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arxiv: 2606.17842 · v1 · pith:OGNIMUMAnew · submitted 2026-06-16 · ❄️ cond-mat.str-el · cond-mat.mtrl-sci

The antiferromagnetic transition in the frustrated bixbyite β-Fe₂O₃ magnet

Chenjun Tang (1) , Ond\v{r}ej Malina (2 , 3) , Ji\v{r}i Tu\v{c}ek (4) , Francois Fauth (5) , Marti Gich (1) , Jose Luis Garcia-Mu\~noz (1) ((1) Institut de Ci\`encia de Materials de Barcelona (ICMAB-CSIC). Cerdanyola del Vall\`es. Spain. , (2) Regional Centre of Advanced Technologies
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Materials. CATRIN. Olomouc. Czech Republic. (3) Nanotechnology Centre-CEET VSB-TUO. Ostrava. Poruba Czech Republic. (4) Research Centre of FEEI-UPCE. Pardubice. Czech Republic. (5) CELLS-ALBA Synchrotron. Cerdanyola del Vall\`es (Barcelona). Spain)
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Pith reviewed 2026-06-26 22:50 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mtrl-sci
keywords antiferromagnetic transitionbeta-Fe2O3bixbyite structuregeometric frustrationnoncollinear magnetismneutron diffractionirreducible representation
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The pith

The antiferromagnetic transition in β-Fe₂O₃ proceeds via activation of the mH₁⁺ irrep at the H-point with antitranslation symmetry, yielding a primitive magnetic cell of non-polar type-IV symmetry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the magnetic structure that appears below the Néel temperature in β-Fe₂O₃ using diffraction data. It shows that a noncollinear antiferromagnetic order sets in through a specific irreducible representation at the wavevector (1,1,1) accompanied by an antitranslation, resulting in two interpenetrating cubic subcells with inverted moments. This arrangement makes every iron-oxygen-iron exchange antiferromagnetic and generates substantial geometric frustration quantified by an index of about 7.6, which is high for binary oxides. Readers interested in frustrated magnetism would see how the bixbyite lattice supports noncollinear order and potential functional elements like Ising-like ions in hexagonal rings.

Core claim

A noncollinear antiferromagnetic structure sets in abruptly via activation of irrep mH₁⁺ at the H-point [k=(1,1,1)] together with antitranslation (1'|½,½,½). Below T_N, the magnetic cell becomes primitive (P_I a3̄), yielding two interpenetrating primitive cubic subcells with inverted moments and non-polar type-IV symmetry. All Fe³⁺-O-Fe³⁺ exchanges are antiferromagnetic, and the bixbyite structure promotes geometric frustration and noncollinear magnetism through coexisting magnetic sublattices with distinct symmetries and easy axes.

What carries the argument

The mH₁⁺ irreducible representation at the H-point combined with the antitranslation symmetry operation, which determines the noncollinear spin arrangement and reduces the magnetic cell to primitive symmetry.

If this is right

  • All Fe³⁺-O-Fe³⁺ superexchange interactions are antiferromagnetic.
  • The magnetic structure has non-polar type-IV symmetry.
  • The frustration index reaches f ≃ 7.6, among the highest for binary magnetic oxides.
  • Distorted Fe2O₆ octahedra form hexagonal rings in {111} planes that could host switchable magnetic states via central Fe1 ions with Ising-like anisotropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The high frustration may imply degenerate ground states or slow relaxation that could be tested with low-temperature susceptibility or muon spin rotation measurements.
  • Analogous noncollinear structures and frustration might occur in other bixbyite compounds containing multiple distinct magnetic sublattices.
  • The identified symmetry and anisotropy features could be used to engineer magnetoelectric coupling or tunable magnetic switching in related oxide lattices.

Load-bearing premise

The neutron and synchrotron diffraction intensities uniquely select the mH₁⁺ irrep and the associated antitranslation symmetry, with no significant contribution from other irreps or magnetic domains that could produce equivalent or better fits to the same data.

What would settle it

A different magnetic structure model that fits the observed neutron and synchrotron diffraction intensities at least as well as mH₁⁺ plus the antitranslation, or the detection of additional Bragg peaks inconsistent with the proposed propagation vector and symmetry.

Figures

Figures reproduced from arXiv: 2606.17842 by (2) Regional Centre of Advanced Technologies, 3), (3) Nanotechnology Centre-CEET VSB-TUO. Ostrava. Poruba, (4) Research Centre of FEEI-UPCE. Pardubice. Czech Republic., (5) CELLS-ALBA Synchrotron. Cerdanyola del Vall\`es (Barcelona). Spain), Chenjun Tang (1), Czech Republic., Francois Fauth (5), Ji\v{r}i Tu\v{c}ek (4), Jose Luis Garcia-Mu\~noz (1) ((1) Institut de Ci\`encia de Materials de Barcelona (ICMAB-CSIC). Cerdanyola del Vall\`es. Spain., Marti Gich (1), Materials. CATRIN. Olomouc. Czech Republic., Ond\v{r}ej Malina (2.

Figure 1
Figure 1. Figure 1: Rietveld refinement (black line) of the room temperature D1B@ILL (λ = 1.28 Å) neutron diffraction pattern (red circles) of -Fe2O3 at 300 K (Ia-3). Structural parameters are given in Table I [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Room-temperature crystal structure of -Fe2O3. Projections of the Ia-3 structure refined at 300 K for different unit-cell orientations. The two crystallographically distinct Fe1 (red) and Fe2 (green) sites, together with the O atoms (blue), are shown in the figure. For each orientation, three representations are provided: (a) the complete structure, (b) only the regular Fe1O6 octahedra (red), and (c) only … view at source ↗
Figure 3
Figure 3. Figure 3: Magnetic susceptibility. (a) FC and ZFC susceptibility curves of the -Fe2O3 sample (H=2 kOe). (b) Inverse susceptibility and Curie-Weiss fit above the antiferromagnetic transition. Inset: extrapolated Curie temperature (C = – 847 K) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Contour maps of -Fe2O3 showing the 2θ-T projection of the temperature evolution of the neutron diffraction intensity (D1B@ILL, λ = 1.28 Å). The low-Q region highlights the onset and temperature evolution of the main magnetic reflections below the antiferromagnetic transition [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the magnetic order. Temperature evolution of the integrated neutron intensity of the {120} magnetic reflection [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rietveld refinement (black line) of the neutron diffraction pattern measured at 10 K (red circles). An expanded view of the low-Q region is also shown, highlighting the main magnetic reflections associated with the magnetic ordering. The 10 K pattern was refined using the primitive cubic PIa-3 symmetry and the magnetic propagation vector k=(1,1,1). As reference magnetic atoms we can consider Fe1 (8b) at (0… view at source ↗
Figure 7
Figure 7. Figure 7: Antiferromagnetic order in -Fe2O3 (MSG PIa-3 [No. 205.36], with k=(1,1,1)). Several projections of the magnetic structure are provided for clarity: (a) [001], (b) [111], and (c) [110]. The left column shows all magnetic atoms. For clarity, the center and right columns display only the magnetic moments at the Fe1 (red) and Fe2 (green) sites, respectively. Note that the magnetic moments of the Fe1 atoms loc… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of the unit-cell across the magnetic transition, obtained from synchrotron and neutron diffraction data. Inset: derivative da/dT (synchrotron XRPD). This β–Fe2O3 compound can be considered of Heisenberg type. In a previous Mössbauer investigation of this powder iron oxide, the temperature dependence of the multiplet components provided a critical exponent β of the antiferromagnetic phase transiti… view at source ↗
Figure 9
Figure 9. Figure 9: Magnetic order in the layer of Fe1 and Fe2 octahedra perpendicular to [1 1 -1]. In this plane, distorted Fe2O₆ octahedra (green) form a network of hexagonal rings interconnected by triangular units. Each Fe2 atom is part of a hexagonal ring oriented perpendicular to one of the ⟨111⟩ directions. Likewise, each Fe1 atom is located at the center of one hexagonal ring [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Magnetic order and main exchange interactions. Intra-layer: (a) the Fe2 hexagonal rings; (b) the Fe2 triangles. Inter-layer: (c) Fe1–Fe2 bonds between successive [111] layers. (left) Front view. (center) Front view showing Fe1-O-Fe2 exchange bonds (for clarity, the in-plane Fe2 hexagonal ring has been omitted). Fe1, Fe2 and O atoms are shown in red, green and blue colors, respectively. The exchange intera… view at source ↗
read the original abstract

Although Fe$_2$O$_3$ compounds are among the most extensively studied transition-metal oxides, the magnetic properties of $\beta$-Fe$_2$O$_3$ remain poorly characterized. Using neutron and synchrotron X-ray diffraction, we investigate the temperature-driven magnetic transition in $\beta$-Fe$_2$O$_3$. A noncollinear antiferromagnetic structure sets in abruptly via activation of irrep $mH_1^{+}$ at the H-point [$\mathbf{k}=(1,1,1)$] together with antitranslation $(1'|\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$. Below $T_{\mathrm{N}}$, the magnetic cell becomes primitive $(P_I a \bar{3})$, yielding two interpenetrating primitive cubic subcells with inverted moments and non-polar type-IV symmetry. All Fe$^{3+}$-O-Fe$^{3+}$ exchanges are antiferromagnetic, and the bixbyite structure promotes geometric frustration and noncollinear magnetism through coexisting magnetic sublattices with distinct symmetries and easy axes. Its frustration index $f \simeq 7.6$ is among the highest reported for binary magnetic oxides. In $\{111\}$ planes, distorted Fe2O$_6$ octahedra form hexagonal rings interconnected by triangular units. Notably, hexagonal Fe2 rings host a central Fe1 ion with strong Ising-like anisotropy, which could act as a switching element for the rings' magnetic state. These features point to routes for functional design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses neutron and synchrotron X-ray diffraction to characterize the antiferromagnetic transition in β-Fe₂O₃. It reports that a noncollinear AFM structure onsets abruptly below T_N via activation of the mH₁⁺ irrep at the H-point k=(1,1,1) together with the antitranslation (1'|½,½,½), producing a primitive magnetic cell (P_I a3̄) consisting of two interpenetrating cubic subcells with inverted moments and type-IV symmetry. All Fe³⁺-O-Fe³⁺ superexchanges are antiferromagnetic; the bixbyite lattice yields geometric frustration (f ≃ 7.6) with distinct Fe1/Fe2 sublattice symmetries, easy axes, and {111} hexagonal rings that may be switched by central Fe1 Ising anisotropy.

Significance. If the irrep assignment holds, the work supplies a well-documented example of high-frustration noncollinear order in a binary oxide, with one of the largest reported f values, and identifies concrete structural motifs (interpenetrating subcells, ring-triangular units) that could guide functional design. The use of combined diffraction plus standard symmetry analysis is a standard but useful contribution to the catalog of frustrated magnets.

major comments (2)
  1. [Magnetic structure section] Magnetic structure section: the claim that intensities uniquely select mH₁⁺ plus the stated antitranslation (producing P_I a3̄) is load-bearing for all subsequent statements about sublattice symmetries and frustration. The manuscript must supply explicit goodness-of-fit metrics (χ², R_wp, or equivalent) comparing this model against other H-point irreps, linear combinations, and domain-averaged alternatives; without those comparisons the uniqueness step is not demonstrated.
  2. [Results on temperature dependence] Results on temperature dependence: the abruptness of the transition and the reported frustration index f ≃ 7.6 rely on the chosen propagation vector and basis; if alternative models fit the same Bragg intensities within error, both the cell description and the numerical value of f become model-dependent.
minor comments (2)
  1. Provide the precise T_N value extracted from the diffraction data together with any observed thermal hysteresis.
  2. Clarify whether the synchrotron X-ray data were used only for nuclear structure or also contributed to magnetic intensity constraints.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for quantitative support of the magnetic structure assignment. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Magnetic structure section] the claim that intensities uniquely select mH₁⁺ plus the stated antitranslation (producing P_I a3̄) is load-bearing for all subsequent statements about sublattice symmetries and frustration. The manuscript must supply explicit goodness-of-fit metrics (χ², R_wp, or equivalent) comparing this model against other H-point irreps, linear combinations, and domain-averaged alternatives; without those comparisons the uniqueness step is not demonstrated.

    Authors: We agree that explicit quantitative comparisons are required to substantiate the uniqueness of the mH₁⁺ assignment. In the revised manuscript we will add a dedicated paragraph and table in the magnetic structure section that reports χ² and R_wp values for the mH₁⁺ + antitranslation model versus the other H-point irreps (mH₂⁺, mH₃⁺), their linear combinations, and domain-averaged alternatives. These fits will be performed on the same neutron data set used in the original analysis. revision: yes

  2. Referee: [Results on temperature dependence] the abruptness of the transition and the reported frustration index f ≃ 7.6 rely on the chosen propagation vector and basis; if alternative models fit the same Bragg intensities within error, both the cell description and the numerical value of f become model-dependent.

    Authors: The frustration index f ≃ 7.6 is obtained from the ratio |Θ_CW|/T_N, where Θ_CW is determined solely from bulk susceptibility data via a Curie-Weiss fit above T_N; this quantity is therefore independent of any magnetic structure model. The abruptness of the transition is directly observed in the temperature dependence of the integrated intensity of the magnetic Bragg peaks, which appear discontinuously below T_N. Nevertheless, the detailed description of the magnetic cell and sublattice symmetries does depend on the irrep choice. The goodness-of-fit comparisons we will add (see response to the first comment) will demonstrate that alternative H-point models do not reproduce the observed intensities within error, thereby removing model dependence from the cell description. revision: partial

Circularity Check

0 steps flagged

No circularity: experimental structure solution from diffraction data

full rationale

The paper determines the magnetic structure (irrep mH₁⁺ plus antitranslation) directly from neutron and synchrotron intensities via standard symmetry analysis. No derivation chain reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or ansatz by construction. The frustration index is a standard empirical ratio computed from measured T_N and θ_CW. All load-bearing steps are external data fits, not internal redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard crystallographic tools for magnetic structure determination but introduces no new free parameters, ad-hoc axioms, or postulated entities beyond the measured diffraction intensities.

axioms (1)
  • standard math Standard representation theory of magnetic space groups can be used to label the observed ordering as irrep mH₁⁺ plus antitranslation.
    Invoked to assign the symmetry of the magnetic structure from the H-point propagation vector.

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Works this paper leans on

40 extracted references

  1. [1]

    C. J. Tartaj, P., Morales, M.P., Gonzalez-Carreño, T., Veintemillas-Verdaguer, S. and Serna, The Iron Oxides Strike Back: From Biomedical Applications to Energy Storage Devices and Photoelectrochemical Water Splitting, Adv. Mater. 23, 5243 (2011)

    2011

  2. [2]

    Pullar, Hexagonal Ferrites: A Review of the Synthesis, Properties and Applications of Hexaferrite Ceramics, Prog

    Robert C. Pullar, Hexagonal Ferrites: A Review of the Synthesis, Properties and Applications of Hexaferrite Ceramics, Prog. Mater. Sci. 57, 1191 (2012)

    2012

  3. [3]

    J. H. Jeong, H. H. Kim, J. Lee, and Y. Baek, Epsilon Iron Oxide ( ε-Fe2O3 ) as an Electromagnetic Functional Material : Properties , Synthesis , and Applications, 31, 465 (2024)

    2024

  4. [4]

    Zboril, M

    R. Zboril, M. Mashlan, and D. Petridis, Iron ( III ) Oxides from Thermal Processes s Synthesis , Chem. Mater. 14, 969 (2002)

    2002

  5. [5]

    Danno, D

    T. Danno, D. Nakatsuka, Y. Kusano, H. Asaoka, M. Nakanishi, T. Fujii, Y. Ikeda, and J. Takada, Crystal Structure of β-Fe2O3 and Topotactic Phase Transformation to α- Fe2O3, Cryst. Growth Des. 13, 770 (2013)

    2013

  6. [6]

    Malina et al., Magnetic Ground State of Nanosized β-Fe2O3 and Its Remarkable Electronic Features, RSC Adv

    O. Malina et al., Magnetic Ground State of Nanosized β-Fe2O3 and Its Remarkable Electronic Features, RSC Adv. 5, 49719 (2015)

    2015

  7. [7]

    E. R. Bauminger, L. Ben-Dor, I. Felner, E. Fischbein, I. Nowik, and S. Ofer, Mössbauer Effect Studies of β-Fe2O3, Phys. B+C 86–88, 910 (1977)

    1977

  8. [8]

    Sakurai, A

    S. Sakurai, A. Namai, K. Hashimoto, and S. I. Ohkoshi, First Observation of Phase Transformation of All Four Fe2O 3 Phases (γ → ε → β → α-Phase), J. Am. Chem. 21 Soc. 131, 18299 (2009)

    2009

  9. [9]

    Tuček, L

    J. Tuček, L. Machala, S. Ono, A. Namai, M. Yoshikiyo, K. Imoto, H. Tokoro, S. I. Ohkoshi, and R. Zbořil, Zeta-Fe2O3 - A New Stable Polymorph in Iron(III) Oxide Family, Sci. Rep. 5, 15091 (2015)

    2015

  10. [10]

    J. L. García-Muñoz, A. Romaguera, F. Fauth, J. Nogués, and M. Gich, Unveiling a New High-Temperature Ordered Magnetic Phase in ϵ-Fe2O3, Chem. Mater. 29, 9705 (2017)

    2017

  11. [11]

    MacHala, J

    L. MacHala, J. Tuček, and R. Zbořil, Polymorphous Transformations of Nanometric Iron(III) Oxide: A Review, Chem. Mater. 23, 3255 (2011)

    2011

  12. [13]

    Grau-crespo, A

    R. Grau-crespo, A. Y. Al-baitai, and I. Saadoune, Vacancy Ordering and Electronic Structure of γ -Fe 2 O 3 ( Maghemite ): A Theoretical Investigation, J. Phys. Condens. Matter 22, 255401 (2010)

    2010

  13. [14]

    Lebrun et al., Anisotropies and Magnetic Phase Transitions in Insulating Antiferromagnets Determined by a Spin-Hall Magnetoresistance Probe, Commun

    R. Lebrun et al., Anisotropies and Magnetic Phase Transitions in Insulating Antiferromagnets Determined by a Spin-Hall Magnetoresistance Probe, Commun. Phys. 2, 50 (2019)

    2019

  14. [15]

    Lebrun, A

    R. Lebrun, A. Ross, O. Gomonay, V. Baltz, U. Ebels, A. L. Barra, A. Qaiumzadeh, A. Brataas, J. Sinova, and M. Kläui, Long-Distance Spin-Transport across the Morin Phase Transition up to Room Temperature in Ultra-Low Damping Single Crystals of the Antiferromagnet α-Fe2O3, Nat Commun 11, 6332 (2020)

    2020

  15. [16]

    Greaves, A Powder Neutron Diffraction Investigation of Vacancy Ordering and Covalence in γ-Fe2O3, J

    C. Greaves, A Powder Neutron Diffraction Investigation of Vacancy Ordering and Covalence in γ-Fe2O3, J. Solid State Chem. 49, 325 (1983)

    1983

  16. [17]

    W. E. Henry and M. J. Boehm, Intradomain Magnetic Saturation and Magnetic Structure of γ-Fe2O3, Phys. Rev. 101, 1253 (1956)

    1956

  17. [18]

    G. A. Ferguson and M. Bass, Magnetic Structure and Vacancy Distribution in γ-Fe203 by Neutron Diffraction, Phys. Rev. 112, 1130 (1958)

    1958

  18. [19]

    Tronc, C

    E. Tronc, C. Chanéac, and J. P. Jolivet, Structural and Magnetic Characterization of ε- Fe2O3, J. Solid State Chem. 139, 93 (1998)

    1998

  19. [20]

    Gich et al., High- and Low-Temperature Crystal and Magnetic Structures of ε-Fe 2O3 and Their Correlation to Its Magnetic Properties, Chem

    M. Gich et al., High- and Low-Temperature Crystal and Magnetic Structures of ε-Fe 2O3 and Their Correlation to Its Magnetic Properties, Chem. Mater. 18, 3889 (2006)

    2006

  20. [21]

    Bonnevie Svendsen, Beta-Fe2O3- Eine Neue Eisen(Lll) Oxyd-Struktur, Die Natur- Wissenschaften 542 (1958)

    M. Bonnevie Svendsen, Beta-Fe2O3- Eine Neue Eisen(Lll) Oxyd-Struktur, Die Natur- Wissenschaften 542 (1958)

    1958

  21. [22]

    Wiarda and G

    D. Wiarda and G. Weyer, Mössbauer Investigations of the Antiferromagnetic Phase in 22 the Metastable β–Fe2O3, Int. J. Mod. Phys. B 7, 353 (1993)

    1993

  22. [23]

    Fauth, R

    F. Fauth, R. Boer, F. Gil-Ortiz, C. Popescu, O. Vallcorba, I. Peral, D. Fullà, J. Benach, and J. Juanhuix, The Crystallography Stations at the Alba Synchrotron, Eur. Phys. J. Plus 130, 160 (2015)

    2015

  23. [24]

    J. M. Perez-Mato, S. V. Gallego, E. S. Tasci, L. Elcoro, G. de la Flor, and M. I. Aroyo, Symmetry-Based Computational Tools for Magnetic Crystallography, Annu. Rev. Mater. Res. 45, 217 (2015)

    2015

  24. [25]

    J. M. Perez-Mato, D. Orobengoa, and M. I. Aroyo, Mode Crystallography of Distorted Structures, Acta Cryst. A66, 558 (2010)

    2010

  25. [26]

    Elcoro, B

    L. Elcoro, B. J. Wieder, B. A. Bernevig, and B. Bradlyn, Magnetic Topological Quantum Chemistry, Nat. Commun. 12, 5965 (2021)

    2021

  26. [27]

    J. M. Perez-Mato, S. V. Gallego, L. Elcoro, E. Tasci, and M. I. Aroyo, Symmetry Conditions for Type II Multiferroicity in Commensurate Magnetic Structures, J. Phys. Condens. Matter 28, 286001 (2016)

    2016

  27. [28]

    B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, ISODISPLACE: A Web- Based Tool for Exploring Structural Distortions, J. Appl. Crystallogr. 39, 607 (2006)

    2006

  28. [29]

    Rodríguez-Carvajal, Recent Advances in Magnetic Structure Determination by Neutron Powder Diffraction, Phys

    J. Rodríguez-Carvajal, Recent Advances in Magnetic Structure Determination by Neutron Powder Diffraction, Phys. B Phys. Condens. Matter 192, 55 (1993)

    1993

  29. [30]

    Momma and F

    K. Momma and F. Izumi, VESTA 3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data, J. Appl. Crystallogr. 44, 1272 (2011)

    2011

  30. [31]

    Supplemental Material, (2026)

    2026

  31. [32]

    Ikeda, M

    Y. Ikeda, M. Takano, and Y. Barmo, Formation Mechanism of Needle-like α-Fe₂O₃ Particles Grown Along the c Axis and Characterization of Precursorily Formed β- Fe₂O₃, Bull. Inst. Chem. Res. 64, 249 (1986)

    1986

  32. [33]

    A. P. Ramirez, Strongly Geometrically Frustrated Magnets, Annu. Rev. Mater. Sci. 24, 453 (1994)

    1994

  33. [34]

    J. M. D. \thinspac. Coey, Magnetic Oxides and Other Compounds, in Handbook of Magnetism and Magnetic Materials, edited by M. Coey and S. Parkin (Springer International Publishing, Cham, 2020), pp. 1–76

    2020

  34. [35]

    V Belov, N

    N. V Belov, N. N. Neronova, and T. S. Smirnova, Shubnikov Groups, Kristallografiya 2, 315 (1957)

    1957

  35. [36]

    V Gallego, J

    S. V Gallego, J. Etxebarria, L. Elcoro, E. S. Tasci, and J. M. Perez-Mato, Automatic Calculation of Symmetry-Adapted Tensors in Magnetic and Non-Magnetic Materials: A New Tool of the Bilbao Crystallographic Server, Acta Crystallogr. Sect. A 75, 438 23 (2019)

    2019

  36. [37]

    Kanamori, Superexchange Interaction and Symmetry Properties of Electron Orbitals, J

    J. Kanamori, Superexchange Interaction and Symmetry Properties of Electron Orbitals, J. Phys. Chem. Solids 10, 87 (1959)

    1959

  37. [38]

    J. B. Goodenough, Magnetism and the Chemical Bond ((New York: Interscience (Wiley-Interscience)), 1963)

    1963

  38. [39]

    Motida and S

    K. Motida and S. Miyahara, On the 90° Exchange Interaction between Cations (Cr3+, Mn2+, Fe3+ and Ni2+) in Oxides, J. Phys. Soc. Jpn. 28, 1188 (1970)

    1970

  39. [40]

    Ohkoshi, A

    S. Ohkoshi, A. Namai, and S. Sakurai, The Origin of Ferromagnetism in ε-Fe2O3 and ε-GaxFe2-XO3 Nanomagnets, Society 11235 (2009)

    2009

  40. [41]

    Knížek, P

    K. Knížek, P. Novák, and Z. Jirák, Exchange Interactions in ϵ-Fe2O3: GGA + U Calculations, J. Phys. Condens. Matter 33, 155502 (2021). 24 Table I . Refined crystal structure (cell parameters, atomic coordinates and isotropic displacement parameters (B )) for β-Fe₂O₃ obtained from Rietveld refinement of neutron powder diffraction data at 300 K and 10 K (sp...

    2021