MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 173 



all effects which ions exert upon each other are neglected. The equation 

 for the distribution function of ionic velocities is then linear instead of 

 quadratic. It is clear that this simplification presents great advantages 

 from the point of view of calculation. 



In deriving a criterion for the validity of this assumption we must 

 distinguish two types of effects of the ions upon each other. The first 

 is the space charge effect. In this effect the ions at large distances make 

 the major contribution. Its magnitude depends on apparatus dimensions. 

 The criterion for no space charge distortion of the field E is 



where n is the number density^of the ions and X a suitable length chosen 

 from apparatus dimensions. Inequality (2) is quite stringent because 

 it predicts field distortions at values of n of the order of 10^ cm~l This 

 is the value at which it will become impossible, or at least difficult, to 

 make significant experimental measurements. But from the point of view 

 of theory this criterion is not relevant. Space charge does not change 

 the character of the velocity distribution of the ions because the type 

 of ion-ion interaction producing the space charge field is long range and 

 creates only a smooth modification of the electric field which we may 

 presume to have been included in the original field. What we are con- 

 cerned with here are ion-ion interactions which have a random character 

 and thus are apt to upset a velocity distribution derived from the "low 

 concentration" theory. From this point of view neighboring ions are 

 most effective because their relative location fluctuates rapidly, and 

 hence, the Coulomb force between them will induce mutual scattering. 

 The magnitude of this force is of the order eV^^ where n is the number 

 density of the ions. It is known from theory^ that the effect of a Coulomb 

 force is preferably not represented by discrete "coUisons" but by a 

 continuous bending of the entire path. Thus we come to the conclusion 

 that random ion-ion forces have no effect if the force given above cannot 

 produce a significant deflection in one mean free path. This means 



eW^ <<C mean ion energy (3) 



According to whether we are in the high or low field region we get differ- 

 ent criteria from this. At low field the thermal energy predominates 

 and we get 



''' « pa (3a) 



e^n 



3Mott and Massey, The Theory of Atomic Collision, Oxford Press 1933, 

 Chapter III. 



