Lattice dynamics with second neighbor interactions

Abstract

Hitherto, the study of lattice dynamics has been based primarily upon the approximation of harmonic forces, and nearest neighbor interactions. Some attention has been given to the influence of surfaces and defects, although they are known not to influence the bulk properties very greatly. While retaining the harmonic approximation, we have devised a method to take into account far neighbor interactions, surfaces and defects, by the application of the method of transfer matrices to the band diagonal matrices which arise in such systems. Only one-dimensional chains are considered. It is found that the normal modes depend upon a sum of 2n travelling waves, when there are nth neighbor interactions between all particles, and that the simultaneous consideration of this entire bundle of waves results in a much more systematic theory. Likewise, there is a maximum n-fold degeneracy present in the presence of nth neighbor interactions, which makes itself felt whenever it is possible to embed the chain in a ring or infinite periodic chain. The ordering of frequencies by a wave number which is characteristic of first-neighbor chains is lost, in part because there are n wave numbers to consider. Often one of these will be dominant, and it is possible to trace tendencies in a dispersion diagram, which in any event permits one to analyze the structure of the normal modes. The lowest frequency normal mode is always nodeless, but when far neighbor interactions predominate the second lowest mode may have the maximum possible number of nodes. When n = 1, the results are those of the Sturm-Liouville theory. Many of the accustomed properties of the first-neighbor lattices persist for far neighbors, although in somewhat modified form. Point defects produce localized modes of high frequency, diatomic lattices show an optical and an acoustical band although they may overlap in some regions and the optical band may shrink in others, and nonuniform chains show localized high frequency modes. Often one can obtain the properties of a chain for mild values of some parameter by interpolation of the properties with extreme values. We base our analysis on lattices with second neighbor interactions to present the results in a readily intelligible form, but the general form of the results is usually evident. Finally, it is noted that a number of other, nonrelated systems also lead to band-diagonal matrices, and therefore most of our conclusions are applicable to all such systems.