520 BELL SYSTEM TECHNICAL JOURNAL 



f = J, (3) 



where t is the thickness in milHmeters and K is now 1.96 X 10^ 

 It will be noted that this constant differs from that found in the case 

 of the perpendicular cut crystal. Moreover the temperature coefficient 

 of this frequency is positive. 



These facts lead one to believe that this is not a simple longitudinal 

 vibration. Cady ^ has pointed out that if it be considered as a shear 

 vibration in the X- Y plane the frequency can be calculated using the 

 appropriate shear modulus.^" 



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

 to the electric or X axis, and is given by the same expression and 

 constant as the frequencies of the perpendicular cut plate. It has 

 the same characteristic negative temperature coefficient. 



For these parallel cut plates there are then two possible major 

 modes which, however, differ in type of vibration and sign of tempera- 

 ture coefficient. 



Limiting this discussion to the high-frequency region, it is seen that 

 these parallel and perpendicular cut plates have different frequency- 

 thickness constants and temperature coefficients of opposite sign. On 

 closer examination it is found that there is an additional difference 

 which involves the variation of the magnitudes of these frequency- 

 thickness constants and temperature coefficients with the ratio of 

 width to thickness of the plate. 



For the perpendicular cut plate the frequency-thickness constant 

 changes but little with the size of the crystal. The same is true for 

 the temperature coefficient, and from recent measurements on a 

 number of sizes of plates the magnitude of this coefficient lies between 

 minus 20 and minus 35 cycles in a million per degree centigrade. 



The parallel cut plate, on the other hand, has a frequency-thickness 

 constant which for any but thin plates of large area varies considerably 

 with the width. The temperature coefficient also varies with the 

 width, and is in addition a function of the temperature. This 

 coefficient has a wide range of values whose limits are approximately 

 plus 100 cycles in a million per degree centigrade and minus 20 cycles 

 in some special instances, with all possible intermediate values in- 

 cluding zero. Then, as has been mentioned before, these parallel cut 



9 Cady, Phys. Rev., 29, p. 617, 1927. 



" If it could be shown that the shear modulus of this plane had a positive tempera- 

 ture coefficient it would substantiate this assumption, but there is no information 

 at present available regarding the effect of temperature on the elastic constants 

 other than for the two values of Young's modulus. 



