The restored electron theory of metals and thermionic formulœ

In a recent paper Sommerfeld has completely reinstated the electron theory of metallic conduction and other thermo-electric effects in good conductors by applying the Fermi-Dirac statistics to an atmosphere of free electrons in the metal, of a concentration of the order of one per atom. It is the purpose of this note to call attention to an application of this theory which Sommerfeld does not discuss, namely, the calculation of the density of evaporated electrons in equilibrium with the heated metal. This calculation, of course, bears on the well-known formula for thermionic emission and gives it a form on which I wish to comment. It is for some reasons desirable to establish the vapour-pressure formula on the new basis without particularised assumptions as to the mechanism of emission and absorption of electrons. To do this we have only to consider the equilibrium state of an assembly of electrons in two phases—one a volume outside the metal in which the energies of the electrons are the characteristics of the Schrödinger waves proper to such an enclosure in which the potential energy of an electron is zero; the other a similar volume inside the metal, with similar characteristics, but now, of course, all diminished by a large (constant) negative potential energyχ 0 . Outside the density is low and the degeneracy is negligible, so that we have (practically) classical distributions. Inside the electrons are almost tight-packed and the degeneracy is of dominant importance. It follows at once by the usual statistical arguments that if N- is the average number of electrons in either phase, εσ the characteristics and gσ the weights for electrons in that phase, then N- =μ ∂/∂μ ∑σ g σ log(1 +μe ─εσ /k T). (1)