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

Carpenter, M.A.; Harrison, R.J.; Shaw-Stewart, J.; Adachi, K.; Senn, M.S.; Howard, C.J.

doi:10.1107/S2052520625004779

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ISSN: 2052-5206

Volume 81| Part 4| August 2025|

https://doi.org/10.1107/S2052520625004779

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

Michael A. Carpenter,^a ^* Richard J. Harrison,^a James Shaw-Stewart,^b Kanta Adachi,^c Mark S. Senn ^d and Christopher J. Howard ^e

^aDepartment of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom, ^bEveroze Partners Ltd, Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, United Kingdom, ^cGraduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, Japan, ^dDepartment of Chemistry, University of Warwick, Coventry CV4 7AL, United Kingdom, and ^eSchool of Engineering, University of Newcastle, University Drive, Callaghan, New South Wales 2308, Australia
^*Correspondence e-mail: [email protected]

Edited by M. Dusek, Academy of Sciences of the Czech Republic, Czechia (Received 22 December 2024; accepted 26 May 2025; online 8 July 2025)

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.

Keywords: magnetite; Verwey transition; charge order; order parameter coupling; spontaneous strain.

1. Introduction

The Verwey transition in magnetite, Fe₃O₄, has attracted intense interest since the first report of a large, discontinuous change in resistivity at T_v ∼125 K (Verwey, 1939 ). Recent structural studies have resolved many of the issues relating to the driving mechanism for the transition by confirming that it involves a combination of electronic effects, including Jahn–Teller distortion of sites containing Fe²⁺, with charge ordering such that Fe atoms with different nominal charges adopt an ordered arrangement on the octahedral sites (Wright et al., 2001 , 2002 ; Goff et al., 2005 ; Senn et al., 2012a , 2012b , 2013 , 2015 ; Perversi et al., 2016 ; Attfield, 2022 ). Below T_v the ordered structure is in crystallographic space group Cc with lattice parameters $[\sqrt 2]$ a × $[\sqrt 2]$ a × 2a, where a represents the cell dimension of the parent cubic structure (Iizumi & Shirane, 1975 ; Yoshida & Iida, 1979 ; Iizumi et al., 1982 ; Senn et al., 2012a; Perversi et al., 2016). The complex electronic structure has been found to involve highly structured three state polarons known as trimerons (Senn et al., 2012a, 2012b, 2015; Perversi et al., 2016; Attfield, 2022).

Rather than dwelling on these details, the purpose of the present study was to consider the Verwey transition from a different perspective, as an important case of linear/quadratic coupling between two order parameters with different symmetries. This form of coupling has specific consequences for the evolution of crystals with multiple instabilities (Cano et al., 2010 ; Salje & Carpenter, 2011 ), as has been shown for electronic and magnetic order parameters in Ba(Fe_1–xCo_x)₂As₂ (Cano et al., 2010; Böhmer & Meingast, 2016 ; Carpenter et al., 2019 ), NdCo₂ (Driver et al., 2014 ), and Pr_0.48Ca_0.52MnO₃ (Carpenter et al., 2010a , Carpenter et al., 2010b ), for structural and magnetic order parameters in Fe_1–xO (Carpenter et al., 2012 ) and for electronically driven transitions with multiple order parameters in alloys with martensitic transitions such as Ti₅₀Pd_50–xCr_x (Driver et al., 2020 ) and the Heusler compound Ni_50+xMn_25–xGa₂₅ (Salazar Mejía et al., 2018 ).

We make use of the software suite ISOTROPY from Brigham Young University (Stokes et al. ISOTROPY Software Suite, https://iso.byu.edu) to identify the symmetry properties of two order parameters. One of these, Q_e, is used to represent purely electronic aspects of the transition and a second, Q_co, to represent cation charge ordering. Following other group theoretical treatments (Piekarz et al., 2007 ), we focus on the role of the linear/quadratic coupling term $[\lambda Q_{\rm e}Q^{2}_{\rm o}]$ in a Landau expansion, where λ is a coupling coefficient. ISOVIZ from the ISOTROPY package is used to illustrate the possible patterns of cation charge order that might be represented by Q_co.

As a test of this description of the transition, lattice parameter data from the literature (Wright et al., 2000 ; Senn et al., 2015; Pachoud et al., 2020 ) are used to follow the temperature dependence of spontaneous strains which couple with each of Q_e and Q_co. Symmetry-adapted atomic displacement data obtained using ISODISTORT and reported by Senn et al. (2015), are used to explore relationships between order parameters with different symmetries that relate to the cation charge ordering.

2. Symmetry analysis

Magnetite, Fe₃O₄, at room temperature is an inverse spinel, that is the structure is cubic in space group Fd3m. Tetrahedral sites, Wyckoff 8a, nominally contain eight Fe³⁺ while octahedral sites, 16d, nominally contain eight Fe²⁺ + eight Fe³⁺. Fd3m symmetry evidently requires that Fe²⁺ and Fe³⁺ have a disordered arrangement on the octahedral sites. In order to respect developments in the literature and to illustrate a simpler case first, we follow Yamauchi et al. (2009 ) in initially treating the transition as being from Fd3m to P2/c and then considering the refined structure in space group Cc.

2.1. P2/c ( $[{\bi a}/{\bf \sqrt 2}]$ × $[{\bi a}/{\bf \sqrt 2}]$ × 2 $[{\bi a}]$ )

Piekarz et al. (2007) showed that no single order parameter can yield a structure in space group P2/c with unit cell $[a/\sqrt 2]$ × $[a/\sqrt 2]$ × 2a, where a is the lattice parameter of the parent cubic structure. A combination of order parameters with the symmetries of at least two irreducible representations (irreps) of space group Fd3m is required to achieve this reduction in symmetry. The full list of irreps is given in Table 1 of Piekarz et al. (2007) and in Table S1 of the supporting information that goes with the present paper. Pairwise combinations of irreps that can give the required symmetry change are listed in Table S1. As a requirement of symmetry, the transition must be first order in character.

Evidence of which two irreps to choose is provided initially by softening of the single crystal elastic modulus C₄₄ as T_v is approached from above. This is based on softening reported for magnetite by Schwenk et al. (2000 ) and Kozłowski et al. (2000 ) which is characteristic of bilinear coupling between the strain component e₄ and an order parameter with symmetry $[{{\Gamma}}_5^ +]$ . The atomic scale mechanism could, in principle, involve a classical soft acoustic mode, an electronic instability or some combination of the two. For present purposes the essential point is that it has the symmetry of $[{\Gamma}_{5}^{ +}]$ . (Note that, for the benefit of readers who may not be so familiar with group theory, we use the informal description `has the symmetry of' in place of the more formal expression `transforms as'.)

The symmetry of the second order parameter must account for the pattern of cation charge order. Out of the full list of irreps given in Table S1, only $[{\Gamma}_{5}^{ +}]$ , Δ₅, X₁, and X₃ lead to ordering in the octahedral sites of the P2/c structure. The patterns of ordering for each of these have been generated using ISOVIZ and are illustrated using different sized circles in Figs. S1–S4 in the supporting information for the succession of eight layers within the unit cell (z = 0, 1/8c, 2/8c, 3/8c, …). Each illustration shows three different sized circles, some increased in size, some reduced in size and some unchanged by the symmetry operator.

The requirement that there are Fe with just two different formal charges (represented by circles of only two different sizes), present in equal proportions is not met by any single irrep, so at least two must operate in combination. One of the active irreps has to be Δ₅ since it is the only one that gives doubling of the c-repeat with respect to the parent cubic structure. Combinations of Δ₅ ordering in a 1:1 ratio with each of $[{{\Gamma}}_5^ +]$ , X₁, and X₃, i.e. Δ₅ + $[{{\Gamma}}_5^ +]$ , Δ₅ + X₁ and Δ₅ + X₃, give patterns of order which meet the requirement of equal proportions of two differently charged cations. The pattern shown in Fig. 1 is the ordering scheme of Yamauchi et al. (2009) which arises from a combination of Δ₅ with X₁. Layers at z = 0, 2c/8, 4c/8, 6c/8 alternate in having only one or other of the differently charged cations. Layers at z = c/8, 3c/8, 5c/8, 7c/8 each contain equal proportions of the two differently charged cations alternating in rows. Alternative patterns of order given by Δ₅ + $[{{\Gamma}}_5^ +]$ and Δ₅ + X₃ can be visualized by inspection using Figs. S2 + S1 and S2 + S4.

Figure 1
Schematic representation of cation charge ordering in octahedral sites for ordered arrangements arising from equal proportions of Δ₅ and X₁. Small and large circles represent two different formal charges. Δ₅ gives alternation in rows within the layers at z = c/8, 3c/8, 5c/8, 7c/8, but has no effect on atoms in the layers at z = 0, 2c/8, 4c/8, 6c/8 (Fig. S2). X₁ leads to alternation of layers at z = 0, 2c/8, 4c/8, 6c/8, with only small or only large circles in each, but has no effect in the layers at z = c/8, 3c/8, 5c/8, 7c/8 (Fig. S3). The ordering scheme for Δ₅ gives doubling of the c-repeat. The pattern has been laid out to show that it is the same as shown for the P2/c structure illustrated in Figure 2 of Yamauchi et al. (2009

). By itself, it would conform to space group Pbcm.

By itself the ordering scheme of Yamauchi et al. (2009), as illustrated in Fig. 1, would conform to space group Pbcm. This can be reached from the parent space group, Fd3m, by taking Δ₅ as the symmetry of the active order parameter, q_Δ, because a component q_X1, with the symmetry of irrep X₁, is then present as a secondary order parameter. The full output from ISOTROPY for this symmetry change is given in Table S2. Coupling between order parameter components with symmetries Δ₅ and X₁ would have the form $[\lambda {q_{{\rm X}1}}q_{{\Delta}}^2]$ in lowest order but biquadratic coupling, $[\lambda q_{{\rm X}1}^2q_{{\Delta}}^2]$ , is always allowed. The requirement of a 1:1 ratio of Fe atoms with different formal charges on the octahedral sites appears to imply a rigid dependence of q_X1 on q_Δ. In the case of the 1:1 ordering resulting from Δ₅ + X₃ the same argument would apply to ensure a strict dependence of q_X3 on q_Δ. In other words, it is assumed that the favoured ordering scheme depends, effectively, on q_Δ alone.

Coupling between the order parameter with symmetry $[{{\Gamma}}_5^ +]$ , q_Γ, and the order parameter with symmetry Δ₅, q_Δ, is linear/quadratic, $[\lambda {q_{{\Gamma}}}q_{{\Delta}}^2]$ . On this basis the generalized coupling behaviour for electronic and cation charge order contributions to the P2/c structure is λQ_eQ_co². Quotation marks have been added here to emphasize that the form of coupling depends on symmetry arguments rather than specifics of the atomic scale driving mechanisms. Here and throughout, we use upper case Q to represent the generic coupling without reference to symmetry and lower case q to represent order parameters or order parameter components with specific symmetry.

2.2. Cc ( $[\sqrt 2]$ a × $[\sqrt 2]$ a × 2a)

The full list of irreps for Fd3m → Cc includes eight with symmetry of the Γ point (0,0,0), three with symmetry of the Δ point (0,1/2,0), four with symmetry of the X point (0,1,0) and two with symmetry of the W point (1/2,1,0) (Table S3). Just as for Fd3m → P2/c, a combination of order parameters with the symmetries of at least two irreducible representations (irreps) of space group Fd3m is required to achieve the reduction in symmetry. The combinations listed in Table S3 automatically generate an order parameter with symmetry $[{{\Gamma}}_5^ +]$ . By symmetry, the transition has to be first order in character.

Different cation charge ordering schemes on the octahedral sites can be generated by irreps $[{{\Gamma}}_5^ {+}, {\Delta}_{5}]$ , X₁ and X₃ with, in comparison with the P2/c structure, additional degrees of freedom implied by the increased number of independent components in each of Δ₅, X₁ and X₃. Two further irreps, W₁ and W₂, are also permissive of ordering and there are two different patterns of ordering associated with each of these (Figs S5–S8). The two configurations of W1 differ by an offset of 1/4c at layers z = 0, 2c/8, 4c/8, 6c/8 with respect to layers at z = c/8, 3c/8, 5c/8, 7c/8 (compare Fig. S5 with Fig. S6). The same applies to the two configurations of W2 (Figs. S7 and S8).

Arrangements for ordering of equal proportions of Fe with two different formal charges on the octahedral sites still require the operation of Δ₅ with some fixed combination of the other irreps. There are multiple ways in which ordering schemes of different irreps could be combined in attempts to reproduce the pattern of ordering shown in Figure 3 of Yamauchi et al. (2009) for the Cc structure, and no attempt has been made here to explore these in full. The key symmetry argument remains that the essential coupling term for the transition as a whole has the form $[\lambda {q_{{\Gamma}}}q_{{\Delta}}^2]$ .

3. Strain analysis

Spontaneous strains presented in symmetry-adapted form provide an indirect means of evaluating the evolution of different order parameters in systems with phase transitions that depend on multiple order parameters (McKnight et al., 2009 ; Carpenter et al., 2005 ; Carpenter & Howard, 2009a , 2009b , 2010b; Eckstein et al., 2022 ). In the case of the P2/c model structure of magnetite, there is a minimum of three order parameters to consider and these appear to have the symmetry of $[{{\Gamma}}_5^ +]$ , Δ₅ and X₁. Following the argument in the previous section in relation to the requirement that, for an equal proportion of only two Fe cations with different formal charges on the octahedral sites, it is assumed that there is only one independent order parameter for the charge order component of the transition. The single order parameter used here is q_Δ, with the understanding that the pattern of ordering depends on some combination of Δ₅ and X₁ (and/or X₃) ordering schemes in fixed proportions. The same argument applies to the Cc structure; i.e. it is assumed that X₁, X₃, W₁ and W₂ contributions to the charge ordering at octahedral sites do not vary independently with respect to q_Δ.

The volume strain, e_a (= e₁ + e₂ + e₃) has the symmetry of $[{{\Gamma}}_1^ +]$ . e_t [ $[ = {1 \over {\surd 3}}\left({2{e_3} - {e_1} - {e_2}} \right)]$ ] and e_o (= e₁ − e₂) are tetragonal and orthorhombic shear strains, respectively, and have the symmetry of $[{{\Gamma}}_3^ +]$ . The remaining shear strains, e₆, e₄ and e₅, have the symmetry of $[{{\Gamma}}_5^ +]$ . The $[{{\Gamma}}_5^ +]$ order parameter for the P2/c and Cc structures has three components, q_Γ1 ≠ q_Γ2 = q_Γ3. The Δ₅ order parameter has two components, q_Δ1 ≠ q_Δ2 ≠ 0 for Cc as the subgroup symmetry and one, q_Δ1 ≠ 0, q_Δ2 = 0, for subgroup P2/c. When treated as driving two separate but coupled instabilities, the temperature dependence of these two order parameters and their coupling with strain would be expected to conform to solutions of a Landau expansion with the form:

$[\eqalignno{G =& {{1}\over{2}}{a}_{\Gamma }\left(T-{T}_{c\Gamma }\right)\big({q}_{\Gamma 1}^{2}+2{q}_{\Gamma 2}^{2}\big)+{{1}\over{3}}{b}_{\Gamma }{q}_{\Gamma 1}{q}_{\Gamma 2}^{2}\cr &+{{1}\over{4}}{c}_{\Gamma }\big(4{q}_{\Gamma 2}^{4}\!+\!4{q}_{\Gamma 1}^{2}{q}_{\Gamma 2}^{2}\!+\!{q}_{\Gamma 1}^{4}\big)+{{1}\over{4}}{c}_{\Gamma }^{\prime}\big(5{q}_{\Gamma 2}^{4}\!+\!6{q}_{\Gamma 1}^{2}{q}_{\Gamma 2}^{2}\!+\!{q}_{\Gamma 1}^{4}\big)\cr &+{\lambda }_{\Gamma 1}{e}_{\rm a}\big({q}_{\Gamma 1}^{2}\!+\!2{q}_{\Gamma 2}^{2}\big) + {\lambda }_{\Gamma 2}{e}_{\rm t}\big({q}_{\Gamma 1}^{2}-{q}_{\Gamma 2}^{2}\big)\cr &+{\lambda }_{\Gamma 3}\big({e}_{6}{q}_{\Gamma 1}\!+\!{e}_{4}{q}_{\Gamma 2}\!+\!{e}_{5}{q}_{\Gamma 2}\big)+{{1}\over{2}}{a}_{\Delta }\big(T\!-\!{T}_{{\rm c}\Delta }\big)\big({q}_{\Delta 1}^{2}\!+\!{q}_{\Delta 2}^{2}\big)\cr &+{{1}\over{4}}{b}_{\Delta }{\big({q}_{\Delta 1}^{2}\!+\!{q}_{\Delta 2}^{2}\big)}^{2} + {{1}\over{4}}{b}_{\Delta }^{\prime}\big({q}_{\Delta 1}^{4} + {q}_{\Delta 2}^{4}\big)\!+\!{\lambda }_{\Delta 1}{e}_{\rm a}\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)\cr &+{\lambda }_{\Delta 2}{e}_{\rm t}\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)+{\lambda }_{\Delta 3}{e}_{6}\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)\cr &+{\lambda }_{\Delta 4}\big({e}_{6}^{2}+{e}_{4}^{2}+{e}_{5}^{2}\big)\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)+{\lambda }_{\Gamma \Delta 1}{q}_{\Gamma 1}\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)\cr &+{\lambda }_{\Gamma \Delta 2}\big({q}_{\Gamma 1}^{2}+2{q}_{\Gamma 2}^{2}\big)\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)+{\lambda }_{\Gamma \Delta 3}{q}_{\Gamma 2}^{2}\big({q}_{\Delta 1}^{2}+{q}_{\Delta 2}^{2}\big)\cr &+{{1}\over{6}}\big({C}_{11}^{\rm o}+2{C}_{12}^{\rm o}\big){e}_{\rm a}^{2}+{{1}\over{4}}\big({C}_{11}^{\rm o}-{C}_{12}^{\rm o}\big)\big({e}_{\rm o}^{2}+{e}_{\rm t}^{2}\big)\cr &+{{1}\over{2}}{C}_{44}^{\rm o}\big({e}_{4}^{2}+{e}_{5}^{2}+{e}_{6}^{2}\big). &(1)}]$

Here a, b, c are standard coefficients with subscripts to identify which order parameter they refer to. T_cΓ and T_cΔ represent critical temperatures for the two instabilities. λ_Γ1, λ_Γ2, etc. are coupling coefficients and $[C_{ik}^{\rm o}]$ , i, k = 1–6, represents bare elastic moduli, i.e. excluding the influence of the phase transition. The lowest order term for direct coupling between the two order parameters is linear/quadratic, $[{\lambda _{{\Gamma \Delta }1}}{q_{{{\Gamma}}1}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ , but there are also biquadratic terms, $[\lambda _{\Gamma \Delta 2}\left({q_{{\Gamma}1}^{2} + 2q_{{{\Gamma}}2}^2} \right)\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ and $[\lambda _{\Gamma \Delta 3}q_{{{\Gamma}}2}^2\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ . The P2/c and Cc structures both have e_a ≠ 0, e_t ≠ 0, e_o = 0, e₆ ≠ e₄ = e₅.

Setting the equilibrium conditions $[\partial G/\partial {e_{\rm a}} = \partial G/\partial {e_{\rm t}} = \partial G/\partial {e_4} = \partial G/\partial {e_5} = \partial G/\partial {e_6} = 0]$ gives

$[{e_{\rm a}} = {{ - {\lambda _{{{\Delta}}1}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right) - {\lambda _{{{\Gamma}}1}}\left({q_{{{\Gamma}}1}^2 + 2q_{{{\Gamma}}2}^2} \right)} \over {{1 \over 3}\left({C_{11}^{\rm o} + 2C_{12}^{\rm o}} \right)}} \eqno(2)]$

$[{e_{\rm t}} = {{ - {\lambda _{{{\Gamma}}2}}\left({q_{{{\Gamma}}1}^2 - q_{{{\Gamma}}2}^2} \right) - {\lambda _{{{\Delta}}2}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)} \over {{1 \over 2}\left({C_{11}^{\rm o} - C_{12}^{\rm o}} \right)}} \eqno(3)]$

$[{e_4} = {e_5} = {{ - {\lambda _{{{\Gamma}}3}}{q_{{{\Gamma}}2}}} \over {C_{44}^{\rm o} + 2{\lambda _{{{\Delta}}4}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)}} \approx {{ - {\lambda _{{{\Gamma}}3}}{q_{{{\Gamma}}2}}} \over {C_{44}^{\rm o}}} \eqno(4)]$

$[{e_6} = {{ - {\lambda _{{{\Gamma}}3}}{q_{{{\Gamma}}1}}-{\lambda _{{{\Delta}}3}}\left({q_{{{\Delta}}1}^2\!+\!q_{{{\Delta}}2}^2} \right)} \over {C_{44}^{\rm o} + 2{\lambda _{{{\Delta}}4}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)}} \approx {{ - {\lambda _{{{\Gamma}}3}}{q_{{{\Gamma}}1}}-{\lambda _{{{\Delta}}3}}\left({q_{{{\Delta}}1}^2\! +\!q_{{{\Delta}}2}^2} \right)} \over {C_{44}^{\rm o}}}. \eqno(5)]$

Fig. 2(a) includes values for the shear strains extracted from lattice parameter data given in Figure 3 of Senn et al. (2015), using expressions listed in Table S4 for the crystallographic orientation shown in Fig. S9. Values of the original lattice parameters are shown in Fig. S10. e₄ is expected to scale with q_Γ2 [equation (4)] and displays the strongly first order character of the transition. e_t and e₆ both arise in part by coupling with $[\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ [equations (3) and (5)] and show a less abrupt temperature dependence as T_v is approached from below.

Figure 2
Variations of shear strains determined from lattice parameters given in Figure 3 of Senn et al. (2015

) for a synthetic magnetite sample with T_v ∼123 K, and in Figure 1 of Pachoud et al. (2020

) for a synthetic sample with estimated composition x = 0.0228 in Fe_3–xZn_xO₄. Pachoud et al. quoted T_v = 92 K for their sample but the discontinuity in lattice parameters occurred between 85 and 90 K in their lattice parameter data. Original lattice parameters and expressions for calculation of the strains are given in Fig. S10 and Table S4, respectively, with respect to the reference system shown in Fig. S9. Coloured dashed lines in (b), (c), (d) represent discontinuities at T ∼ T_v.

If there was only one intrinsic instability, the three shear strains would evolve in a more closely similar manner because the order parameters representing electronic and charge order contributions would vary in fixed proportions with each other. In particular, if the transition depended only on $[\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ , e₆ would be a linear function of e_t, passing through the origin. Fig. 2(b) shows that this dependency is not observed, consistent with a formulation in terms of coupling between two separate instabilities. Variations of e₄ with respect to e₆ and e_t are shown in Fig. 2(c) for completeness.

By itself, an electronic order parameter with q_Γ1 = q_Γ2 = q_Γ3 would produce a rhombohedral strain, e₄ = e₅ = e₆, with e_i ∝ q_i. Ordering on the basis of the Δ₅ order parameter with q_Δ1 ≠ 0, q_Δ2 = 0 alone would yield a structure in space group Pbcm. Addition of the orthorhombic strains to the rhombohedral strains produces the monoclinic distortion, e₆ ≠ e₄, in space group P2/c and is accompanied by relaxations attributable to the difference between q_Γ1 and q_Γ2. Ordering on the basis of the Δ₅ order parameter with q_Δ1 ≠ q_Δ2 ≠ 0 alone would also yield an orthorhombic structure, this time in space group Pmc2₁. The combination of orthorhombic and rhombohedral strains would then give the same monoclinic distortion, but in space group Cc.

The relationship between e₄ ( $[ \propto]$ q_Γ2) and e₄ − e₆ {≃ [λ_Γ3(q_Γ1 − q_Γ2) + λ_Δ3(q_Δ1² + q_Δ2²)](C₄₄^o)⁻¹} in Fig. 2(d), shows that e₄ remains nearly constant in comparison with the contributions from relaxations due to (q_Γ1 − q_Γ2) and $[\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ within the stability field of the monoclinic structure.

Determination of values for the volume strain, e_a, would require sufficient lattice parameter data in the stability field of the high-symmetry phase to allow extrapolation of a baseline into the stability field of the low-temperature phase. In the absence of data for the cubic structure of magnetite above T_v, it has not been possible to determine values of e_a for the synthetic magnetite sample of Senn et al. (2015) (T_v ∼123 K). However, data reported for a wider temperature interval by Wright et al. (2000), and reproduced in Fig. S11, show that the transition is accompanied by a small increase in volume for a synthetic sample with T_v ∼ 108 K. A similar positive volume strain is evident in the lattice parameter data shown in Fig. 6 of Kozlowski et al. (1999 ) for a synthetic crystal with T_v ∼ 120 K.

Lattice parameters of Wright et al. for the low temperature structure were refined under rhombohedral symmetry (e₄ = e₅ = e₆). Values of the strains e_a and e₄ obtained from these are shown in Fig. 3(a) using expressions given in Table S4. Although the magnitude of e₄ at low temperatures is closely similar to that shown in Fig. 2(a), it has a steeper temperature dependence as T_v is approached from below. If the transition was driven only by the electronic component, as represented by q_Γ1 (= q_Γ2 = q_Γ3), the two non-zero strains would be expected to vary as e_a ∝ e₄² ∝ q_Γ1² [equations (2) and (4)]. As seen in Fig. 3(b), e_a and e₄² display a linear correlation but the straight line fit to the data does not pass through the origin. The simplest explanation is that there are two contributions to the volume strain [equation (2)] and that deviations from ideal stoichiometry cause shear strain coupling with long range charge order to be suppressed. Non-zero values of e_a between T_v and ∼150 K are indicative of precursor effects arising from one or both of the electronic and charge order contributions. For comparison with data for the crystal with T_v = 123 K, spontaneous strain values determined from lattice parameter data given in Fig. 1(c) of Pachoud et al. (2020) have been added to Fig. 2. The sample of Pachoud et al., Fe_3–xZn_xO₄, was described as having T_v = 92 K and estimated composition x = 0.0228. The original lattice parameters from refinement in space group Cc are included in Fig. S10. All the shear strains have values which are lower than for the pure magnetite sample of Senn et al. (2015) but, apart from a change in sign for values of e_t, the overall pattern of their temperature dependence is similar. Pachoud et al. reported data for the cubic lattice parameter between T_v and 300 K (Fig. S10), from which values of e_a have been determined using values of a_o extrapolated into the stability field of the monoclinic phase. As shown in Fig. 3(a), they are not distinguishable from zero.

Figure 3
Variations of spontaneous strains determined from lattice parameters given in Figure 3 of Wright et al. (2000

) for a synthetic magnetite sample with T_v = 108 K. The original lattice parameters are reproduced in Fig. S11 and expressions for the strains are given in Table S4. Also included in (a) is the volume strain, e_a, determined from lattice parameter data of Pachoud et al. (2020

) for a synthetic magnetite crystal with estimated composition x = 0.0228 in Fe_3–xZn_xO₄. (Pachoud et al. (2020

) reported T_v = 92 K for this sample; the small discontinuity in e_a deternined from their data is shown here as being at 88 K). The dashed line in (b) represents the first order discontinuity at ∼T_v. The solid line is a fit to data at T < T_v.

Heat capacity measurements have shown that increased doping leads to a change from first order character towards second order character (Kozłowski et al., 1996 , 1997 , 2000). The data in Fig. 2 for the sample with T_v ∼ 88 K show a distinct first order step in the shear strain rather than a tendency to become more nearly second order, however. This could be an issue of experimental resolution arising from the difficulty in following changes in lattice parameters when distortions from cubic geometry become very small.

4. Symmetry-adapted atomic displacements

One means of characterizing the evolution of different components of a symmetry change involving as many possible secondary irreps as occurs in the present case is to follow the pattern of symmetry-adapted atomic displacement distances from structure refinements. Advantage is taken here, therefore, of values of symmetry-adapted displacement modes reported by Senn et al. (2015) for the refined Cc structure, with respect to the high symmetry (Fd3m) structure. ISODISTORT from the ISOTROPY software suite had been used to follow a total of 168 modes. While the amplitudes of these do not provide direct information on the pattern of cation charge ordering, the temperature dependence of displacement modes with symmetry $[{{\Gamma}}_5^ +]$ , Δ₅, X₁, X₃, W₁ and W₂ will be indicative also of the temperature dependence of ordering on the basis of the same symmetry elements.

In the case of $[{{\Gamma}}_5^ +]$ , there are six symmetry-adapted displacements, d_i (i = 1–6), and the average of their amplitudes is |d_Γ5+| = (|d₁| + |d₂| + |d₃|…)/6. The same treatment for |d_Δ5| (22 modes), |d_X1| (28 modes), |d_X3| (15 modes), |d_W1| (20 modes), and |d_W2| (22 modes) yielded the values shown in Fig. 4(a). All show essentially the same pattern of a discontinuity at T_v followed by a small increase before levelling off below ∼80 K. Each irrep contains multiple components and, in principle, the average displacement |d_Γ5+| should show the same temperature dependence as the average $[{{\Gamma}}_5^ +]$ shear strain, (e₄+e₅+e₆)/3. The numbers are small but the data in Fig. 4(b) are consistent with this. However, individual sets of values of d_i show variations in their temperature dependences from flat to significantly curved, with the implication that using average values hides potential information about the variations of individual modes belonging to each irrep. It has already been shown by the evolution of e₄ (∝ ∼q_Γ2) in Fig. 2(a) that the electronic aspect of the order parameter shows very little temperature dependence, so the relatively steep temperature dependences can be ascribed predominantly to the cation charge ordering process. The values of d_i shown in Fig. 4(c) were picked out as displaying the steepest temperature dependence and, hence, as potentially most revealing of order parameter components that relate to the ordering.

Figure 4
Variation of symmetry-adapted atomic displacements, |d|, with respect to the parent cubic structure, derived from original data of Senn et al. (2015

) refined in space group Cc. Three different crystals, c8 (red), c11 (blue), c17 (green), were from the same original batch as used for determination of the temperature dependence of lattice parameters shown in Fig. S10a. (a) Temperature dependence of average values of displacements with symmetry $[{{\Gamma}}_5^ +, ]$ Δ₅, X₁, X₃, W₁ and W₂. (b) Comparison of the average displacement amplitude |d_Γ5+| with variations of e₄ and e₆, showing that it has the same temperature dependence as the average value of shear strains belonging to irrep $[{{\Gamma}}_5^ +]$ . (c) Temperature dependence of values of individual displacement modes with symmetry $[{{\Gamma}}_5^ +, ]$ Δ₅, X₁, X₃, W₁ and W₂. $[\left| {{d_{6{{\Delta}}5}}} \right|]$ is mode 37 in the list of modes from Senn et al. (2015

), $[\left| {{d_{6{\rm X}1}}} \right|]$ is mode 43, $[\left| {{d_{5{\rm X}3}}} \right|]$ is mode 78, $[\left| {{d_{3{\rm W}1}}} \right|]$ is mode 61, $[\left| {{d_{6{\rm W}2}}} \right|]$ is mode 68. (d) Variations of average |d| values for different symmetries with respect to variation of |d_Δ5|: all are approximately linear but straight lines through the data do not pass through the origin. (e) Comparison of individual |d| variations selected as displaying the largest temperature dependences for different symmetries with the variation of the individual values $[\left| {{d_{6{{\Delta}}5}}} \right|]$ : straight lines through the data were constrained to pass through the origin showing that the X₁, W₁ and X₃ modes can be understood as scaling linearly with the Δ₅ mode. (f) A straight line through data, constrained to pass through zero, shows that the W₂ mode can be understood as scaling with the square of the Δ₅ mode.

In their simplest form, allowed terms for coupling of ordering on the basis of irrep X₁ with ordering on the basis of irrep Δ₅ are $[\lambda {q_{{\rm X}1}}q_{{\Delta}}^2]$ and $[\lambda q_{{\rm X}1}^2q_{{\Delta}}^2]$ . If the lowest order coupling term is dominant, a scaling q_X1 ∝ $[q_{{\Delta}}^2]$ would be expected. If the cation charge ordering on the octahedral sites is rigidly constrained by the 1:1 ratio of Fe²⁺ to Fe³⁺, a fixed dependence from the higher order term as q_X1 ∝ q_Δ is more likely. Fig. 4(d) demonstrates that average values of |d_i| with different symmetries each vary approximately linearly with |d_Δ5| but straight lines through them do not pass through the origin. A plot of |d_i| averages against |d_Γ5+|² (not shown) gives essentially the same result - straight lines that do not pass through the origin. On the other hand, data from single modes showing the maximum temperature dependence plotted in the same way [Fig. 4(e)] show linear dependences that, for X₁, W₁ and X₃ values, include the origin. The variation of values for the W₂ mode are permissive of a W₂ component scaling with values of the square of the Δ₅ mode [Fig. 4(f)]. The numbers are small and spread over only narrow ranges, but Fig. 4(e), at least, is permissive of the rigid dependence of X₁ ordering on Δ₅ ordering assumed when developing the scheme shown in Fig. 1.

Notwithstanding the assumptions involved in interpreting the symmetry-adapted displacements, Fig. 5 reveals a consistent pattern of evolution for the two principal order parameters referred to previously in a generic manner as $[{q_{{\Gamma}}}]$ and q_Δ. For this, the average values |d_Γ5+| with |d_Δ5| and values for single modes with the same symmetry were scaled so that they would each extrapolate to a value of Q = 1 at 0 K. A similar scaling of values of e₄ and e₄-e₆ from Fig. 2(d) has been added. Scatter is greatest for the individual displacement modes and least for the strain variations, reflecting the relative degrees of experimental uncertainties. The shear strains and displacement amplitudes do not depend on exactly the same combinations of order parameter components but they show the same pattern of a discontinuity at T_v from zero to ∼0.85 for $[{q_{{\Gamma}}}]$ and ∼0.95 for q_Δ. Below T_v, $[{q_{{\Gamma}}}]$ varies only slightly in comparison with a wider variation of q_Δ, in a non-linear manner that is consistent with the evolution of two driving order parameters that are coupled but have separate critical temperatures.

Figure 5
Proxies for the variation of order parameters with symmetry $[{{\Gamma}}_5^ +]$ and Δ₅, each scaled to Q = 1 at 0 K. The average and individual symmetry-adapted displacements do not depend on the same order parameter components as the shear strains but all three sets of data show the same pattern of evolution: a large discontinuity in the electronic and cation charge order components of the transition at T_v (broken line) is followed by small variations predominantly in the cation charge order component with further reduction in temperature.

5. Discussion

5.1. Linear/quadratic coupling for two instabilities with different critical temperatures

Consideration of the Verwey transition in magnetite from the perspective of multiple order parameters brings into focus coupling according to $[\lambda {q_{{\Gamma}}}q_{{\Delta}}^2]$ . Although this does not require that the microscopic mechanisms are described, q_Γ has been discussed in terms of a (zone centre) electronic instability and q_Δ in terms of (zone boundary) ordering of Fe²⁺ and Fe³⁺ on octahedral sites. Coupling between the two order parameters could be direct, as expressed through the term $[{\lambda _{{{\Gamma \Delta }}1}}{q_{{{\Gamma}}1}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ , or indirect through the common strain, e₆, due to the terms $[{\lambda _{{{\Gamma}}3}}{e_6}{q_{{{\Gamma}}1}}]$ and $[{\lambda _{{{\Delta}}3}}{e_6}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ .

Following the generic treatment of Salje & Carpenter (2011) (and see, also, Cano et al., 2010), linear/quadratic coupling $[\lambda Q{P^2}]$ between order parameters Q and P can lead to two substantially different outcomes depending on the relative values of the critical temperatures, T_cP and T_cQ. For T_cQ > T_cP, the expectation would be for two discrete transitions and a sequence of structures with falling temperature as Q = P = 0 → Q ≠ 0, P = 0 → Q ≠ 0, P ≠ 0. For T_cP > T_cQ only one transition would be expected, Q = P = 0 → Q ≠ 0, P ≠ 0, with Q and P displaying different dependences on temperature through the stability field of the low-symmetry structure. It has been shown here that experimental data for symmetry-adapted strains and atomic displacements which scale with the two proposed order parameters in magnetite are consistent with different temperature dependences below a single, discrete transition at T_v. Details of the Landau expansion [equation (1)] are more complicated than the generic expression used by Salje & Carpenter (2011), but the implication is that T_cΔ is most likely greater than T_cΓ.

5.2. Chemical doping and local strain heterogeneity

Changes in oxidation state balanced by increasing vacancy concentrations on the cation sites and substitution of Zn, Ti or Al for Fe by up to ∼3.5% causes T_v to reduce from ∼125 K to ∼80 K (Kozłowski et al., 1996, 1997, 1999, 2000; Kołodziej et al., 2012 ; Goto & Lüthi, 2003 ; Kąkol et al., 2012 ). However, nonlinear softening of C₄₄ as T → T_v from above, diagnostic of a pseudoproper ferroelastic transition, is unaffected by doping with Zn (Fig. 6 of Schwenk et al., 2000). The softening is described by C₄₄ = $[C_{44}^{\rm o}]$ (T − $[T_{{\rm c}\Gamma }^{*}]$ )/(T − T_cΓ), where $[T_{{\rm c}\Gamma }^{*}]$ is the value of T_cΓ renormalized by bilinear coupling of q_Γ2 with e₄ (at q_Δ1 = q_Δ2 = 0) and $[C_{44}^{\rm o}]$ is the value of C₄₄ in the absence of the instability.

Schwenk et al. (2000) obtained the same values of $[T_{{\rm c}\Gamma }^{*}]$ and T_c, 66 K and 56 K, respectively, for three different compositions, x = 0, 0.02, 0.032 in Fe_3–xZn_xO₄, implying that the Γ-point instability is not suppressed by doping and that the strength of bilinear coupling, as represented by the value of λ_Γ3 in equation (1), is also unaffected. The dominant influence of changes in composition at this level appears to be related predominantly to the cation charge order component of the phase transition, therefore. Given that almost identical lowering of T_v is seen as a function of composition for different substituting cations (Figure 2 of Kąkol et al. 2012), the effect is more likely to have an effectively physical rather than purely chemical origin. A simple explanation in this context relates to local strain heterogeneity accompanying the substitution of small spheres for large spheres, or vice versa, in a more or less elastic matrix.

In the case of silicate solid solutions, hard mode infrared spectroscopy has shown that cations of different sizes are accommodated by the development of local strain heterogeneity on a length scale of a few unit cells (Atkinson et al., 1999 , 2024 ; Boffa Ballaran et al., 1998 ; Carpenter et al., 1999 ; Carpenter & Boffa Ballaran, 2001 ). The influence of such local strain heterogeneities on phase transitions is seen most clearly in the plateau effect, whereby the transition temperature for a displacive transition in a pure crystal is unaffected by chemical substitutions at the lowest concentrations. The temperature of a thermodynamically continuous transition only starts to change once some critical doping level has been reached, corresponding to the point at which strain fields round individual replacement cations start to overlap. For example, the plateau of nearly constant temperature for the displacive transition in NaAlSi₃O₈ extends to ∼2% substitution of K⁺ (∼1.5 Å) for Na⁺ (∼1.0 Å) implying that the strain fields around individual K⁺ ions have diameters of ∼20–40 Å (Carpenter et al., 1999). In the case of La³⁺ (∼1.03 Å) substitution for Pr³⁺ (∼0.99 Å) in the perovskite PrAlO₃, the plateau for a transition at ∼150 K extends to 1.6 ± 0.2%, implying that individual strain fields around La³⁺ have a diameter of ∼16–18 Å (Carpenter et al., 2009 ).

Strain heterogeneities must exist in doped magnetite but T_v for the Verwey transition does not show a discrete plateau with increased doping. This could be due to overlapping of strain fields at smaller doping levels in a close packed oxide structure, and/or to the complication of two order parameters interacting at a first order transition. Nevertheless, there is a break in the trend of decreasing T_v with composition at ∼1.3% substitution of Fe by Zn, Ti, Al or vacancies [as summarized in Figure 4 of Attfield (2022)], which correlates with the plateau limit in other materials and hints at some analogous influence of local strain fields on the transition.

5.3. Local strain heterogeneity and suppression of macroscopic strain

An additional consequence of local strain heterogeneity in systems with cation ordering is that coherent macroscopic strains become suppressed by the introduction of site disorder. For example, crystals of the perovskite La_0.6Sr_0.1TiO₆ can be prepared with either ordered or disordered distributions of vacancies on the A-cation site. The disorder is accommodated, in part at least, by the development of local strain heterogeneity that is eliminated when the vacancies evolve to an ordered configuration. The same displacive phase transition occurs in samples with disordered vacancies as when they are ordered but the macroscopic spontaneous strain is almost entirely suppressed when the crystals are prepared with a disordered state (Howard et al., 2007 ). Comparison in Fig. 2(a) of results for the undoped crystal (T_v = 123 K) and the Zn-doped crystal (T_v = 92 K) shows that, in addition to lowering T_v, the effect of Zn-doping is to reduce the magnitude of the shear strains, consistent with the experience from perovskites. The volume strain is suppressed to essentially zero [Fig. 3(a)].

Evidence from the evolution of C₄₄ at T > T_v indicates that bilinear coupling $[{\lambda _{{{\Gamma}}3}}\left({{e_6}{q_{{{\Gamma}}1}} + {e_4}{q_{{{\Gamma}}2}} + {e_5}{q_{{{\Gamma}}2}}} \right)]$ is not affected significantly by changes in composition, implying that the suppression of shear strain e₆ by the introduction of extraneous cations or vacancies amounts primarily to lowering of values of the coupling coefficient λ_Δ3 in the term $[{\lambda _{{{\Delta}}3}}{e_6}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ . If $[{\lambda _{{{\Delta}}3}}]$ is reduced, the contribution of coupling via the common strain e₆ to the effective linear/quadratic coupling between q_Γ and q_Δ will also be reduced. This in turn is likely to account for suppression of the transition, both in terms of lowering values of T_v and reducing the magnitude of q_Γ1.

Suppression of the volume strain by doping [Fig. 3(a)] implies that either or both of the coupling coefficients in $[{\lambda _{{{\Gamma}}1}}{e_{\rm a}}\left({q_{{{\Gamma}}1}^2 + 2q_{{{\Gamma}}2}^2} \right)]$ and $[{\lambda _{{{\Delta}}1}}{e_{\rm a}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ is/are reduced. A standard result of order parameter coupling with volume strain, as in the case of the α–β transition in quartz (Carpenter et al., 1998 ), is that the fourth order coefficient in a Landau expansion is renormalized such that strong coupling drives the transition towards first order character. Suppression of this coupling in magnetite must be a contributory factor to the change from first order character towards second-order character with increased doping seen in the heat capacity measurements of Kozłowski et al. (1996, 1997, 2000). By symmetry the transition is first order in character due to the presence of the third-order term, $[{1 \over 3}{b_{{\Gamma}}}{q_{{{\Gamma}}1}}q_{{{\Gamma}}2}^2]$ . However, b_Γ is a property of the material and can be small, as appears to be the case if the transition becomes close to second order rather than simply being smeared over some temperature interval in doped samples.

The third significant difference in strain evolution between the pure and doped samples is the change in sign of e_t [Fig. 2(a)]. According to equation (3), the implication is that λ_Γ2 and λ_Δ2 in $[{\lambda _{{{\Gamma}}2}}{e_{\rm t}}\left({q_{{{\Gamma}}1}^2 - q_{{{\Gamma}}2}^2} \right)]$ and $[{\lambda _{{{\Delta}}2}}{e_{\rm t}}\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ have opposite sign. Reduction in the value of one of the coefficients would change the sign of the sum of strain contributions to the overall value of e_t, as would differential reduction in the values of $[q_{{{\Gamma}}1}^2 - q_{{{\Gamma}}2}^2]$ and $[\left({q_{{{\Delta}}1}^2 + q_{{{\Delta}}2}^2} \right)]$ . If this is the case, biquadratic coupling of the two order parameters via e_t as a common strain would be unfavourable, contributing to rather complicated patterns of evolution of the order parameters below T_v with and without doping.

5.4. Cc structure

The structure reported by Yamauchi et al. (2009) in space group P2/c indicated that, if the cation charge ordering depended only on distributing large Fe²⁺ ions and small Fe³⁺ ions on octahedral sites in a ratio of 1:1, the preferred pattern by itself would have symmetry Pbcm. In this case the Δ₅ irrep has only one component and the ordering is permitted by irreps Δ₅ and X₁ in fixed proportions. The pattern obtained from their calculations in space group Cc has differences due, presumably, to stabilization achieved by clustering to form the trimerons identified by Senn et al. (2012a). Symmetry-adapted atomic displacement amplitudes are permissive of linear dependence of ordering on the basis of X₁ symmetry with ordering on the basis of Δ₅. They also provide some indication of which additional symmetry components should be investigated as being most significant in this context. The overall picture is of the discontinuity at T_v being to a well ordered structure with only small increases in the degree of order with further falling temperature.

6. Conclusion

Equation (1) provides a practical description for the Verwey transition using the minimum number of independent order parameters required to give the observed symmetry change. It serves to emphasize that the transition belongs to an important class of phase transitions in multiferroic materials where linear/quadratic coupling between two order parameters defines the form of interaction between separate instabilities. As with other examples of linear/quadratic coupling referred to in Introduction, the characteristic features are multiple order parameters, strong coupling via common strains, diverse patterns of elastic constant variations and diverse patterns of behaviour with changing composition, depending on how the critical temperatures of the two instabilities vary.

As in previously described examples, the zone centre instability is electronic. In general, there are many possibilities for the zone boundary instability, including magnetism and structural changes such as octahedral tilting in perovskites. In magnetite both order parameters are related to changes in electronic structure in the sense that the cation charge order arises from the cooperative Jahn–Teller instability of Fe²⁺. As with phase transitions in all these materials, the role of strain at both local and macroscopic length scales is fundamental in controlling the overall structural evolution and microstructure.

7. Related literature

The following references are cited in the supporting information: Carpenter (2007 ); Meyer et al. (2000 ); Meyer et al. (2001 ); Salje et al. (1991 ); Sondergeld et al. (2000 ).

Supporting information

Tables S1-S4, Figs. S1-S11. DOI: https://doi.org/10.1107/S2052520625004779/dk5138sup1.pdf

Acknowledgements

MSS acknowledges the Royal Society for a fellowship (UF160265 & URF\R\231012).

Funding information

The following funding is acknowledged: Royal Society (grant No. UF160265 to Mark Senn; grant No. URF\R\231012 to Mark Senn).

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