Stability of the Ferromagnetic State in the Band Model

An exact expression is given for the coefficient $C$ in the energy ${\mathrm{CK}}^2$ of a spin wave of small wavevector $K$ in a ferromagnetic cubic metal, assuming complete spin alinement in the ground state. This is given in a form which applies to a one‐band model with short‐range interparticle interactions and is compared with the well‐known r.p.a. formula. The main feature of the new result is a modified superexchange term which, in particular, takes account of two‐particle correlations of the Kanamori type and of magnon‐electron scattering. A necessary condition for the ferromagnetic state to be stable is $\text{C\hspace{0.17em}>\hspace{0.17em}0}$ and this is investigated for some special cases. Some doubt is thrown on Nagaoka's demonstration of stability in an infinite crystal for the case of a narrow, almost half‐filled, band.