98 BELL SYSTEM TECH NIC A L JOURNA L 



+ ^2^2 + (tzEz + a^Je 



5m = (iimTl + dirnT-i. + d^mTz + dimTi + d^^Th + d^^Te 



+ |l£, + ^|l £, + !pi £3 + /'Ic/e (58) 

 47r 47r 47r 



</^ = 9 (/o- = 6[ai Ti + Q!2 7^2 + af Ts + af 7^4 + af Ts + af rej 



+ eiplE, + Pa'^Es + plE,] + />C^(/e. 



w = 1 to 6, m = 1 to 3 



The superscripts E, 0, and T indicate respectively constant field, constant 

 temperature and constant stress for the measurements of the respective 

 constants. It will be noted that the elastic compliance and the piezo- 

 electric constants d^n are for isothermal conditions. The a^ constants are 

 the temperature expansion constants measured at constant field, while the 

 p^ constants are the pyroelectric constants relating the ratio of 5 == D/47r 

 to increase in temperature ^6, measured at constant stress. Since there is 

 constant stress, these constants take into account not only the "true" pyro- 

 electric effect which is the ratio of 5 = Z>/47r to the temperature at constant 

 volume, but also the so called "false" pyroelectric effect of the first kind 

 which is the polarization caused by the temperature expansion of the crystal. 

 This appears to be a misnomer. A better designation for the two effects 

 is the pyroelectric effect at constant strain and the pyroelectric effect at 

 constant stress. Cp is the specific heat at constant pressure and constant 

 field. 



If we substitute these equations into equation (42), the total free energy 

 becomes 



!^ = E Z s^nTmTn + 2 ^^ Xl d'toT^Eo 4- 2 i; a'„Tje 



n = l 0=1 

 3 T,e 



+ Z E ^ £o£, + 2 E PoEpde + ^^ ^e. 

 0=1 p=i 47r 0=1 t) 



Equation (58) can then be obtained by partial derivatives of the sort 



at/ _ d£ _dQ dU 



(59) 



dTn' dEp' e d(de)' 



By tensor transformations the expression for U in (59) can be shown to 

 be equal to the expression for U in (55). 



The adiabatic equations holding for a rapidly vibrating crystal can be 



