Crystallographic Symmetry Restrictions to Piezoelectricity

Generally, the piezoelectric effect could exist just in non-centrosymmetrical crystallographic symmetry classes. Mechanical stress/strain as a second-rank symmetrical tensors are basically centrosymmetrical external fields. If the materials crystallographic symmetry include cerrtre of symmetry operation, the resulting symmetry of material subjected to such field is also cerrtrosymmetrical (see Neuman s Law in Nye (1985)). Therefore, piezoelectric effect is excluded. Centrosymmetrical crystal stays centrosymmetrical even after the application of the mechanical stress and no polar direction for the polarization vector might exist in such stmcture. 

Among the 32 crystallographic classes in Table 2.1 only 21 of them are non-cen-trosymmetrical. Moreover in cubic non-cerrtrosymmetrical class 432, the piezoelectric coefficients are all equal zero because of symmetry. Totally, 20 piezoelectric crystallographic classes might exhibit non-zero piezoelectric coefficients. 

In polar crystals, the piezoelectric polarization generated as a result of mechanical stress application will contribute to the spontaneous polarization existing previously. In polar-neutral crystals, the polar directions are mutually compensated . As a result of mechanical stress application, singular polar direction appears in such crystals. Piezoelectric polarization is generated in that direction and crystal is piezoelectrically polarized. The only exception among the polar-neutral classes is cubic 432 class, where all piezoelectric coefficients are identically equal zero because of symmetry (Zheludev 1975). 

Crystallographic symmetry results in some constraints to the tensor components of any material property of the crystal. Tensor components transformation, corresponding to any symmetry element, must not resnlt in any change of the material property tensor. Piezoelectric effect is described by third-rank tensor. According to the general transformation mle (see Eq. (2.39)) applied to piezoelectric tensor 

The independerrt componerrt s stmctrrre of the piezoelectric coefficient is more clearly seen in rrratrix than in tensor notation. Symmetry of the piezoelectric tensor reflects syrrrmetry of mechanical stress/strain (they are secorrd-rarrk syrrrmet-rical tensors). Piezoelectric coefficient is therefore third-rarrk terrsor syrrrmetrical with respect to the permutation of two indexes. Piezoelectric coeffiderrts satisfy following relations between tensor dyk and matrix di coefficients 

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