422 BELL SYSTEM TECHNICAL JOURNAL 



If i is expressed in amperes per cm. 2, the value of U'\s\ 20 amperes/cm.^ 

 ° K.^ The quantities W, w, or <p are called the work function ; the 

 difference between them is merely one of units. <p is expressed in 

 volts, IV in degrees Kelvin, and Pm, K and W in ergs. Pm is called the 

 outer work function and K, the inner work function. Oftentimes it is 

 convenient to refer to Pm, K and W as if they were expressed in volts. 

 c. Treatment in terms of energies. For many purposes it is con- 

 venient to have expressions for the distribution in energies instead of 

 in velocities. We can then express these energies in equivalent volts 

 and obtain numerical values which are more familiar. Let 



E = (m/2){u^ -\-v~-{- W-) ; £„ = {m/2)u\ 



" the normal component of the energy "; F„ = En/e; n(E)dE = the 

 number of electrons per cm.^ having energies in the range {E, dE) ; 

 similarly for n{En)dEn\ N{Vn)dVn is the number striking a cm.^ of 

 surface per second having normal component of energies in the range 



{Vn,dVn). 



Then 



nmdE = ^ (2.)^ 1 + exp. [g - X)/.r] ^^' ^''^ 



n{Er,)dE„ = ^^ -^, In 



1 + exp 



(K - E„) 



N{En)dEn = Y^ In 



iTrGemkT , 



(kT) 

 kT' 



j dEn, (26) 



1 + exp. ('^T#^)] dE^' (27) 



N{Vn)dV„ 



h' 



l+exp.( ^ ,^/"' )]dl\. (28) 



Equations (25) and (26) are readily derived from equation (18); 

 while equations (27) and (28) follow from equation (19). It is also 

 instructive to compare equation (28) with the corresponding equation 

 which is based on classical statistics, namely, 



iV(Fn)<f Fn = nieyiTrmkT)^ exp. (- Vne/kT)dV„. (29) 



This is readily derived from equation (8). 



d. Comparison between classical and quantum-mechanical treatment. 

 Comparison between equations (28) and (29) is best brought out by a 

 graph such as Fig. 1 which shows log N(Vn) versus Vn for the two 

 cases. It is to be remembered that N{V„)dV„ is the number of elec- 

 trons in the metal which strike 1 cm.^ of surface per second whose 



