Hard core phonon frequency at transition temperature

A simple theoretical derivation is given to show that one may expect a hard core for the phonon frequency at the transition temperature. This approach implies that the frequency of the phonon will follow ω ∝ {[1 + γ(T - T _C)] V ₀} ^{1}/{2} instead of that predicted by soft mode theory. Our expression should be valid whenever a Brownian sublattice exists in the crystal.