PIEZOELECTRIC CRYSTALS IN TENSOR FORM 



S9 



Si = 511^1 + 512^2 + SuTz + SuTi + 51575 + Sy^Ti, 



Si = S21T1 -\- S22T2 + 523^3 + S^iTi + 5257^6 + 526^6 



53 = S31T1 + 532^2 + 533^3 + 53474 + 53575 + 53676 



54 = 54i7i + 54272 + 54373 + 5447i + 54575 + 54676 

 '^'5 = S^iTi -\- Sf,iTl -\- 55373 + 55474 + 55575 + ^6676 

 Si = 56l7i + 56272 + 56373 + 56474 + 56575 + 56676 



(26) 



Inhere 



i+i 



Sii = 



_(-i)'"^A:y 



(27) 



for which A*^ is the determinant of the dj terms of (28) and'A^y the minor 

 obtained by suppressing the ith andjth columo 



A'^ = 



<"ll Ci2 Ci3 Cu '"15 <^16 



^12 ^22 <r23 Cu C25 ^26 



Cl3 C23 ^33 C34 <"36 ^36 



ri4 C24 C34 f44 C45 C46 



^15 ^25 <"35 Cib Cbb ^56 



^16 <^26 ''36 ^46 C{,( Ce6 



(28) 



Since c.-y = cy, it follows that 5,y = 5y,. The potential energy can be 

 expressed in the form. 



27£ = 5ii7? + 2S12T1T2 + 25i37\73 + IsuTiTi + 25i57i76 + 25i67i76 



+ 52272 + 2S23T2T3 + 25247274 + 2S26T2T5 + 2S2iT2T ^ 

 + •^3373 + 253^X3X4 -\- 2S3bT3Tb + 25367376 



+ 54474 + 25457475 + 25467476 (29) 



+ 55575 + 2sb%Ti,Ti 

 -\- SbbTe- 

 The relations (26) can then be derived from expressions of the type 



5i = 



dPE 



S, = 



dPE 



(30) 



dTi ' ' "" 576 



1.4 Isothermal and Adiabatic Elastic Constants 



We have so far considered only the elastic relations that can be measured 

 statically at a constant temperature. The elastic constants are then the 

 isothermal constants. For a rapidly vibrating body, however, there is no 



