Determination of branch fraction and minimum dimension of mass-fractal aggregates

Particles of micrometer to nanometer size often aggregate to form branched structures. Such materials include metals and metal oxides as well as biological and polymeric materials (considering the persistence length as a primary unit). Characterization of such structures is difficult since they typically display disordered, irregular features in three dimensions. Branched aggregates display two limiting size scales: that of the primary particle, R1 and that of the aggregate, R2. The mass-fractal model is often used to describe such structures where the aggregate mass, z=M2/M1, is related to the aggregate size, r=R2/R1, through a scaling relationship z=alpha r (d(f)), where the lacunarity alpha is close to 1 and may depend on the growth mechanism. Scattering of x rays, light and neutrons yields a direct measure of the mass-fractal dimension since I(q) approximately q(-d(f)) for 1/R2