The density of states for thermal vibrations on a fractal is calculated with careful attention paid to the normalization condition. It is found that at the crossover between Debye-type excitations (long wavelength) and "fracton" excitations (short-length scale) the density of states is discontinuous. The size of the discontinuity is related to the ratio of the fracton dimensionality to the Euclidean dimensionality. Application is made to percolating structures. A set of missing modes is identified which may be the origin of the two-level systems hypothesized for amorphous structures. The specific heat of epoxy resin exhibits a crossover from a Debye-type region ($T<\mathrm{}8\mathrm{}$ K) to a region (8-50 K) where the vibrational density of states depends linearly on the frequency. Over the same frequency regime, the thermal conductivity exhibits an effective phonon mean free path of the order of (or less than) a lattice constant. We interpret this behavior in terms of quantized fractons, with an energy range 8-50 K, and we suggest that these fracton states are localized. This is consistent with the usual interpretation of a precipitous drop in the phonon mean free path at the crossover energy of 8 K. Analogous behavior is argued for the thermal properties of glasses which exhibit a similar structure in the thermal conductivity. Recent neutron-irradiated quartz experiments tend to confirm this interpretation.

• Received 28 March 1983

DOI:https://doi.org/10.1103/PhysRevB.28.4615