Scaling Approach for the Structure Factor of a Generalized System of Scatterers

A scaling approach for understanding the broad, general features of scattering of waves from nanoscale systems is presented. The approach uses the concept of a system of scatterers with arbitrary length scales, mass and surface fractal dimensions, and correlations between scatterers. It is based on comparing q−1, where q is the magnitude of the scattering wave vector, to the various length scales of the system of scatterers to determine whether the waves are scattered in phase or not. This comparison along with the fact that only fluctuations in the density of the scatterers scatter waves, yields power laws and cross over points which make up the structure factor. The system of scatterers can represent single spheres, fractal aggregates, or ensembles of such entities in a scattering volume. Hence a large range of experimental situations can be described and are unified under this comprehensive description.