The Sommerfeld free-electron theory

The simple Drude-Lorentz theory described earlier in this chapter pictures the valency electrons in a metal as free to move in a potential well of the form shown in fig. 5.06. Within the metal, from B to C, the potential is uniform, but at the surface a potential difference, V (of the order of 10 V), prevents electron escape. If the electrons are assumed to obey the laws of classical mechanics their energies will correspond to the Boltzmann distribution appropriate to the temperature of the specimen. At room temperatures a quite negligible fraction of the electrons will have energies sufficient to surmount the potential 

This difficulty was resolved by Sommerfeld, who treated the electrons in a metal as obeying the laws of the quantum theory instead of those of classical mechanics. In terms of wave mechanics an electron in motion has a wave-like character and its behaviour must be associated with that of a wave which is propagated in the same direction as the electron and whose wavelength is given by 

Equation (5.04) still represents the relationship between E and k, and this relationship is still parabolic, as shown in fig. 5.070, but this figure no longer represents a continuous variation of E now E can assume only 

If we insert in equation (5.04) the k values given by (5.06) we have for the permitted energies 

Analogous arguments applied to the case of a three-dimensional specimen in the form of a cube of side L give for the corresponding permitted energies the expression 

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