The phonon-dispersion relations for cubic PbTiO${}_3$ ($T_c=763$ 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 PbTiO${}_3$ are nearly isotropic, in contrast to the distinct $q$ dependence of the TO branches for tetragonal PbTiO${}_3$. The zone-center TO mode energy softens with decreasing temperature from 1173 to 793 K. Moreover, the transverse acoustic (TA) branch along [$\xi ,\xi ,\xi$] shows significant softening around $\xi =0.25$ 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 PbTiO${}_3$ are discussed in connection with the simple perovskite oxides and the Pb-based relaxors.

• Received 2 July 2012

DOI:https://doi.org/10.1103/PhysRevB.86.134306