Robustness of the Fractal Regime for the Multiple-Scattering Structure Factor

In the single-scattering theory of electromagnetic radiation, the {\it fractal regime} is a definite range in the photon momentum-transfer $q$, which is characterized by the scaling-law behavior of the structure factor: $S(q) \propto 1/q^{d_f}$. This allows a straightforward estimation of the fractal dimension $d_f$ of aggregates in {\it Small-Angle X-ray Scattering} (SAXS) experiments. However, this behavior is not commonly studied in optical scattering experiments because of the lack of information on its domain of validity. In the present work, we propose a definition of the multiple-scattering structure factor, which naturally generalizes the single-scattering function $S(q)$. We show that the mean-field theory of electromagnetic scattering provides an explicit condition to interpret the significance of multiple scattering. In this paper, we investigate and discuss electromagnetic scattering by three classes of fractal aggregates. The results obtained from the TMatrix method show that the fractal scaling range is divided into two domains: 1) a genuine fractal regime, which is robust; 2) a possible anomalous scaling regime, $S(q) \propto 1/q^{\delta}$, with exponent $\delta$ independent of $d_f$, and related to the way the scattering mechanism uses the local morphology of the scatterer. The recognition, and an analysis, of the latter domain is of importance because it may result in significant reduction of the fractal regime, and brings into question the proper mechanism in the build-up of multiple-scattering.