Acoustic anomalies in terbium molybdate near the improper ferroelastic-ferroelectric phase transition

The anomaly in the longitudinal elastic constant ${\tilde{C}}_{11}$ of terbium molybdate (TMO) associated with the ferroelastic-ferroelectric phase transition at 160°C was investigated by Brillouin scattering spectroscopy. An analysis is presented, based on the phenomenological theory of Levanyuk, which describes both the thermodynamic anomaly due to induced bilinear coupling in the low-symmetry phase and the effects of anharmonic coupling of strain to critical fluctuations of the order parameter which occurs in both phases. Soft-mode dispersion curves from neutron scattering experiments and the temperature-dependent magnitude of the order parameter deduced from x-ray and birefringence studies were used as input for the data-fitting procedures. In our analysis of the data in the low-temperature ferroelectric phase, we consider four models for the second $A_1$ soft mode which is the partner of the 5.7-meV mode. With the assumption that the coupling constant $K_2$ between an acoustic phonon and a pair of soft optic phonons is independent of the soft-mode wave vector, only the fourth model produced a self-consistent fit to our data over the range $20°\mathrm{C}<~T<~400°\mathrm{C}$ which is also consistent with previous studies of the soft mode.