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ZERO TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 377 



All the compliance constants are now determined except S4a , Su and 

 ^12 . From the relations for a crystal in the quartz class^ 



B E 



^^4 ^ ~T~E i2 y ^*6 = 2(5fi — Su) = -^—i F2 (2^) 



C44C66 — Cu Cu ^66 — Cu 



the remaining constants can be obtained. Inserting the values of cu , ct% 

 and cu from (12), we find 



Su = 1.986 X ia-^2[i + 201 X 10-«(r - 50°) 



+ 200 X ia-^(r - 50)2 _ 26 x io-i2(r - 50)^ + • • •] 



sli = 2.89 X 10-i2[l - 138 X 10-«(r - 50°) 



- 18 X 10-»(r - 50)2 -f 3 X 10-i2(r - 50)^ + • • •] 

 Su = -0.1005 X 10-i2[l - 678 X lOr^T - 50°) 



- 2110 X 10-9(r - 50)2 _^ 510 X 10-i2(r - 50)^ + • • •] 

 Su = -0.174 X 10-i2[l - 1270 X 10-6(r - 50°) 



- 575 X io-9(r - 50)2 _ 215 x io-i2(r - soy + • • •] 



It is sometimes desirable to use the c values as a function of tempera- 

 ture. The remaining values can be obtained from the relations valid for 

 quartz^ 



r, E SS3 , Su r^ E _ SSS Su E _ SlZ E _ Sll + ^12 /rf^\ 



ZCii "I ^~ J ^^12 — -^ ', Cu — — ',Czz— \Zi ) 



a p a p a a 



where 



E / E i E \ rk B^ a E / E E \ rt E^ 



oi == S33 {Sii -\- Su) — 2su ; p = Su {sii — Su) — ^Su 

 dz = 104.8 X 10+io[l - 165 X 10-6(r - 50) 



- 187 X 10-\T - 50)2 _ 410 x 10-i2(r - 50)^ + • • •] 

 cfg = 9.6 X 10+io[l - 510 X 10-6(r - 50) 



- 2000 X 10-9(r - 50)2 + 600 X 10-i2(r - 50)^ H ] 



cn = 86.75 X 10+i°[l - 53.5 X 10-«(r - 50) 



- 75 X lOr'iT - 50)2 _ 15 x 10-i2(r - 50)^ + • • •] 

 cu = 6.15 X 10i«[l - 3030 X 10-6(r - 50) 



- 1500 X 10-9(r - 50)2 _j_ 1910 X 10-i2(r -50)' + • • •] 

 5 See Piezoelectric Crystals and Their Application to Ultrasonics, page 207. 



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