Scattering of fractons, the Ioffe-Regel criterion, and the (4/3) conjecture

Abstract

The relaxation time for scattering of vibrational modes by structural irregularities in d-dimensional random systems is shown to cross over from 1/τ∼${\mathrm{\omega }}^{\mathrm{d}+1\mathrm{}}$ ${\mathrm{\omega }}_{\mathrm{c}}^{\mathrm{-}\mathrm{d}}$ for phonons (ω${\mathrm{\omega }}_{\mathrm{c}}$, the crossover frequency between phonon and fracton vibrational excitations) to 1/τ∼${\mathrm{\omega }}^{5\mathrm{-}3\mathrm{d}\mathrm{̃}}$ for fractons (d̃ is the fracton dimensionality). The Ioffe-Regel criterion for localization, ωτ∼1, and a scaling Ansatz, then lead to the Alexander-Orbach value, d̃=(4/3), and 1/τ∼ω for all ω>${\mathrm{\omega }}_{\mathrm{c}}$. .AE