378 THE BELL SYSTEM TECHNICAL JOUENAL, APRIL 1951 



VI. Predicted Angles for CT and DT Face Shear 

 Crystals 



The other two cuts of primary interest for frequency controlled oscillators 

 are the CT and DT low frequency face shear modes. An exact solution 

 for the frequency vibration of a face shear mode has not yet been obtained, 

 but Hight and Willard^ •'' have pointed out an empirical relation that agrees 

 with the measured frequencies over the entire range of angles of rotated Y 

 cut crystals. This relation is for a square crystal 



1.23 /~r^ 



f=-ri/-^ (29) 



where / is one edge dimension and ^^5 the shear elastic constant pertaining 

 to the face shear mode. In terms of the orientation angle^ 



^f5 = -^^4 cos d + ^fe sin 6 + 4^su sin 6 cos 6 (30) 



Introducing the values of su , sf% and su from equations (24) and (26) 

 the frequency becomes 



fi X io-^« = 



14.27 



[(1.986 cos^ d + 2.89 sin^ 6 - 1.802 sin 6 cos 6) 



+ (399 cos^ e - 398 sin^ d - 251.5 sin 6 cos 6) (31) 



X 10"'(r - 50) + (397 cos' ^ - 52 sin' 

 - 72 sin ^ cos 6) X 10~^(r - 50)' + (-52 cos' 6 

 + 8.7 sin' + 98 sin $ cos 6) X 10~" (T - 50)' + • • • 



Since the formula is very approximate the small correction due to tem- 

 perature expansions has been neglected. With this equation the indicated 

 angles for zero temperature coefficient — which are obtained by setting the 



*A Simplified Circuit for Frequency Substandards Employing a New Type of Low 

 Frequency Zero-Temperature- Coejficient Quartz Crystal, S. C. Hight and G. W. Willard, 

 Proc. LR.E., Vol. 25, No. 5, pp. 549-563, May 1937. The factor 1.23 agrees better with 

 experiment than the value 1 .25 given in the paper. 



' Since this paper was written a much more nearly exact solution of a face shear mode 

 vibration has been obtained by R. D. Mindlin and H. T. O'Neil. This solution is an ex- 

 tension of the thickness shear vibration of a crystal published by Mindlin (Journal of 

 Applied Physics, probably March issue 1951). For a square plate there are two solutions 

 which are very close in freauency. For case A which corresponds to / of equation (29) ly- 

 ing along the X direction the empirical factor F becomes 



F = 1.2718 - .03471g - .03727^2 where g ^ —^, 



and sn' and s^ are the elastic compliances corresponding to the rotated cuts. For the B 

 case which corresponds to / lying along z' the same formula holds but g => t^ — -,. 



