PIEZOELECTRIC CR YSTA LS IX TENSOR FORM 125 



and transforming to a rotated system of axes whose direction cosines are 

 given by (142), the resulting equation becomes 



(147) 



, _ £ dx'i dx'j dXk dxt rp' , J dXi dx, dXm j^i 

 »'■ ~ ^''''^ ^. ^ f)r[ f)r'. ^^ '"* ax- ax- 'ax' ' 



OXi OXj OXk OXf UXt VXj UXm 



./ _ emn dXn dXm 77' i j , ^^n dXk dx( f 

 47r daPn OX,n OXn OXk dX( 



All the stresses except Tn can be set equal to zero and all the fields except 

 Ez vanish. Furthermore, all the strains are dependently related to ^n . 

 Hence for a thin longitudinal crystal the equation of motion becomes 



, _ £ dx[ dx'i dXk dxt rp' , . dx[ dxi dx„ / 



"^*'^'^ dx- dx- dx'y dx^ '"" dx-dX-dx{ ' 



. , , (148) 



./ _ c^ ^ 5^ p' a: J dxzdxkdx( , 



47r 5x;, ^jcs " 5x„ 5a;i dx'i 



In terms of the two index symbols for the most general type of crystal, we 

 have 



E' E' £ /)4 I /^ E I E \ i)2 2 1 /T £ I E \ el 2 



51111 = ^11 = SiiW + (2^12 + 566)^1^^1 + (2^13 + 55b)4Wi 



+ 2{Sii + 5f6)^iWl«l + Isf^Vh + 25f6AWi + 5^2^! 



+ /0 £ I jB \ 2 2 I r. E 3 , r,/ E , E\ Iff 



(isiz + summi + isufmni + 2(^25 + 546)wi^i?h 



2s26fniCi + 533W1 + IsziHinii + 2536^1^1 



+ 2(5^6 + 5f5)«iAwi 



(149) 

 ! din = dn = dn^sd + du^ml + ^is^^i + dutzmiiti + dif^t^itii 



I + dwtilinii + diinizli + doomm + d^sntsfii + dumsmifh 



I + </25«3A"i + di^niztinii 4- c?3i"3^i + dsiUzml + dzs,mn\ 



i + dummini + dziUzkni + dz&nz^inii 



\ €33 = «ii4 + leiitzmz + 2i.iz(znz + €22^3 + 2€23W3W3 + €33^3 



I Hence by cutting 18 crystals with independent direction cosines 9 elastic 

 constants and 6 relations between the remaining twelve constants can be 



I determined. All of the piezoelectric constants and all of the dielectric 

 constants can be determined from these measurements. These constants 

 can be measured by measuring the resonant and antiresonant frequencies 



\ and the capacity at low frequencies. The resonant frequency Jr is deter- 



I mined by the formula 



h = Yi V^ ^^^^ 



^^ y psii 



