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The transverse correlation length for randomly rough surfaces

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Abstract

It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx 3=ζ(x 1), where the surface profile function ζ(x 1) 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(x 1)ζ(x 1′)〉/〈ζ2(x 1)〉=W (|x 1x 1′|) has the Lorentzian formW(|x 1|)=a 2/(x 1 2 +a 2) we find that 〈d〉=0.9080a; when it has the Gaussian formW(|x 1|)=exp(−x 1 2 /a 2), we find that 〈d〉=1.2837a; and when it has the nonmonotonic formW(|x 1|)=sin(πx 1/a)/(πx 1/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(|x 1|)] decays to zero with increasing |Q|. We also obtain the functionP(itx 1), which is defined in such a way that, ifx 1=0 is a zero of ζ′(x 1),P(x 1)dx 1 is the probability that the nearest zero of ζ′(x 1) for positivex 1 lies betweenx 1 andx 1+dx 1.

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Maradudin, A.A., Michel, T. The transverse correlation length for randomly rough surfaces. J Stat Phys 58, 485–501 (1990). https://doi.org/10.1007/BF01112758

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Key words

  • Transverse correlation length
  • rough surfaces