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The results obtained in the previous chapters now may be applied to expanding the understanding of the physical properties of the solid state. This leads to the development of the Sommerfeld theory of metals, the Fermi-Dirac statistics, the band theory, and the Brillouin zone theory of solids. These elementary presentations are not intended to be comprehensive, nor are they used here to describe all of the important properties of solids. They are considered to be sufficient to provide a reasonable, elementary basis for the understanding of the properties of thermoelectric materials and a foundation for further study.