Diffuse x-ray scattering in perovskite ferroelectrics

Diffuse x-ray scattering measurements on the cubic paraelectric phase of single-crystal ${\mathrm{PbTiO}}_3$ are presented. No diffuse scattering sheets are found, in contrast to sheets observed in the cubic paraelectric phases of ${\mathrm{BaTiO}}_3$ and ${\mathrm{KNbO}}_3$. A quantitative analysis of the diffuse scattering indicates that the soft-mode branch contribution to sheets is negligible in all three ferroelectric perovskites. Differences in the diffuse scattering are explained by differences in the disordering of the local displacements, thus resolving a long-standing controversy.

Figure 1

Phonon dispersion of the lowest energy phonon modes along $\left[g00\right]$ measured by inelastic neutron scattering in the cubic phases of (a) ${\mathrm{PbTiO}}_3$ (Ref. 13), (b) ${\mathrm{KNbO}}_3$ (Ref. 14), and (c) ${\mathrm{BaTiO}}_3$ (Ref. 15). The lines are merely guides to the eye.

Figure 2

(Color online) Diffuse scattering profiles from cubic ${\mathrm{PbTiO}}_3$ $\left(T=800\mathrm{K}=1.05T_C\right)$ around the (400) and (501) reflections plotted on a logarithmic scale. (a) The ($h$, $k$) plane at $l=0$. (b) The ($h$, $k$) plane at $l=1$. The “hot spot” at (5.05, $-0.25$, 1) is due to a weak resolution-limited peak, presumably caused by a multiple-diffraction effect. Note that no diffuse sheet is present along $h=5$.

Figure 3

Comparison between theory and experiment along $l$ at $h=4$ and $k=0$ of cubic ${\mathrm{PbTiO}}_3$, using the measured dispersion of the TA-phonon modes (Ref. 13). Two sets of measurements are presented.

Figure 4

(Color online) Calculated TDS profiles using the continuum approximation for the measurements presented in Fig. 2. The calculated profiles are quantitatively accurate only within $0.2\mathrm{r.l.u.}$, beyond which the calculated profiles do not fall off as rapidly as the experimental measurements, due to neglect of phonon dispersion in the calculation.