Wave localization in binary isotopically disordered one-dimensional harmonic chains with impurities having arbitrary cross section and concentration

Institute for Physical Science and Technology and Department of Physics, University of Maryland(Dated: July 20, 2011)The localization length for isotopically disordered harmonic one-dimensional chains is calculatedfor arbitrary impurity concentration and scattering cross section. The localization length dependson the scattering cross section of a single scatterer, which is calculated for a discrete chain havinga wavelength dependent pulse propagation speed. For binary isotopically disordered systems com-posed of many scatterers, the localization length decreases with increasing impurity concentration,reaching a mimimum before diverging toward infinity as the impurity concentration approaches avalue of one. The concentration dependence of the localization length over the entire impurityconcentration range is approximated accurately by the sum of the behavior at each limiting concen-tration. Simultaneous measurements of Lyapunov exponent statistics indicate practical limits forthe minimum system length and the number of scatterers to achieve representative ensemble aver-ages. Results are discussed in the context of future investigations of the time-dependent behaviorof disordered anharmonic chains.