204 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



It comes out to be 



IM sin^ x\./^ sin^ x\./^ sin^ X + 4m(l - cos x)\ /^^qn 



This shows up immediately the equipartition property for small m/M 

 and the overwhelming preponderance of the motion in the field direction 

 for large mJM. As an analysis of the random motion, however, equation 

 (58) is deficient because the three directions become only comparable 

 after the square of the drift velocity (52) is subtracted out of the 2;-com- 

 ponent. We find 



(if + mf /J^«i^'x + 2m(l-cosx)\ 



/ 2\ / \2 _ \ O/T^ / (^()\ 



~ Mm / 3^ sin X + 4m(l - cos x) \/ l - cos xV 



and from this a more refined partition formula which only counts random 

 motion 



. . * /sin^ x\./sin^ x\./-^ sin^ X + 2m(l - cos x)\ /^^^ 

 e,.e,.e. = ^__/.^__/.^ (M + m)r / ^^^^ 



For small m/M this result does not differ essentially from (58) but if 

 m/M is large the z component of the random energy does not grow in- 

 definitely the way the total energy does. Instead it stops at a value which 

 is about four times one of the other two values. 



A discussion of these expressions for special models will be delayed 

 until the equations are extended to intermediate field conditions. This 

 will be done in Part III. 



lie. THE CASE OF LARGE MASS RATIOS : ELECTRONS OR HEAVY IONS 



The distribution of velocities for a small value of m/M is treated 

 in the literature because it applies to electrons.*'^ However, for the sake 

 of completeness the derivation will be carried out here for the high 

 field case. In this derivation all features of the law of scattering are left 

 open, except that conservation of the kinetic energy is assumed. 



The development in spherical harmonics carried out in Section HA 

 is the suitable starting point for small m/M, because, in this case, the 

 distribution is almost spherically symmetrical and the expansion in 

 spherical harmonics is also an expansion in powers of m/M. If we keep 

 only haic) and hi{c) in the system (47) and treat m/M as small we get 

 two equations, one for »/ = and the one f or y = 1. We may then use 



