202 BELL SYSTEM TECHNICAL JOURNAL 



— Zx = SbiXx -\- ^52 Yy ~l~ S^sZz -f- Sa Yg -\- Si^Zx 



-\- ^66 Xy — ai6 Ex — 025 Ey — ^36 Ez 



— Xy = S6l Xx + ^62 Yy + ^63 ^z + ^64 Yz + S^^Zx 



-\- ^66 Xy — ai6 Ex — ^26 Ey — dsG Ez 



Px = — d\i Xx — dn Yy — di3 Zz — du Yz — d^ Zx — die Xy -\- ki Ex 



Py = — 021 Xx — dil Yy — a23 Zz — 024 Y z — «25 Zx — d%e Xy -\- K2 Ey 



Pz = —dziXx — dn Yy — d^zZz — dsi Yz — d^^Zx — dze,Xy + kz Ez 



where Xx ,yy , Zz are the three longitudinal strains, yz iZx ■, Xy the three shear- 

 ing strains, Xx ,Yy , Zz the three longitudinal stresses Yz , Zx , Xy the three 

 shearing stresses; Px , Py , Pz the x, y and z components of the polarization, 

 and Ex ,Ey , Ez the x, y and z components of the electric field, ^fi , • • • , s^e 

 are the 36 elastic compliances. The superscript E is added to show that 

 they must be measured when the field E is zero or the crystal plated and short 

 circuited. As shown from section C of this appendix they can be measured 

 from the resonances of completely plated crystals. From the principle of 

 conservation of energy it can be shown that there is the general relation 

 between the elastic compliances 



Sij = Sji (A. 2) 



so that the greatest number of compliance moduli is 21. In equation (A.l) 

 the dij are the piezoelectric constants measured by observing the propor- 

 tionality between the strains and the applied fields in the absence of external 

 stresses. Ki are the "free" susceptibilities of the crystals in the three space 

 directions measured in the absence of stress. The susceptibilities are related 

 to the "free" dielectric constants Ki by the equation 



Ki = 1 + ^TTKi (A.3) 



In addition to these equations we have also that the charge per unit area Q 

 on the surface is related to the field and polarization by 



Q. = ^ + P. 

 At 



ExKi 

 At 



dn Xx — di2 Yy — diz Zz — du Yz — dii Zx — di6 A j 



(A.4) 



0- = l + ^' 



E K 



— — dzi Xx — dz2 Yy — dzz Zz — dzi Yz — dz5 Zx — dz6 Xy 



At 



