The distribution of electrons in a metal

Abstract
1. The Statistics of a Degenerate Gas . An interesting application of the new statistics of Fermi and Dirac has been made by Thomas (and also independently by Fermi) to the distribution of electrons in heavy atoms. The basic idea of these researches is that the “electron gas” surrounding a nuclues “degenerate,” so that every cell of extension h 3 of a six dimensional phase space contains two electrons, one electron spinning in one direction and another in the opposite direction. An upper limit to the possible translational energies of the electrons is imposed by the condition that electrons shall not have enough energy to escape from the field of the nucleus, viz:— ϵ≼ e . V, (1) where ϵ and e represent the energy and charge of the electron and V is the potential field. The possible range of momentum co-ordinates (neglecting relativity considerations) is then limited by p ≼(2 me V) ½ (2) and the total number of cells of extension h 3 in the phase space is given by 4π(2 me V) 3/2 /3 h 3 (3) for every unit of ordinary (co-ordinate) space available. If every cell contains two electrons, the density is given by ρ = —8π e (2 me V) 3/2 /3 h 3 . (4) Since, however, the potential V is determined by the nuclear charge and the distribution of electrons, we have ∇ 2 = —4πρ = (32π 2 e (2 me V) 3/2 /3 h 3 ,(5) an equation to determine V, subject to V → 0 as r → ∞ V r as r → 0}, (6) where E is the charge on the nucleus.

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