Localization of acoustic vibrations in aerogels

In materials that are homogeneous at length scales larger than a correlation length ξ, acoustic phonons of wavelength λ smaller than ~2πξ do not exist. The acoustic vibrations at these high frequencies are highly localized and have been called fractons. The usual ultrasonic measurements cannot be used at frequencies beyond the phonon-fracton crossover. However, Brillouin-scattering measurements allow the determination of the fracton-structure factor S(q ,ω) in the crossover regime. Such measurements have been performed on silica aerogels, which are excellent model fractals, and they have been complemented by Raman scattering over the entire fracton frequency range, and by inelastic neutron-scattering measurements of the density of fracton states. The results allow the determination of both the fractal and spectral dimensions. The former characterizes the scaling of mass, and the latter that of the vibrational density of states