Vibrational Spectrum of a One-Dimensional Chain with Randomly Distributed Impurity Springs

The frequency spectrum of a one-dimensional lattice containing randomly distributed impurity springs has been evaluated to first order in the concentration q of impurity springs. It is shown that, with some mathematical manipulation, the solution can be placed into correspondence with the solution of Langer for the analogous problem of isotopic impurities. The virtual crystal approximation which yields the correct elastic constants requires an effective spring constant γ which is given, in terms of normal spring constant γ and the impurity spring constant γ^' by the relation 1γ=[(1-q)γ]+qγ^'.