656 BELL SYSTEM TECHNICAL JOURNAL 



quency examples of a dispersion of this kind have not as yet been 

 observed in dielectrics which are not piezo-electric.^® It follows then 

 that dielectrics behave as though the inertia of the particles which 

 contribute to dielectric polarization is small enough that the inertia 

 reaction mor can be neglected in the electrical frequency range. This 

 is an empirical result; the possibility of a polarization of the type kz 

 occurring in the electrical frequency range is not excluded by the 

 general theory of dispersion. The higher the frequency of an im- 

 pressed field the greater should be the likelihood of encountering the 

 type of frequency-variation described by kz (or equation (12)), because 

 the prominence of the moP' term increases with the square of the 

 frequency. 



The preceding discussion shows that we can write equation (14) 

 in the form 



k = k, + ka, (19) 



where k is the total polarizability, ki is the sum of the instantaneous 

 polarizabilities and ka the sum of the absorptive polarizabilities, that is, 

 of the polarizabilities which vary with frequency according to equation 

 (16). If for simplicity we take the case in which the dielectric has only 

 one representative of ki and one of ka, we obtain by substituting the 

 values of ki and ka given respectively in (15) and (16), 



k = ^ + ,. ""', (20) 



/i (^^2W + /a) 



as an expression for the total polarizability. 



Defining r' by t' = r/f, and dropping the subscripts in (20) to make 

 the notation simpler, we obtain 



K Ki ~\ p 



1 



1 + iu^r' 



(21) 



which is the total polarizability per unit volume for a dielectric having 

 two types of polarization, the one represented in (21) by the instan- 

 taneous polarizability ki and the other by the absorptive polarizability 



1'^ Piezo-electric crystals such as quartz and Rochelle salt form exceptions, but for 

 them dielectric polarization is coupled to macroscopic mechanical strains in the 

 material and the mass reactance is due to the flexing or extension of the entire crystal. 

 The dielectric constant of such a crystal as measured in almost any direction, 

 shows an increase with increasing frequency, followed by anomalous dispersion. This 

 is the behavior required by equation (12), or rather by an equation for the dielectric 

 constant derivable from equation (12). This dispersion, however, depends upon the 

 size and shape of the crystal, the nature of the electrodes and the manner of supporting 

 the crystal during the measurements, and the exact interpretation of such measure- 

 ments is a rather complex procedure. See, for example, W. P. Mason, Proc. L R. E., 

 23, 1252-1263 (1935). 



