A linear/quadratic order parameter coupling description of the Verwey transition in magnetite, Fe3O4.

The change in symmetry Fd3m → Cc at the Verwey transition in magnetite puts it in a class of phase transitions with linear/quadratic coupling between two separate order parameters. Direct coupling between an order parameter Qe, to represent an electronic instability, and an order parameter Qco, to represent cation charge ordering, has the form λQeQco2, with Tc(Qe) < Tc(Qco), but there must also be indirect coupling through the common strain, e6, due to strain coupling terms λe6Qe and λe6Qco2. Qe has the symmetry of irrep Γ5+ while the pattern of cation charge ordering of Fe2+ and Fe3+ on octahedral sites depends on some combination of irreps Δ5, X1, X3, W1 and W2. The software package ISOVIZ has been used to show how reported patterns of order for the simplified structure in space group P2/c can be understood in terms of a linearly dependent mix of patterns with symmetry Δ5 and X1, so that Qco can be treated in the first instance as though it has the symmetry of Δ5. Spontaneous strains calculated from published lattice parameters and symmetry-adapted atomic displacements from previous structural refinements in space group Cc have been used to confirm that the two order parameters have different temperature dependences, consistent with this phenomenological treatment. The effect of chemical doping can be understood in terms of the development of local strain heterogeneity which acts to suppress the macroscopic strains and which appears to have a greater influence on charge ordering than on the electronic structure.