First and Second Order Equations for Piezoelectric 

 Crystals Expressed in Tensor Form 



By W. P. MASON 



Introduction 



AEOLOTROPIC substances have been used for a wide variety of elastic 

 piezoelectric, dielectric, pyroelectric, temperature expansive, piezo- 

 optic and electro-optic effects. While most of these effects may be found 

 treated in various publications there does not appear to be any integrated 

 treatment of them by the tensor method which greatly simplifies the method 

 of writing and manipulating the relations between fundamental quantities. 

 Other short hand methods such as the matrix method can also be used for 

 all the linear effects, but for second order effects involving tensors higher 

 than rank four, tensor methods are essential. Accordingly, it is the purpose 

 of this paper to present such a derivation. The notation used is that agreed 

 upon by a committee of piezoelectric experts under the auspices of the Insti- 

 tute of Radio Engineers. 



In the first part the definition of stress and strain are given and their inter- 

 relation, the generalized Hookes law is discussed. The modifications caused 

 by adiabatic conditions are considered. When electric fields, stresses, and 

 temperature changes are applied, there are nine first order effects each of 

 which requires a tensor to express the resulting constants. The effects are 

 the elastic effect, the direct and inverse piezoelectric effects, the temperature 

 expansion effect, the dielectric effect, the pyroelectric effect, the heat of 

 deformation, the electrocaloric effect, and the specific heat. There are 

 three relations between these nine effects. Making use of the tensor trans- 

 formation of axes, the results of the symmetries existing for the 32 types of 

 crystals are investigated and the possible constants are derived for these 

 nine effects. 



Methods are discussed for measuring these properties for all 32 crystal 

 classes. By measuring the constants of a specified number of oriented cuts 

 for each crystal class, vibrating in longitudinal and shear modes, all of the 

 elastic, dielectric and piezoelectric constants can be obtained. Methods 

 for calculating the properties of the oriented cuts are given and for deriving 

 the fur.damental constants from these measurements. 



1 For example Voigt, "Lehrl)uch der Kiistall Physik," B. Tcul)ner, 1910; Wooster, 

 "Crystal Physics," Cainl)ridge Press, 1938; Cady "Piezoelectricity" McGraw Hill, 1946. 

 * The matrix method is well described 1)V W. L. Bond "The Mathematics of the Ph\sical 

 Properties of Crystals," B. S. T. J., Vol. 22, pp. 1-72, 1943. 



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