218 BELL SYSTEM TECHNICAL JOURNAL 



transformation of axes, the components of the rotated tensors are easily- 

 calculated and the elastic and piezoelectric constants for rotated crystals 

 determined. 



The variation of Young's modulus as a function of orientation was first 

 worked out by Voigt. In terms of the IRE angles specifying the orientation 

 of a crystal plate, the s compliance modulus (inverse of Young's Modulus) 

 is given by the equation 



E' E / 2 rt 2,1 • 2 ,\2 I /T I £ \ . 2 /, 2 , 



•^H — Sn{cos 0COS i/' + sni \f/) + (2^13 + 544) sm ^ cos \f/ 



X (cos 6 cos xj/ + sin" i/-) + 533 sin 9 cos \p — 2sfi sin 6 sin \f/ cos \p (x'\.58) 



X [3 (cos (f cos 6 cos ip — sin (p sin \p) — (sin (p cos d cos xp — cos </? sin yp)"] 



As discussed in Chapter II by W. L. Bond, the IRE angles are meas- 

 ured as follows: Taking the X' axis along the length of the crystal, the Y' 

 along the width, and the Z' along the thickness, the angle 6 is the angle 

 between the Z or optic axis and Z'. (p is the angle between the projection 

 of the Z' axis on the XY plane and the X axis, while \{/ the skew angle is 

 the angle between the length and the tangent to the great circle which con- 

 tains the Z and Z' axes and the length of the crystal X'. A crystal having 

 its thickness along the ,Y axis (Y-cut crystal) will have the angles 



6 = 90°; (p — 0°; \p variable but equal to 90° when the length coincides 

 with the Y axis. Under these conditions 



E' E • 4 , . /o I E \ ■ 2 , 2 , 



Sn = Sn sni xp + (2.^13 + Su) sm \p cos \p 



E 3 (A.59) 



-\- S33 cos \p — 2sn sin \p cos \p 



This equation has been made use of in evaluating the elastic constants of 

 quartz as shown by equations (A. 20). For this equation A2 was measured 

 from the F axis rather than from the Z as in the IRE angle and 



A2^ xp - 90° (A. 60) 



Since from equation (A. 19) the frequency of a long thin crystal in longitudi- 

 nal motion is known to be 



/ = f./l/^ (A.19) 



the longitudinal frequency of any oriented crystal can be calculated from 

 equations (A.58) and (A. 19). 



It is the purpose of this section to show also that the temperature coeffi- 

 cient of the longitudinal frequency of any oriented crystal can be calculated 

 provided we know the temperature coefficient of the six elastic constants of 



1" Methods for Specifying Quartz Crj'stal Orientation and their Determination by 

 Optical Means," this issue of the B. S. T. J. 



