QUARTZ CRYSTAL CIRCUIT ELEMENTS 455 



temperature coefficient requiring close temperature regulation and a 

 troublesome secondary frequency spectrum. Assuming that the 

 temperature coefficient of the desired frequency could be materially 

 reduced, the effect of any secondary spectrum must also be minimized 

 before temperature regulation can be abandoned. In fact, from the 

 standpoint of satisfactory production and operation of these crystal 

 plates, it is perhaps more important that the secondary spectrum be 

 eliminated than that the temperature coefficient be reduced. 



The Secondary Spectrum 



The secondary spectrum of these plates, as has been indicated 

 above, is caused by vibrations of the same or of other types than the 

 wanted vibration taking place in other directions of the plate and 

 coupled to the wanted vibration mechanically. This condition of 

 affairs exists in all mechanical vibrating systems but is complicated 

 in the case of quartz by the complex nature of the elastic system 

 involved. 



Considering specifically the case of the F-cut plate the desired 

 vibration is set up through the medium of an Xy strain. Hence any 

 coupled secondary vibrations must be set in motion through this Xy 

 strain. Referring to the following elastic equations for quartz (in 

 these equations X, Y and Z are directions coincident with the crystallo- 

 graphic axes; see Fig. 1 and Appendix), 



Xx = CuXs + Cnyy + CnZ^ + Cuy^ 



Yy = Cl2Xj; + Ciljy + C^^Zz + C^iJ z 



Zz = CxzXx + Ci^Jy + CzzZz 



Yz = CnXx + Ciiyy + cajz 



Zx = ~r Css^^x -\- c^^Xy 



Xy = + CseZx ~t" C^^Xy 



(2) 



it will be seen that by reason of the constant C56 an Xy strain will set 

 up a stress in the Zx plane which in turn will produce a Zx strain. 

 Hence the .y^ and z^ strains are coupled together mechanically, the 

 value of the constant c^& being a measure of that coupling. 



High order overtones of vibrations resulting from this Zx strain 

 constitute the major part of the secondary frequency spectrum of 

 these plates. 



The technique for dealing with this secondary spectrum in the past 

 has been the proper choice of dimensions. At high frequencies these 

 overtones occur very close together and when one set is moved out 

 of the range by grinding a given dimension another set will appear. 

 Some benefit is obtained with the clamped holder, which tends to 

 inhibit certain types of transverse vibrations; but as indicated above. 



