Phonon-Fracton Crossover and Self-Similarity in Real Fractals

Abstract

Recent experimental studies on the structure and dynamics of silica aerogels are reviewed. These tenuous materials can be prepared with porosities ranging typically from 80% to 99%. Small-angle neutron scattering indicates that these monolithic solids can have fractal structures extending over more than two orders of magnitude in length. The structural picture which emerges is that aerogels are made of dense particles of mean size a, forming a fractal assembly up to a correlation length χ, beyond which the materials become macroscopically homogeneous. To these three structural regions correspond distinct vibrational regimes. With increasing frequency, and decreasing length scale, these are the phonons, fractons, and particle modes. We show how Brillouin, Raman, and inelastic neutron scattering give information on the dynamics of aerogels over a frequency range sufficiently extended to cover all three regimes. Evidence is given of a crossover from phonons to fractons, and the results allow the determination of both the fractal and spectral dimensions. The former dimension characterizes the scaling of mass, and the latter that of the vibrational density of states. © 1989 IOP Publishing Ltd.