Neutron-Scattering Study of the Ferroelectric Phase Transformation in ${\mathrm{Tb}}_2$ ${\left(\mathrm{M}\mathrm{o}{\mathrm{O}}_4\right)}_3$

We establish a first example of a ferroelectric phase transformation where a Brillouin-zone-boundary soft mode [at the $M$ point (½½,0)] rather than a Brillouin-zone-center mode of the parent paraelectric phase (PE) is responsible for the transition. By inelastic scattering of neutrons, we have measured low-frequency phonon-dispersion relations in ${\mathrm{Tb}}_2$ ${\left(\mathrm{M}\mathrm{o}{\mathrm{O}}_4\right)}_3$ for symmetry directions emanating from the $M$ point in PE. For $T>\mathrm{}{T}_0=159\mathrm{}°$C the frequency ${\omega }_M$ of a doubly degenerate mode at $M$ follows a Curie-Weiss law ${\omega }_M^2=A(T-T_C)$, with $A=0.0165\mathrm{}$ me${\mathrm{V}}^2$/°C and $T_C=149\mathrm{}°$C. With the help of group theory, the symmetry properties of soft modes which lead from the PE symmetry (tetragonal $P\overline{4}{2}_{1}m$) to the symmetry (orthorhombic $\mathrm{Pba}2\mathrm{}$) of the ferroelectric (FE) phase were determined. The soft-mode eigenvectors contain parameters which are not fixed by symmetry and their "static" values can be obtained from existing x-ray-structure data. "Dynamic" values are determined here from the integrated inelastic-scattering intensity of the soft mode measured in PE at various $M$ points. The "static" and the "dynamic" values are in good agreement. The condensation of such an antiferroelectric soft mode cannot directly produce the spontaneous polarization $P_z$ in FE. As has been suggested theoretically, our measurements show that the antiferroelectric static displacements constitute the order parameter, which couples to a shear strain $u_{\mathrm{xy}}$, which in turn produces the polarization by piezoelectric coupling. The spontaneous polarization and the spontaneous strain in FE are shown to be proportional to the square of this order parameter. From the initial slopes of acoustic branches we derive a set of elastic constants in PE.