The Quantum Mechanics of Electrons in Crystals

A general solution is developed for the motion of electrons in the potential field of the nuclei in a crystal lattice. As usual the energy interaction terms due to nuclear vibration and to the presence of other electrons are neglected; they are to be included later by approximation methods.It is shown for low energies the wave function becomes a linear combination of the atomic wave functions, the allowed energies approximating the discrete atomic levels; and for high energies the wave function approaches that of the free electron, with the allowed energies a nearly continuous range. However, for electrons coming into the crystal from outside, the crystal becomes impenetrable for those electronic wave-lengths and directions analogous to the beams producing Bragg and Laue beams in x-rays.The solution is computed for a simple form of potential lattice, and the results are shown to be in quantitative agreement with the experimental results of Davisson and Germer. The phenomenon they call anomalous dispersion is shown to be a natural consequence of the characteristics of the wave function.