596 BELL SYSTEM TECHNICAL JOURNAL 



zero at the potential minimum has the following result: 



dVaY ^ 2hm r 

 (dx) e J„ , 



(u — u')n{Uc)dUc + 



Ahm 





(dv^y 2hm r , ^^ ^ ^. 



, , \ = I (u — u )n{Uc)dUc, 



[dx) e J^, 



un(uc)dur, (8) 



(9) 



where u' is the electronic velocity at the potential minimum, i.e., 

 (u'y = Uc~ - (lefhm) v. 



At this point the analysis departs for the first time from the classic 

 analyses of Fry " and Langmuir,^ through the introduction of the 

 concept that the instantaneous rate of emission may be expressed as the 

 sum of an average rate of emission plus an instantaneous deviation. 

 That is, 



n{uc) = no{Uc) + d(uc), (10) 



transforms (8) and (9) into the following equations: 



(dVaY 2hm f" , /N / Nj 



— I (« — 11 )no{Uc)dUc 



(dx) 



where 



and 



where 



+ 



4hm 



XUc' 

 uno{uc)dUc + a(5), (11) 



2htH I Ahfft I ^ 

 a(b) = I ill — u')b{uc)duc H I u8{uc)duc 



{dVsY _2hm C (,, _ ^/)„^(^,^)^^^. _^ ^(5)^ (12) 



(dx) 



m = 



2hm 





u — u')b{ti,)duc. 



Since the average rate of emission may be expressed by the Max- 

 wellian relation, 



Wo(w,.) = 2aA^Mce~'*"'^ 

 where 



_ hm 

 " ~ 2kf' 



the indicated integrations in fll) and (12) have as a result, 



(kTy (dvaY _ Nhm jw^_^, 

 e \ a 



(e) (dx) 



X 



e^ - 1 i-e■^P(^lv) - 2M 



+ «(6) (13) 



