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BELL SYSTEM TECHNICAL JOURNAL 



the dependence of conductivity on frequency in terms of the displace- 

 ment and velocity of a single bound ion. As we are not concerned 

 here with very high frequencies, we may employ the abbreviated 

 equation of motion given in equation (17) of the preceding paper to 



AMPLITUDE OF DISPLACEMENT, S , OF ION 

 IN AN IMPRESSED FIELD, F = Fq COS U)t 



A- LOW FREQUENCY ((jj«T-') 



e'=eo y'=o 



DISPLACEMENT OF ION FROM EQUILIBRIUM POSITION 



Fig. 3 — The mechanism of anomalous dispersion illustrated by a simple model. 



The model consists of a single bound ion. The potential energy of this ion in- 

 creases when it is displaced from its equilibrium position 0. The ion also experiences 

 a frictional force proportional to its velocity, as if it were an ion in solution. The 

 upper part of the diagram shows the way in which the amplitude (5) of displacement 

 of the ion by a given applied field varies with the frequency. It has its maximum 

 amplitude at low frequencies {A in the diagram) and a comparatively negligible 

 amplitude at high frequencies (C in the diagram). In this model the amplitude is a 

 measure of the dielectric constant. The limiting value Tcd of the conductivity 

 prevails under the conditions C where the amplitude is comparatively negligible. 



discuss the motion of the bound ion of this model ; this equation is 



rv -\- js = eF, 

 where v — ds/dt. 



When a d-c voltage Vi is applied to this model, it establishes a field 

 Fi which displaces the bound ion to a new equilibrium position Si if the 

 field is allowed to act upon the ion for a sufficient time. The new 



