MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 217 



to h db. Thus even if all c,'s were equally probable the probability for 

 Cf would vary as c/ dc/ . Actually very small d's may be specially probable 

 as the theorem states and this fact may or may not increase the prob- 

 ability for small c/. This means that P{cf)dcf probably varies as C/dc/, and 

 may perhaps even contain a smaller power of C/ . When such a probability 

 function is plotted in velocity space it will vary as 1/c/ . Thus we know 

 that the distribution function (p{c) for ions immediately following a col- 

 lision has a singularity at the origin at least as 1/c/ . The actual distri- 

 bution function h(c) is derived from this one by spreading each point 

 out in the forward direction as shown in Fig. 11. For the mean free time 

 case we can write this out explicitly in the form 



1 r - 



He) = - / <p{c - Sit) e~' dt (76) 



T Jo 



For the case of a mean free path or other laws the formula is more awk- 



_i 

 ward but they all differ from the above only in replacing e ^ by a more 



complicated weight function. The singularity of h arises out of the singu- 

 larity of <p which contains at least a factor l/Vcl + Cy + {Cz — at)^. 

 Along the c^-axis, this become a factor l/{cz — at); this factor makes the 

 integral diverge for all positive Cz ; as we approach the origin from nega- 

 tive Ca's the distribution function will become infinite at least as /n 1/c, . 



The reasoning given is intrinsicly classical because of the use of "in- 

 finitely small" impact parameters. We should not hasten to conclude, 

 however, that the quantization of the angular momentum will necessarily 

 remove the singularity. Indeed we know that the only mechanical 

 information which has to be put into the Boltzmann equation (31) is the 

 differential cross-section for scattering. If this quantity does not differ 

 essentially in the 180° direction from a classical cross section then it 

 will not modify the conclusion we have reached. 



Conclusions which are more informative, but less "anschaulich" may 

 be obtained from a study of Boltzmann's equation either in its closed 

 form (34) or (40) or its 'Xegendre" form (46) or (47). In view of the 

 proof given we will give only an outline of the reasoning. First we can 

 remove the second term in (40) by the substitution 



.(c) = exp[-/\^J.*(c) 



The exponential is easily seen to be always positive and finite for finite c. 

 The Boltzmann equation then takes the form 



exp f- f" 4^1 a ^* = i- (an integral) 

 L •'0 Ctr(c)J dCz C 



