QUARTZ CRYSTAL APPLICATIONS 



221 



The first region would be a Z-cut crystal with its length somewhere in the 

 XY plane and would result in a temperature coefficient of two parts per 

 million negative. Such a crystal is not of much interest since there is no 

 piezoelectric constant for driving it. The other region \^ — ^ 90° would also 

 result in the length being near the AT crystallographic plane, but would 

 allow the major surface to be made perpendicular to the X axis and hence 

 would allow the crystal to be driven piezoelectrically. By allowing \p to 

 be slightly greater than 90°, the fourth term in the numerator can be made 

 slightly negative and of a value greater than the two positive terms. This 

 results in the +5° X-cut crystal having nearly a zero coefficient and this 

 angle is the most favorable one for a low coefficient longitudinal mode of 

 motion. All other directions have a negative temperature coefficient. 



The remaining temperature coefficients of the six elastic constants can be 

 evaluated from Fig. 1.12, and equation (A. 22). The frequency temperature 

 coefficient can be expressed by the equation: 



Tf = 3.9 + 6.5 cos- d 



^ 1 pfe Taj, sin' d + cf4 Tcf, cos' d + T^f, cf, sin 2^ 1 (A.71) 

 2 L C66 sin" 9 + Cii cos' 6 + cfi sin 26 



since in terms of the IRE angles the series of crystals is given by (p = —90- 

 6 = 90- A2;^P = 90°. Taking the A T, BT, and F-cut, whose coefficients 

 have accurately been determined, we have 



From these data and equation (A.71), the three temperature coefficients can 

 be evaluated as 



Tef, = 164.2; T,f, = 165.7; T,f^ = +90.2 



(A.72) 



To convert these into compliance temperature coefficients we have to 

 make use of the relations of equations (A. 8) 



E _ n,f E E K 



See — ^{Sii — S12) — 



C44 



E E £2 » 



E 

 •Si 4 



E 



— Cu 



rf / E E £;2v » 



E 



•S44 — 



^66 



BE E^ 



Cii ^66 — Cu 



