Piezoelectric Components in Anisotropic Systems

From the definitions of the piezoelectricity (see 8.41 and 8.42), and from the Curie principle, it follows that d/ jf. will transform as the product of Xj, Xy Accordingly, in a system with inversion symmetry, transformations like x -X, y -y, and z -z, would require that d for any i,j,k, which 

In general, the number of the possibly nonzero components can be determined for each crystallographic group. In listing these, we treat separately those that allow, and those that exclude polar order. We will follow the so-called Schoenfliess notation.  

Those symmetry groups which do not permit polar order are the following  

For those who are not familiar with the Schoenfliess, or other notations, we recommend for example E.A. Wood, Crystals and Light, 2nd ed., Dover Publications, New York, 1977. 

The polar groups have ten elements. Four of them are the Cy Cy Cj, C4 and Cg, corresponding to systems with two-fold, three-fold, four-fold and six-fold symmetry axes (say, in the z direction), respectively. In the classes = 1/ 2,3,4, 6) the xz plane is the plane of symmetry, and in z is the perpendicular to the plane of symmetry. 

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