Neutron diffuse scattering from polar nanoregions in the relaxor Pb ( Mg 1 / 3 Nb 2 / 3 ) O 3

We have studied the diffuse scattering in the relaxor $\mathrm{Pb}({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}$ (PMN) using triple-axis neutron scattering techniques. The diffuse scattering first appears around the Burns temperature ${T}_{d}\ensuremath{\approx}620\mathrm{K},$ indicating that its origin lies within the polar nanoregions (PNR's). While the relative intensities of the diffuse scattering around (101), (200), and (300) are consistent with those previously reported by Vakhrushev et al., they are, surprisingly, entirely different from those of the lowest-energy transverse-optic (TO) phonon. This observation led Naberezhnov et al. to claim that this TO mode could not be the ferroelectric soft mode. However, a recent neutron study by Gehring et al. has unambiguously shown that the lowest-energy TO mode does soften on cooling and that the relative intensities are similar to those of ${\mathrm{PbTiO}}_{3}.$ If the diffuse scattering in PMN originates from the condensation of a soft TO mode, then the atomic displacements of the PNR must satisfy the center-of-mass condition. But, the atomic displacements determined from diffuse scattering intensities do not fulfill this condition. To resolve this contradiction, we propose a simple model in which the total atomic displacement consists of two components ${\ensuremath{\delta}}_{c.m.}$ and ${\ensuremath{\delta}}_{\mathrm{shift}}.$ Here ${\ensuremath{\delta}}_{c.m.}$ is created by the soft-mode condensation and thus satisfies the center-of-mass condition. On the other hand, ${\ensuremath{\delta}}_{\mathrm{shift}}$ represents a uniform displacement of the PNR's along their polar direction relative to the surrounding (unpolarized) cubic matrix. Within the framework of this model, we can successfully describe the neutron diffuse scattering intensities observed in PMN.