LOW TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 89 

 and set 





the three solutions will be 



'A , ifA'+fs'+fc') 



1 = ^/2VPcos| + 



/,,3=^-2VPcos(|±^)+(/^1+41±M. (1 



8) 



From these equations and equation (2), the frequencies and tempera- 

 ture coefficients of all three modes of motion have been calculated by 

 Bechmann. Based on these calculations the angles of zero coefficient 



140 



130 



120 



)I0 



70 



60 



50 



40' 



II 



-30 -20 -10 



20 30 40 

 VALUE OF e 



50 



60 



70 



80 



Fig. 10 — Angles of cut for zero temperature coefficient high-frequency 

 shear crystals for two rotations. 



are shown on Fig. 10 for the angular placement of the direction of 

 propagation adopted on Fig. 11. 



Using the empirical formula (10) for the low-frequency shear vibra- 

 tion a surface of zero coefficient low-frequency shear vibrating crystals 

 can be calculated. ^^ For this crystal three angles are required to 



" Multiple orientation low- and high-frequency shear crystals are discussed in 

 British Patent 491,407 issued to the writer on September 1, 1938. 



