  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.
