LOW TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 85 

 formula was developed for the frequency which is 



where d = x — z' [{ the plate is square and d = (x -\- z')/2 if only 

 nearly square. The elastic constant 555' depends on the orientation 

 angle 6 according to the equation, 



555' = Sii cos^ 6 + ^66 sin^ 9 + 45i4 sin 6 cos 6. (11) 



Figure 7 shows the measured values of frequency and the values 

 calculated from equations (10) and (11). Agreement is obtained 

 within 2 per cent. 



From equations (10) and (11) the temperature coefficient of fre- 

 quency of a shear vibrating plate should be for a square crystal 



Tf=- (1/2) 



'^ c;n2 /:) J_ IcT a\r, ft nr,a ft 1 



(12) 



£44^844 cos^ d + s^eTsee sin^ 9 + 4:SuTsu sin d cos d 



Sii cos^ d + 566 sin^ 6 + 45i4 sin 6 cos 6 



The temperature coefficient of length along the optic axis is about 

 7.8 parts per million (per degree centigrade) while that perpendicular 

 to the optic axis is 14.3 parts per million. For any other direction 



Ti = 7.8 + 6.5 cos2 d, (13) 



where 6 is the angle between the length and the optic axis. Hence 



n = 14.3; T/ = 7.8 + 6.5 cos^ d, 

 and 



Tp = - 36.4 per degree C. (14) 



The temperature coefficients of the six elastic constants were evaluated 

 in a former paper. ^^ Since then they have been slightly revised so 

 that the best values now are 



n„ - + 12, Tc^ = -54.0, 



Ts,, = - 1,265, this Tc,, = - 2,350, 



rs,3 = - 238, results in Tc,, = - 687, 



r«,4 = + 123, r,., = + 96, (15) 



r.33 = + 213, r.33 = - 251, 



r.,, = + 189, Tr,, = - 160, 



r.,, = - 133.5, r.,, = + 161. 



^^ "Electric Wave Filters Employing Quartz Crystals as Elements," VV. P. Mason, 

 B. S. T. J., 13, p. 446, July 1934. 



