  Antiferromagnetic insulators on the diamond lattice are candidate materials
to host exotic magnetic phenomena ranging from spin-orbital entanglement to
degenerate spiral ground-states and topological paramagnetism. Compared to
other three-dimensional networks of magnetic ions, such as the geometrically
frustrated pyrochlore lattice, the investigation of diamond-lattice magnetism
in real materials is less mature. In this work, we characterize the magnetic
properties of model A-site spinels CoRh2O4 (cobalt rhodite) and CuRh2O4 (copper
rhodite) by means of thermo-magnetic and neutron scattering measurements and
perform group theory analysis, Rietveld refinement, mean-field theory, and spin
wave theory calculations to analyze the experimental results. Our investigation
reveals that cubic CoRh2O4 is a canonical S=3/2 diamond-lattice Heisenberg
antiferromagnet with a nearest neighbor exchange J = 0.63 meV and a Neel
ordered ground-state below a temperature of 25 K. In tetragonally distorted
CuRh2O4, competiting exchange interactions between up to third nearest-neighbor
spins lead to the development of an incommensurate spin helix at 24 K with a
magnetic propagation vector k = (0,0,0.79). Strong reduction of the ordered
moment is observed for the S=1/2 spins in CuRh2O4 and captured by our 1/S
corrections to the staggered magnetization. Our work identifies CoRh2O4 and
CuRh2O4 as reference materials to guide future work searching for exotic
quantum behavior in diamond-lattice antiferromagnets.
