186 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



as is evident from (16) ; in addition the multipliers of the first two com- 

 ponents are equal. We may write therefore 



j(r,0=-O)^ (24) 



where (2)) is a tensor which is axially symmetric about the field direction ; 

 its two components which we shall call the longitudinal diffusion coeffi- 

 cient D\\ and the transverse coefficient Dx are computed entirely from 

 the unperturbed velocity distribution /(c) . This makes D\\ and Dj. inde- 

 pendent of the density or its gradient; it is to be noted, however, that 

 they do depend on the electric field as a parameter because this quantity 

 enters several times in the course of the computation. 



ID. DIMENSIONAL ANALYSIS 



Dimensional analysis is a convenient tool in a qualitative discussion 

 of (13) and (20). In order to get results the situation has to be 

 schematized somewhat, but not so much as to impair its usefulness. 

 In the first place it is convenient to keep in mind the two limiting cases 

 of high and low field, as discussed in the introduction. In addition some 

 assumption must be made about a{y) and 11 (x) occurring under the in- 

 tegral sign. The most convenient way to dispose of 11 (x) is to take it as 

 independent of 7. This happens to be true for the two models treated 

 in detail later, the polarization force model, and the hard sphere model. 

 Actually n(x) can be taken as approximately independent of 7 in a wider 

 sense. The forces which produce scattering are either repulsive or short 

 range attractive, that is, long range attractice forces are absent. As long 

 as this is the case the scattering is roughly isotropic and hence can 

 change but little with 7.^* 



A more drastic assumption is needed to dispose of o-(7). We must 

 assume 



<r(y)-y° = r (25) 



where a and F are taken to be constants. This assumption contaiils two 

 important special cases in it. They arise respectively by taking a = 

 and a = 1. The case a = is the case of a constant mean free path as 

 exemplified by the hard sphere model. The case a = 1 is the case of 

 constant mean free time; it is applicable to the polarization force as dis- 



^* This statement is checked in detail in Section IIIB for the polarization 

 force. This is the attractive force with the longest range which can arise in this 

 field. 



