712 BELL SYSTEM TECHNICAL JOURNAL 



upper part of the energy-range, where the first term is much the 

 larger. Writing then z<-exp \^— (e — Wi)lkT'^ for the integrand, 

 we integrate with ease by well-known formula^, and get: 



i = ^^^^ (y^r)-V-<"'«-" •)/*^. (76) 



The experimental test consists in plotting (log i — 2 log T) against 

 1/7"; we should get a straight line provided that Wa does not vary 

 with temperature. The experiments do lead to precisely this result. 

 The slope of the line varies from metal to metal, and depends on the 

 state of the metal surface; identifying it with {Wa — Wi)/k, one finds 

 that Wa exceeds Wi by amounts ranging from one equivalent volt 

 upward to five or six; so the approximation just mentioned is abun- 

 dantly justified, even up to temperatures of incandescence. 



The contrast with the predictions of the classical theory is peculiarly 

 interesting. Assuming the Maxwell distribution for the interior 

 electrons, one arrives easily (the reader can do it by substituting the 

 value of A from (64) into (73)) at the formula: 



{Iirm) 



jjrAkryi^e-''-'-"^^. (77) 



On testing this formula by plotting (log i — ^ log 7") against 1/T, it 

 is found that the experiments yield lines as beautifully straight as those 

 obtained by plotting (log i — 2 log T). Indeed — as everyone knows 

 who has dabbled in thermionics — a function of the type exp (— c/T) 

 varies so exceedingly rapidly with l/T that it makes no perceptible 

 difference to the graph whether or not the function is multiplied by a 

 constant or by any modest power of T. One cannot then use the 

 graphs to distinguish between the theories, even if one could be sure 

 that Wa is not a function of T. But the classical theory proposes 

 that we identify the slope of the aforesaid line with Wa/k; and it 

 has been the custom so to do. 



Now if the Fermi distribution-function is the right one, physicists 

 have been underestimating the work-function all along. They have 

 plotted experimental curves which agreed in shape with (76) and (77) 

 and from these they have evaluated the constant figuring in the 

 exponent, a constant which they have denoted usually by — bjk; 

 and then they have equated b to W„, whereas if it is right to apply 

 the new statistics they should have added Wi to b and then equated 

 the sum to Wa- Nor is the alteration slight: for if there are as many 

 free electrons in the metal as there are atoms, then Wi is six volts 



