Wave localization in binary isotopically disordered one-dimensional   harmonic chains with impurities having arbitrary cross section and   concentration
  The localization length for isotopically disordered harmonic one-dimensional chains is calculated for arbitrary impurity concentration and scattering cross section. The localization length depends on the scattering cross section of a single scatterer, which is calculated for a discrete chain having a wavelength dependent pulse propagation speed. For binary isotopically disordered systems composed of many scatterers, the localization length decreases with increasing impurity concentration, reaching a mimimum before diverging toward infinity as the impurity concentration approaches a value of one. The concentration dependence of the localization length over the entire impurity concentration range is approximated accurately by the sum of the behavior at each limiting concentration. Simultaneous measurements of Lyapunov exponent statistics indicate practical limits for the minimum system length and the number of scatterers to achieve representative ensemble averages. Results are discussed in the context of future investigations of the time-dependent behavior of disordered anharmonic chains. 
K. A. Snyder and T. R. Kirkpatrick