Fracton interpretation of vibrational properties of crosslinked polymers, glasses, and irradiated quartz
Abstract
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 Debyetype excitations (long wavelength) and "fracton" excitations (shortlength 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 twolevel systems hypothesized for amorphous structures. The specific heat of epoxy resin exhibits a crossover from a Debyetype region (T<8 K) to a region (850 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 850 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 neutronirradiated quartz experiments tend to confirm this interpretation.
 Publication:

Physical Review B
 Pub Date:
 October 1983
 DOI:
 10.1103/PhysRevB.28.4615
 Bibcode:
 1983PhRvB..28.4615A