationary, stochastic, Gaussian process, that the transverse correlation length a of the surface roughness is a good measure of the mean distance 〈d〉 between consecutive peaks and valleys on the surface. In the case that the surface height correlation function 〈ζ(x1)ζ(x1′)〉/〈ζ2(x1)〉=W (|x1-x1′|) has the Lorentzian form W(|x1|)=a2/(x12+a2) we find that 〈d〉=0.9080 a; when it has the Gaussian form W(|x1|)=exp(-x12/a2), we find that 〈d〉=1.2837 a; and when it has the nonmonotonic form W(|x1|)=sin(πx1/a)/(πx1/a), we find that 〈d〉=1.2883 a. These results suggest that 〈d〉 is larger, the faster the surface structure factor g(|Q|) [the Fourier transform of W(|x1|)] decays to zero with increasing |Q|. We also obtain the function P(itx1), which is defined in such a way that, if x1=0 is a zero of ζ′(x1), P(x1)dx1 is the probability that the nearest zero of ζ′(x1) for positive x1 lies between x1 and x1+dx1. © 1990 Plenum Publishing Corporation.