Magnetic Properties of $A$-site Antiferromagnetic Spinels CoRh$_2$O$_4$ and CuRh$_2$O$_4$

We characterize the magnetic properties of the $A$-site spinels CoRh$_2$O$_4$ and CuRh$_2$O$_4$ by means of thermodynamic and neutron scattering measurments and perform group theory analysis, Rietveld refinement, mean-field theory, and spin wave theory calculations to analyze our results. Our investigation reveals that CoRh$_2$O$_4$ is a canonical $S=3/2$ diamond-lattice Heisenberg antiferromagnet with a nearest neighbor exchange $J=0.63$ meV and a N\'eel temperature of 25 K. In CuRh$_2$O$_4$, the distorted diamond lattice leads to the development of an incommensurate helical order at 24 K. Strong reduction of the ordered moment is observed for the $S=1/2$ spins and reproduced by $1/S$ corrections to spin-wave theory.