Piezoelectric tensor

The linear piezoelectric tensor relating polarization and stress in x-cut quartz is... 

This crystal, which belongs to the space group monoclinic C2, is associated with the piezoelectric tensor (chain axis = c-axis) ... 

Example 15.4-3 Both the piezoelectric effect and the Pockels effect involve coupling between a vector and a symmetric 7(2). The structure of K is therefore similar in the two cases, the only difference being that the 6 x 3 matrix [rqi is the transpose of the 3x6 matrix [diq where i 1, 2, 3 denote the vector components and q= l,. .., 6 denote the components of the symmetric 7(2) in the usual (Voigt) notation. Determine the structure of the piezoelectric tensor for a crystal of C3v symmetry. 

Determine the independent terms of the piezoelectric tensor, the vectors Cj and 2 of the coordinate system should be chosen parallel to the twofold axes and 63 parallel to the 4 axis. If a square crystal plate of KH2PO4 is cut in such a way that the sides are parallel to the twofold axes and an electric field of 3(K)0 V/cm is applied to 3, i.e. perpendicular to the plate, determine the deformation of the plate. 

Let us calculate the nanowire piezoelectric reaction to electric field Es applied along polar axes z. Using the calculated elastic field, one could find piezoelectric reaction as dkij = duij /dEk. One of fhe nonfrivial consequences of fhe flexoeffect is the local appearance of new piezoelectric tensor components, related to the unit cell deformation (see Fig. 4.23a), absent in the bulk system ... 

Here s are the elastic strains, a are the stresses, E are the electric potentials and D are the electrie displacements. [C] represents the adiabatic elastic compliance tensor at constant electric filed, [d] is the adiabatic piezoelectric tensor and [p] is the adiabatie electrie permittivity at constant stress. From these constitutive equations, it is readily known that the pie-zoelectrie element generates eleetrie signals due to mechanical motions and vice versa. 

Symmetry point group Dielectric tensor Elastic tensor ors , Piezoelectric tensor dimn Elasto-optic tensor Pmn Electrooptic tensor rmk Nonlinear susceptibility tensors x > 1 X< > Table number... 

General Static dielectric tensor Elastic stifEhess tensor Piezoelectric tensor Elastooptic tensor... 

General Static dielectric tensor Dissipation factor Elastic stiffness tensor Piezoelectric tensor EUastooptic tensor... 

A deformation of the smectic C structure should produce variation in the tilt angle, 0, and therefore in the polarization, P. Thus, a piezoelectric response is generated and can be measured. To date, only two components of the piezoelectric tensor have been studied for FLCPs (Fig. 23) the cl coefficient, which describes the electric response in the z direction to the acoustically induced deformation along... 

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... 

Primary as well as secondary pyroelectric effects are not allowed in polar-neutral crystals due to the symmetry. We can demonstrate this situation on secondary pyroelectric effects for a-quartz, where the thermal expansion as well as piezoelectric tensor is non-zero. Using the form of material tensors for thermal expansion and piezoelectricity for symmetry class 32 we can get... 

The third-order permittivity is described by the third rank tensor Its symmetry properties are similar to those of the piezoelectric tensors. In particular all tensor coordinates vanish for all centro-symmetric crystal classes and the class 432. However, all three indices are interchangeable. As a consequence of this either the first two or the last two indices may be contracted to a single matrix index which allows representation in a 3 by 6 matrix. This offers advantages for tabulation of data but has httle merit otherwise since there is no corresponding matrix index for the respective variables. 

Calculation of the desired crystal cut orientation is simple. Let us suppose quartz cut (XYa)f (working in transversal mode) with the lengthy-axis rotated by an angle I about the crystallographic x-axis. The rotated components of the piezoelectric tensor can be calculated using Eqs. (5.10) and (5.15)... 

There is only one independent piezoelectric coefficient for -quartz. Structure of the piezoelectric tensor components is following... 

While the piezoelectric tensor of the point symmetry group 622 contains indeed four non-zero components, the longitudinal piezoeffect is missing. 

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