# Ferromagnetism and spin waves in the band theory

## Abstract

Intra-atomic exchange (Hund's rule mechanism) and Heisenberg nearest-neighbor exchange are examined for their role in the ferromagnetism of metals with degenerate bands. We examine the ground state, and find there is ferromagnetism once the largest eigenvalue j00 of the exchange matrix exceeds 1/2×No. of atoms/density of states at the Fermi surface. We then find several spin-wave spectra, of which one "acoustic" and at least one "optical" spectrum have infinite lifetime in the random phase approximation. The initial parabolic behavior of the acoustic spectrum yields Bloch's T32 low at low temperature. There is a maximum wave vector beyond which no spin-wave solutions exist, corresponding to a minimum wavelength of at least several atomic distances. Formulas are given, and the copious numerical results calculated by W. Doherty on the IBM-7094 computer are summarized in graphs and tables. The ferromagnetic ground state is stable versus antiferromagnetic states only so long as umklapp is neglected. Because umklapp is most important in half-filled bands, we find qualitative agreement with previous calculations that antiferromagnetism can result in this case. © 1963 The American Physical Society.