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BELL SYSTEM TECHNICAL JOURNAL 



The other branch contains the impedance 



}fw^ 



Sn 



2v 



dn . (^ 

 . **" 2. J 



cgs units = 



-j^tsu"^ X 9 X 10'' 

 2v 



Oily, I d\\ tan — - 

 2v 



ohms (A.50) 



This branch will have a zero impedance or will resonate when the tangent is 

 infinite or when 



(A.51) 



ir~ 2 '''■^" ~ 2l~ U^ 



E 



Hence for a fully plated crystal it is the zero field elastic constant that 

 determines the resonant frequency. 



Near this resonant frequency, the impedance of equation (A.50) can be 

 represented by a series capacitance and inductance having the values 



Ci = — 5 



u = 



£2 



pSu 



X9X W 



ft 7r2 5fi X 9 X 10" ' 

 Taking the ratio between Co and Ci we have 



Sf. dn 



(A.52) 



Co 

 Ci 





8 \ 47r Ji 





1 - 



(A.53) 



Xl 5ii 



where k the coefficient of electromechanical coupling is equal to 



k = d 



' y KUl, 



(A.54) 



These values are used inequations (A.2 7) and (A. 28) to evaluate the piezo- 

 electric constants of quartz. 



A.4. Use of Voigt's Relations in Locating Regions of Low Tempera- 

 ture Coefficient Crystals for Simple Modes of Motion 



In Section 1.5 of the text, the statement is made that all longitudinally 

 vibrating crystals of quartz have a zero or negative temperature coefficients. 

 This can be proved from \'oigt's relations for quartz and a knowledge of 

 the temperature coefficients of the six elastic constants of quartz. Since 

 the same method can be used to locate the regions of low temperature coeffi- 

 cient for other simple modes of motion a short discussion of the method is 

 given here. 



The Voigt relations given in equation (A. 6) give the values of the piezo- 



