Determination of fluctuations in local symmetry and measurement by convergent beam electron diffraction: applications to a relaxor-based ferroelectric crystal after thermal annealing

Single crystals of Pb(Mg_1/3Nb_2/3)O₃–31%PbTiO₃ (PMN–31%PT) are known for their complex domain structures at the nanometre scale. While their average symmetry has been studied by X-ray, neutron and electron diffraction methods, there is little knowledge about variations in symmetry at the local scale. Here, direct evidence is provided for the volume dependence and spatial dependence of symmetry fluctuations by using quantitative convergent beam electron diffraction and energy dispersive X-ray spectroscopy. Fluctuations in symmetry were determined by using different electron beam probe sizes ranging from ∼2 to 25 nm from a crystal ∼62 nm thick. The symmetry of PMN–31%PT was found to increase linearly as the average volume increased, and the local symmetry fluctuated from one location to another at the nanoscale. Energy dispersive X-ray spectroscopy indicates that chemical fluctuations are significant when the probe size decreases to ∼2 nm. The symmetry fluctuation is attributed to locally varying composition-dependent ionic displacements and spontaneous polarization.