Stability of Crystal Lattices

The condition that a given lattice model be stable against all small deformations is that all the eigenvalues of all the dynamical matrices be positive. The stability of several lattices is studied by calculating the dynamical matrices for a large number of wave vectors in the Brillouin zone. Central potential interactions represented by Lennard-Jones and Rydberg forms are used, and the nearest-neighbor distance $\epsilon$ is allowed to vary throughout a range of ±10% of the value ${\epsilon }_0$ which minimizes the static lattice potential. The fcc and hcp lattices are stable for all central potentials and all values of $\epsilon$ studied; the bcc is stable for all values of $\epsilon$ for long-range central potentials; and the diamond is stable for a range of $\epsilon >{\epsilon }_0$ for all central potentials studied. The sum of the static lattice-potential plus the harmonic zero-point energy is minimized as a function of $\epsilon$, and it is found for all stable models except diamond that this procedure increases $\epsilon$ from ${\epsilon }_0$ by about ${\epsilon }_0\kappa$, where $\kappa =\frac{\hslash }{M^{\frac{1}{2}}{D}^{\frac{1}{2}}{\epsilon }_0}$, with $M=\mathrm{mass}\mathrm{of}\mathrm{ions}$ and $D=\mathrm{static}\mathrm{lattice}\mathrm{binding}\mathrm{energy}$. For diamond, however, there is no minimum in the range of $\epsilon$ for which the lattice is stable, for physically reasonable values of $\kappa$. Born has suggested that if a lattice is stable for long waves, it is stable for short waves; a counterexample has been found in the present study for the diamond lattice. A comprehensive table of accurate lattice sums is given in an Appendix.