Classical Theory of Spin Configurations in the Cubic Spinel

It is shown that the Yafet-Kittel triangular spin-configurations in the cubic spinel do not minimize the classical Heisenberg exchange energy. (Only nearest-neighbor $A-B$ and $B-B$ interactions, $J_{\mathrm{AB}}$ and $J_{\mathrm{BB}}$, are included; one spin-magnitude $S_A$ for the $A$ sites, and one, $S_B$, for the $B$ sites is assumed.) A theory of the classical ground state more general than that of Yafet and Kittel is investigated. This consists of first determining the largest value, $y_0$, of $y\equiv \frac{J_{\mathrm{BB}}{S}_B}{J_{\mathrm{AB}}{S}_A}$ for which the Néel configuration is stable with respect to arbitrary small spin-deviations. ($y_0$ is roughly 10% smaller than the value of $y$ found by Yafet and Kittel for the breakdown of the Néel configuration.) A perturbation method for finding the minimum energy configuration when $y-y_0$ is small and positive is then employed. It is concluded, (1) that equilibrium configurations exist which have nonzero angles between spins on the $A$ sites simultaneously with angles between those on the $B$ sites, in contrast with the Yafet-Kittel results; and (2) that there will be long-range-ordered, canted spin configurations in the cubic spinel, contrary to Anderson's suggestion. These conclusions are discussed in connection with experiments on Mn${\mathrm{Cr}}_2$ ${\mathrm{O}}_4$ and ${\mathrm{Mn}}_3$ ${\mathrm{O}}_4$.