68 BELL SYSTEM TECHNICAL JOURNAL 



BC cuts. The 556 constant determines the coupHng between the face and 

 thickness shear modes. As shown in Fig. 6.7 this constant passes through 

 zero at two values, namely +31° and —59° and the resulting angles have 

 been termed the AC and BC cuts. These angles are very close to the AT 

 and BT cuts and hence they also possess the benefits of low coupling between 

 modes. In addition to making the cross coupling constants zero, a rotation 

 of the crystal plate with respect to the crystallographic axes also results in a 

 change in the e.xtensional ?.nd shear elastic constants. Notice that these pass 

 through maxima and minima at the zero values for the cross coupling con- 

 stants. This of course affects the resonant frequencies of isolated modes. 

 Changes as great as 50% increase in frequency constants may be obtained 

 by choosing the proper rotations. The equations relating the elastic con- 

 stants as functions of orientation are given in appendix B for more com- 

 plete use. 



6.4 Coupling between Modes of Motion 



As pointed out in the previous section, the frequency equation of a given 

 mode of motion will give accurate results only in the case where the mode of 

 motion is isolated. This is very rarely the case since most quartz crystals in 

 common use are in the form of plates where the frequency determining di- 

 mension is not large in comparison with all other dimensions. Only in the 

 case of a long thin rod vibrating in length-thickness flexure of the first order 

 would this be true. It was also shown that the coupling between different 

 modes of motion could be related to the mutual clastic constants (.9,-; and c,/) 

 and that some of these could be made zero by the proper choice of orientation 

 of the finished crystal plate. The elastic constants s^ and dj only relate to 

 the coupling between the extensionals, the shears and the extensional to the 

 shear. For example 523 relates to the coupling between the extensional 

 modes along the Y and Z axes, 556 relates to the coupling between the low 

 and high frequency shear modes of a F cut plate and .^24 relates to the cou- 

 pling between an extensional mode along the Y axis and a shear mode in 

 the YZ plane. One other important coupling condition occurs and that is 

 between the flexure and the shear modes. There is at present no mathe- 

 matical theory relating this form of coupling except from simple assumptions 

 that may be drawn from the fact that the shear modulus enters as a control- 

 ling factor in determining the frequency of a bar vibrating in flexure and from 

 the similarity of the two types of motion near the boundaries. Since it is 

 possible to have a definite coupling between extensional and shear modes 

 there must be coupling between the extensional and flexure modes. It 

 would be expected that it would be proportional to the coupling between the 

 extensional and shear "modes. 



