128 BELL SYSTEM TECH NIC A L JOURNA L 



effect. Finally the third equation of (155) shows that there is a reciprocal 

 efifect in which a stress or a displacement generates a change in temperature 

 even in the absence of added heat dQ. These effects can be called the strain 

 temperature and charge temperature effects. 



The second form of the piezoelectric equations given by (58) are more 

 familiar. In tensor form these can be written 



Sij = sfjktT.cl + dZijEm + afy do 



8n = dlk( Tk( + '4^E^ + pi dQ (156) 



47r 



dQ = eda ^ QatcTut + QplErr, + pCl dS 



The afy are the temperature expansion coefficients measured at constant 

 field. In general these are a tensor of the secjnd rank having six com- 

 ponents. The constants pn are the pyroelectric constants measured at 

 displacements which relate the increase in polarization or surface charge 

 due to an increase in temperature. They are equal to the so-called "true" 

 pyroelectric constants which are the polarizations at constant volume caused 

 by an increase in tempeiature plus the "false" pyioelectric effect of the 

 first kind which represents the polarization caused by a uniform temperature 

 expansion of the crystal as its temperature increases by dQ. As mentioned 

 previously it is more logical to call the two effects the pyroelectric effects 

 at constant stress and constant strain. By eliminating the stresses from 

 the first of equations (156) and substituting in the second equation it is 

 readily shown that 



Pn = Pn — OC^,enij (157) 



Hence the difference between the pyroelectric effect at constant stress and 

 the pyroelectric effect at constant strain is the so-called "false" pyroelectric 

 effect of the first kind a^je^a . 



The first term on the right side of the last equation is called the heat of 

 deformation, for it represents the heat generated by the application of the 

 stresses TkC ■ The second term is called the electrocaloric effect and it 

 represents the heat generated by the application of a field. The last term 

 is p times the specific heat at constant pressure and constant field. 



The temperature expansion coefficients a.-y form a tensor of the second 

 rank and hence have the same components for the various crystal classes 

 as do the dielectric constants shown by equation (134). 



The pyroelectric tensor pn and /?'„ are tensors of the first rank and in 

 general will have three components pi , p2 , and Ps . In a similar manner 

 to that used for second, third and fourth rank tensors it can be shown that 

 the various crystal classes have the following comi)onents for the first rank 

 tensor />,. . 



