It is found that five distinct distant-neighbor interactions must be considered in order to characterize adequately the Heisenberg energy in normal cubic spinels having diamagnetic $A$-site ions. The classical theory of the ground spin state is presented in terms of these interactions, which span a five-dimensional parameter space. It is demonstrated that the ground state consists of a spiral spin configuration with $\mathrm{k}=(0, 0, l)$ if and only if the parameter values fall inside a limited region, which is determined rigorously by means of the Luttinger-Tisza method. Because of the circumscription of this spiral ground-state region, agreement with the experimental findings for Zn${\mathrm{Cr}}_2$${\mathrm{Se}}_4$ requires that at least one of the distant-neighbor interactions be ferromagnetic. Furthermore, these interactions can be analyzed into their constituent superexchange terms, which can in turn be related by standard superexchange theory. The resulting criteria for physical reasonability place additional limits upon the acceptable range of values for the interactions. This range is found to exclude those sets of interactions which have previously been proposed in the literature. A simplified superexchange model is used to illustrate the determination of physically reasonable fits to the data and to study their dependence upon the degree of $A$-site covalence.

• Received 6 June 1967

DOI:https://doi.org/10.1103/PhysRev.163.435