QUARTZ CRYSTAL APPLICATIONS 207 



Equations (A. 10) might also have been obtained directly from equations 

 (A. 8) by substituting the charges from the last three equations in terms of 

 the fields. This substitution yields the additional relations 



(A.12) 



(A. 2). Values or the Elastic and Piezoelectric Constants 



The first and one of the best determinations of the elastic constants of 

 quartz was made by Voigt. Using static deformations of unplated crystals 

 he determined the elastic constants to be 



Cn = 85.1 X IQio dynes/cm2; cjo = 6.95 X IQi"; 



c,3 = 14.1 X IQi"; Cu = 16.8 X W 



C33 = 105.3 X 10'"; Cu = 57.1 X 10'"; (A-13) 



(Cn — Cii\ 



C66 = ( " 2 ) = ^^-^ ^ ^^ 



From these the moduli of compliance can be calculated and are 



511 = 129.8 X 10-1* cmVdyne; ^12 = -16.6 X lO-^"; 

 5i3 = -15.2 X 10-'4; 514 = -43.1 X 10"'^ 

 533 = 99.0 X 10-14; su = 200.5 X lO-i^ 

 566 = 2(5,1 - ^12) - 292.8 X 10-1". 



(A.14) 



Whether these are zero field or zero charge constants is not known. If 

 they were measured in a room with high humidity, the polarization produced 

 by strain would soon be annulled by a current flow through the leakage re- 

 sistance of the adsorbed moisture, and the constants would be dj or Sij . 

 On the other hand if the displacements were measured in a very dry room, 

 the leakage resistance is very small and it may take hours to annul the polari- 

 zation through a leakage current flow. In that case the constants measured 



