Electrons in Perturbed Periodic Lattices

It is shown that the motion of an electron in a periodic potential, such as is found in a solid, plus a slowly varying perturbative potential, can be derived from the energy in the periodic lattice alone, as a function of momentum or wave number. A Schr\"odinger equation is set up, in which the Hamiltonian is the sum of this energy in the periodic lattice---the momentum being replaced by a differential operator---and of the perturbative potential energy. The resulting wave function modulates atomic functions to provide a solution of the perturbed problem. This method is applied to give proofs of simple theorems in conduction theory, to discuss the energy levels of impurity atoms in a semiconductor, and to consider excitons; all are problems which have been considered before, but which are treated more straightforwardly by the present method. Applying the method statistically, the combined Poisson's equation and Fermi-Dirac statistics is set up for impurities in metals and semiconductors, and for the theory of rectifying barriers.