The Theory of Ferromagnetism: Lowest Energy Levels

The theory of ferromagnetism is formulated both by the method of energy bands and the method of spin waves, and the connection between them is investigated. The particular problem considered is that where $N-1\mathrm{}$ electrons in a band have one spin, one electron the opposite spin. Each method yields a set of unperturbed wave functions, each of which can be expressed in terms of the other. Neither represents the true situation, which must be found by perturbation theory. The final result is that below the continuum of levels indicated by the energy band theory there is a set of discrete levels. The lowest of these is essentially a spin wave, and is the state ordinarily occupied. For investigating the temperature variation of magnetization, the spin wave theory should be used. On the other hand, the energy band theory leads to approximately correct conclusions as to which elements should be ferromagnetic. The formulation is in terms of orthogonal atomic functions, which have advantages compared with the nonorthogonal functions usually employed. In the limit of broad energy bands, and nonferromagnetic substances, the discrete levels approach the continuum, and their wave functions represent a situation in which the electron of negative spin can wander to a considerable distance from the positive ion which it has left, but not to infinite distance.