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\a), , Band Theory of Solids, , ——___—____—, , > 5.1. Introduction, , The free electron model of metals assumes that the conduction electrons, within a metal move in a region of constant potential and so are completely, free to move about in the crystal, restrained only by the surface of the crystal, and the electrons obey Fermi-Dirac statistics. The theory certainly explains, several electronic properties of metals e.g.,specific heat, paramagnetism etc.,, yet there are several other properties of solids for which the free electron, model helps in on way. For example it does not help to distinguish between, a metal and an insulator; the resistivity of a good conductor at low temperatures, , may be of the order of 107 '° ohm-metre and that of a good insulator may, , be as high as 10°” ohm-metre. Further the theory does not explain the, behaviour of semi-conductors. Therefore the free electron theory must be, modified., , One way to modify the free electron theory is to take into account of, the periodic potential with periodicity of the lattice. In one electron-model, of a solid, the periodic potential may be thought of as arising due to periodic, charge distribution associated with ion cores situated on the lattice sites plus, the constant (average) contribution due to all other free electrons of the, crystal. The latter contribution to the potential accounts, in an average sense,, the interaction effects of a single electron with all others., , ‘A one dimensional representation of periodic potential with a period of, lattice constant ‘a’ is shown in Fig. 5.1. The potential at the surface is, , rire, , Surface

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dense hevels are almost continuols, ‘These energy bands are in general, separated by regions:, , allowed energy levels. These regions are called forbidden, ‘paps or band gaps: In some crystals, the adjacent enengy, fnummber of levels in such a merged band is equal to the sum, of the Jevels into which both levels of the atoms split up., amount of splitting is not the same for different levels. Those, the valence electrons in an atom are disturbed mast, while tho, inner electrons are disturbed only slightly. Fig. 5.3 represents., of levels as a function of distance + between atoms.

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‘The other values of cuz will be in accessible. The consequence, , can be understood with reference to Fig. 5.6 which represents the, side PERS scout) of (6) a «ection of fer te