CXLI. Self trapped electrons in a Fermi gas

It is shown that Hartree's equations for the negative hydrogen ion have several solutions, the usual one of which does not minimize the energy. This non-uniqueness is also shown to exist for a Fermi gas where the usual plane waves of the Sommerfeld model do not even give a stationary value for the energy. The energy of the gas can be lowered, by describing some of the electrons by wave functions that decrease exponentially. The energy is lowered by about 1 ev per electron of this type. If the electron density is sufficiently low (band width E < 0·5 ev) all the electrons can be so trapped and the metal becomes an insulator ; this provides an a priori reason for Mott's rejection of the Bloch wave functions for nickel oxide. For higher electron densities it is possible to trap a fraction of the electrons, so saving considerable energy ; this is compared with Wigner's correlation energy. It is suggested that Friedel, in his explanation of the fine structure of an x-ray absorption edge, has used several solutions of Hartree's (or Fock's) equations that correspond to only one exact excited wave function, so invalidating his explanation of this effect.