QUARTZ CRYSTAL APPLICATIONS 



219 



quartz, and that regions of low temperature coefficient crystals can be located 

 for this and other simple modes of motion for which the frequency can be 

 calculated in terms of the elastic constants. 



Dififerentiating equation (A. 19) with respect to t the temperature 



dsu 



+ - 



dt 



E' 



Sn J 



or 



(A.61) 



where Ta the temperature coefficient of the quantity a is defined as the rate of 

 change of a with temperature divided by the value of a. The temperature 

 coefficient of the length / = A'' is 7.8 parts per million per degree centigrade 

 along the optic axis, and 14.3 parts per million perpendicular to it. For a 

 general orientation, the temperature coefficient of length varies as 



Tf = 14.3 - 6.5(sin'^cos'^) 



(A.62) 



Since the total mass remains the same when the crystal expands, the tem- 

 perature coefficient of the density is the negative of the sum of the coefficients 

 of the three axes or 



Tp = -36 A 

 Hence the temperature coefficient of frequency becomes 



/ dsn 

 Tf = 3.9 + 6.5 sin' d cos' lA - ; ^^ 



E' 



Sn 



(A.63) 



(A.64) 



Dififerentiating equation (A. 58) we have as the temperature coefficient of a 

 general orientation 



Tf = 3.9 + 6.5 sin2 d cos^ ^p 



Sn Tsf ^{cos^ d cos^i/' + sin')/')' + (2513^^,3 + sfiTsf,) X 



sin 6 cos i/'(cos 6 cos i/' + sin" xj/) + Ss^T,^^ X 

 sin 6 cos )/' — IsiiTgf^ sin 6 sin \p cos xp X 

 [3 (cos <p cos 6 cos i/' — sin (^ sin yp)' — (sin <p cos 6 cos i/' + cos (p sin ^) ] 



5ii(cos 6 cos i/' + sin xl/)" + (2^13 + Su) sin' 6 cos \p X 



(cos 6 cos xj/ + sin" xp) -\- 533 sin 6 cos xp 

 — 2su sin 6 sin xj/ cos i/'[3(cos <p cos 9 cos xp — sin (p sin xp) 



— (sin <p cos 6 cos xp + cos tp sin i/') ]. 



(A.65) 



