Stability of Ferro- and Antiferromagnetism under thes−fInteraction

A system of spins interacting with each other through the $s-f$ indirect exchange interaction can exhibit ferro- and antiferromagnetism alternately as a function of the Fermi momentum $k_F$ of conduction electrons and the distance $a$ between rare-earth atoms. This alternate behavior, however, depends critically on the $s-f$ exchange parameter ${\mathcal{J}}_{\mathrm{sf}}\mathrm{}\left(q\right)\mathrm{}$ and, if ${\mathcal{J}}_{\mathrm{sf}}\mathrm{}\left(q\right)\mathrm{}$ decays too fast, only ferromagnetism appears. The stability condition of these states is investigated in detail, and the following conclusion is obtained: In the regions of $k_{F}a$ where the ferromagnetic energy is lower than any of the classical energies of antiferromagnetic states, ferromagnetism is all that can appear at all temperatures, while in other regions of $k_{F}a$ many states, including the ferromagnetic state, may be metastable, and magnetic transitions among them may be observed, but the ground state is one of antiferromagnetic states. The calculation of the stability condition resembles that of the Kohn anomalies, and hence the magnetic structures of the system is shown to be closely related to the Fermi surface.