658 



BELL SYSTEM TECHNICAL JOURNAL 



The constants ne^jf and ki are not present in (26), being replaced by- 

 two special values of the polarizability, the zero-frequency value and 

 the infinite-frequency value. However, it is not the polarizability 

 but the dielectric constant which is directly observed in measurements 

 on dielectrics, so it is desirable to replace ^o and ka, by their equivalents 

 in terms of the dielectric constant. But, as the earlier discussion has 

 indicated, the relation between dielectric constant and polarizability is 

 different for different types of dielectrics; three alternative expressions 

 analogous to (22a), (226) and (22c) will therefore be derived. 



For materials to which equation (22a) (or the equivalent and simpler 

 relation (5)) applies 



h = 



Co 



1 



1 



eo+ 2 



+ 2 



9(60 - ej 



47r(eo + 2)(6co + 2) 



(27) 



where to is the zero-frequency dielectric constant and 6co is the infinite- 

 frequency dielectric constant. Then equation (26) can be replaced by 



47r _ €o — 1 



+ 



eo 



1 



eo + 2 



+ 2 



1 



1 + ic 



(28) 



By rationalizing and using the second expression given for ^o — > 

 equation (27) we can write equation (28) in the alternative form 



m 



~3" 



- 1 



+ 2 



+ 



3( 



«o 



.) 



1 



(€o + 2)(eoo + 2)J 1 -f coV' 

 3(eo - too) 



— I 



(€0+ 2)(€co + 2) J 1 + o^V' 



(29) 



Equation (29) is the complex polarizability per unit volume multiplied 

 by the factor 47r/3 and expressed in terms of observable values of the 

 dielectric constant and the relaxation-time r'. The relaxation-time 

 can also be expressed in terms of the reciprocal of a special value of the 

 frequency; this permits all of the theoretical constants such as ne^jf 

 and t' to be replaced by certain special values of the dielectric constant 

 and a critical value of the frequency. 



A simpler expression for the polarizability is obtained in the case of 

 gases, or whenever equation (6) gives the relation between polariz- 

 ability and dielectric constant. Equation (26) then gives 





- 1 + 



«o 



1 + iwr' 



(30) 



And for materials to which the relation (cf. equation (9)) proposed by 



