QUARTZ CRYSTAL APPLICATIONS 215 



Solving these equations for A and B and substituting in (A.43) we have 



or 



-yy = d. 



Yy= - 



11 £i 



coc . coy , coy 



tan -- sin ^^ + cos 



2v V 



'^y-^y- 



+ ^11 E^ 



dnEj, 



S\i 



cos 



<^{y - m) 



U>1 



cos — 



2v 



(A.46) 



The electrical impedance measured at the terminals of a plated crystal is 

 then determined by substituting the value of Yy in the last of equations 

 (A.v34) and integrating the charge Q over the whole surface. The current 

 into the crystal is then 



Jo 



Kx _ dn 1. cos CO (y — (jly 

 4t 



Sxx 



Oil 

 COS — 



2v 



dy 



= jOiiEx 



= joiEJl 



rrr J 2 / tan 



Aa _ flii_ . _ 2v 



4r 



S\\ 



2v 



(A.47) 



'LC 



Oil 



2 tan — 



^r _^ dn 3 2v 

 47r sn w 



2v J 



where A'l' = Ki — 



Sn 



is called the longitudinally clamped dielectric 



constant, i.e. the dielectric constant that would be measured if we suppress 

 the longitudinal strain along the y axis but not the other strains. The 

 admittance of the crvstal then is 



(A.48) 



This consists of two terms which represent parallel branches in the equiva- 

 lent circuit. One of these is the capacitance 



P PT/'^'^ P PTr^^ 



IwlJ^l ., IwlJ^l r J 



Co = — , — , — cgs units = T-. ^ . . .^.; farads 



47r^, 



4x^,9 X IQii 



(A.49) 



