Experimental evidence for nonspherical magnetic form factor in ${\mathrm{Ru}}^{3+}$

The Mott insulator $\alpha \text{−}{\mathrm{RuCl}}_3$ has generated great interest in the community due to its possible field-induced Kitaev quantum spin liquid state. Despite enormous effort spent trying to obtain the form of the low-energy Hamiltonian, there is currently no agreed upon set of parameters which is able to explain all of the data. A key piece of missing information lies in the determination of the magnetic form factor of ${\mathrm{Ru}}^{3+}$, particularly for a true quantitative treatment of inelastic neutron scattering data. Here we present the experimentally derived magnetic form factor of ${\mathrm{Ru}}^{3+}$ in the low spin $4d^5$ state using polarized neutron diffraction within the paramagnetic regime on high-quality single crystals of $\alpha \text{−}{\mathrm{RuCl}}_3$. We observe strong evidence of an anisotropic form factor, expected of the spin-orbit coupled $j_{\text{eff}}=\frac{1}{2}$ ground state. We model the static magnetization density in increasing complexity from simple isotropic cases, to a multipolar expansion, and, finally, ab initio calculations of the generalized $j_{\text{eff}}=\frac{1}{2}$ ground state. Comparison of both single ion models and inclusion of ${\mathrm{Cl}}^{-}$ anions support the presence of hybridization of ${\mathrm{Ru}}^{3+}$ with the surrounding ${\mathrm{Cl}}^{-}$ ligands.