1 1 2 BELL SYSTEM TECH NIC A L JOURNA L 



result in a reduction of the number of dielectric, piezoelectric, and elastic 

 constants. 



Since the tensor equation is easily transformed to a new set of axes by 

 the transformaion equations (76) this form is particularly advantageous 

 for determining the reduction in elastic, piezoelectric and dielectric con- 

 stants. For example consider the second rank tensors, c^^ and ak( for the 

 dielectric constant and the expansion coefficients. Ordinarily for the most 

 general symmetry each tensor, since it is symmetrical, requires six inde- 

 pendent coefficients. Suppose however that the X axis is an axis of twofold 

 or binary symmetry, i.e., the properties along the positive Z axis are the 

 same as those along the negative Z axis. If we rotate the axes 180° about 

 the A' axis so that -f Z is changed into — Z, the direction cosines are 



(113) 



(114) 



Applying (113) to (114) summing for all values of k and / for each value of 

 i, and J we have the six components 

 ' ' _ ' _ ' _ ' _ ' _ ('1 1 -\ 



€11 — CU ; «12 — ~ €12 ; tl3 — — ei3 ; €22 — €22 ; ^23 — ^23 ', ^33 — ^33 • \ll^) 



Since a crystal having the A' axis a binary axis of symmetry must have the 

 same constants for a -\-Z direction as for a — Z direction, this condition 

 can only be satisfied by 



€12 = €13 = 0. (116) 



The same condition is true for the expansion coefficients since they form a 

 second rank tensor and hence 



«12 = «13 = 0. (117) 



In a third rank tensor such as dijk , enk , gnh , I' nk , we similarly find that 

 of the eighteen independent constants 



hm = //le ; //ii3 = //i5 ; /?2ii = /'2i ; //222 = /'22 ; //223 = hi ; 



(118) 



//233 = /'23 ; /'311 = //31 ', /'322 = /'32 ', Ihi^i — ll'M ', //333 — "33 • 



are all zero. The same terms in the dijk , ^nk , gnk tensors are also zero. 



