CHAPTER VI 



Modes of Motion in Quartz Crystals, the Effects of Coupling and 

 Methods of Design 



By R. A. SYKES 

 6.1 Introduction 



WITH the recent extended use of Quartz crystals in oscillators and 

 electrical networks has come a need for a comprehensive view of the 

 various types of crystal cuts. In addition there has been a need for illus- 

 tration of some of the methods employed in choosing the proper cut for a 

 given requirement, the manner in which quartz crystals vibrate and the basic 

 principles governing the choice of a design to use certain cuts most advan- 

 tageously. In particular one of the greatest problems associated with the 

 recent large scale production of cr3^stals for oscillator purposes has been that 

 of obtaining crystals the activity and frequency of which would not vary to 

 any large degree over a wide range in temperature. 



It is the intention of this chapter to present a physical picture of the man- 

 ner in which quartz crystals vibrate in their simplest forms and then to show 

 what has been learned from these simple forms that will apply to the more 

 complex combinations of motion. The motion of a bar or plate is deter- 

 mined almost wholly b}' its dimensions and the particular type of wave gen- 

 erated, or frequency applied, and very little upon the driving system if the 

 coupling to the driving system is small. In the case of quartz the coupling 

 between the electric and mechanical system is small and hence we may study 

 the motion of rods and plates without always considering the effect of changes 

 due to the method of excitation (i.e., piezo-electric). However the ease of 

 exciting and measuring a particular mode does depend on the piezo-electric 

 constant driving it. Basically only three t}-pes of motion will be considered ; 

 flexural, extensional and shear. These three types of motion or combina- 

 tions of these can be considered to represent most of the cases with which 

 we will concern ourselves. In additioft, the frequency equations will be 

 given for common types of motion and the effect of coupling between various 

 modes of motion. Finally the general rules relating to the dimensioning of 

 oscillator plates wiU be presented. 



6.2 Types of Motion in Quartz Rods and Plates 



6.21 Flexural 



The motion associated with flexure will be discussed first because this is 

 the type of motion that we see more commonly in nature. This motion is 



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