LOW TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 91 



shown that the 555' constant is given by the equation 



Sbb = (su — 2si3 + -^33) cos^ i// sin^ 2(p + 566 sin^ 95 sin^ i// 

 + 4^14 sin (p [sin 36 cos (p (cos^ \l/ cos^ (p — sin^ \f/) 

 + cos 3^ sin i// cos \}/ (cos 2^ + cos^ (ip)J 



+ 544 (cos^ \}/ cos^ 2^ + sin^ i// cos^ ^), (19) 



where the angle 7 is given in terms of a new angle xp and (p by the 

 equation 



cos 7 = sin 97 cos \p. " (20) 



If we introduce this expression into equation (12) and introduce the 

 numerical values of equation (15), the expression for the temperature 

 coefficient of a low-frequency shear crystal cut at any angle becomes 



Tr = 



.c,on^-'> 2,, •'^L^- 5877.5 cos^ 4. sin^ 2^ 



4.5 + 2.9 (sm^ (p cos^ x}/ + cos- (p) + ' 



195cos2i/'sin2 295 



+ 15790 sin^ (p sin^ \p + 10340 sin (^ [sin 3d cos 99 (cos^ \p cos 2y; — sin^ i^) 

 + 292,8 sin^ cp sin^ \p — 172 A sin ^ [sin 3d cos 99 (cos^ xj/ cos 2^9 — sin^ i/') 



+ cos 3^ sin \l/ cos 1^ (cos 295' + cos^ <p)^ 

 + cos 3^ sin \p cos i/' (cos 2^? + cos^ <p)'} 



— 19,525 (cos^ \p cos^ 2<p + sin^ \l/ cos^ 99) 

 + 200.5 (cos^ \j/ cos^ 2^9 + sin^ ij/ cos^ 99) 



(21) 



Figure 13 gives a contour map of the location of the angles of zero 

 temperature coefficient. The dotted lines indicate the paths for which 

 the piezo-electric constant is a maximum and hence for which the 

 crystal is most easily excited. 



IV. A New Crystal Cut, Labeled the GT Crystal, Which Has A 

 Very Constant Frequency for A Wide Temperature Range 



All of the zero temperature coefficient crystals so far obtained have a 

 zero temperature coefficient only for a specified temperature, while on 

 either side of this temperature the frequency either increases or de- 

 creases in a parabolic curve with the temperature. This is well illus- 

 trated by Fig. 14 which shows a comparison of the frequency stability 

 of the standard zero temperature coefficient crystals over a wide tem- 

 perature range. What is plotted is the number of cycles change in a 

 million from the zero coefficient temperature. These curves show 

 that for a 50° C. change from the zero coefficient temperature the fre- 

 quency of standard zero temperature coefficient crystals may change 

 from 30 to 140 parts per million. The curves are usually nearly para- 



