40 BELL SYSTEM TECHNICAL JOURNAL 



so that: 



S° - S = 4Trdck~'d (12.5) 



The change in strain caused by rendering the surfaces conducting is: 



e° - e= (S° - S)X = 4Td,.k~'dX (12.6) 



If the crystal be now insulated and the stress removed, an induction of 

 opposite sign will occur and because of the assumed linear dependence of D 

 on X the new induction will be equal to the negative of the previous one. 

 The induction D = —dX indicates an electric field: 



E = Airk'^D = 4Tk~'dX (12.7) 



Also, the strain will alter by an amount —e", where, since the action takes 

 place with non-conducting surfaces: 



e" = SX 

 This leaves a strain on the crystal, of amount: 



e' = e° - e" = (S° - S)X (12.8) 



From (12.6), (12.7) and (12.8) it follows that: 



e' = dcE (12.9) 



As the medium is in just the condition that an electric field E would put the 

 unstressed medium, (12.9) is the equation of the converse piezo-electric 

 effect. It is to be noted that the set of constants that relates polarization 

 and stress is the conjugate of the set that relates electric field and strain. 

 For convenience in notation the converse effect will be written as 



, e = gE (12.10) 



where 



g = dc (12.11) 



Rewriting (13) as a7 ^ = («7 ga^ )aE we see that 



e' = g'E' 

 where 



g' = c-'ga (12.12) 





