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BELL SYSTEM TECHNICAL JOURNAL 



for any long thin crystal vibrating longitudinally. Hence when the density 

 is known, Sn can be calculated from the resonant frequency and the length 

 of the crystal. Using the values of Sn obtained for 15 independent orienta- 

 tions enough data is available to solve for the constants of the first of 

 equations (149). The capacities of the different crystal orientations meas- 

 ured at low frequencies determine the dielectric constant 633 and si.x orienta- 

 tions are sufficient to determine the six independent dielectric constants 

 tmn ■ The separation between resonance and antiresonance Af = /a — Jr 

 determines the piezoelectric constant dn according to the formula 



d\i = 



;1/ 



£33 

 4^ 



^11 



(151) 



The \-alues of dn measured for 18 independent orientations are sufficient 

 to determine the eighteen independent piezoelectric constants. 



The remaining six elastic constants can be determined by measuring long 

 thin crystals in a face shear mode of motion. Since this is a contour mode 

 of motion, the equations are considerably more complicated than for a 

 longitudinal mode and involve elastic constants that are not constant field 

 or constant displacement constants. It can be shown that the fundamental 

 frequency of a crystal with its length along x\ , width (frequency determining 

 direction) along .Vo and thickness (direction of applied field) along xs , will be 



1 / c.E I c,E , a// c.E c,E\2 1 . c. 



{ = — i/ ^ 22 -\- C66 ± V (C22 — ^66 ) + 4C26 



^ 2C y 2p 



(152) 



where the contour elastic constants are given in terms of the fundamental 

 elastic constants by 



E E £2 



c.E -^ll •^66 ■^16 



C21 = ; 



E E E E 



c,E -^12 ■^16 •^11 -^26 



C26 = 1 



E E £2 



c.B _ SnS22 ^12 



C66 — : 



(153) 



where A is the determinant 



A = 



(154) 



Since all of the constants except svi and ^ee can be determined by measure- 

 ments on longitudinal crystals and the value of (25f2 + ^ee) has been de- 



' This is proved in a recent paper "Properties of Dipotassium Tartrate (DKT)- Crys- 

 tals," Phys. Rev., Nov., 1946. 



