Crystal stability, thermal vibration, and vacancies

The stability of crystals is studied with use of a model which includes atomic vibration and thermal creation of vacancies. The vibrational dynamics is treated within an Einstein approximation to the self-consistent phonon theory including cubic anharmonic contributions. The force constants in the dynamics are reduced when vacancies appear and the thermal equilibrium number of vacancies is determined by the dynamics to form an integrated, consistent model. The predicted instability temperatures $T_I$ and instability volumes ${\rho }_I$ lie within 10-20% of those "observed" in computer simulations of crystal stability. The predicted instability temperatures $T_I$ lie within 20-40% of observed melting temperatures $T_M$, depending upon the crystal. This suggests that vacancies and vibrational dynamics will induce instability not far above $T_M$ and may be largely responsible for instability in real crystals. In agreement with much previous work, we find that the Lindemann melting rule holds for thermal melting in classical crystals. At quantum melting or instability, however, the Lindemann ratio $\delta$ takes a wide range of values ($0.04\le \delta \le 0.35$). This supports the view that the Lindemann rule is an empirical expression of the vibrational amplitude at which the classical free energy of the solid phase exceeds that of the fluid.