# Piezoelectricity

### From Online Dictionary of Crystallography

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Piezoélectricité (*Fr*). Piezoelectrizität (*Ge*). Piezoelectricidad (*Sp*). Piezoelettricità (*It*)

## Definition

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 connverse piezoelectric effect. The piezoelectric effect and the converse efect are described by third-rank tensors:

- For a small stress, represented by a second-rank tensor,
*T*, the resulting polarization, of components P_{ij}, is given by:_{k}

*P*=

_{k}*d*

_{kij}T_{ij}where *d _{kij}* is a third-rank tensor representing the direct piezoelectric effect.

- For a small applied electric field, of components
*E*, the resulting strain, represented by a second-rank tensor,_{k}*S*, is given by:_{ij}

*S*=

_{ij}*d*+

_{ijk}E_{k}*Q*

_{ijkl}E_{k}E_{l}where the first-order term, *d _{ijk}*, represents the inverse piezoelectric effect and the second-orer term,

*Q*, 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.

_{ijkl}The matrices associated to the coefficients *d _{kij}* and

*d*of the direct and converse piezoelectric effects, respectively, are transpose of one another.

_{kij}## 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, 2*mm*,

3, 32, 3*m*, 4, [math]{\bar 4}[/math],422, 4*mm*, [math]{\bar 4}[/math]2*m*, 6, [math]{\bar 6}[/math],622, 6*mm*, [math]{\bar 6}[/math]2*m*

23, [math]{\bar 4}[/math]3*m*

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 on 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*.*

## See also

An introduction to crystal physics (Teaching Pamphlet of the *International Union of Crystallography*)

Section 10.2 of *International Tables of Crystallography, Volume A*

Section 1.1.4 of *International Tables of Crystallography, Volume D*