204 BELL SYSTEM TECHNICAL JOURNAL 



Equation (A. 5) is not the only way of relating the elastic and elec- 

 tric quantities. For example if we substitute the values of the fields of the 

 last three equations of (A.6) in the first six equations, we can write 



-X, = s^nX:. + sl2Yy + SuZ, + sU\ - gllQ. 



-yy = sl^X, + s^nYy + SnZ, - s'LW + gnQ, 



— Zz = SizXx + SizYy -\- SziZ^ 



— yz = 514X1 — sii^Yy + si^Y:, — guQx 

 Zx = SuZx + 2snXy -\- guQy 



-Xy = Isi.Zx + 2(5?i - s%:)Xy + IgnQy (A.7) 



-E, = -^ (3, + gnXx - gn Yy + gu F. 

 Ai 



47r 

 ^v = -;7^Qy - gii Zx - 2^11 Xy 



47r 





where 



oO — o^ ^trdu Q B I '^irdu q e ^irdn du 



Sn - 5ii — -— ^ , 5i2 = 5l2 + -— ^ ; ^14 = 5i4 — J. 



J^i rii Ki 



Q _ B 4:irdu 4Trdn 4:Tdu 



544 - 544 — -—J- ; ^11 = —nr ', gu 



The superscript Q is added to show that these are the elastic compliances 

 that will be measured when the free charge on the surface is zero. These 

 elastic constants are the ones measured when an unplated crystal is put in 

 an airgap holder with a large air-gap since then no charge can flow to the 

 surface of the crystal. The difference between the zero field and zero charge 

 elastic constants for quartz is less than 1 per cent. For rochelle salt, how- 

 ever, they may differ by a factor of 4. For rochelle salt the principal piezo- 

 electric constant dx^ and the "free" dielectric constant A'f vary widely in 

 value and phase angle with variations in temperature and frequency, whereas 

 the piezoelectric constant gi4 which is proportional to the ratio of these two 

 is nearly a constant for all frequencies and temperatures, so that the formu- 

 lation of equation (A.7) is more advantageous than that of equation (A.6). 

 For quartz, however, both forms are reasonably constant. Furthermore 

 the elastic constants of equation (A.6) are those for a plated crystal which 

 are usually the ones of interest for a crystal employed in an oscillator or filter. 

 Hence this formulation has been used in this appendix. 



