MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 183 



The function m(c) is not the solution of our problem because of the 

 presence of the second and third term on the left which arise from an 

 electric field and a density variation respectively. These disturbances 

 will be assumed of different relative importance. The density variation 

 will be assumed sufficiently small so that the third term can be treated 

 by perturbation theory; the field term, on the other hand will be taken 

 so large that the equilibrium distribution (11) no longer represents a 

 first approximation to the solution. In consequence, the equation is 

 solved in two stages. In the first, only the second term on the left is 

 retained, and the resultant equation is treated rigorously; in the second, 

 the full equation (10) is used, but the new terms are taken as 

 perturbations. 



The first stage describes those properties of the ion gas which it po- 

 sesses when assumed of uniform density. Since the field is also assumed 

 uniform and not changing in time, the dependence on r and t drops out. 

 We may then write 



die r, t) = n/(c) (12) 



where n is a constant and /(c) is a velocity distribution function. The 

 equation for / reads 



a.| = ^ // |M (C')/(c') - M(C)/(c) l7<r(7)n(x) da,, dC (13) 

 with the side condition 



j f{c)dc = 1 (14) 



As a result of solving (13) we shall obtain the distribution function 

 /(c) as a function of the electric field contained in a. This distribution 

 differs essentially from the Maxwellian one in that it is not symmetric 

 about the origin. The vectorial mean of the velocity is therefore not zero 



(c) = /*/(c)c dcT^O (15) 



This is the drift velocity of the ion in the field which is reached as a 

 compromise between the acceleration a and the frictional losses caused 

 by the ion-atom collisions. From the structure of equation (13) there is 

 one general prediction that can be made concerning this velocity, namely 

 that it depends on the gas density and the field only through a/N; 

 this is the well known E/po of the experimental analysis. This type of 



