AB2X4 spinel structures: similarity and difference between the centrosymmetric, Fd-3m, and non-centrosymmetric, F4132, space groups

Alla Arakcheeva; Arnaud Magrez; Gervais Chapuis

arxiv: 2605.24198 · v1 · pith:5IAN2XJSnew · submitted 2026-05-22 · ❄️ cond-mat.mtrl-sci

AB2X4 spinel structures: similarity and difference between the centrosymmetric, Fd-3m, and non-centrosymmetric, F4132, space groups

Alla Arakcheeva , Arnaud Magrez , Gervais Chapuis This is my paper

Pith reviewed 2026-06-30 14:57 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords spinel structuresspace groupsFd-3mF4132X-ray diffractioncentrosymmetryanharmonic displacementsAB2X4

0 comments

The pith

The centrosymmetric Fd-3m and non-centrosymmetric F4132 space groups are equivalent for X-ray structure determination of AB2X4 spinels.

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

Many spinel compounds are traditionally assigned to the centrosymmetric Fd-3m space group, yet some display physical properties incompatible with inversion symmetry. The paper shows that the non-centrosymmetric F4132 space group produces identical results to Fd-3m in structure determination and refinement from X-ray diffraction data when standard harmonic displacement models are used. This equivalence allows loss of centrosymmetry without any atomic displacements from the positions in Fd-3m. Distinguishing the two space groups requires an anharmonic model of atomic displacements, provided that model is consistent with the data. The work addresses the mismatch between reported structures and observed properties in these materials.

Core claim

The centrosymmetric Fd-3m and non-centrosymmetric F4132 space groups are equivalent for structure determination and refinement based on X-ray diffraction data. The loss of inversion symmetry can therefore occur without atomic displacements. Only the use of an anharmonic model of atomic displacements can distinguish these space groups if such a model is consistent with experiment.

What carries the argument

Equivalence of the Fd-3m and F4132 space groups under harmonic atomic displacement models during X-ray refinement

If this is right

Spinel structures can lose centrosymmetry without requiring atomic displacements from Fd-3m positions.
Compounds traditionally refined in Fd-3m may in fact belong to F4132.
Physical properties that rule out inversion symmetry can be explained by F4132 assignment.
Harmonic models alone cannot resolve the symmetry choice in these compounds.

Where Pith is reading between the lines

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

Refinement protocols for spinels should routinely compare both space groups once anharmonic terms are included.
The equivalence may prompt checks of other reported structures where properties indicate broken symmetry.
Similar hidden equivalences could exist between other pairs of space groups in different structure types.

Load-bearing premise

Standard harmonic models of atomic displacements are used in the X-ray structure refinement.

What would settle it

A dataset in which an anharmonic displacement model produces a statistically significant improvement in refinement fit for F4132 over Fd-3m would show the space groups are distinguishable.

read the original abstract

Many compounds belonging to the spinel AB2X4 structure play an important role due to their wide range of practical applications. Most of them are traditionally assigned to the centrosymmetric space group Fd-3m. However, the physical properties of some spinels are incompatible with centrosymmetry. This discrepancy is often accounted for by reducing the symmetry to the non-centrosymmetric space group F-43m, allowing thus small atomic displacements from their original position in Fd-3m. In this work, we demonstrate that the loss of the inversion symmetry can occur without any atomic displacements, since the centrosymmetric Fd-3m and non-centrosymmetric F4132 space groups are equivalent for structure determination and refinement based on X-ray diffraction data. If consistent with experiment, only the use of an anharmonic model of atomic displacements can distinguish these space groups. This study aims to clarify certain misconceptions regarding the structural symmetry and physical properties of spinel type compounds.

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.

Desk Editor's Note private letter to a colleague

The paper claims Fd-3m and F4132 are equivalent for X-ray refinement of spinels but supplies no explicit structure-factor check to support it.

read the letter

The main takeaway is that the authors argue the centrosymmetric Fd-3m and non-centrosymmetric F4132 space groups produce identical X-ray diffraction patterns for standard spinel sites when only harmonic displacements are used. This would let researchers assign non-centrosymmetric symmetry without atomic shifts, with anharmonic terms as the sole distinguisher.

They correctly flag that some spinel properties clash with inversion symmetry and that the common fallback to F-43m plus displacements may not be the only option. The discussion of how refinement practices can mask this choice is the useful part.

The soft spot is the missing step. Fd-3m requires h+k+l=4n while F4132 allows unmixed indices, so all-odd reflections are permitted in the latter. For the two descriptions to be indistinguishable, the structure factor on every Fd-3m-forbidden reflection must be identically zero in the F4132 model with the usual 8a/16d/32e positions. The abstract states equivalence without showing the calculation or a numerical check. If the full paper contains that derivation it would fix the gap; otherwise the central claim rests on assertion rather than demonstration.

This is for crystallographers and materials scientists working on AB2X4 compounds. A reader already wrestling with space-group assignment in spinels could get value from the framing. It deserves peer review because the topic is practical and the claim, if backed by the missing calculation, would affect how people interpret data on batteries and magnets.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that for AB2X4 spinel compounds the centrosymmetric space group Fd-3m and the non-centrosymmetric space group F4132 are equivalent for X-ray diffraction structure determination and refinement (no atomic displacements required), and that only an anharmonic model of atomic displacements can distinguish the two descriptions.

Significance. If the equivalence is demonstrated, the result would affect how spinel structures are assigned and refined from XRD data, potentially resolving apparent conflicts between observed physical properties and assumed centrosymmetry without invoking displacements or lower-symmetry groups such as F-43m.

major comments (1)

[Abstract] Abstract (final paragraph) and main text: the assertion that the two space groups are equivalent under harmonic ADPs requires an explicit demonstration that the structure factor F(hkl) is identically zero for every reflection violating the Fd-3m condition h+k+l=4n when the standard 8a/16d/32e sites are occupied in the F4132 description; no derivation, algebraic identity, or numerical verification of this cancellation is supplied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the claimed equivalence. We address the major comment below and agree that the manuscript requires revision to include the requested demonstration.

read point-by-point responses

Referee: [Abstract] Abstract (final paragraph) and main text: the assertion that the two space groups are equivalent under harmonic ADPs requires an explicit demonstration that the structure factor F(hkl) is identically zero for every reflection violating the Fd-3m condition h+k+l=4n when the standard 8a/16d/32e sites are occupied in the F4132 description; no derivation, algebraic identity, or numerical verification of this cancellation is supplied.

Authors: We agree with the referee that an explicit demonstration is necessary to support the assertion of equivalence under harmonic ADPs. The manuscript currently states the result without providing the algebraic identity or numerical check. In the revised version we will add a short derivation (or equivalent numerical verification) showing that the structure factor contributions from the symmetry-related 8a, 16d and 32e sites in F4132 cancel for all reflections with h+k+l eq 4n when only harmonic displacement parameters are used. This addition will be placed in the main text immediately after the statement of equivalence. revision: yes

Circularity Check

0 steps flagged

No circularity detected; equivalence claim rests on standard diffraction theory without self-referential reduction

full rationale

The paper's central claim is that Fd-3m and F4132 are equivalent for X-ray structure determination/refinement under harmonic ADPs, with anharmonic terms as the sole distinguisher. No quoted derivation reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain. The assertion is presented as a demonstration from crystallographic principles rather than an input renamed as output. This matches the default case of a self-contained argument against external benchmarks, warranting score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no free parameters, ad-hoc axioms, or invented entities are identifiable. The work relies on standard assumptions of X-ray crystallography (harmonic displacements, kinematic diffraction) that are not enumerated here.

pith-pipeline@v0.9.1-grok · 5723 in / 1061 out tokens · 44262 ms · 2026-06-30T14:57:30.730965+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references

[1]

Determining the correct space group symmetry is essential for predicting, understanding and tuning their physical properties

Introduction The spinel structure type is commonly adopted by many AB2X4 compounds, including those with important practical applications (Srikala, 2024; Wang et al., 2023; Rafi et al., 2025; Arshad et al., 2024; He et al., 2023; Song et al., 2023; Shan et al., 2023; Xu et al., 2023; Tsurkan et al., 2021; Narang & Pubby, 2021; Szablowski et al., 2025; and...

2024
[2]

translationengleiche

among others. Furthermore, two reports refer to the non-centrosymmetric space group F4132 for AB2X4 compounds. One of these concerns ZnFe2O4 (Dronova et al., 2022), where the observed h00 reflection condition (h ≠ 4n with integer n) is inconsistent with this space group. The other case involves LiMn1.5Ni0.5O4, in which A = Li, B2 = (Mn1.5Ni0.5) and X4 = O...

2022
[3]

Thus, based on structural parameters alone, no distinction can be made between the Fd3̄m and F4132 space groups. In terms of lattice complexes (Fischer & Koch, 2002), the crystal structures are also identical in both cases: A – 8a 4̅3m Fd3̄m a D; B – 16d .3̄m Fd3̄m c T; X – 32e .3m Fd3̄m e ..2D4xxx. 2.2. Reflection conditions The general reflection condit...

2002
[4]

Here again we would expect small shifts in the atomic parameters

to P4132 with a maximal subgroup of index k4 (k for klassengleich). Here again we would expect small shifts in the atomic parameters. In reality the large difference in the atomic coordinates results from the arbitrariness of the position of the origin of P4132 described in the International Tables for Crystallography. Thus, while comparing different stru...

2002
[5]

Space Group Symmetry

Table 1 Comparison of the Fd3̅m and F4132 space groups describing the structure of spinel AB2X4. Characteristics of space groups are taken from the International Tables for Crystallography (2002), Vo l u m e A “Space Group Symmetry”, Fifth edition). Space group Fd3̄m (no. 227); origin choice 1 at 4̅3m F4132 (no.210); origin choice at

2002
[6]

& (-2 -4 -8) 99.7 and 72.6 10.7 51.6 and 47.6 8.1 Table 5 Atomic coordinates of Li(Mn,Ni)2O4 in Fd3̅m and F4132 space groups after Amin et al. (2020). Crystal system, space group Cubic, Fd3̅m (no

2020
[7]

CRSII5-171003)

in F4132 Origin choice 2 at .3̄m in Fd3̄m, accordingly at .32 in F4132 Origin shift (000) (½ ½ ½) (1/8 1/8 1/8) (1/8 1/8 1/8) + (½ ½ ½) = (5/8 5/8 5/8) Coordinate shift (000) (-½ -½ -½) (-1/8 -1/8 -1/8) (-5/8 -5/8 -5/8) A 8a: (0,0,0); (¼,¼,¼) 8b: (½,½,½); (¾,¾,¾) 8a: (7/8,7/8,7/8); (1/8,1/8,1/8) 8b: (3/8,3/8,3/8); (5/8,5/8,5/8) B 16d: (5/8,5/8,5/8) 16c: (...

2020
[8]

& Verma, M

Srikala, D., Kaushik, S.D. & Verma, M. (2024). Physics of the Solid State, 66, 327-340. Szablowski, L., Wojcik, M. & Dybinski, O. (2025). Energy, 316, 134540. Trueblood, K.N., Bürgi, H.-B., Burzlaff, H., Dunitz, D., Gramaccioli, C.M., Schulz, H.H. & Abrahams, S.C. (1996). Acta Cryst., A52, 770-781. Tsurkan, V ., Krug von Nidda, H.-A., Delsenhofer, J., Lun...

2024

[1] [1]

Determining the correct space group symmetry is essential for predicting, understanding and tuning their physical properties

Introduction The spinel structure type is commonly adopted by many AB2X4 compounds, including those with important practical applications (Srikala, 2024; Wang et al., 2023; Rafi et al., 2025; Arshad et al., 2024; He et al., 2023; Song et al., 2023; Shan et al., 2023; Xu et al., 2023; Tsurkan et al., 2021; Narang & Pubby, 2021; Szablowski et al., 2025; and...

2024

[2] [2]

translationengleiche

among others. Furthermore, two reports refer to the non-centrosymmetric space group F4132 for AB2X4 compounds. One of these concerns ZnFe2O4 (Dronova et al., 2022), where the observed h00 reflection condition (h ≠ 4n with integer n) is inconsistent with this space group. The other case involves LiMn1.5Ni0.5O4, in which A = Li, B2 = (Mn1.5Ni0.5) and X4 = O...

2022

[3] [3]

Thus, based on structural parameters alone, no distinction can be made between the Fd3̄m and F4132 space groups. In terms of lattice complexes (Fischer & Koch, 2002), the crystal structures are also identical in both cases: A – 8a 4̅3m Fd3̄m a D; B – 16d .3̄m Fd3̄m c T; X – 32e .3m Fd3̄m e ..2D4xxx. 2.2. Reflection conditions The general reflection condit...

2002

[4] [4]

Here again we would expect small shifts in the atomic parameters

to P4132 with a maximal subgroup of index k4 (k for klassengleich). Here again we would expect small shifts in the atomic parameters. In reality the large difference in the atomic coordinates results from the arbitrariness of the position of the origin of P4132 described in the International Tables for Crystallography. Thus, while comparing different stru...

2002

[5] [5]

Space Group Symmetry

Table 1 Comparison of the Fd3̅m and F4132 space groups describing the structure of spinel AB2X4. Characteristics of space groups are taken from the International Tables for Crystallography (2002), Vo l u m e A “Space Group Symmetry”, Fifth edition). Space group Fd3̄m (no. 227); origin choice 1 at 4̅3m F4132 (no.210); origin choice at

2002

[6] [6]

& (-2 -4 -8) 99.7 and 72.6 10.7 51.6 and 47.6 8.1 Table 5 Atomic coordinates of Li(Mn,Ni)2O4 in Fd3̅m and F4132 space groups after Amin et al. (2020). Crystal system, space group Cubic, Fd3̅m (no

2020

[7] [7]

CRSII5-171003)

in F4132 Origin choice 2 at .3̄m in Fd3̄m, accordingly at .32 in F4132 Origin shift (000) (½ ½ ½) (1/8 1/8 1/8) (1/8 1/8 1/8) + (½ ½ ½) = (5/8 5/8 5/8) Coordinate shift (000) (-½ -½ -½) (-1/8 -1/8 -1/8) (-5/8 -5/8 -5/8) A 8a: (0,0,0); (¼,¼,¼) 8b: (½,½,½); (¾,¾,¾) 8a: (7/8,7/8,7/8); (1/8,1/8,1/8) 8b: (3/8,3/8,3/8); (5/8,5/8,5/8) B 16d: (5/8,5/8,5/8) 16c: (...

2020

[8] [8]

& Verma, M

Srikala, D., Kaushik, S.D. & Verma, M. (2024). Physics of the Solid State, 66, 327-340. Szablowski, L., Wojcik, M. & Dybinski, O. (2025). Energy, 316, 134540. Trueblood, K.N., Bürgi, H.-B., Burzlaff, H., Dunitz, D., Gramaccioli, C.M., Schulz, H.H. & Abrahams, S.C. (1996). Acta Cryst., A52, 770-781. Tsurkan, V ., Krug von Nidda, H.-A., Delsenhofer, J., Lun...

2024