On the problem of one-dimensional disordered chain

Conclusion

Some properties of a one-dimensional disordered homogeneous chain were studied in this paper. Using standard techniques of probability theory, expressions for the frequency distribution function (2) and the localization length (6) were derived. Having considered only pair correlations between atoms, both these expressions contained only one unknown function — the joint probability distribution of the massm _n and the ratiot _n ^± = −ku _n±1/u _n which could be found as a solution of the integral equation (5). Our approach to the problem was applied on the ideal lattice and the lattice with low concentration of impurities. In these cases the solutions of the integral equation (5) reduced to the functional form (7) were found analytically. Using these solutions, old well-known results for ϱ(ω ²) and the local vibration of impurities were derived by this method.

Derivation of all equations in this paper is straightforward from the equations of motion. The quantities we deal with have a clear physical meaning, which facilitated, for instance to find the solutions of functional equation (7) in the special case of the ideal crystal. This is what we consider to be the advantages of our approach.