QUARTZ CRYSTAL CIRCUIT ELEMENTS 457 



the elimination of the effects of the coupled secondary frequency 

 spectrum over a wide temperature range is a difhcult matter. 



Another method has been developed recently for dealing with these 

 coupled vibrations."* This consists of reducing, by a change in 

 orientation, the magnitude of the elastic constant responsible for the 

 coupling. If the orientation of the crystal plate be shifted with 

 respect to the crystallographic axes then, in general, the elastic con- 

 stants with reference to the axes of the plate will vary. The direct 

 constants (ci/, C22', •••) which represent the longitudinal and shear 

 moduli will of course vary in magnitude only, while the cross constants 

 (ct/, • • •) will vary both as to magnitude and sign. There is a possi- 

 bility therefore that the proper choice of orientation of the plate will 

 reduce Ci,e to zero without at the same time introducing other couplings. 



Figure 2 shows graphically the variation of c^e and C5& as a function 

 of rotation about the X axis. These have been calculated by means 

 of the equations given in the appendix. It will be seen that at ap- 

 proximately + 31° and — 60° rse' becomes zero. Here then are two 

 orientations for which the coupling between the Xy and Zx strains 

 should be zero. 



In shifting the orientation of the plate the necessity for exciting the 

 wanted vibration piezo-electrically must not be lost sight of. Hence 

 in addition to computing the values of the elastic constants the 

 variation of the piezo-electric moduli as a function of orientation must 

 also be examined. Figure 2 also shows the effect of rotation about 

 the X axis on d26 (the constant connecting the Ey' electric field with 

 the Xy strain). It will be seen that at both + 31° and — 60° the Xy 

 vibration can be excited piezo-electrically but it is to be expected that 

 a plate cut at — 60° will be relatively inactive,^ for di^' at this point 

 is only 20 per cent of its value for the F-cut plate. On the other 

 hand at -f 31° a plate would be practically equivalent to the F-cut 

 as far as activity is concerned. The frequency of the Xy vibration 

 for these special orientations can be calculated by means of equation 

 (1) substituting for Cee the value of Cee' for the given angle as read 

 from the curve of Fig. 2. 



* The expression of the coupling between two modes of vibration in quartz in 

 terms of the elastic constants was first suggested in 1930 by Mr. W. P. Mason of 

 the Bell Telephone Laboratories. 



* The word "activity" is a rather loose term used by experimenters in this field 

 to describe the ease with which a given vibration can be excited in a particular circuit. 

 It is often spoken of in terms of the grid current that is obtained in that circuit 

 or the amount of feedback necessary to produce oscillation. It can better be ex- 

 pressed quantitatively as the coupling between the electrical and mechanical systems 

 (not to be confused with the mechanical coupling between different vibrations 

 described above) which is a simple function of the piezo-electric and elastic moduli 

 of the vibration involved and the dielectric constant of the crystal plate. 



