Spin wave energies in the band theory of ferromagnetism

A detailed investigation is made of the coefficient C in the energy CK2 of a spin wave of small wave vector K in a ferromagnetic cubic metal. It is assumed that the spins of all particles, within the energy band system considered, are alined in the ground state. An exact expression is obtained for C on the basis of one plausible assumption, that the inverse lifetime of a spin wave tends to zero faster than K2 as K → 0. This expression is examined in detail for a one-band model with short-range inter-particle interactions and it consists of three terms. The first two terms are closely related to the well known r. p. a. formula for C although the second term is modified in an important way. In particular this modification is shown to take account of two-particle correlations of the type considered by Kanamori, but adapted to the ferromagnetic situation, and of magnon-electron scattering. The third term in C is completely new but is generally expected to be smaller than the second term. In an Appendix contact is made with Nagaoka’s work on spin wave energies in a nearly half-filled band.