596 BELL SYSTEM TECHNICAL JOURNAL 



zero at the potential minimum has the following result: 



, ■ ° — I (m — u')n{uc)duc -\ • I un{uc)dur, (8) 



{dx) e Jj,^, e Jp 



{dx) e 



I (m — u')n{uc)duc, (9) 



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

 {u'Y = Ue- - {2e/hm) 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, 



«(Mc) = Wo(m<,-) + 5(Mc), (10) 



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



— I (m — ic )n(i{u,)dUc 



f ''uc' 



{dx) 



where 



and 



where 



4hm C""' 



H uno{u,)duo + a{d), (11) 



e Jo 



"Zhttt I AhtH I 

 a(8) = I (w — u')8{uc)dUc H I u8{Uc)dUc 



{dV^ ^2hm r ^^_ ^/)„^(^^)^^^ _^ ^(5)^ (12) 



{dx) e J„/ 



^(5) ^ ^ r (^ _ u')5{Ur)dUr. 



e Juc' 

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

 wellian relation, 



«o(«.) = laNUce-""'', 

 where 



_ hm 

 " ~2kf' 



the indicated integrations in (11) and (12) have as a result, 



{kry {drra)^ ^ Nhm It _^, 

 {e) {dx) € \a 



X [e"- 1 +e''P(Vr7) - 2^^ 



+ «(5) (13) 



