418 BELL SYSTEM TECHNICAL JOURNAL 



given by Maxwell's law 



'»i/7 5 r — 'MI I •»/ - —1— *j)- —I— '7^1-1 



diidvdw, (7) 



, X , , , nrn^ \ — ni(u- + v~ + w'-) 

 n{u, V, w)dudvdw = py—^-^e\p. jtj^ 



where u, v, w are the velocities in the x, y, z directions respectively ; 

 n{u, V, w) is the number of electrons per cm.' having u values in the 

 range (w, du), i.e., between u and u + du, v values in the range iv, dv), 

 and w values in the range (w, dw) ; n is the total number of electrons 

 per cm.' ; m is the electron mass ; and k is Boltzmann's constant. The 

 number of electrons having u components of velocities in the range 

 (u, du) is obtained by integrating equation (7) with respect to v and 

 w from — oo to + 00 . 



n{u)du = nim/lirkTy exp. (— mu-/2kT)du. (8) 



The total number of particles striking a cm.- of surface per second is 

 given by 



Nt = j un{u)du = n{kT/2Trm)K (9) 



But only those electrons whose values of u exceeds Wo = {2p/m)^ 

 will escape from the surface. The quantity p, called the work func- 

 tion, represents the potential energy of the electron outside the metal ; 

 it is the work that must be done to take an electron at rest in the metal 

 and transport it across the surface to a distance at which the surface 

 forces are negligible. The total number, TV, of electrons w^hich can 

 escape from one cm.^ of surface in one second is then 



N = I un{u)dn 





n{m/2TrkT)hi exp. { — mu-/2kT)du 



= nikT/2Trmy exp. (- p/kT). (10) 



The emission current in amperes per cm.^ is 



i= Ne= AT^ exp. (- p/kT) (11a) 



or 



log i - ^ log T = log A' - P/2MT, (lib) 



where 



A' = ne{k/2Trm)K (12) 



e = charge on the electrons in coulombs = 1.59 X 10' 



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