Bethe-Sommerfeld Model

FEM is a model, not an exact theory. One advantage with the simplification is that the Fermi-Dirac statistics may be applied to obtain the occupancy of the levels at any temperature. This is also true for the three-dimensional version of FEM, the Bethe-Sommerfeld model. 

The quantum numbers n, Uy, and n are natural numbers. We assume that there is one valence electron per metal atom and atoms along the edge of the cube. The energy levels with 

The energy difference between the HOMO and the LUMO may be calculated as the derivative dE/dN, where N is the number of electrons. For the HOMO, we have from Equation 16.26 

The HOMO-LUMO gap decreases inversely proportional to the number of electrons. 

The Fermi level is located between the HOMO and the LUMO. In the present case, the Fermi level is thus localized in a continuous band of energy levels. The HOMO-LUMO gap is referred to as the gap at the Fermi level. In the case of a metal, the Fermi gap is equal to zero for all practical purposes. 

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In 1933, Arnold Sommerfeld and Hans Bethe revised the Drude model. Their more complete, quantum mechanical theory goes under the name of the free electron... 

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