Simultaneous softening of acoustic and optical modes in cubic PbTiO ₃

The phonon-dispersion relations for cubic PbTiO₃ (T _c=763 K) have been determined along the [1,0,0], [1,1,0], and [1,1,1] directions at T=793 K using inelastic neutron scattering. A set of the transverse optical (TO) branches drops sharply toward the zone center. The TO phonon curves for cubic PbTiO₃ are nearly isotropic, in contrast to the distinct q dependence of the TO branches for tetragonal PbTiO₃. The zone-center TO mode energy softens with decreasing temperature from 1173 to 793 K. Moreover, the transverse acoustic (TA) branch along [ξ,ξ,ξ] shows significant softening around ξ=0.25 and 0.5. The anomaly around the midpoint suggests a tendency toward forming a fourfold periodicity along the [1,1,1] direction, whereas the anomaly toward the R point results from the softening of the oxygen octahedron rotation. These two anomalies persist up to 1173 K and are weakly temperature dependent. The TA branches along [1,0,0] and [1,1,0] soften in the entire q range as the temperature approaches T _c. Although the phonon softening occurs simultaneously, the softening of the zone-center TO mode plays an important role in the single phase transition. The phonon dispersion relations for PbTiO₃ are discussed in connection with the simple perovskite oxides and the Pb-based relaxors.