Journal of Statistical Physics

, Volume 58, Issue 3, pp 485–501

The transverse correlation length for randomly rough surfaces

  • A. A. Maradudin
  • T. Michel

DOI: 10.1007/BF01112758

Cite this article as:
Maradudin, A.A. & Michel, T. J Stat Phys (1990) 58: 485. doi:10.1007/BF01112758


It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx3=ζ(x1), where the surface profile function ζ(x1) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha 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 (|x1x1′|) has the Lorentzian formW(|x1|)=a2/(x12+a2) we find that 〈d〉=0.9080a; when it has the Gaussian formW(|x1|)=exp(−x12/a2), we find that 〈d〉=1.2837a; and when it has the nonmonotonic formW(|x1|)=sin(πx1/a)/(πx1/a), we find that 〈d〉=1.2883a. These results suggest that 〈d〉 is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x1|)] decays to zero with increasing |Q|. We also obtain the functionP(itx1), which is defined in such a way that, ifx1=0 is a zero of ζ′(x1),P(x1)dx1 is the probability that the nearest zero of ζ′(x1) for positivex1 lies betweenx1 andx1+dx1.

Key words

Transverse correlation lengthrough surfaces

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • A. A. Maradudin
    • 1
  • T. Michel
    • 1
  1. 1.Department of Physics and Institute for Surface and Interface ScienceUniversity of CaliforniaIrvine