Interatomic forces, phonons, the Foreman-Lomer Theorem and the Blackman Sum Rule

TL;DR

This paper generalizes the Foreman-Lomer theorem for estimating interatomic forces from phonon data, extends it to complex lattices, and clarifies its relation to the Blackman sum rule, providing exact relations in terms of atomic force constants.

Contribution

It derives a generalized form of the Foreman-Lomer theorem applicable to various lattice types and relates it to the Blackman sum rule, with explicit formulas involving atomic force constants.

Findings

Generalized the Foreman-Lomer theorem for complex lattices.

Extended the method to hexagonal close packed and diamond lattices.

Derived exact relations for deviations from the Blackman sum rule.

Abstract

Foreman and Lomer proposed in 1957 a method of estimating the harmonic forces between parallel planes of atoms of primitive cubic crystals by Fourier transforming the squared frequencies of phonons propagating along principal directions. A generalized form of this theorem is derived in this paper and it is shown that it is more appropriate to apply the method to certain combinations of the phonon dispersion relations rather than to individual dispersion relations themselves. Further, it is also shown how the method may be extended to the non-primitive hexagonal close packed and diamond lattices. Explicit, exact and general relations in terms of atomic force constants are found for deviations from the Blackman sum rule which itself is shown to be derived from the generalized Foreman-Lomer theorem.