We report a detailed single-crystal and powder neutron diffraction study of ${\mathrm{Co}}_2{\mathrm{TiO}}_4$ and ${\mathrm{Co}}_2{\mathrm{SnO}}_4$ between the temperature 1.6 and 80 K to probe the spin structure in the ground state. For both compounds the strongest magnetic intensity was observed for the (111)${}_{\text{M}}$ reflection due to ferrimagnetic ordering, which sets in below $T_{\text{N}}=48.6$ and 41 K for ${\mathrm{Co}}_2{\mathrm{TiO}}_4$ and ${\mathrm{Co}}_2{\mathrm{SnO}}_4$, respectively. An additional low intensity magnetic reflection (200)${}_{\text{M}}$ was noticed in ${\mathrm{Co}}_2{\mathrm{TiO}}_4$ due to the presence of an additional weak antiferromagnetic component. Interestingly, from both the powder and single-crystal neutron data of ${\mathrm{Co}}_2{\mathrm{TiO}}_4$, we noticed a significant broadening of the magnetic (111)${}_{\text{M}}$ reflection, which possibly results from the disordered character of the Ti and Co atoms on the $B$ site. Practically, the same peak broadening was found for the neutron powder data of ${\mathrm{Co}}_2{\mathrm{SnO}}_4$. On the other hand, from our single-crystal neutron diffraction data of ${\mathrm{Co}}_2{\mathrm{TiO}}_4$, we found a spontaneous increase of particular nuclear Bragg reflections below the magnetic ordering temperature. Our data analysis showed that this unusual effect can be ascribed to the presence of anisotropic extinction, which is associated to a change of the mosaicity of the crystal. In this case, it can be expected that competing Jahn-Teller effects acting along different crystallographic axes can induce anisotropic local strain. In fact, for both ions ${\mathrm{Ti}}^{3+}$ and ${\mathrm{Co}}^{3+}$, the $2t_g$ levels split into a lower $d_{xy}$ level yielding a higher twofold degenerate $d_{xz}/d_{yz}$ level. As a consequence, one can expect a tetragonal distortion in ${\mathrm{Co}}_2{\mathrm{TiO}}_4$ with $c/a<1$, which we could not significantly detect in the present work.

We elucidate the magnetic interactions and the role of spin (electron) correlation in determining the ground state of the honeycomb compound $\mathrm{M}{\mathrm{n}}_4\mathrm{T}{\mathrm{a}}_2{\mathrm{O}}_9$, by neutron powder diffraction, inelastic neutron scattering (INS), specific-heat ($C_P$) measurements, and electronic-structure calculations. The antiferromagnetic long-range order with moments along $c$ occurs at 102 K with strong exchange striction and small anisotropy. It is described by the three-dimensional Ising model. Diffuse magnetic scattering has been observed above $T_N$, which is attributed to the two-dimensional spin correlations within the $\mathrm{M}{\mathrm{n}}^{2+}$ honeycombs. This is confirmed by the calculated exchange constants. INS experiments and spin-wave simulations together with $C_P$ measurements reveal two gapped modes on the $ab$ plane, originating from the rotation of the spins away from the easy axis $c$. The magnetic anisotropy is mainly determined by an electron-correlation-assisted dipole-dipole interaction. This work provides insight into the competing origins of the magnetic anisotropy, which leads to different magnetic ground states in the family of honeycomb compounds.