FIEZOELECTKIC CRYSTALS IN TENSOR FORM 129 



Class 1 : components pi , p-i , ps ■ 



Class 3 : I' axis of binary symmetry, components 0, p2 ,0 (158) 



Class 4: components pi , 0, ps . 



Classes 7, 10, 14, 16, 20, 23, and 26: components 0, 0, pz ; and Classes 

 2, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 19, 21, 22, 24, 25, 27, 28, 29, 30, 31, and 

 32: components 0, 0, 0, i.e., /> = 0. 



For a hydrostatic pressure, the stress tensor has the components 



Tn = T22= Tss^ —p = pressure; T12 = Tn = ^23 = (159) 

 Hence the displacement equations of (156) can be written in the form 



K = '4^ Em- <^np + pldQ (160) 



where 



<^np = dnlJn + d n22T22 + <^n33?'3.3 



that is the contracted tensor d nkkTkk ■ This is a tensor of the tirst rank 

 which has the same components as the pyroelectric tensor pn for the various 

 cPv'stal classes. 



7. Second Order Effects in Piezoelectric Crystals 

 We have so far considered only the conditions for which the stresses and 

 tields are linear functions of the strains and electric displacements. A 

 number of second order effects exist when we consider that the relations are 

 not linear. Such relations are of some interest in ferroelectric crystals such 

 as Rochelle salt. A ferroelectric crystal is one in which a spontaneous 

 polarization exists over certain temperature ranges due to a cooperative 

 effect in the crystal which lines up all of the elementary dipoles in a given 

 "domain" all in one direction. Since a spontaneous polarization occurs in 

 such crj'stals it is more advantageous to develop the equations in terms of 

 the electric displacement rather than the external field. Also heat effects 

 are not prominent in second order effects so that we develop the strains and 

 potentials in terms of the stresses and electric displacements D. By means 

 of McLaurin's theorem the first and second order terms are in tensoi form 



_ dSij dSij 1 r d'^Sij 



^'' ~ dTkf ^'^ ^ a5„ ^" + 21 IdTkCdT^r ^'^^'^ 



+ 2 „„ „j TkCdn + rr^r 6„5o + ■ • • higher terms 



d'E„ 





(161) 



dTktdTn 



TklT^r 



d^Em d^E„, 1 



+ 2 ^^T^T ^i<^^" + ^777 5„5o + • • • higher terms 



dTktdSn d5„d5o 



whereas before 8 = D/4ir 



