QUARTZ CRYSTAL APPLICATIONS 209 



relatively thick pieces of quartz, and determining the asymptotic value for 

 high order harmonics, they obtained the elastic constants 



cn = 87.55 X IQio dynes/cm-; cn = 6.07 X W; Cn = 13.3 X W; 



cu = -cu = 17.25 X 10'"; rgg = 106.8 X 10'"; c^ = 57.19 X 10'". 



(A.16) 



In addition they came to the conclusion that Cb6 had a value of 18.4 X 10'", 

 which was different from the value of cu as required by theory. Their 

 measurements were made with high harmonics in air gap holders so that the 

 values measured should determine the rfy constant. To explain the dis- 

 crepancy found, Lawson^ has suggested that the rfj constants 



c?j = cfj + ■iireueij/Ki (A. 17) 



do not obey the same symmetry relations as the cfy constants. This sugges- 

 tion does not seem to be borne out by equations (A. 10), from which the sym- 

 metry relations of the cfy constants can be determined. If we start with a 

 generalized form of these equations applicable to any crystal and apply the 

 symmetry relations for quartz, we find that it is still necessary to satisfy the 

 symmetry relations between the constants found previously and in particular 



cfe = c?4 (A.18) 



In order to investigate this matter further, and to obtain more reliable 

 values of the elastic constants, an analysis has been made of a number of 

 measurements previously obtained for oriented crystals. In particular two 

 families of oriented crystals were investigated. One family was a set of 

 oriented X cut crystals vibrating longitudinally. They were cut with their 

 major faces normal to the X axis and with their lengths at angles ^2 of from 

 +43° to —79° with respect to the Y or mechanical axis. They were 

 oriented similarly to the +5° and —18.5° filter crystals shown by Fig. 1.9. 

 When these crystals are 7 to 10 times as long as they are wide or thick it 

 has been shown previously^ that their length resonances are determined 

 very accurately by the equation 



- /« = 97i/^ (^-19) 



^^y Y pS22' 



where 4 is the length of the crystal, p the density and 522' the inverse of 

 Young's Modulus along the length for a plated crystal. This is related to 

 the angle of cut ^2 by the equation 



^22' = Sii cos A2 + S33 sin .42 + 2su cos .42 sin .42 



+ (2^13 -f Sii) sm .42 cos .42 



3 A. W. Lawson, Phys. Rev., 59, 838 (1941). 



^ "Electrical Wave Filters Employing Quartz Crystals as Elements." \V. P. Mason, 

 B. S. T. J., Vol. XIII, pp. 405-452, Jul\' 1934. See Figs. 25, 31 and 32. 



