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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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.

STRUCTURAL SCIENCE
CRYSTAL ENGINEERING
MATERIALS

ISSN: 2052-5206

Volume 81| Part 4| August 2025|

https://doi.org/10.1107/S2052520625004779

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