On the vibrational spectrum of at three-dimensiional lattice

Abstract All those properties of a crystal which depend on the heat motion of the constituent particles, require for their detailed explanation a knowledge of the actual form of the vibrational spectrum; about this very little is known, even qualitative features being difficult to obtain. It was supposed at first (Born and v. Kármán 1913) that the v2 law proposed by Debye formed a good approximation to the truth, but the experimental evidence, which gradually accumulated, tended to show that the specific heat did not vary exactly as T3 at temperatures where this law was expected to hold. Theoretical investigations (Blackman 1935) of the properties of a two-dimensional lattice showed that the spectrum could have markedly different features from those of a continuum distribution. It was furthermore found that large variations of θD with temperature could occur and that spurious T2 regions were possible. On account of the similarity of the two- and three-dimensional cases as regards the frequency equation and in certain particular features, it was assumed that the spectrum would not be very different in the three-dimensional case; with this assumption it was possible to explain the rise in the θD value of substances like KCl at low temperatures, and the discrepancies between elastic and thermal data. The theoretical predictions have been confirmed to some extent by the recent experimental work of Keesom and Clark (1935). They find that the rise of the θD curve stops at helium temperatures, as had been expected, but that the values decrease at still lower temperatures. Whether this last effect is real or not does not appear to be definitely settled. A possible theoretical explanation will be considered below.