202 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



there are two notable exceptions to this rule, however, which make the 

 recurrence method possible, the equations v = have no left leg and the 

 equations s = v have no right leg. Starting out with the average 

 s = 0, V = 0, which equals unity by definition one can thus proceed 

 systematically as shown in Fig. 10, to get other averages. The averages 

 reached are the ones for which s and p are non-negative integers of equal 

 parity with the restriction s ^ v. One verifies easily that this set is equiva- 

 lent to the set of all products of integer powers of the velocity compo- 

 nents. 



The first three relations one uses in the path outlined in Fig. 10, are 

 the simplified forms of the three equations (50). We find 



2p / XV _ 4(M + my 



(c P, (cos t?)) - ^^ / 3ji^ si^2 ^ ^ ^^(^ _ ^^^ ^^ y ^ _ ^^^ ^ ^ 



\ ar A ar I 



or, more conveniently with the help of (53) 



{M + mf / ^ sii^' X 4- 4m(l - cos x) \ 



lr^\ = \ ^ /__ C54) 



^'^ 3P^ / 3M sin^ X + 4m(l - cos x) \/ l - cos x V 



\ ar /\ ar / 



The three equations (52), (53) and (54) give the drift velocity, the 

 total energy, and the energy partition of the travelling ion. Equation 

 (52) gives a constant mobility and can actually be derived from a low 

 field theory. Formula (52) thus states that for problems involving a 

 constant mean free time the high field and low field mobilities are numeri- 

 cally identical. One would suspect that the intermediate field value would 

 have to fall in line too. This is indeed the case as will be shown in Section 

 IIIA. 



A convenient interpretation of (53) may be had by combining (52) 

 and (53) in the following way 



(,mc^) = m{c^)'' -h M<c.)' (55) 



The left side is essentially the total energy of the ion, the first term on 

 the right is the energy visible in the drift motion; it follows therefore 



