MODES OF MOTION IN QUARTZ CRYSTALS 87 



- 163.514 , •, , ^ oo 



fx» = — ^ — Wx, kilocycles 6.28 



Upon substitution of the value of Css for a ^T-cut in equation 6.22 there 

 results 



u ^p 1 /^ 1 , /30.3 X 1010 



/f*X-C=-i/— = -/*/ ^-55 = 169.0 kilocycles - cm. 



which is 3.3% greater than that observed in equation 6.28 and 1.6% greater 

 than that shown in equation 6.27. The apparent difference in the observed 

 shear modulus in the X and Z' directions for the ^T-cut can be explained 

 from the fact that Young's modulus is quite different in the two directions 

 for the BT-cut while it is nearly the same for the AT-cnX. as verified by 

 equation 6.24 and 6.25. 



From the discussion in this section it can be seen that a single theory that 

 would relate all the now known resonances in quartz plates together with the 

 effects of coupling would be prodigous indeed. In order to reduce the design 

 of quartz plates to a simple engineering basis it is necessary to take specific 

 examples and investigate the region in the vicinity of the frequency to be used 

 based on general theory and then apply approximations that fit the specific 

 cases. 



6.5 Methods for Obtaining Isolated Modes of Motion 

 6.51 GT Type Crystals 



In the case oi GT type crystals the modes that cause the greatest concern 

 are flexure modes in the two planes of the length and thickness and the width 

 and thickness. The desired mode is that of an extensional mode along the 

 width. To produce a low temperature coefl&cient it is also necessary that 

 this mode be coupled to an extensional mode along the length, a fixed fre- 

 quency difference from it. Therefore it will be necessary to prevent flexure 

 modes from occurring at either of these two frequencies. Fig. 6.21 shows 

 the frequency of various flexure modes that would be observed in Gr-cut 

 plates for different ratios of thickness to length. In the case of the Gr-cut 

 the elastic constants in the length and width directions are the same and 

 therefore it is only necessary to determine the flexures in one plane to get a 

 determination in both. From the plot of frequencies shown in Fig. 6.21, 

 it would be very easy to determine the proper thickness for any given GT 

 plate. Since in all practical cases there is a definite relation between the 

 length and width of this type of plate, it would be necessary to examine 

 the flexures in these two directions as a function of the change in thickness. 



