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

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