Dynamic Displacement Disorder of Cubic ${\mathrm{BaTiO}}_3$

The three-dimensional distribution of the x-ray diffuse scattering intensity of ${\mathrm{BaTiO}}_3$ has been recorded in a synchrotron experiment and simultaneously computed using molecular dynamics simulations of a shell model. Together, these have allowed the details of the disorder in paraelectric ${\mathrm{BaTiO}}_3$ to be clarified. The narrow sheets of diffuse scattering, related to the famous anisotropic longitudinal correlations of Ti ions, are shown to be caused by the overdamped anharmonic soft phonon branch. This finding demonstrates that the occurrence of narrow sheets of diffuse scattering agrees with a displacive picture of the cubic phase of this textbook ferroelectric material. The presented methodology allows one to go beyond the harmonic approximation in the analysis of phonons and phonon-related scattering.

Figure 1

Diffuse scattering of a cubic ${\mathrm{BaTiO}}_3$ single crystal at $T_C+100\mathrm{K}$ within the (a) $hk0$ and (b) $hk1.5$ planes of the momentum space. The left semicircles show experimental data; the right ones are calculated (see the text). Darker contrast means higher intensity. The vertical and horizontal diffuse lines are caused by the short-range Ti-ion displacement correlations. The two highlighted segments relate to the paths explored in Figs. 3 and 4. The sharp, tiny spots in noninteger positions in the left semicircle of (a) come from $\lambda /3$ contamination of the beam. A larger version of this figure is deposited in the Supplemental Material [44].

Figure 2

Distribution of the $y$ components of the relative $\mathrm{Ti}\text{−}{\mathrm{O}}_6$ displacement within a selected $xy$ layer of the simulated crystal box: (a) Instantaneous displacements. (b) Displacements averaged over a 500 fs period. (c) Displacements averaged over a 4 ps period. Panel (d) shows a fraction of the simulated atomic structure with arrows showing the magnified vector of the local Ti displacement. (e) Isosurface plot of probability distribution for the Ti atom position with respect to the center of the oxygen octahedron. The average magnitude of the displacement equals 0.147 Å, and the eight most likely positions are marked as corners of the superposed cube with about 0.3-Å-long edges.

Figure 3

Dynamical analysis of the diffuse scattering intensity: (a) Map of $S(q,\omega )$ in the [$q$ 0.5 1.5] direction. (b) X-ray scattering intensity determined from integration of $S(q,\omega )$ over the indicated frequency intervals.

Figure 4

Dispersion maps in the [3 $q$ 0] direction: (a) Total, displayed using a log scale. (b) Linear combination of partial $S(q,\omega )$: $S_{\text{Ba}}+S_{\text{BaTi}}+S_{\text{Ti}}$, that unveils the optic part of the signal. (c) Optic spectra at different $q$ values indicated by arrows on the map in (b), and a fit to a damped harmonic oscillator described in the text.