Piezoelectricity is the property presented by certain materials that exhibit an electric polarization when submitted to an applied mechanical stress such as a uniaxial compression. Conversely, their shape changes when they are submitted to an external electric field; this is the converse piezoelectric effect. The piezoelectric effect and the converse effect are described by third-rank tensors:
- For a small stress, represented by a second-rank tensor, Tij, the resulting polarization, of components Pk , is given by:
where dkij is a third-rank tensor representing the direct piezoelectric effect.
- For a small applied electric field, of components Ek, the resulting strain, represented by a second-rank tensor, Sij, is given by:
where the first-order term, dijk, represents the inverse piezoelectric effect and the second-order term, Qijkl, a symmetric fourth-rank tensor, the electrostriction effect. The sense of the strain due to the piezoelectric effect changes when the sign of the applied electric field changes , while that due to electrostriction, a quadratic effect, does not.
The matrices associated to the coefficients dkij and dkij of the direct and converse piezoelectric effects, respectively, are transpose of one another.
Piezoelectric point groups
The geometric crystal classes for which the piezoelectric effect is possible are determined by symmetry considerations (see Curie laws). They are the non-centrosymmetric classes, with the exception of 432. The 20 piezoelectric crystal classes are therefore:
1, 2, m, 222, 2mm,
3, 32, 3m, 4, ,422, 4mm, 2m, 6, ,622, 6mm, 2m
Quartz, of point group 32, is the most widely used piezoelectric crystal.
It is considerations of symmetry that led the brothers Jacques (1855-1941) and Pierre Curie (1859-1906) to the discovery of piezoelectricity in materials such as tourmaline, quartz, boracite, sodium chlorate, Rochelle salt [Curie J. and Curie P. (1880), C. R. Acad. Sci. Paris, 91, 294-295,Développement, par pression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées]. The inverse piezoelectric effect was predicted by Lippmann G. [(1881), Ann. Chim. Phy. 24, 145-178, Principe de conservation de l'électricité] and discovered by Curie J. and P. [(1881), C. R. Acad. Sci. Paris, 93, 1137-1140, Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées].
Section 10.2 of International Tables of Crystallography, Volume A
Section 1.1.4 of International Tables of Crystallography, Volume D