PIEZOELECTRIC CR VST A LS IN TENSOR FORM 113 



In a fourth rank tensor such as Cijk(, Sijkt, applying the tensor trans- 

 formation equation 



_ dXi dXj dXk dxe . . 



'^*^tn ^'^n v'V'o ""vp 



and the condition (113) we similarly find 



Cl6 = Cl6 = ^25 = C26 = C35 = C36 = C45 = Ca = 0. (120) 



If the binary axis had been the Y axis the corresponding missing terms 

 can be obtained by cyclically rotating the tensor indices. The missing 

 terms are for the second, third and fourth rank tensors, transformed to 

 two index symbols. 



Cu , Cl6 , C24 , C26 , C34 , C36 , C45 , C55 . 



Similarly if the Z axis is the binary axis, the missing constants are 



ei3 , fi2 ; hn , hn , Ihz , hn , hi , h^ , ha , A26 , hzi , hzf, ; 



(121) 



(122) 



Cu , CiB , C2A , C25 , Czi , C35 , C46 , Cb6 • 



Hence a cr>'stal of the orthorhombic bisphenoidal class or class 6, which 

 has three binary axes, the X, Y and Z directions, will have the remaining 

 terms, 



Cu , ^22 , ^33 ; hu , ^25 , ^'36 ', Cn , Cn , Cl3 , C21 , C23 , C33 , C44 , C55 , Cee (123) 



with similar terms for other tensors of the same rank. Rochelle salt is a 

 crystal of this class. 



If Z is a threefold axis of symmetry, the direction cosines for a set of 

 axes rotated 120° clockwise about Z are, 



f I = --- = - .5 ; wi = -— = - .866 ; «i = t— = 

 oxi 0X2 dXz 



^3 = ^^ = .866; m2=^=-.5; «2 = ^^ = (124) 



0x1 0x2 0x3 



, dx'z dx'z ^ dx'z 



4 = — -=0; m3=-— = 0; riz = ^— = 1. 



dxi 0X2 0X3 



Applying these relations to equations (114) for a second rank tensor, we 

 find for the components 



€11 = .25eii+ .433ei2+ •75e22 ; ei2 = —. 433 cu + .25 €12 + .433^22 



ei3 = — -Seis — .866e23 ; €22 = .75€u — .433ei2 + .25c22 (125) 



€23 = .866 en — .5e2j ; €33 = €33 • 



