Mechanical Stability of Crystal Lattices with Two-Body Interactions

The stability criteria developed by Max Born are applied to investigate the mechanical stability of body-centered-cubic (bcc) and face-centered-cubic (fcc) Morse-function crystal lattices {i.e., lattices in which the atoms interact via the morse interatomic potential energy function $\varphi \mathrm{}\left(r\right)=D[e^{-2\mathrm{}\alpha (r-r_0)}-2\mathrm{}{e}^{-\alpha (r-r_0)}]$}. It is shown that the conditions for stability can be expressed uniquely as a function of $\alpha a$, where $a$ is the lattice parameter of the crystal. The fcc lattice is stable for all values of $\alpha a$, while the bcc lattice is stable only for values of $\alpha a$ which are less than 4.8. The possibility of using Morse-function lattices to represent cubic crystals with particular values of elastic moduli $C_{11}$ and $C_{12}$ is investigated. The Morse function can serve quite well for this type of representation for fcc crystals. For bcc crystals, however, the ratio $\frac{C_{11}}{C_{12}}$ does not exceed about 1.36; thus the representation is inherently fairly poor.