QUARTZ CRYSTAL APPLICATIONS 



If we make the assumption that there are seven elastic constants 

 diflfers from Cn , the frequency of this series of crystals will be'' 



f = L A Aee' 



211 



and C56 



where Cee' = c^e cos^ A2 + ^^4 sin" . I2 — cfg sin 2.42 (A.22) 



The determination for the Y cut gives directly 



fee = 40.5 X 10 dynes per square cm. 



The other two cuts give the values 



cfe = 18.2 X 10'"; Cii = 58.65 X lO'" 



To test out the hypothesis that cfe differs from cf* or .yfe from 

 can make use of equation (A. 8) writing cfg in place of cf4 . Then 

 these equations simultaneously we find 



544 — 



^66 



~Cs6 



Cu 



f E E E^-. ■ 



yCii Cc,6 C56 ) 



/ E E E^\' 



Table II 



/■ E E B^N 



(A.23) 



(A.24) 



2^14 we 

 solving 



(A.25) 



Substituting in the values from (A.23) and (A.24) we find 



54*4 = 197.8 X 10"'* cmVdyne; 4 = -89.0 X 10"''; 

 s!, = 2(sn - su) = 286.5 X 10"''. 



(A.26) 



Comparing the value of 556 with 2^14 given in equation (A.21) we see that 

 they are equal within the experimental error, so that these measurements do 

 not indicate that there are seven elastic constants but only the customary 

 six. Using these values all the elastic constants can be evaluated as shown 

 by Table III. 



Measurements have also been made to determine accurately the piezo- 

 electric constants. This was done by using the ratios of capacities of two 

 standard rotated A" cut crystals for which these ratios have been accurately 

 determined. As shown by section C of this appendix, the ratio of capacities 

 r of a crystal is related to the piezoelectric constant dn , the elastic constant 

 522' and the free dielectric constant Ki by the equation 



. , . . / /I - A 



r = ratio or capacities = -- ( — — — I 



8 \ «- / 



(A.27) 



