176 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



total energy. In order to test whether an offset Maxwellian distribution 

 is a satisfactory approximation we have to go one step further and 

 examine the partition of the energy among the three degrees of freedom. 

 There we find Hershey's distribution in error, for he assumes equiparti- 

 tion for the random motion, while, in reality, the random energy parallel 

 to the field is much higher than at right angles,^ giving the distribution a 

 decided *'ridge'* structure. This discrepancy could be taken into account 

 by the use of an elliptically distorted Maxwellian distribution, shown in 

 Fig. 1(b), and this may prove to be convenient in some applications. 



For a detailed knowledge of the distribution function it is necessary 

 to specify the interaction between an ion and a molecule. This interaction 

 can be, broadly speaking, summarized under three headings: (a) the 

 polarization force, (b) the short distance repulsion, and (c) symmetry 

 effects. The polarization force arises because an ion, when passing close 

 to a molecule, induces on it a dipole moment; this moment is then at- 

 tracted by the charge of the ion. The attractive force F resulting from 

 this is 



' - ^-? 



where P is the polarizability of a gas molecule and e the charge of the 

 ion. The force varies inversely as the fifth power of the distance p; for 

 such a force the cross section o- varies inversely as the speed of encounter 

 7. Whenever the cross section shows this type of variation it is advantage- 

 ous to define a mean free time r rather than a mean free path X. The 

 formula is 



There is a standard difficulty which arises when one tries to make use 

 of a formula of the type (5). For most force laws, a total cross-section a 

 cannot be defined; a differential cross section per unit solid angle always 

 exists, but it becomes infinite in the forward direction because of small 

 deflections suffered by particles passing by each other at a large distance. 

 Thus equation (5) is, strictly speaking, meaningless. This is actually 

 never a difficulty in the computation of a physical quantity. However, 

 equation (5) is convenient for prder-of-magnitude thinking and the 

 question arises how it can be reasonably interpreted. The general method 

 of salvaging (5) — excluding a small forward cone from consideration — 

 is of little value for this purpose. An analysis of the inverse fourth power 



