QUARTZ CRYSTAL APPLICATIONS 217 



electric and elastic constants for crystals with their three edge dimensions 

 along the three crystallographic axes. Most low-coefficient crystals, how- 

 ever, are oriented crystals with one or more of their edges lying along 

 directions not parallel to the crystallographic axes. The theory of elasticity, 

 however, provides methods for calculating the values of the constants for 

 rotated axes. If the rotated axes X' , Y' , Z' are related to the crystallo- 

 graphic axes X, Y, and Z by the relation 



XY Z 



X' 

 Y' 

 Z' 



4 ^3 nz 



where A , • • • , Ws are the direction cosines between the axes indicated, the 

 theory of elasticity provided relations between the stresses of the rotated 

 axes and the stresses of the crystallographic axes, between the strains of 

 the rotated axes and the strains of the crystallographic axes, and between 

 the field, polarizations, or charges of the rotated axes and the same quantities 

 for the crystallographic axes. Then if we express the relation between the 

 stress, strain and fields for the rotated axes, the elastic and piezoelectric 

 constants are determined. 



Two shorthand methods are also available for calculating the constants 

 of rotated crystals. One method is the matrix method which is based 

 upon the fact that relations in (A. 6) can be expressed in a matrix equation 



-e = s^X-\-dE (A.56) 



where e are the strain components, X the stress components, s^ the elastic 

 compliance matrix, d the piezoelectric matrix and E the field components. 

 By applying the rules of matrix multiplication the .y and d matrices can be 

 transformed to rotated axes having the direction cosines of equation (A. 57) 

 with respect to the crystallographic axes. The other method is the method 

 of tensor analysis. Equations (A. 6) can be expressed in the form 



— iij = SijaffXaQ ~r dijyXLv (A. 57) 



where e,-,- is the second rank strain tensor, Xa^ the second rank stress tensor, 

 Sija& the fourth rank compliance tensor, E^ the field vector, and (/i,v the third 

 rank piezoelectric tensor. By employing the geometric rules for tensor 



^ This method of determining the constants for rotated axes is discussed in a former 

 paper "Dynamic Measurements of the Constants of Rochelle Salt," Phys. Rev., April 15, 

 1939, Appendix I. 



* This method is discussed in a recent paper by W. L. Bond, "The Mathematics of 

 The Physical Properties of Crystals," B. S. T. /., Jan. 1943. 



' The tensor method of writing the elastic and piezoelectric relations is discussed by 

 Atanasoff and Hart and by Lawson. See references (2) and (3). 



