454 BELL SYSTEM TECHNICAL JOURNAL 



where Cee = the elastic constant for quartz connecting the Xy stress 

 with an Xy strain = 39.1 X 10^" dynes per cm.^ 

 P = the density of quartz = 2.65 gms. per cm.^ 

 / = the thickness in cm. 



On substituting the numerical values in equation (1), a frequency- 

 thickness constant of 192 kc. per cm. is obtained which checks within 

 3 per cent the value of this constant found by experiment. 



This Xy shear vibration is not appreciably affected when the 

 plate is rigidly clamped, the clamping being applied either around the 

 periphery if the plate is circular, or at the corners if square. Hence a 

 mechanically rigid holder arrangement is possible which is particularly 

 suitable for mobile radio applications.^ 



The temperature coefficient of frequency of this vibration is approxi- 

 mately + 85 parts/million/C.°, which means that for most applications 

 it must be used in a thermostatically controlled oven. In operation, 

 this comparatively large temperature coefficient is responsible for a 

 major part of any frequency deviations from the assigned value. 



Another important characteristic of the F-cut crystal is the secon- 

 dary frequency spectrum of the plate. This secondary spectrum con- 

 sists of overtones of low frequency vibrations which are mechanically 

 coupled to the desired vibration and cause discontinuities in the 

 characteristic frequency-temperature and frequency-thickness curves 

 of the crystal. In some instances these coupled secondary vibrations 

 can be utilized to produce a low temperature coefffcient over a limited 

 temperature range.' But in general, at the higher frequencies (above 

 one megacycle) this secondary spectrum is a source of considerable 

 annoyance, not only in the initial preparation of a plate for a given 

 frequency but in the added necessity for some form of temperature 

 control. In practice, these plates are so adjusted that there are no 

 discontinuities in the frequency-temperature characteristic in the 

 region where they are expected to operate, but at high frequencies it 

 is difficult to eliminate all of these discontinuities over a wide temper- 

 ature range. If, then, for any reason the crystal must be operated 

 without the temperature control, a frequency discontinuity with 

 temperature may cause a large frequency shift greatly in excess of 

 that to be expected from the normal temperature coefficient. 



From the above considerations it may be concluded that the 

 standard F-cut plate has two distinct disadvantages: namely, a 



2U. S. Patent No. 1883111, G. M. Thurston, Oct. 18, 1932. "Application of 

 Quartz Plates to Radio Transmitters," O. M. Hovgaard, Proc. I. R. E., 1932, p. 767. 



3 "Observations on Modes of Vibrations and Temperature Coefficients of Quartz 

 Plates," F. R. Lack, Proc. L R. E., 1929, p. 1123; Bell Sys. Tech. Jour., July, 1929. 



