QUARTZ CRYSTAL APPLICATIONS 203 



These equations hold for the most general type of crystal. In addition 

 Voigt showed that if there was any symmetry existing in the crystal, a num- 

 ber of the constants were zero and certain relations existed between other 

 constants. For e.xamplc quartz has a trigonal symmetry about the Z or 

 optic axis, and three digonal axes of symmetry (the three X or electrical 

 axes) about which it is necessary to turn through an angle of 180° before 

 the original pattern is restored. Voigt showed that by expressing the rela- 

 tions (A.l) in terms of rotated axes and imposing the symmetry condi- 

 tions, the following relations existed between the elastic and piezoelectric 

 coefhcients 



E E E E E E E E E r^ 



Sib — ■^16 — -^25 — -^'ie — •^34 — -^35 — -^36 — -^45 — •^46 — vJ 



E B E E E E E E 



522 — 5ii; 523 — 5l3 ; 524 — 514) 544 — •^555 



E r, E E / ^ ^ \ 



556 — ^5i4; 566 — -^(.Sil 5i2J 



di3 = ^16 = di(, = dn = d-iz = ^23 = d-ii = ^31 = dz2 (A. 5) 



= d33 = du = ^36 = C?36 = 



di2 = —dn] d-ib = —du; d2& = —2dn 



F F 



Kl = K2 



Hence the relations between the stresses, strains, polarizations and fields for 

 quartz reduce to the simpler forms 



— Xx = SuXj, + 51^2^^ + SisZ: + sfiY^ — dnE^ 



— yy = SnXx -f SnYy + 5i3Zj — sf4Yz + duEx 



— Zx = SnXx + 5i3Fj, + 533Zj 



— yz = Si^Xx — Si^Yy + 544 F2 — d\\Ex 

 — Zx = 544Z1 -\- IsiiXy -\- duEy 



— Xy = 2sfiZx + 2(5fi — Sn)Xy -f 2dnEy (A.6) 



Qx = —. — — ^11 Xx + ^11 Yy — du Yz 

 47r 



E K"" 



Qy = ;, + duZx -\- 2dn Xy 



47r 



_ EzKz 



47r 



The superscripts have been left ofiF the constants 513 , 533 and K3 since it 

 will be shown that their values are not affected by the way in which they 

 are measured. 



