Phonon-fracton anharmonic interactions: The thermal conductivity of amorphous materials

Abstract

The anharmonic interaction (third-order elastic Hamiltonian) between fractons (short-length-scale vibrational excitations) and phonons (long-length-scale vibrational excitations) is introduced. The relationship between phonon-fracton scattering rates and the thermal conductivity \ensuremath{\kappa} is developed. Two relevant anharmonic lifetimes are calculated: two phonons combining into a single fracton, and one phonon combining with a fracton to produce a fracton. The former is important to the behavior of \ensuremath{\kappa} in the ``plateau'' regime; the latter to \ensuremath{\kappa} at temperatures above the plateau regime. The latter is interpreted in terms of phonon-assisted fracton hopping, and gives rise to an extra heat conductance which increases linearly in the temperature T. This behavior appears to be in agreement with recent high-temperature measurements of \ensuremath{\kappa} in epoxy resin by de Oliveira and Rosenberg.