Piezoelectricity linearly relates an induced polarization to an applied stress, as shown in (3.1), (3.1) Where σ jk is the applied stress, P i is the induced polarization, and d ijk is the piezoelectric charge coefficient. Einstein notation is used, where repeated indices are summed. Because piezoelectricity is a third-rank tensor property, a good starting point to understanding the crystal chemistry of piezoelectric materials is to consider the impact of symmetry on such a property. Neumann's law states that the geometrical representation of any physical property contains the symmetry of the point group of the material. As shown in Fig. 3.1, of the 32 crystallographic point groups, only 21 are noncentrosymmetric. Odd-rank tensor properties are symmetry forbidden in centrosymmetric structures, making piezoelectricity a null property for such materials. In the same way, in point group 432, the combination of symmetry elements eliminates piezoelectricity. The remaining 20 point groups are potentially piezoelectric. Of these 20 point groups, ten are polar, that is, they have a vector direction in the material that is not symmetry-related to other directions. Such materials can have a spontaneous polarization, which is typically a function of temperature. Thus, these materials are pyroelectric. Ferroelectric materials are a subset of pyroelectric materials in which the spontaneous polarization can be reoriented between crystallographically-defined directions by a realizable electric field. Thus, all ferroelectric materials are both piezoelectric and pyroelectric.
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