On anomalous vibrational spectra

Abstract

In previous work on the vibrational spectrum of simple lattice models, it has been shown (Blackman 1937) that the spectrum becomes anomalous when special values are chosen for the force constants entering into the description of the model. As an example we may take the case of a square lattice of lattice distance d containing one particle per cell, in which the force constants α (for particles at a distance α) and γ (for particles at a distance d √2) were used. When γ / α tended to zero it could be shown that the spectrum changed from the two-dimensional to that of the linear chain. A similar result holds in the corresponding three-dimensional case. Although one would not expect any actual crystal to correspond to a limiting case (it would not be stable) it is conceivable that there should be crystals which approach the limiting case. In all cases which have hitherto been discussed, the anomaly is associated with the behaviour of the low-frequency end of the spectrum. This suggests that one could trace the effect to some property of the elastic continuum. An examination of the lattice mentioned above, for which the elastic continuum has two elastic constants c 11 and c 12 = c 44 , shows that the transition to the limiting case ( γ / α = 0) can be put in the form c 44 / c 11 → 0.