172 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



The limits of such a procedure can easily be estimated. In the case 

 of an electric field, perturbation techniques apply if the kinetic energy 

 acquired by the ion from the field is small compared to thermal energy. 

 This means at least that the energy acquired in one mean free path be 

 small, i.e., 



eE\ « kT 



where e is the electronic charge, E the electric field, k Boltzmann's con- 

 stant, T the absolute temperature, and X the mean free path. Actually 

 the situation is not even that favorable. If the mass of the ions and the 

 molecules is very different, the energy transferred upon collision is 

 small, and hence the ions possess the ability to store the acquired energy 

 through many collisions; for this reason, the inequality reads more 

 properly 



(M m\ 

 \m] "^ Mj 



eE\ « kT, 



where m is the mass of the ions and M the mass of the gas molecules. 

 After some substitutions this estimate becomes 





eE « p<T, (1) 



where p is the true gas pressure and a the collision cross-section. Taking 

 as an example an ion travelling in the parent gas we find 



E .,^a ^ 47r.l0-'' _ .„_6 



_ « 2 - ^ 2- ^ ,, ,- = 5-10 e.s.u. 



p e 5-10-1^ 



or in commonly employed units 



E 



— <^ 2 volt/cm (mm Hg) . 



P 



It is clear that this limit is often surpassed in experimental situations. 



The cases in which the limit (I) is applicable are of no further interest 

 here because they are well covered in the literature.^ A field will be called 

 "low" when it satisfies the criterion (1) and "high'^ when the inequality 

 is reversed. It is important to notice that a fixed field at a fixed gas 

 density may shift from "low" to "high" through a drop in temperature. 



All calculations to follow will contain the assumption of "low ion con- 

 centration" which is often made in studies of this sort. It means that 



* See for instance: A. M. Tyndall, The Mobility of Positive Ions in Gases, 

 Cambridge University Press, 1938, Chapter IV. 



