MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 



41 



SECTION 13 

 The Converse Piezo-electric Effect as a Non-Linear Function 



If the strain of a crystal is not strictly a linear function of the electric field 

 causing it we must relate the components of strain to field terms of the sec- 

 ond power as well as to first power terms. That is, the equation e = gE 

 (which gives the strain e in terms of the electric field E through the 18 

 constants g) must be modified to include terms EiEj . All such terms are 

 included in the symmetric matrix (E Ec). 



A transformation a that replaces E by a,-E' also replaces Ec by Eca so 

 that (£ Ec) is replaced by (a<.£ EcO), that is (£ Ec) being self-conjugate, 

 transforms similarly to the stress matrix. We may rearrange this as a one 

 column matrix similar to the stress matrix X, as follows: 



{E Ec) = 



E{ 



El 



El 



EiEz 



EiEi 



E1E2 



E 



(13.1) 



We may now relate the strain to E and E through the two matrices g 

 and G: 



e = gE + GE (13.2) 



If transformations permitted by the symmetry of the crystal are per- 

 formed, g' must equal g and G' must equal G, this allows us to simplify the 

 matrices; g is no different than before and hence vanishes for all types having 

 centers of symmetry (and for the pentagonal icositetrahedral class). 



Rewriting (1) as aj ^ = («7 gcT )aE -f aj GaT^aE we see that 



g'E' -f G'E 



where 



= oio^ga 



G' = ac Gc 



(13.3) 



The matrix G transforms as the elastic modulii matrix does but G,,- 5^ 

 Gji . Applying G' = UrGa we arrive at the set of matrices that follow 



