  Holographic particle characterization uses quantitative analysis of
holographic microscopy data to precisely and rapidly measure the diameter and
refractive index of individual colloidal spheres in their native media. When
this technique is applied to inhomogeneous or aspherical particles, the
measured diameter and refractive index represent properties of an effective
sphere enclosing each particle. Effective-sphere analysis has been applied
successfully to populations of fractal aggregates, yielding an overall fractal
dimension for the population as a whole. Here, we demonstrate that holographic
characterization also can measure the fractal dimensions of an individual
fractal cluster by probing how its effective diameter and refractive index
change as it undergoes rotational diffusion. This procedure probes the
structure of a cluster from multiple angles and thus constitutes a form of
tomography. Here we demonstrate and validate this effective-sphere
interpretation of aspherical particles' holograms through experimental studies
on aggregates of silica nanoparticles grown under a range of conditions.
