Observations on Modes of Vibration and Temperature 

 Coefficients of Quartz Crystal Plates ^ 



By F. R. LACK 



The characteristics of piezo-electric quartz crystal plates of the per- 

 pendicular or Curie cut are compared with parallel or 30-degree cut plates 

 with reference to the type of vibration of the most active modes, the fre- 

 quency of these modes as a function of the dimensions, and the magnitude 

 and sign of the temperature coefficients of these frequencies. 



It is pointed out that the two principal modes of the perpendicular 

 cut plate appear to be of the longitudinal type, the high-frequency mode 

 being a function of the thickness while the low-frequency mode is a function 

 of the width (along the electric axis). Both modes have a negative tempera- 

 ture coefficient of frequency. Of the two corresponding modes of the 

 parallel cut plates a shear vibration is responsible for the high frequency. 

 This frequency has a positive temperature coefficient. The low-frequency 

 mode is of the longitudinal type and has a negative temperature coefficient. 



Considering only the high-frequency vibration of these plates it is ob- 

 served that there are characteristic variations of the frequency and tempera- 

 ture coefficient with the ratio of dimensions of the plate and the temperature, 

 which are peculiar to the parallel cut plate. These variations can be 

 attributed to a coupling of the shear and longitudinal modes. 



It is then shown that if the parallel cut plate be treated as a group of 

 coupled oscillatory systems with appropriate temperature coefficients the 

 usual coupled system analysis will explain the curves of frequency vs. 

 dimensional ratio, frequency vs. temperature, and temperature coefficient 

 vs. dimensional ratio that are characteristic of this plate. This analysis 

 offers an explanation of the low temperature coefficients which can be 

 produced by a proper choice of the dimensional ratios. 



WITH the increasing demands of the radio industry for a high 

 degree of carrier-frequency stabihty, considerable attention 

 has been focused recently on the piezo-electric quartz crystal as a 

 circuit element in frequency generating systems. The low damping 

 of these mechanical oscillators, combined with their piezo-electric 

 properties makes them particularly suitable for frequency control 

 where a high degree of constancy is required. The frequency stability 

 of the quartz plates prepared in the usual manner, is however, often 

 not sufficient for many of the demands for constant frequency. For 

 instance such a crystal plate does not compare favorably as a sub- 

 standard of frequency with a good astronomical clock. To meet the 

 demands for frequency substandards as well as many other practical 

 problems concerning frequency generation in the communication art, 

 it becomes necessary to devise methods for improving the frequency 

 stability of these crystal systems. This involves a study of the 

 many factors upon which this stability depends. 



A crystal plate constitutes an extremely complex vibration system 

 with a large number of degrees of freedom which are for the most 



1 Presented April 3, 1929, before Institute of Radio Engineers. 



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