Quantum Theory of the Defect Solid State

Much has been written about the failure of classical theory in interpretating the specific heat of metals and the subsequent development of a quantum theory of the perfect solid state. It would be quite impossible to give an adequate account of this development here, and the 

The major observations which a theory of the solid state must explain are (1) the freedom of motion of electrons through a metal and their inability to move in an insulating crystal and (2) the breadth of the allowed energy states (as deduced from spectroscopic observations on solids) relative to those of the free atoms or ions of which the crystal is composed. 

The quantum mechanical problem involves the solution of the Schrddinger equation in its appropriate form, given a potential energy function V x) which descnbes the system concerned. In its simplest (one-dimensional) form the Schrddinger equation is 

For metals a reasonable approximation is obtained by assuming a uniform potential energy (- F) for the electrons within the crystal referred to a zero of energy for the free electron at rest outside the crystal. The separation of the allowed energy levels is very small if 

The problem in which we are more interested is the quantum mechanics of periodic lattices which have been perturbed by the presence of defects. There is considert able experimental evidence for the association of discrete localized states with lattice defects of one sort or another the introduction of a perturbation into the quantum-mechanical problem should lead naturally to the prediction of these states. Quite recently Slater (18) has generalized a theorem used by Wannier for the discussion of excited states of crystals and through its use has clarified the whole problem of electronic motions in perturbed periodic lattices. It is possible to give an essentially non-mathematical discussion of Slater s treatment, as it is one which lends itself to simple graphical illustration. 

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