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

TL;DR

This paper calculates the localization length in one-dimensional harmonic chains with isotopic disorder, showing how it varies with impurity concentration and scattering cross section, and discusses implications for disordered anharmonic chains.

Contribution

It introduces a method to compute localization length for arbitrary impurity concentrations and scattering cross sections in disordered harmonic chains, extending previous models.

Findings

Localization length decreases with impurity concentration, then diverges near full concentration.

The concentration dependence is accurately modeled by summing limiting behaviors.

Practical limits for system size and scatterer number are identified for ensemble averaging.

Abstract

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.