Long-range pair correlation and its role in small-angle scattering from fractal clusters

Abstract

The short- and long-range structures of computer-grown random-mass fractal clusters are described using exponential, Gaussian, and two different power-law pair-correlation functions. For all of the correlation functions the short-range structure is determined by the fractal dimension D and the long-range structure is expressed with a size parameter. One power correlation function has an additional shape parameter. Closed-form expressions are derived for the small-angle x-ray scattering for each of the four correlation functions. Clusters are grown using diffusion-limited-aggregation, Eden, dielectric-breakdown (DBM), ballistic, and random-polymer models. The Debye sum is used to calculate the small-angle scattering for each cluster. The parameters in the correlation functions are adjusted to provide the best fit to the Debye-sum scattering. The power laws reproduce the short- and long-range structural information much more accurately than the exponential or Gaussian models, which lack definitive size cutoffs and fractal scaling at intermediate- and long-range distances. In many cases the two-parameter power function produces fits that are as good as those with a third parameter. This indicates that the long-range shape parameter in the three-parameter power correlation function is simply related to the fractal dimension. The power fits accurately give the fractal dimensions and the radii of gyration for the clusters. Expressions are derived for the Guinier and the fractal-region scattering for each correlation function. Asymptotic formulas are used to explain large-q (fractal-region) scattering intensity that varies as ${\mathit{q}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$, where 1\ensuremath{\le}4. It is shown for D=2 systems that the fractal scattering is independent of the size and shape parameters. The extensions of this work to the scattering by multifractals are discussed. An efficient method is also presented to calculate large DBM clusters with noninteger growth exponents.