PIEZOELECTRIC CR VST A LS IN TENSOR FORM 99 



obtained by setting dQ equal to zero in the last of equations (58) and elim- 

 inating dQ from the other nine equations. The resulting equations are 



Bm = dim Ti + d^m T2 + dzm Ti + dim Ti (60) 



+ d,m n ^ d^T,+ '^ El + ^' £2 + '-^^ £3 

 47r 4t 47r 



where the symbol a for adiabatic is understood and where the relations 

 between the isothermal and adiabatic constants are given by 



E E (^ B .T f^ T,a T,Q l.T .T r\ 



Hence the piezoelectric and dielectric constants are identical for isothermal 

 and adiabatic conditions provided the crystal is not pyroelectric, but differ 

 if the crystal is pyroelectric. The difference between the adiabatic and 

 isothermal elastic compliances was discussed in section (1.4) and was shown 

 to be small. Hence the equations in the form (60) are generally used in 

 discussing piezoelectric crystals. 



Two other mixed forms are also used but a discussion of them will be 

 delayed until a tensor notation for piezoelectric crystals has been discussed. 

 This simplifies the writing of such equations. 



3. General Properties of Tensors 



The expressions for the piezoelectric relations discussed in section 2 can 

 be considerably abbreviated by expressing them in tensor form. Further- 

 more, the calculation of elastic constants for rotated crystals is considerably 

 simplified by the geometrical transformation laws established for tensors. 

 Hence it has seemed worthwhile to express the elastic, electric, and piezo- 

 electric relations of a piezoelectric crystal in tensor form. It is the purpose 

 of this section to discuss the general properties of tensors applicable to 

 Cartesian coordinates. 



If we have two sets of rectangular axes (Ox, Oy, Oz) and (Ox', Oy' , Oz) 

 having the same origin, the coordinates of any point P with respect to the 

 second set are given in terms of the first set by the equations 



x' — (iX -\- miy -\- Jhz 



y' = lix -\- m^y + «22 (61) 



z' = I3X + m^y -\- HiZ. 



