Crystal and electronic structure and NMR of magnetite
The crystal structure for magnetite at temperatures below the Verwey transition temperature was determined by Attfield and coworkers in 2012 by x-ray diffraction. In a recent paper I presented results of hybrid density functional theory calculations on magnetite using the structure from Attfield and coworkers. Conduction electrons in magnetite all have the same (minority) electron spin orientation and there is one conduction electron per pair of Fe ions in octahedral (or B) sites. There are 112 ions in the low temperature unit cell of magnetite - 16 Fe ions in tetrahedrally coordinated A sites, 32 Fe ions in octahedrally coordinated B sites and 64 O ions. It belongs to the monoclinic Cc space group. The unit cell contains four types of rows of Fe B sites - a B1 row a B2 row and two types of mixed B3/B4 rows. The figure below shows an isosurface for the conduction electrons which localise on Fe B sites in the B rows.
Charge density isosurface for minority spin electrons in the conduction bands of Cc Fe3O4
Fe ions on B sites in Fe3O4 are arranged in tetrahedra which are shown in blue above. Close inspection of the conduction electrons localised on these sites shows that in some cases single electrons are shared by two sites connected by the edge of a tetrahedron and others belong to a zig-zag chain which passes through B1 rows.
Orbital order in magnetite can be probed using nuclear magnetic resonance (NMR). The Fe isotope with an atomic mass of 57 amu has a nuclear spin of 1/2 as do the conduction electrons in magnetite. Conduction electrons in magnetite produce a magnetic hyperfine field at Fe nuclei. The hyperfine field contains an isotropic Fermi contact part and an anisotropic part which depends on the orbital order around the nucleus. In an NMR experiment on magnetite by Mizoguchi in 2001 a spherical magnetite crystal was placed an a strong, uniform magnetic field and the resonance frequency at which RF radiation was absorbed by Fe nuclei was measured for various orientations of the crystal in the field.
Nuclear spins placed in a strong, uniform magnetic field are 'flipped' by RF radiation and the resonance frequency at which this occurs is determined mainly by the Fermi contact part of the hyperfine field and, to a lesser extent, by the anisotropic part. However, the anisotropic part of the hyperfine field determines how the flipping frequency depends on orientation of the crystal in a magnetic field applied to the crystal. In the same paper I also calculated how NMR resonance frequencies in Mizoguchi's measurements should vary with crystal orientation in a magnetic field using the charge density shown opposite. Variation of resonance frequencies with alignment of the magnetic field with the a, b or c crystal axes depends on the orbital order on a particular Fe A or B site. An electron in a Fe 3dxy orbital shows a strong dip in frequency when the field is along the crystal c axis. An electron in a Fe 3dxz or 3dyz orbital induces splitting into pairs of frequencies when the field lies in the bc or ab planes. This is because each Fe ion in the unit cell has a symmetry equivalent ion. Fe ions in B3/B4 rows have smaller conduction electron populations and are in 3dzz orbitals or are designated Fe3+ when the conduction electron density is very low.
This work was sponsored by Science Foundation Ireland under grant number RFP/09/MTR2295