QUARTZ CRYSTAL APPLICATIONS 



197 



coefficient must have been caused by the coupHng between the two modes. 

 As a result of this observation it follows that if we have a shear vibrating 

 crystal with a positive temperature coefficient and cut another crystal at 

 45 degrees from this crystal, the strong coupled mode which corresponds to 

 the shear vibration will also have a positive temperature coefficient. As 

 we grind down on the side, the two modes become farther apart in frequency 

 and less closely coupled. Then, since they both will have a negative coeffi- 

 cient if separated far enough, it follows that for some ratio of axes, one of the 

 modes will have a zero coefficient. This was tested out for a series of orien- 

 tations near the CT and DT with the results shown in Fig. 1.16. Positive 

 angle crystals had zero coefficients at ratios of axes varying from 1 to .855 



Fig. 1.15 — Relation between a face shear mode and two coupled longitudinal modes 



depending on the angle while negative angle crystals had zero coefficients 

 at ratios from .64 to 1.0. For positive angle crystals it was the higher fre- 

 quency mode that was the stronger and could be given the zero coefficient, 

 while for the negative angle crystals it was the lower frequency mode that 

 was the stronger and corresponded to the face shear mode. 



Several of the positive angle crystals were measured over a temperature 

 range with the results shown by Fig. 1.17. For angles above 51 °-30' the 

 curvature was positive, while for angles below 51°-30' the curvature was 

 negative. Right at 51°-3()' the large square law curvature term disappeared 

 and the frequency was constant to one part in a million over a 100-degree 

 Centigrade range centered at 50°C. as shown by Fig. 1.18. Some further 

 experiments showed that this fiat range could be moved around a bit by 



