Numerical studies of second- and fourth-order correlation functions in cluster-cluster aggregates in application to optical scattering

Two- and four-point density correlation functions p₂(r) and p₄(r) are studied numerically and theoretically in computer-generated three-dimensional lattice cluster-cluster aggregates (CCA) with the number of particles N up to 20 000 in application to the light scattering problem. The ``pure'' aggregation algorithm is used, where subclusters of all possible sizes are allowed to collide. We find that large CCA clusters demonstrate pronounced multiscaling. In particular, the fractal dimension determined from the slope of p₂(r) at small distances differs from that found from the dependence of the radius of gyration on the number of monomers (according to our data, 1.80 and 1.94, respectively). We also consider different functional forms for p₂ and their general properties and applicability. We find that the best fit to the numerical data is provided by the generalized exponential cutoff function with coefficients depending on N. The latter dependence is a manifestation of multiscaling. We propose some theoretical approaches for calculating p₄(r), assuming p₂(r) is known. In particular, we find the small-r asymptote for the p₄(r) and verify it numerically. In addition, we find that p₄(r) cannot be represented by a scaling dependence with a cutoff function, like p₂(r). Instead, p₄(r) is given by an expansion in terms of integer powers of r^2D-3, where D is the fractal dimension (≈1.8 for CCA clusters).