Abstract
Localized vibrational states above a mobility band (ω>ωc) are thought to characterize amorphous and glassy structures. The cross-over from a phonon (extended) to a fracton (localized) nature of lattice vibrations on a percolating network is a realization of such a cross-over. Lattice anharmonicity can connect extended and localized states, affecting dispersion and lifetimes for both species. We have examined these effects using the phonon-fracton picture to extract explicit numerical results. In particular, we have calculated, for phonons, the inelastic scattering lifetimes and the temperature and frequency dependences of the velocity of sound and, for the fractons, inelastic scattering lifetimes and the phonon-assisted hopping rates. These scattering processes are all related. They are simply different orderings of the scattering vertex. As a consequence, experimental determination of one of these quantities allows estimation of each of the others without significant undetermined parameters.
We have used the phonon inelastic scattering rate extracted from thermal conductivity experiments to make a quantitative estimate of the contribution κhop to the thermal conductivity arising from phonon-assisted fracton hopping. Applying our model to vitreous silica, we find an upper limit for κhop(T) which can be compared with experiment (where T is in kelvins):
Working backwards from these expressions results in a third-order anharmonic coupling coefficient between extended and vibrational states about two orders of magnitude larger than that which would have been extracted from extended-state interactions only. This suggests that the open nature of amorphous structures allows for much larger lattice displacements than in the homogeneous (long-length scale) regime.
We have also calculated the frequency and temperature dependences of the velocity v s of sound using the same approach. For vitreous silica we find that
We find a frequency dependence only to order (ω/ωc)2≈10−6 Remarkably, our predictions form the basis for the previously unexplained ‘Bellessa effect’ found in nearly all amorphous and glassy structures.
These calculations, based on a fractal model, provide quantitative agreement with experimental situations over a wide temperature range, suggesting that amorphous and glassy structures exhibit a comparable dynamical vibration structure.