LOW TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 81 



where / is the thickness of the plate. Hence for the A type vibration 

 which corresponds to the first velocity Cj, the frequency will be 



_ 1 /c66 cos^ 6 + C44 sin^ d — Icn sin 6 cos ^ _ 1 jcee 



(8) 



The solid curve of Fig. 4 shows a plot of this equation while the 

 measured values are shown by dots. 



The frequencies of the other two modes of motion are given by 



2t\ p 2tM p VX22X33 



where 



This formula is the same as that for the frequencies given by two 

 coupled modes ^ and hence can be interpreted as a mode of vibration, 

 determined by X33, and a mode of vibration, determined by the 

 constant X22, coupled together through the coupling compliance X23. 

 For an isotropic medium one of these modes would be a pure shear and 

 the other a longitudinal mode, but in a crystalline medium the motions 

 are not strictly along or perpendicular to the direction of motion 

 The A type vibration which is an Xy' shear vibration is not coupled 

 to the other two since the coupling elasticities X12 and X13 are equal to 

 zero. For a more general rotation, however, they will not necessarily 

 be equal to zero and hence the general solution of equation (1) will 

 represent two shear like vibrations, the Xy and jz', and a nearly 

 longitudinal y/ vibration all mutually coupled together. 



The Christofel formula is only valid for a plate of thickness / which 

 extends to infinity in all other directions and hence this solution does 

 not show the coupled frequencies due to the contour dimensions which 

 occur in a finite plate. In general there are two types of vibration 

 which couple strongly to the A type vibration, the low-frequency 

 shear modes and the flexure modes in which bending occurs in the 

 xy' plane. As pointed out by Lack, Willard and Fair,^ both the AT 

 and the BT occur near angles of cut for which the coupling to the 

 Zx' low frequency shear mode vanishes. Hence one would expect that 

 these crystals would have fewer subsidiary resonances and this 

 expectation is verified by experiment. A practical result is that the 



* "Electrical Wave Filters Employing Quartz Crystals as Elements," W. P. 

 Mason, July 1934, page 444, B. S. T. J. 



'"Some Improvements in Quartz Crystal Circuit Elements," B. S. T. J., July 

 1934. The problem of couplings is discussed in more detail in the U. S. Patent 

 2,173,589, Sept. 19, 1939 issued to R. A. Sykes and the writer. In this patent the 

 AC cut and the — 18.5° A' cut crystals are described. 



