Vibrational stability and melting

The self-consistent phonon (SCP) theory is used to study the mechanical stability of crystals to vibration as temperature is increased. The temperature at which instability occurs, $T_I$, depends sensitively on the approximation made to the SCP theory. In the self-consistent harmonic approximation, $T_I$ is ∼10-200 times the observed melting temperature $T_M$, depending on the crystal considered. In higher approximations $T_I$ is reduced to $\sim \mathrm{}(3-4)\mathrm{}{T}_M$ in all cases. Although the explicit character of the instability depends somewhat on the approximation, the instability is sudden, and takes place at a critical vibrational amplitude reminiscent of the Lindemann model of melting.