Ferromagnetism in a Narrow, Almost Half-FilledsBand

We consider conduction electrons in a narrow $s$ band with a strong repulsive potential which acts when two electrons are at the same atomic site. It is assumed that the electron-transfer matrix elements are nonvanishing only between nearest-neighbor sites, and that the band is almost half-filled, or $|N-N_e|\ll N$, $N$ and $N_e$ being, respectively, the number of atoms and electrons in the crystal. Then it is proved quite rigorously that, if the repulsive potential is sufficiently strong, the ferromagnetic state with the maximum total spin is the ground state for simple cubic and body centered cubic structures as well as for face centered cubic and hexagonal closed packed structures with $N_e>N$, and is not the ground state for face centered cubic and hexagonal closed packed structures with $N_c<N$. For the former case, it is also shown that it is not the ground state if the repulsive potential is weaker than some critical value which is of the order $\left(\mathrm{bandwidth}\right)\times \frac{N}{|N-N_e|}$.