Inelastic extended-electron localized-vibrational-state scattering rate

We calculate the inelastic scattering rate of extended-electronic states off localized-vibrational states. We perform the calculation for electrons on a fractal lattice interacting with either fractons, or with phonons. We show that the respective scattering rates have quite different temperature dependences, allowing one to distinguish between the two forms of excitations. We calculate the temperature dependence of the inelastic electronic scattering rate off fractons for both degenerate and nondegenerate electron statistics. The temperature dependence of the inelastic electronic scattering rate for degenerate statistics is proportional to T(2d-theta)/(2theta), where theta is the exponent involved in the range dependence of the force constant, and d is the Euclidean or embedding dimension. This rate in d=2 varies as (approximately) T8/7 for fractons on a percolation network with scalar forces, and as T5/19 with central and bending forces, as opposed to T2 for phonon excitations (both extended and localized). For nondegenerate electron statistics, the inelastic electronic scattering rate in the fracton regime is proportional to Tkd-2-theta, where k is the electronic wave vector. For T1/2, this leads to a scattering rate proportional to T(d/2)-thetaT1/5 for a percolating network in d=2 with scalar forces, and as T-7/4 for central and bending forces, as opposed to the linear temperature dependence in d=2 obtained for phonons. For the case of scattering off phonon excitations, the inelastic electronic scattering rate for degenerate statistics is proportional to Tdc(1-d)theta/(2theta). For nondegenerate electron statistics, it is proportional to Tkd-2theta. Here, c is the crossover frequency separating the phonon regime from the fracton regime, and is the corresponding characteristic length. The results are conveniently expressed in the form of scaling relations.