Neutron scattering study of the tetragonal-to-incommensurate ferroelastic transition in barium sodium niobate

The static and dynamic characteristics of the incommensurate phases of barium sodium niobate have been investigated by means of elastic and inelastic neutron scattering between room temperature and 660°C. The observation of two incommensurate phases on heating from room temperature is consistent with previous x-ray results. Phase II, which is stable up to $T_{\mathrm{II}}=250\mathrm{}°$C, is nearly commensurate, the modulation vector being $\overrightarrow{\mathrm{k}}=\left[\right({\overrightarrow{\mathrm{a}}}^{*}+{\overrightarrow{\mathrm{b}}}^{*})\frac{(1+\delta )}{4}+\frac{{\overrightarrow{\mathrm{c}}}^{*}}{2}]$, with $\delta \approx 1$%. Phase I is incommensurate with the same direction of modulation, and $\delta$ varies linearly on heating from 8% at $T_{\mathrm{II}}$, up to 12.5% at $T_{\mathrm{I}}\simeq 288°$C. Above $T_{\mathrm{I}}$, crystalline phases with tetragonal symmetries are stable, $T_{\mathrm{II}}$ is discontinuous, and $T_{\mathrm{I}}$ continuous. On cooling down from $T_{\mathrm{I}}$ a large thermal hysteresis is noted in the modulation wavelength and the satellite intensities. The variations are smeared and no discontinuity similar to $T_{\mathrm{II}}$ is observed. The precursor effects of $T_{\mathrm{I}}$ consist, in the tetragonal phase, of a soft phonon and a central peak. The soft mode is underdamped only above ∼385°C. The static susceptibility deduced from the total scattered intensity diverges at $T_{\mathrm{I}}$, with ${\omega }_0^2\sim \frac{1}{\chi }$, varying linearly between 300°C and 360°C. The central peak, which is observed even at 660°C, diverges at $T_{\mathrm{I}}$ while its width remains always equal to the instrumental width. The dispersion surface of the soft mode in the ($\eta \xi \frac{1}{2}$) plane has a valley shape, the bottom of which is perpendicular to the modulation vector. It gives rise to diffused scattering forming rods along $\left[1\mathrm{}\overline{1}0\right]$. At a microscopic level the modulation is due to displacements consisting of a collective shearing of the oxygen octahedra in the structure. The dynamics of this motion involve strong correlations in planes containing the modulation vector and the polar $c$ axis. The order parameter of the observed sequence of transitions has four components. This unusual dimension is responsible for the breaking of the macroscopic symmetry in the incommensurate phases and the onset of improper ferroelastic properties. However, the incommensurate behavior deriving from it is expected to be the same as that previously studied for two-component order parameters. Thus the corresponding extended Landau theory is not likely to account for the unusual hysteresis observed below