Maxwell relations of thermodynamics. For those piezoelectric crystals for which the polarization is of the crystal-field induced type, a formalism has been worked out that allows for the calculation of piezoelectrical coefficients dij from electrostatic lattice constants or higher-order Madelung constants. Crystal classes Of the 32 crystal classes, 21 are non-centrosymmetric (not having a centre of symmetry), and of these, 20 exhibit direct piezoelectricity (the 21st is the cubic class 432). Ten of these represent the polar crystal classes, which show a spontaneous polarization without mechanical stress due to a non-vanishing electric dipole moment associated with their unit cell, and which exhibit pyroelectricity. If the dipole moment can be reversed by applying an external electric field, the material is said to be ferroelectric. The 10 polar (pyroelectric) crystal classes: 1, 2, m, mm2, 4, , 3, 3m, 6, . The other 10 piezoelectric crystal classes: 222, , 422, 2m, 32, , 622, 2m, 23, 3m. For polar crystals, for which P ≠ 0 holds without applying a mechanical load, the piezoelectric effect manifests itself by changing the magnitude or the direction of P or both. For the nonpolar but piezoelectric crystals, on the other hand, a polarization P different from zero is only elicited by applying a mechanical load. For them the stress can be imagined to transform the material from a nonpolar crystal class (P = 0) to a polar one, having P ≠ 0. Materials Many materials exhibit piezoelectricity. Crystalline materials Langasite (La3Ga5SiO14) – a quartz-analogous crystal Gallium orthophosphate (GaPO4) – a quartz-analogous crystal Lithium niobate (LiNbO3) Lithium tantalate (LiTaO3) Quartz Berlinite (AlPO4) – a rare phosphate mineral that is structurally identical to quartz Rochelle salt Topaz – Piezoelectricity in Topaz can probably be attributed to ordering of the (F,OH) in its lattice, which is otherwise centrosymmetric: orthorhombic bipyramidal (mmm). Topaz has anomalous optical properties which are attributed to such ordering. Tourmaline-group minerals Lead titanate (PbTiO3) – Although it occurs in nature as mineral macedonite, it is synthesized for research and applications. Ceramics Ceramics with randomly oriented grains must be ferroelectric to exhibit piezoelectricity. The occurrence of abnormal grain growth (AGG) in sintered polycrystalline piezoelectric ceramics has detrimental effects on the piezoelectric performance in such systems and should be avoided, as the microstructure in piezoceramics exhibiting AGG tends to consist of few abnormally large elongated grains in a matrix of randomly oriented finer grains. Macroscopic piezoelectricity is possible