Symmetry determination of piezoelectric (1-x)Pb(Mg$_{1/3}$Nb$_{2/3})$O$_{3}$-xPbTiO$_{3}$ single crystal near morphotropic phase boundary

The symmetry of (1-x)Pb(Mg$_{1/3}$Nb$_{2/3})$O$_{3}$-xPbTiO$_{3}$ single crystal (PMN-xPT) is investigated at the morphotropic phase boundary with x=31{\%}. The XRD result indicates that the average symmetry of annealed PMN-31{\%}PT single crystal is in close agreement with the monoclinic M$_{C}$ (Pm) phase, but with strain. The local symmetry of PMN-31PT are then investigated at three different directions of [001]$_{C}$, [010]$_{C}$, [011]$_{C}$ and [111]$_{C}$ by using convergent electron diffraction (CBED) technique with a probe size of 2 nm. The dynamical theory is used to simulate the CBED patterns for the reported phases of PMN-xPT. The CBED results show that significant deviations of the experimental CBED patterns from the simulations. The deviations in the CBED results are considered as a result from a local distortion induced by the different B cations of Mg$^{2+}$, Nb$^{5+}$ and Ti4$^{+}$ The local symmetry of PMN-31{\%}PT is not defined by any reported phases and the symmetry of PMN-31{\%}PT seen in X-ray diffraction is the average of local structures