Previously noted similarities between the central pair potential, CPP, the DeLaunay angular force, DAF, and the Lehman, Wolfram, and De Wames axially symmetric, AS, models for lattice dynamics coupled with criticisms of the DAF model, not applicable to the CPP model, has resulted in a puzzling and confusing picture. New physical arguments are used to develop a self-consistency condition, essentially unrecognized previously, which must be included in the use of all these force constant models. In this event, all of the above three models are proven to be identical. We then go on to examine the Clark–Gazis–Wallis, CGW, model. A systematic procedure is presented allowing a practical extension of the expression for the force constants from the current first and second nearest neighbour interactions to include third nearest neighbour interactions. For fcc systems we show that the previous expressions are incomplete and, consequently, overly restrictive and misleading and, hence, causing a current controversy. Also, we point out that the CGW model reproduces the general tensor force, GTF, model, at least to third nearest neighbour interactions. It is important to note that the application of the above mentioned self-consistency condition restores the Cauchy relation in all the models. Finally, numerical results are presented for the phonon spectra of the bcc metals Na and Nb and the fcc metals Cu and Au in order to demonstrate the practical effect of the self-consistency condition. This effect is seen to be larger for metals having larger Cauchy discrepancies. It is inferred that the force constant models, and hence the self-consistency condition, must be modified to yield the Cauchy discrepancy.
On the cosmological solutions in Weyl geometry