Theory of the Work Function. II. The Surface Double Layer

An expression for the work function of a monovalent metal, previously obtained by E. Wigner and the author, included a term which represents the energy required to move an electron through an electrostatic double layer at the surface of the metal. In order to determine the moment of this double layer, it is necessary to make an explicit calculation of the electronic charge density at the surface of the metal, which is attempted in the present paper. In the model taken for this calculation, the density of positive electricity is assumed constant on the negative side of the $\mathrm{YZ}$ plane, and is zero in the positive half-space. The electrons are so distributed as to neutralize the positive charge density in the interior, and the metal as a whole is neutral. An approximate self-consistent solution of the Fock equations gives the charge density at the surface. An attempt is made to go beyond these equations, so as to include the effect of correlation (or polarization) forces on the double layer. The density of positive electricity assumed is the same as that which would exist in Na if the ions were treated as a continuous charge distribution. The final values obtained are 2.35 and 2.0 ev for the work function, and 0.4 and 1.0 ev for the moment of the double layer, where the first values in each are obtained when the correlation forces are included, and the second when these forces are omitted in the calculation. A comparison of these values with the experimental values of the work function of Na is given. It is concluded that the surface barrier is due primarily to exchange and polarization forces, and that ordinary electrostatic forces play a minor role.