Experimental magnetic form factors in ${\text{Co}}_3{\text{V}}_2{\text{O}}_8$: A combined study of ab initio calculations, magnetic Compton scattering, and polarized neutron diffraction

We present a combination of ab initio calculations, magnetic Compton scattering, and polarized neutron experiments, which elucidate the density distribution of unpaired electrons in the kagome staircase system ${\text{Co}}_3{\text{V}}_2{\text{O}}_8$. Ab initio wave functions were used to calculate the spin densities in real and momentum spaces, which show good agreement with the respective experiments. It has been found that the spin polarized orbitals are equally distributed between the $t_{2g}$ and the $e_g$ levels for the spine $\left(s\right)$ Co ions while the $e_g$ orbitals of the cross-tie $\left(c\right)$ Co ions only represent 30% of the atomic spin density. Furthermore, the results reveal that the magnetic moments of the cross-tie Co ions, which are significantly smaller than those of the spine Co ions in the zero-field ferromagnetic structure, do not saturate by applying an external magnetic field of 2 T along the easy axis $a$. In turn, the increasing bulk magnetization, which can be observed by field dependent macroscopic measurements, originates from induced magnetic moments on the O and V sites. The refined individual magnetic moments are $\mu \left({\text{Co}}_c\right)=1.54\left(4\right){\mu }_B$, $\mu \left({\text{Co}}_s\right)=2.87\left(3\right){\mu }_B$, $\mu \left(\text{V}\right)=0.41\left(4\right){\mu }_B$, $\mu \left(\text{O}1\right)=0.05\left(5\right){\mu }_B$, $\mu \left(\text{O}2\right)=0.35\left(5\right){\mu }_B$, and $\mu \left(\text{O}3\right)=0.36\left(5\right){\mu }_B$ combining to the same macroscopic magnetization value, which was previously only attributed to the Co ions.

Figure 1

(Color online) Visualization of the kagome staircase structure of ${\text{Co}}_3{\text{V}}_2{\text{O}}_8$. (a) shows the edge-sharing ${\text{MO}}_6$ octahedra [${\text{Co}}_c{\text{O}}_6$ light blue (light gray), ${\text{Co}}_s{\text{O}}_6$ dark blue (dark gray)] isolated by nonmagnetic ${\text{VO}}_4$ tetrahedra (gray). (b) depicts a single kagome staircase viewed along the $b$ axis with only the magnetic ions on both crystallographic sites [${\text{Co}}_c$ light blue (light gray), ${\text{Co}}_s$ dark blue (dark gray)].

Figure 2

(Color online) Intensity of three different magnetic reflections in dependence of an applied magnetic field revealing primary extinction effects.

Figure 3

(Color online) (a) Crystal structure viewed along the $a$ axis. (b) Experimental and (c) calculated magnetization densities as a projection onto the $b\text{−}c$ plane. Contour lines defining positive values are drawn as solid lines in $0.05{\mu }_B/{\text{Å}}^2$ intervals between $0{\mu }_B/{\text{Å}}^2$ and $0.15{\mu }_B/{\text{Å}}^2$, and in $0.4{\mu }_B/{\text{Å}}^2$ intervals above. Negative isodensities are represented by broken lines in $0.1{\mu }_B/{\text{Å}}^2$ steps.

Figure 4

(Color online) Observed (dots) and calculated (solid lines) normalized directional MCPs (shifted vertically in order to improve clarity, horizontal lines serve as a guide for the eye). The abscissa $p_z$ is taken to be parallel to the respective scattering vector. $\vartheta$ denotes the angle between a respective MCP and the [001] direction.

Figure 5

(Color online) (a) Reconstructed experimental and (b) calculated spin momentum density in the $p_y\text{−}{p}_z$ plane. Contours are drawn in $0.025{\mu }_B/{\left(\text{a}\text{.u}\text{.}\right)}^3$ intervals. White solid lines depict the boundary of the first Brillouin zone.

Figure 6

(Color online) Observed (dots) and calculated (solid lines) MCPs along two different directions. The dotted and dashed lines show the contribution of the ${\text{Co}}_c{\text{O}}_6$ and the ${\text{Co}}_s{\text{O}}_6$ cluster, respectively.

Figure 7

(Color online) ${\text{Co}}_c$ (broken line) and ${\text{Co}}_s$ orbitals (dotted line) as a projection onto different scattering vectors revealing the different directional dependence of inequivalent (upper line, ${\text{Co}}_c{d}_{xz}$ and ${\text{Co}}_s{d}_{yz}$) as well as equivalent orbitals (lower line, $d_{xy}$).

Figure 8

(Color online) Observed (squares) and calculated (dots) magnetic form factors as a function of $\text{sin}\theta /\lambda$. The calculated values represent exclusively the form factors of the ${\text{Co}}_c$ (open symbols) and ${\text{Co}}_s$ (filled symbols), which are accessible due to special reflection conditions. The discrepancies between the magnetic form-factor curves have to be attributed to induced magnetic moments on the V and O sites.