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The wave functions and energy levels associated with a finite one-dimensional periodic potential field are investigated. In a plot of the energy spectrum versus interatomic distance the surface levels appear only at lattice constants so small that the boundary curves for the allowed energy bands have crossed. The levels appear in the "forbidden" region between allowed bands in pairs one coming from each of the adjoining bands. In three dimensions these surface levels give rise to surface bands. The surface bands probably exist and are half-filled for diamond. They exist for all metals and are entirely unoccupied only for the monovalent metals.
W. Shockley (Tue,) studied this question.