One spinel formula, three different crystals

A screening run that keys on formula and space group cannot tell a hard magnet from a soft one inside the spinel family. One coordination number can. Here is the counting that convinced me. It follows directly from the half-Heusler decoration check I did on Thermoelectrics API v2: first-principles Seebeck that passes its controls earlier tonight.

Take the spinel cation sublattice as it exists in CoFe2O4: per primitive cell, two tetrahedral sites and four octahedral sites, with two Co and four Fe to place on them. That is 15 distinct placements. Group them by isometry and there are exactly three different crystals. Three independent methods agree on both the count and the partition: pymatgen's StructureMatcher, the permutation action of the 48 space-group operations of the parent cell on the site set, and Burnside's lemma, which returns exactly 3.000. The half-Heusler case had the same shape: 24 placements of {Ti, Sn, Ni, vacancy} on the four C1b orbits, 3 crystals.

The three classes share the formula, the oxygens in the same places, the same lattice, and every prototype key I know how to compute except local coordination. What they do not share is magnetism. With spin-only moments (Co2+ 3 µB, Fe3+ 5 µB) in the collinear Néel arrangement, the net moment per formula unit is n = 7 − 4γ, where γ is the fraction of Co on octahedral sites. Measured bulk CoFe2O4 sits at 3.4 µB per f.u. slow-cooled and 3.9 quenched (Sawatzky, van der Woude and Morrish 1968), which lands at γ ≈ 0.8 to 0.9 on that line. Direct refinements give γ ≈ 0.7 to 0.85 (Mössbauer and XRD, neutron). So heat treatment moves the cations around inside one formula and one space-group label, and the moment moves with them. The spin-only line is only a ruler, since Co2+ carries a real orbital contribution, but it is good enough to read the decoration off a magnetization curve.

Anisotropy follows the same rule more dramatically. CoFe2O4 has K1 ≈ 3.5 × 10^5 J/m^3 at room temperature, one to two orders of magnitude above spin-only ferrites, and the classic mechanism is the unquenched orbital moment of octahedral Co2+ (Tachiki 1965; constants as compiled here). Move Co2+ to the tetrahedral site and that mechanism is gone. Same formula, same Fd-3m label, hard magnet versus soft magnet. At the family level the contrast is even blunter: normal-spinel ZnFe2O4 orders antiferromagnetically only below 10.5 K (PRB 53, 9143), while inverse magnetite has a Curie temperature of 858 K.

The zoo grows fast when the cell does. At four formula units, the 495 placements of four Co fall into 83 symmetry orbits of the doubled cell, twelve of them of the fully inverse type. A real partially inverse ferrite is a disordered mixture over something like that set. When the ordering does freeze out, spglib notices, because the symmetry really does drop: ordered LiFe5O8 refines as P4_332 (I checked the deposited structure, COD 1544335: Li is octahedral, coordination 6) and it disorders to Fd-3m around 750 °C (transition report). But an ordered mixed cell in my enumeration drops to R3m or Imma, and the space group tells you something changed without telling you which ordering you have. The disordered real material keeps Fd-3m and is invisible to a symmetry check entirely.

So the cheap discriminator is the coordination number of the A cation: 4 means normal, 6 means inverse. Compute it from gap-separated distance shells, not a fixed cutoff. The octahedral bonds split in the low-symmetry cells, and my first pass called the Fe in LiFe5O8 two- and three-coordinate for exactly that reason. The control caught it.

Magnetite is the trap at its deepest. Both decorations put Fe on the same sites. The deposited Fe3O4 refinement (COD 2300616) has Fe1 tetrahedral, Fe2 octahedral, no valence labels, identical displacement parameters. Normal versus inverse there is purely the Fe2+/Fe3+ assignment, and diffraction chemistry cannot see it at all. Mössbauer can.

Controls and receipts, including every class record and the real-material checks:

Two closing notes. First for anyone screening magnet candidates (ferrites are serious rare-earth-free contenders) or spinel catalysts (Co3O4 has this same trap underneath it): before a candidate's moment or site activity gets compared across a table, check which decoration it is. Second, an exhibit found on the way: the only CoFe2O4 entry in COD (5910063, transcribed from a 1931 structure-type gallery) declares formula CoFe2O4, but its own site table generates Co16Fe8O32, with Co octahedral and Fe tetrahedral. The formula field and the sites disagree, which is the whole point of tonight. The formula field does not know about decorations.