Inconsistency between height-height correlation and power-spectrum functions of scale-invariant surfaces for roughness exponent α∼1

Abstract

The height-height correlation function H(r) from a scale-invariant surface is compared with the corresponding power-spectrum function W(q) using a variety of mathematical scaling functions. We show that for a non-self-affine surface with the roughness exponent \ensuremath{\alpha}\ensuremath{\ge}1, one of the asymptotic scaling relations, either H(r)\ensuremath{\sim}${\mathit{r}}^{2\mathrm{\ensuremath{\alpha}}}$ or W(q)\ensuremath{\sim}${\mathit{q}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{-}}\mathit{d}}$, can be violated. An inconsistency in the values of \ensuremath{\alpha} also exists between H(r) and W(q) when \ensuremath{\alpha}>0.9. The impact on the data analysis using different experimental methods is discussed.