522 BELL SYSTEM TECHNICAL JOURNAL 



for these variations. This explanation not only suggests reasons for 

 the above mentioned phenomena, but, what is more important, it 

 indicates the procedure by which it is actually possible to produce 

 crystals having negligible temperature coefficients. Before outlining 

 this theory the experimental facts which served as its foundation will 

 be discussed in detail. 



Frequency-Thickness Constant as a Function of Dimensions 



When work on the production of parallel cut crystals in the broad- 

 cast frequency band was first started, it was found that it was very 

 difficult to grind crystals for certain low frequencies using a 2.5 cm. 

 square plate because of discrete jumps in frequency for a small re- 

 duction of thickness. Fig. 4 is a typical curve showing the wave- 

 length ^^ as a function of the thickness for a 2.5 cm. square crystal. 

 This curve should be a straight line (for from equation (3) it is evident 

 that X = K't) but it will be noted that there are certain discontinuities 



M 



at the upper end. It was found that these discontinuities were 

 present at frequencies that could be identified with harmonics of the 

 frequency the crystal would have if it were vibrating in the direction 

 of its length along the electric axis. 



This was the first definite indication obtained in the Bell Telephone 

 Laboratories that the longitudinal vibration of the crystal in the 

 direction transverse to the applied field could afTect the frequency 

 supposed to depend only on the thickness. It was checked by further 

 work on crystals of other dimensions, and in each case the position of 

 these discontinuities was found to depend on the width of the crystal. 



The presence of a resonant system whose frequency depends upon 

 the width is evidently responsible for this phenomena, this system 

 affecting the frequency of the vibration along the thickness through 

 some form of mechanical coupling. At the suggestion of Mr. R. A. 

 Heising of the Bell Telephone Laboratories, an explanation of these 

 experimental facts was developed based on the treatment of the plate 



" In plotting the change in rate of vibration of a crystal plate as a function of 

 the dimension, it is more convenient to use wave-length instead of frequency because 

 of the direct linear relation between the dimensions and the wave-length. 



