QUARTZ CRYSTAL PLATES 



519 



expression it is found that the same value for K is obtained as that of 

 equation (1). 



The low frequency is a function of the width, the dimension parallel 

 to the Y axis, and is given by the same expression as equation (1) 

 with the same value of K, the width in millimeters being substituted 

 for the thickness. 



For this type of crystal plate there are then two possible major 

 modes which appear to be of the longitudinal type and depend upon 

 the same elastic constant. (Young's modulus in the X- Y or equatorial 

 plane has the same magnitude in any direction.) 



The temperature coefficient of both these frequencies is negative, 

 which is in agreement with the temperature coefficient of Young's 

 modulus for the equatorial plane. ^ 



The Parallel or 30-degree Cut 



When the crystal plate is so cut that its major surfaces are parallel 

 to both the optic and electric axes (the parallel or 30-degree cut, 

 see Fig. 3) this 30-degree shift in orientation from the perpendicular 



OPTIC AXIS 



MECHANICAL 

 AXIS 



Fig. 3 — Orientation of a parallel or 30-degree cut plate with respect 

 to the crystal axes. 



changes the characteristics in some important respects. As before 

 there is a high and a low principal frequency, but in this case the high 

 frequency sometimes occurs as a doublet (two response frequencies a 

 kilocycle or so apart). 



For thin plates of large area the high frequency is a function of 

 the thickness of the plate and is given by the approximate expression 



8 Perrier & Mandrot, Mem. Soc. Vaudoise Set. Nat. (1923), 1, pp. 333-364. 



