Crystalline solid free-electron theory

The molecular orbital treatment of a crystalline solid considers the outer electrons as belonging to the crystal as a whole (10,11). Sommer-feld s early free electron theory of metals neglected the field resulting... 

The success of the simple free electron theory of metals was so staking that it was natural to ask how the same ideas could be apphed to other types of solids, such as semiconductors and insulators. The basic assumption of the free electron theory is that the atoms may be stopped of their outer electrons, the resulting ions arranged in the crystalline lattice, and the electrons then poured into the space between. 

Band theory provides a picture of electron distribution in crystalline solids. The theory is based on nearly-free-electron models, which distinguish between conductors, insulators and semi-conductors. These models have much in common with the description of electrons confined in compressed atoms. The distinction between different types of condensed matter could, in principle, therefore also be related to quantum potential. This conjecture has never been followed up by theoretical analysis, and further discussion, which follows, is purely speculative. 

In the previous chapters, we discussed various models of bonding for covalent and polar covalent molecules (the VSEPR and LCP models, valence bond theory, and molecular orbital theory). We shall now turn our focus to a discussion of models describing metallic bonding. We begin with the free electron model, which assumes that the ionized electrons in a metallic solid have been completely removed from the influence of the atoms in the crystal and exist essentially as an electron gas. Freshman chemistry books typically describe this simplified version of metallic bonding as a sea of electrons that is delocalized over all the metal atoms in the crystalline solid. We shall then progress to the band theory of solids, which results from introducing the periodic potential of the crystalline lattice. 

The quantity we consider in transport theory and call the electron momentum is the pseudo-momentum of the center of mass of the wave packet made up of eigenstates of the Schroedinger equation. I wish to remind you that in the case of a crystalline solid this pseudo-momentum is the eigen value of the operator that describes the translational symmetry of the lattice. In the case of a solid, the range of pseudo-momentum ( k), used in making the wave packet, is small in comparison with the size of the Brillouin Zone. It is a consequence of the effective-mass approximation that the pseudo-momentum of this wave packet behaves like the momentum we use in transport theory. In calculating transport properties we must substitute the free electron mass with the effective mass. This... 

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