The Free Electron Model and Thermionic Emission

The situation of the barrier is different from what it would be with the Boltzmann statistics, as can be seen most clearly from Fig. XXIX-6. Here we have drawn energies, both inside and outside the metal, as in Fig. XXIX-4. The zero of energy is taken to be at the bottom of the picture. Then at the absolute zero there will be filled energy levels up to the energy Wi and empty levels above, the filled ones being shaded in Fig. XXIX-6. We can now see that the potential energy of an electron outside the metal, Wa, is related to Wi and to Lq, the heat of vaporization 

For Lo represents the energy necessary to remove a mole of electrons from the metal at the absolute zero, in equilibrium. For equilibrium, the metal must be left in its lowest state, so that the removed electrons must come from the top of the Fermi distribution, and they must have no kinetic energy after they are removed from the metal. Thus each electron is raised just through the energy Lo/N in the figure. In the Fermi statistics, in other words, the work function represents the difference in energy between the top of the Fermi distribution and space outside the metal. And the result of Sec. 4, Chap. XXVIII, that on account of the contact potential the values of Ea — La and Eb — Lb were equal for two metals at the absolute zero, means graphically that two metals will adjust 

XXIX-6.—Occupied energy levels for the free electron model of a metal, at the absolute zero, illustrating the relation between Wa, W%, and the thermionic work function or latent heat of vaporization of electrons. 

The exact statement of Eq. (4.3), Chap. XXVIII, is that in any two metals in equilibrium, the space outside the metals acquires such a potential that if we subtract from it the amount 

referring to Fig. XXIX-6, we see that an energy lower by the amount 

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