Phonon frequency softening at the paraelectric-ferroelectric phase transition
The Cochran soft mode: at the ferroelectric transition (BaTiO₃, PbTiO₃), a TO phonon mode softens as ω²∝(T−Tc) (Curie-Weiss). Below Tc, the double-well Landau potential V=α(T−Tc)P²+βP⁴ develops two minima — spontaneous polarization P₀=√(α(Tc−T)/2β). The phonon stiffness is zero at the transition (critical slowing down), giving the characteristic soft mode dispersion.