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

Abstract

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 $B\ensuremath{-}B$ 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 ${J}_{1}$, ${J}_{1}{\ensuremath{\sigma}}_{2}$ and ${J}_{1}{\ensuremath{\sigma}}_{3}$): the spin $\mathrm{S}({\mathrm{R}}_{n})$ at lattice point ${\mathrm{R}}_{n}$ is independent of time, is always parallel to one plane, $P$, and the angle made by $\mathrm{S}({\mathrm{R}}_{n})$ with a fixed line in $P$ is of the form $\mathrm{k}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{R}}_{n}$ for ${\mathrm{R}}_{n}$ a cube corner, and of the form $\mathrm{k}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{R}}_{n}+\ensuremath{\pi}$ for ${\mathrm{R}}_{n}$ a body-center position with the vector k determined by the ${\ensuremath{\sigma}}_{i}$. The neutron diffraction pattern for such a "spiral" configuration (with ${\ensuremath{\sigma}}_{2}\ensuremath{\sim}0.6$, ${\ensuremath{\sigma}}_{3}\ensuremath{\sim}0.1$, for example) bears a close relationship with the unusual pattern obtained by Corliss, Hastings, and Weiss with a single crystal of chromium.