Energy Bands in Solids—The Quantum Defect Method

Abstract

This paper describes a procedure for the calculation of electron energy bands in certain solids from spectroscopic data for the corresponding free atom. This method is an improved version of one used by Kuhn and Van Vleck to calculate the energy bands of sodium, potassium, and rubidium. It avoids explicit construction of a one-electron potential to represent the interaction between the valence and core electrons.We assume that the interaction between a valence electron and an ion in the crystal is approximately the same as in the isolated atom. If the interaction is accurately represented by a Coulomb potential outside the ion core, we may express solutions of the radial differential equation in this region as linear combinations of standard Coulomb functions. The combination corresponding to the solution which is well-behaved at the nucleus involves a coupling constant which depends upon the ion potential through a parameter that is a slowly varying function of the energy. At an eigenvalue this parameter can be evaluated from the quantum defect. Hence if the eigenvalue spectrum is known, we may obtain this parameter by extrapolation for arbitrary nearby energies, and the regular solution of the radial equation is consequently determined explicitly, outside the core. This is sufficient information for the calculation of energy bands with available techniques.We establish an approximate formula for the ratio of the amplitude of the wave function near the nucleus to its value at a point outside the core. For an $s$ function this relation involves only the nuclear charge, standard Coulomb functions, and the aforementioned parameter derived from spectroscopic data. It therefore provides a convenient means of calculating ${P}_{F}$ and ${P}_{A}$, the squared amplitudes at the nucleus appropriate to the Knight shift and the atomic hyperfine splitting, respectively. In the latter case our result is identical with a formula given by Fermi and Segr\`e, which gives reasonable agreement with experiment.Arguments are presented in support of the thesis that the quantum defect method takes very general account of exchange and correlation interactions between the valence electron and core electrons. Relativistic effects, including spin-orbit coupling, are also included naturally. We also discuss modifications in the method to take into account deviations of the ion or crystal potential outside the core from Coulomb form. Tables of the essential data, including improved polarization corrections, are given for the alkali metals.