SHOT NOISE IN DIODES 615 



velocity, the term gives the equivalent noise generator, E. Thus 



E=f^{ide-^9 j^e-'9 - 1) (60) 



and the mean-square value of the noise emf. (at a frequency w) is 

 given by : 



£2 = ^^" \ie-^6 ^ Q-i0 _ u\ 



(61) 



The problem is now reduced to finding the mean square value of 

 initial velocity fluctuation, txa^, which corresponds to electrons cross- 

 ing the potential minimum. This may be done by going to (55) 

 which gives the effective value of the instantaneous initial velocity 

 and separating all quantities, including the lower integration limits 

 into d-c. and a-c. components. Thus 



n{uc) = no{tic) + 8(uc) 



uj = Uc + 8Uc 



u' = u^ -{- bu' 



U = Ua + IJLa 



(62) 



The result may be expanded in series form and products of the 5's 

 may be disregarded inasmuch as the a-c. components are small in 

 comparison with the d-c. The indicated operations have as a result 



and 



e f" 



Ma = -^ 1 (U' — Ua)8 



{Uc)dUc 



(63) 



(64) 



The Fourier analysis may be applied to this in the way outlined in 

 connection with (37) and (41) in Part I and gives the mean-square 

 value of velocity fluctuation corresponding to a frequency interval df 

 as follows : 



— 2e2 p 4ekT,,/. 7r\ 



AT = T^«/ (" - Ua)-Uc{Uc)dUc = -f^^-dfi 1—1 

 1 0" ./_ I onm \ 4 / 



(65) 



This may be substituted in (62) giving for the effective noise emf. in 

 the frequency range df 



E/ = 4kTdf\ 



el, 



OT- 



L hmt~ 



1 



Xi[^- + 2 - 2 (cos d -\- 9 sin 6)']. (66) 



