Vibrational spectra are presented for the two-component disordered simple quadratic and honeycomb lattices. These spectra are complex in structure, containing numerous peaks associated with the localized vibrations of clusters of light atoms in predominantly heavy atom local environments. A theorem of Rosenstock & McGill (1962), on the ordering of the displacement eigenfunctions for one-dimensional chains with nearest-neighbour interactions, is generalized.
The kth normal mode of vibration (i.e., the atomic displacements) of a disordered one-dimensional lattice (masses and force constants completely arbitrary) with nearest-neighbor interaction has precisely k − 1 nodes. A fortiori, the same is true for ordered one-dimensional lattices with any number of atoms per unit cell. This theorem exhibits a close relationship between eigenfunctions in monatomic ordered lattices (to which its application has been known for many years) and disordered lattices; a relationship which appears surprising in view of recent demonstrations of the gross differences exhibited in the distribution of eigenvalues. It is thus suggested that some basic concepts of ordered lattice dynamics—propagation vector, phonon momentum, etc.—may retain some simple validity for disordered solids as well. Some numerical examples are given.
It is shown that the eigenfunctions for the disordered two-component chain are strongly localized at intermediate and high frequencies. This result differs from the conclusions drawn by Rosenstock and McGill in 1962, which are from work on very short chains.
A detailed study of the isolated modes due to a variety of defects in one-dimensional two-component vibrating systems is made. A version of an analytic method outlined by Montroll and Potts in 1955 is used to deal with the localized modes associated with the simpler types of defect. More complicated defects can be treated in the same way, but the analysis becomes inconveniently complex, and a numerical approach already extensively used in a similar connection by Dean in 1959 and 1960 is preferred. It is found that a relatively few general rules describe all the results; these rules may be used to obtain much information on the spectra of a wide variety of one-dimensional systems.
