THERMIONIC ELECTRON EMISSION 421 



N{u)du = un{u)du; hence 



N{u)du = T^ w In 1 + exp. ( ^f ) du. (19) 



Now only those having velocities greater than Uc will be able to cross 

 the surface and escape where Uc is given by ^mUc^ = P„,. Pm is the 

 difference in potential energy between an electron at rest inside and 

 outside the metal. Now Pm is about 3/2 times as large as K and 

 therefore ^muc^ > 1.5K. Also for the values of T encountered in 

 thermionic experiments kT is small compared with (hmUc^ — K). 

 Therefore, for values of m > Uc, exp. [_{K — \mu^)/kT^ is a very 

 small quantity and 



In [1 + exp. {{K - hnu'-)/kT)^ = exp. [(A' - lmu-)/kT'] 



to a good approximation. This follows, since 



In (1 + A) = (A - iA2 + lA^ - lA^ + • • .) 



provided A^ < 1. Hence, for u > Uc and the temperatures encoun- 

 tered in thermionic emission 



,,, , , 2ivGmrkT i K — hmu^\ , ..„, 



N{u)du = p u exp. I pp 1 du. (20) 



The number that cross the surface per cm.^ per second is given by 



A' = I N{u)du = Ys I " ^^P- ( UT 1 ^^^ 



lirGmk^T^ (K — \mu 

 U exp 



kT 



lirGmk''- 



h' 



_ / Pm- K \ 



r-exp.(^--^^^j 



(21) 



Finally 



i= Ne= {2TrGmek'/h')T' exp. [- (P,„ - K)/kT:\ 

 = Ur^ exp. (- W/kT) = UT- exp. (- iv/T) 



= UT'exp. (- <pe/kT). (22) 

 where 



U = iTvGmekyji^ (23) 



and 



Pm- K= W = kw= <pe. (24) 



