Unusually High Magnetic Moments Reported for the High-Spin Co²⁺ Ions at Axially Elongated CoO₄X₂ (X = Cl, Br, S, Se) Octahedral Sites: Overestimated Magnetic Moments Caused by Underestimated Magnetic Form Factors

Abstract

An irreconcilable discrepancy between theory and experiment concerning the magnetic moments of the high-spin Co2+ (d7, S = 3/2) ions at the axially elongated CoO4X2 (X = Cl, Br, S, Se) octahedral sites was pointed out in our recent study [Inorg. Chem.2020591831918324]. The magnetic moments μobs of the Co2+ ions in materials containing the CoO4X2 units, determined from refinements of powder neutron diffraction patterns collected in their magnetically ordered states, were found to be notably larger than the spin-only moment of 3 μB (namely, 3.22–4.45 μB). Characteristically, the moments increase almost linearly with the bond-length ratio R = r Co–X/r Co–O, despite the fact that the orbital moments of the Co2+ ions are essentially quenched according to theoretical analyses and DFT + U + SOC computations. In this work, we probe the probable cause for the overestimation of magnetic moments in the ordered magnetic structures. We argue that the overestimated magnetic moments of the Co2+ ions and their increase with the bond-length ratio R originate from the underestimated magnetic form factors of the Co2+ ions at the CoO4X2 octahedral sites. The magnetic form factor F( q ) of the Co2+ ion in each CoO4X2 octahedron is the Fourier transform of the magnetic moment distribution ρ­( r ) around the Co2+ ion, which is the electron density distribution associated with the magnetic orbitals (i.e., the singly occupied d-states) of the CoO4X2 octahedron. The conventional implementation of the magnetic form factor in refinement codes employed an approximation with a q-dependence calculated from the atomic wave function of the isolated ion. Since the spin moment distribution of ρ­( r ) at the axially elongated CoO4X2 (X = Cl, Br, S, Se) octahedral sites is delocalized from the Co2+ sites to its surrounding ligands, the moment distribution becomes anisotropic in shape. Use of the truncated and spherical approximation underestimates the form factor of the Co2+ ion, and its use in neutron diffraction refinements overestimates the magnetic moments. This effect becomes more pronounced as the moment density distribution is more delocalized along the Co–X direction.