Small-angle scattering by fractal systems

TLDR

Fractal structures exhibit self‑similarity and a mass distribution that scales as a power D of length R, with D smaller than the embedding dimension d. The study aims to determine the fractal dimension D of such structures by applying scattering techniques at length scales below 1000 Å. The authors derive corrections to the simple Q⁻ᴰ scattering law near the limits of the fractal range, providing relations useful for aggregation and gelation. They show that the scattering intensity follows S(Q) ∝ Q⁻ᴰ and that polydispersity, particularly in percolation‑model systems, significantly affects the scattering behavior.

Abstract

Fractal structures are characterized by self similarity within some spatial range. The mass distribution in a fractal object varies with a power D of the length R, smaller than the dimension d of the space. When the range of physical interest falls below 1000 Å, scattering techniques are the most appropriate way to study fractal structures and determine their fractal dimension D. Small-angle neutron scattering (SANS) is particularly useful when advantage can be taken of isotopic substitution. It is easy to show that the scattering law for a fractal object is given by S(Q), ~ Q−D, where Q is the magnitude of the scattering vector. However, in practice some precautions must be taken because, near the limits of the fractal range, there are important deviations from this simple law. Some relations are derived which can be applied in relatively general situations, such as aggregation and gelation. The effects of polydispersity, important, in particular, in situations described by percolation models, are also shown.