Phonon dispersion relation in cubic BaTiO$_3$*

The lattice dynamics of cubic BaTiO$_3$ ($T_c=405$ K) has been investigated along the [100], [110], and [111] directions at $T=$453, 700 and 890 K using inelastic neutron scattering. The transverse acoustic (TA) phonon dispersion is relatively isotropic. This indicates that all the measured TA branches are mainly governed by the Ba atom motion against the O atoms. The zone-boundary TA phonon energies for BaTiO$_3$ are approximately twice as large as those for cubic PbTiO$_3$. These differences cannot be explained by the A-site atomic mass alone. The large Ba-O force-constant reflects the ionic nature of Ba atom in BaTiO$_3$. The softening of the transverse optical (TO) $\Delta_5$ and $\Sigma_4$ branches in a wide $q$ range at $T=453$ K is in contrast to the steep softening of the TO $\Lambda_3$ branch toward the zone center. The TO $\Sigma_4$ mode is heavily damped in the range $q\le$0.25 at 453 K, whereas the well-defined TO $\Sigma_4$ peak for $q=$0.25 is observed at 890 K. The low-lying TO $\Delta_5$ and $\Sigma_4$ branches are primary due to the low-frequency Ti-O-Ti chain-like motion along [001]. The soft mode behavior in BaTiO$_3$ is discussed in comparison with the previous study on PbTiO$_3$.