Small-angle scattering by fractal systems

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.