Competing exchanges on a body-centred cubic antiferromagnet ($J_2/J_1=2/3$)

With $H={\sum }_{ij\mathrm{bonds}}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ counting bonds once, type-I and type-II classical collinear energies per spin are $S^2(-4J_1+3J_2)$ and $-3J_2{S}^2$. Their boundary is $J_2/J_1=2/3$. The corresponding ordering reflections have integer indices with odd sum, and three half-integer indices, respectively, in units of $2\pi /a$. This compares classical order, not an exact quantum product-state ground state.