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 Q_e, to represent an electronic instability, and an order parameter Q_co, to represent cation charge ordering, has the form λQ_eQ_co², with T_c(Q_e) < T_c(Q_co), but there must also be indirect coupling through the common strain, e₆, due to strain coupling terms λe₆Q_e and λe₆Q_co². Q_e 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 Q_co 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.