720 BELL SYSTEM TECHNICAL JOURNAL 



frequencies this will be equal to half the mass of the crystal, but at 

 higher frequencies will be less due to the fact that the crystal does not 

 move as a whole. ^ In order to resonate with the compliance Ce at the 

 mechanical resonance frequency of the crystal, Le must equal 



l^E = 5 — . W 



where p is the density of the crystal. The resistances R shown include 

 the dissipation due to internal friction, supersonic radiation from the 

 ends of the crystal, friction at the point of support and all other sources 

 of dissipation. 



If Fo is the force required to keep the crystal from expanding when an 

 electric charge Qi is applied to the crystal then Cm the mutual capaci- 

 tance-compliance of the crystal is equal to 



Cm = QilFo. (3) 



Similarly if Eo is the open circuit voltage for a given expansion {Q2 + Qs) 

 of the crystal then 



Cm = —^^ . (4) 



In order to evaluate Cm in terms of the piezo-electric coefficient d, it 

 is necessary to find the displacement for a free crystal when an electrical 

 potential is applied to the crystal. Short circuiting the network of 

 Fig. 2 on the mechanical ends and setting F = 0, we find 



E 



<22 + (33 = - 



CqCi 

 Cm 



. CoCe 1 — k 



Ek'\CoCE ,-v 



where k is the coefficient of coupling between the electrical and mechan- 

 ical system is defined by the equation 



k = ^CoCe/Cm. (6) 



Use is now made of the piezo-electric equation 



e = dV, (7) 



* Strictly speaking the value of Ce is also a function of frequency, but at the first 

 resonance it can be shown that it differs from its static value by the factor 

 8/[7r2 — ^2(7r2 — 8)] and Le = IJmkp/^- For a highly coupled crystal, this factor 

 does not differ much from unity, and hence in the interest of simplicity, the variations 

 of Ce have been neglected. 



