MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 



43 



Cubic (Classes 28, 29, 30, 

 31 and 32) (5 Constants) 

 Gil Gn Gi2 

 Gi2 Gil Gi2 

 Gi2 Gi2 Gil 

 G44 

 G44 

 G44J 



(13.12) 



Isotropic Bodies 

 (2 Constants) 



Gil G12 Gi2 ^ 

 G12 Gil G12 

 G12 G12 Gu 

 GOO 

 G 

 G 

 G = 2(Gii -G12) 



(13.13) 



According to this analysis, all bodies suffer a change in dimensions when 

 subjected to an electric field. These strains resulting from a field are 

 generally much smaller than those strains e = gE present only in crystals 

 lacking a center of symmetry. For example, quartz has a strain of about 

 6.5 X 10"^ cms/cm/ah volt. Glass in a field of 1000 practical volts per cm 

 has a strain of about 4 X 10"^^ in a 100,000 volt field it has 4 X 10~l 

 Rubber in the 1000 volt field strains by about 7 X 10~* and in the 100,000 

 volt field by about 7 X 10~\ The 1st order quartz strain in these fields 

 would be about 2.2 X 10~ and 2.2 X 10~^ respectively. 



The Second Order Piezo-electric Effect 



If the induction stress relation is not strictly linear one can assume the 

 induction to depend also on second order terms of the stress: 



D= dX-^ p (XX,) 



where (XXc) is a single column matrix formed from the 21 elements of 

 XXc and p is a. matrix of the 63 elements pn.i ■ ■ ■ ^33,3- 



Since X transforms as X' = aX', (XXc) transforms as X'Xc = aXXcac. 

 In the same way that a was formed from a we can form a matrix a that 

 transforms the single column matrix (XXc) through (XXcY = oi{XXc)'- 



aD = ada~^ X + ap{a)~^ a{XXc) or 



D' = d'X' + p'iXXc)' 



where 



d' = ada and p' = ap{a)' 



The first order effect is the same as before. With the relation p' = ap 

 {a)~^ we could perform the operations of symmetry permitted by the 32 

 crystal classes and obtain the reduced matrices. However since a has 484 

 elements we shall limit ourselves to crystals with centers of symmetry. 



As X is unchanged by an inversion through the origin, a is the idemfactor 

 for this transformation and a is —7, also (a) = /. Therefore D' = —D 

 = D so that D vanishes. 



