184 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



dependence does not only hold for (c), but for all averages derivable from 

 /(c), notably the mean energy. 



A more important formal prediction can be made about the second 

 stage of the contemplated calculation. For it will be shown now that the 

 diffusion concept is still applicable in the presence of a strong electric 

 field. It is true, that if we have a variable density in space the primary 

 motion observed is not diffusive but a displacement of the entire density 

 pattern with the drift velocity (c). However, once this dominant com- 

 ponent is subtracted out, then a supplementary current proportional to 

 the density gradient is identified. The constant of proportionality is 

 anisotropic, that is, we have a diffusion tensor rather than a diffusion 

 coefficient. The tensor is axially symmetric about the field direction, 

 yielding a longitudinal and a transverse diffusion coefficient. 



To demonstrate these features it is convenient to assume a special 

 type of variation of ion density in space. As we shall see the velocity 

 distribution is primarily sensitive to the relative density gradient k; 

 we shall therefore assume it to be a constant. In other words we set 



n(r, t) = no exp [k-(r - (c)^] (16) 



The relation can of course not hold everywhere since n increases beyond 

 all bounds in one direction, but we must remember here that we are 

 doing perturbation theory, that is k is assumed small. The inconsistencies 

 in the assumption (16) can then be pushed as far away as we please. 

 Furthermore there is no inconsistency at all in the half space where n 

 decreases. It is to be observed that according to the assumption (16) 

 the spatial distribution is moving unchanged through space with the 

 drift velocity (c). This seems to contradict the program of finding the 

 effect of diffusion upon n(r, t). However, we follow in this simply conven- 

 tional steady state computational methods in which a gradient is as- 

 sumed maintained from an infinitely strong source; the modification 

 appears then as a change in the velocity distribution function, which, in 

 turn, yields a steady diffusion current. We set therefore 



rf(c, r, t) = n(r, Of/(c) + ^(c)] (17) 



where ^(c) is a correction to the solution of (13) which arises from the 

 assumption (16). It follows from the definition of n(r, t) and (14) that 



/ 9(c)dc = (18) 



The consistency of the assumptions (16) and (17) with equation 

 (10) becomes evident when they are substituted into this equation. We 



