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# Piezoelectricity

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:
Pk = dkijTij

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:
Sij = dijkEk + QijklEkEl

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, ${\bar 4}$,422, 4mm ${\bar 4}$2m, 6, ${\bar 6}$,622, 6mm ${\bar 6}$2m

23, ${\bar 4}$3m

Quartz, of point group 32, is the most widely used piezoelectric crystal.

### History

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. Paris91, 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. Paris93, 1137-1140, Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées].

### See also

Section 10.2 of International Tables of Crystallography, Volume A
Section 1.1.4 of International Tables of Crystallography, Volume D