ZERO TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 369 



where / is the thickness of the crystal, p the density, the angle of the 

 normal of the plate measured from the Y axis and cu , cu and Cee three of 

 the seven elastic constants of quartz measured at constant electric field. 

 This equation is valid for an infinite plate but is also a good approximation 

 for a crystal whose cross-sectional dimensions are 30 to 40 times the thick- 

 ness dimensions. Since there are three constants, the two measurements for 

 the AT and BT cuts will give only two relations and we need a measurement 

 for another angle. As discussed in Chapter X, Section 10.2 of "Piezoelectric 

 Crystals and Their Apphcation to Ultrasonics," ^ the remaining cut can 

 be obtained by measuring the thickness shear mode of a F cut plate or the 

 face shear mode of a F cut crystal. The latter mode is considerably easier 

 to dimension in order to obtain a frequency corresponding to the shear 

 mode. Table I shows measurements for the frequency constant of a F 

 face shear mode for a crystal having the following dimensions: Length 

 along the X axis is 36.86 mm, width along the Z axis = 7.625 mm; thickness 

 along the F axis is 0.990 mm. High harmonics were used and the frequency 

 constant was obtained by dividing the frequency by the overtone order. 

 This frequency is controlled by the cu elastic constant according to the equa- 

 tion 



In calculating the Cu constant from the resonant frequencies measured, 

 a correction is introduced by the temperature expansion constants of the 

 crystal. This follows from equation (5) since h the frequency determining 

 axis and p the density both change with temperature. From measurements 

 quoted by Sosman^ for the expansion along the Z axis and perpendicular 

 to the Z axis, we find 



h = /o[l + 7.8 X lOrHT - 50) + 2.8 X 10" ^(r - 50)^ 



- 1.5 X lO-'KT - 50)3 _ ...] 



(6) 

 U,y = /o[l + 14.6 X 10-«(r - 50) + 6.3 X 10-»(r - 50)2 



- 1.9 X ia-i2(r_ 50)3+ ...] 



Multiplying these together the volume expansion is 



V = hlyh = llll + 37 X 10-6(r - 50) + 15.8 X 10-'{T - 50y 



(7) 



- 5 X lor^KT - soy + •••] 



1 Pizoelecteric Crystals and Their Application to Ultrasonics, W. P. Mason, D. van 

 Nostrand Co. 1950, Section 10.2, page 204. 



2 Sosman, The Properties of Sihca, Chemical Catalogue Co., 1927, pp. 386, 387. 



