90 BELL SYSTEM TECHNICAL JOURNAL 



chance for heat to equalize and consequently the elastic constants operative 

 are the adiabatic constants determined by the fact that no heat is added 

 or subtracted from any elemental volume. For gases there is a marked 

 difference between the adiabatic and the isothermal constants, but for 

 piezoelectric cr^'stals the difference is small and can usually be neglected. 



To investigate the relation existing we can write from the first and second 

 laws of thermodynamics, the relations 



dV = [Ti dSi 4- T2 dS2 + T3 dSs 



(31) 

 + T, dSi + Ts dS, + 7^6 dS,] -\-ed(r 



which expresses the fact that the change in the total energy U is equal to 

 the change in the potential energy plus the added heat energy dQ = Q da 

 where is the temperature and cr the entropy. Developing the strains and 

 entropy in terms of the partial differentials of the stresses and temperature, 

 we have 



dS, = ^^ dT, + ?i^ dT, + ^' dTs 



dTi dT2 ST. 



oli 01^ die oQ 



dS, = '^Ut. -h ^^' dT. + §' dn 

 oil 01 2 alz 





(32) 



do = l^ AT. + If AT, + If dT^ 

 all 01 2 01 i 



^^dT, + ^dT, + ^ dT, + ^dQ. 



dTi an - dTe ae 



The partial derivatives of the strains with regard to the stresses are readily 

 seen to be the isothermal elastic compliances. The partial derivatives of 

 the strains by the temperatures are the six temperature coefficients of ex- 

 pansion, or 



dSi dSi ... 



ae ' ae 



To evaluate the partial derivatives of the entropy with re.^pect to the 

 stresses we make use of the fact that U is a perfect difTerential so that 



dS\ da dS^ da ,,.-. 



