On the stability of crystal lattices. I

TLDR

Crystal lattice stability is examined through small‑vibration theory, where stability corresponds to the positive definiteness of macroscopic deformation energy expressed as elastic‑constant inequalities. The authors aim to develop a lattice‑sum method for calculating these stability conditions and apply it to three monatomic lattice types under central forces. The method computes elastic constants as lattice sums assuming central forces and evaluates the face‑centred, simple, and body‑centred lattices. The approach shows that only the long‑wave part of the vibrational spectrum is required, explains the stability of face‑centred lattices versus the instability of simple lattices and the conditional instability of body‑centred lattices, and indicates potential for improving the theory of strength.

Abstract

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.