174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



At high field the ''field" energy predominates and we get 



•■''"■«-K^+i) '*) 



A rough evaluation of inequality (3a) for one mm Hg pressure gives 



2/3 ^^ 10 -41^10 1 irk7 -2 



" ^< 25-10-^ -MO cm 

 n « 10^° particles/cm^ 



This corresponds to a current of about 10^^ particles/cm^ sec or 

 200 /zamps/cm^. At lower pressure the criterion becomes more stringent. 

 Equation (3b) gives similar results. 



It is appropriate to survey at this point the past theoretical work 

 treating the ''low concentration" theory of ionic motion for arbitrary 

 fields. A rather complete body of work exists for electrons where the 

 following three assumptions seem appropriate: (a) that the mass of an 

 "ion" is very small compared to the mass of a molecule, (b) that the 

 total kinetic energy is conserved in each encounter, and (c) that the 

 angular distribution is isotropic in the center of mass system. 



These three assumptions lead to a distribution law given by Chapman 

 and Cowling. The law has considerable flexibility because it permits 

 the substitution of an arbitrary relationship connecting mean free path 

 and speed of encounter. In addition it contains no assumption as to 

 whether we have low or high field. A more specialized and explicit dis- 

 tribution law is obtained if we assume in addition: (d) that the colli- 

 sion cross-section is independent of the speed of encounter (hard sphere 

 approximation) ; and (e) that we deal with the high field case only. The 

 special law resulting in this case is the distribution law of Druyvesteyn. 



If an improvement over the Chapman-Cowling distribution for elec- 

 trons is desired account should be taken of inelastic collisions, that is 

 assumption (b) should be discarded. Work in that direction has been 

 carried out by Smit, Allen^ and others. 



The assumption to be discarded first in theory of ionic motion is, of 

 course, assumption (a). In order to understand what this implies we 

 must understand what advantages assumption (a) has in a calculation. 

 In the limit when the ionic mass is very small the encounters with gas 



* Chapman -Cowling, The Mathematical Theory of Non-uniform Gases, Cam- 

 bridge University Press 1939, Sections 18,7-18.74. Other references are found 

 there. 



» Smit, J. A., Physica, 3, p. 543, 1937 and H. W. Allen, Phys. Rev., 50, p. 707, 

 1937. 



