Classical Spin-Configuration Stability in the Presence of Competing Exchange Forces

It is pointed out that Yafet-Kittel triangular arrangements are not stable in the cubic spinel. The stability criterion used is that the classical Heisenberg energy should not decrease for small, but otherwise arbitrary, spin-deviations from the configuration of interest. It is found that the Yafet-Kittel-Prince configuration can probably be stabilized by a sufficient tetragonal distortion of the pattern of $B-B$ interactions. In addition, the classical ground state is found for the antiferromagnetic body-centered cubic lattice with first, second, and third neighbor antiferromagnetic interactions (with parameters $J_1$, $J_1{\sigma }_2$ and $J_1{\sigma }_3$): the spin $\mathrm{S}\left({\mathrm{R}}_n\right)$ at lattice point ${\mathrm{R}}_n$ is independent of time, is always parallel to one plane, $P$, and the angle made by $\mathrm{S}\left({\mathrm{R}}_n\right)$ with a fixed line in $P$ is of the form $\mathrm{k}·{\mathrm{R}}_n$ for ${\mathrm{R}}_n$ a cube corner, and of the form $\mathrm{k}·{\mathrm{R}}_n+\pi$ for ${\mathrm{R}}_n$ a body-center position with the vector k determined by the ${\sigma }_i$. The neutron diffraction pattern for such a "spiral" configuration (with ${\sigma }_2\sim 0.6$, ${\sigma }_3\sim 0.1$, for example) bears a close relationship with the unusual pattern obtained by Corliss, Hastings, and Weiss with a single crystal of chromium.