Three-dimensional checkerboard spin structure on a breathing pyrochlore lattice

Single crystals of ZnFe₂O₄ were grown from high-temperature molten solution of anhydrous borax [23]. The proportions of ZnO and Fe₂O₃ powders in the initial solution were finely tuned to make the molar ratio of Fe and Zn ions in the grown crystals stoichiometric (within $\pm 0.005$ of 2.000). Furthermore, by growing from different starting temperature, the level of inversion disorder of spinel oxide can be controlled [23].

The field-temperature $H$-$T$ magnetic phase diagram of ZnFe₂O₄ has been characterized by both heat capacity and magnetic susceptibility measurements. Three pieces of stoichiometric crystals in the clean limit were used; all were grown from 850 ^∘C to 750 ^∘C [23], including the one in Supplementary Fig. 1d of Ref. [23]. These samples have a Curie-Weiss temperature $T_{CW}=-25$ K [23]. The magnetization was measured using the vibrating sample magnetometer (VSM) of a Physical Property Measurement System (PPMS) (Dynacool-14T, Quantum Design, Inc.), with temperature varying from 2 K to 400 K and field up to 14 Tesla, applied on two pieces of crystal along the (1, 1, 0) and (1, 0, 0) directions. Heat capacity was measured on two crystals with field direction applied along the $(1,1,1)$ and $(1,0,0)$ directions respectively in a Dynacool-9T PPMS.

Structural and magnetic characterizations of stoichiometric ZnFe₂O₄ were carried out at two neutron elastic scattering beamlines: TOPAZ (BL-12) and CORELLI (BL-9) at the Spallation Neutron Source of Oak Ridge National Laboratory [29, 27, 30, 33, 28]. Both beamlines utilize unpolarized incident neutrons of continuous wavelength over slightly different ranges, from 0.6 to 3.5 Å for TOPAZ and from 0.8 to 2.4 Å for CORELLI. The short wavelength cutoff is set by the focusing limit of the high-flux beam guide, a key component that enables probing sub-mm sized samples. The long wavelength limit is set by the SNS pulse frequency (60 Hz) and the distance between the neutron moderator and the sample position at the beamline [30, 28].

Most of our neutron data were collected from one piece of stoichiometric single crystal that was measured at TOPAZ at 100 K and at CORELLI at 6 K. The single crystal was grown from 1000 ^∘C to 850 ^∘C, weighs 7.4 mg, and is 1.2 mm in diameter (Supplementary Fig. 1c of Ref. [23]). Its magnetic susceptibility was reported earlier [23] with a $T_{CW}=-20$ K. Although CORELLI is specialized for studying elastic diffuse scattering, the 1000 ^∘C-grown single crystal does not demonstrate magnetic diffuse scattering in the ordered phase but instead exhibits a pattern of sharp diffraction spots from both antiferromagnetism and the lattice (Fig. 1, and Ref. [23]). The comparison in Fig. 8 uses magnetic diffuse scattering data taken at CORELLI from a mosaic assembly of stoichiometric crystals (Supplementary Fig. 1e of Ref. [23]); these crystals were grown at temperatures from 1250 ^∘C to 950 ^∘C, resuliting in a high level of inversion disorder.

Despite their similarity of continuous wavelength neutron diffraction, data collections at TOPAZ and CORELLI have different emphases on the coverage of reciprocal space. At both beamlines, single crystal samples are placed at a series of fixed angular positions for a certain amount of time while diffraction happens for neutrons with wavelengths that satisfy Bragg’s law. TOPAZ covers a large 2$\theta$ range from 15^∘ to 155^∘, allowing the same reflection to be measured over a large wavelength range. At TOPAZ, 35 frames were collected for the current sample over a total flux of 94.3 C of proton charge, positioned at largely different $\omega,\chi,\phi$ angular positions. The choices of crystal angular placements emphasized the coverage of many families of reflections in reciprocal space but not necessarily continuous [29]. On the other hand, CORELLI provides a continuous coverage of the reciprocal space in order to reconstruct the diffuse scattering pattern without symmetrization. Typically, the crystal is rotated only along the vertical axis, and in our experiment by 1.5^∘ steps over the whole 360^∘ range for a total of 240 frames. The flux of 0.32 C of proton charge per frame we used at CORELLI is much shorter than that at TOPAZ. Additional details on the data reduction process are given in Appendix A and a discussion of the determination of wavelength-dependent extinction effects is provided in Appendix B.

The basic magnetic wave vector $Q=(1,0,1/2)$ (Fig. 1), expressed in reciprocal space units of the FCC lattice, doubles the magnetic unit cell of ZnFe₂O₄ to $a\times a\times 2a$. The wave vector $(1,0,1/2)$ is located at the high symmetry point $W$ on the first Brillouin zone boundary of the FCC lattice, where $W$ possesses the $\bar{4}2m$ point group symmetry. For the 32 Fe³⁺ spins in the magnetic unit cell, the structure of $Q$ separates them into four blocks of spins with parallel or antiparallel relationships, connected by translation vectors of $(a/2,a/2,0)$, $(0,0,a)$ and $(a/2,a/2,a)$ (Fig. 4) [9, 11]. With the parent pyrochlore lattice, the magnetic unit cell can be simultaneously inspected through both $(1,0,0.5)$ and $(0,1,0.5)$ wave vectors (Fig. 4), and the spin structure is isotropic within the $a$-$b$ plane. This construction in Fig. 4 indicates that time inversion $1^{\prime}$ is a good symmetry operation when it is combined with spatial translations, and the ionic lattice allows the nonmagnetic (grey) type of magnetic point group.

This doubling of the magnetic unit cell assumes a single-$Q$ spin structure. With the presence of three independent wavevectors $(1,0,1/2)$, $(0,1/2,1)$, and $(1/2,1,0)$ and the cubic parent lattice, the antiferromagnetic domains in ZnFe₂O₄ could be either single- or multi-$Q$ type. We first discuss spin structures of the single-$Q$ type, treating the multi-$Q$ scenario below. All three single-$Q$ domains have separated diffraction patterns, and within the cubic parent structure, the only type of twinning is the inversion symmetry breaking which does not affect the diffraction intensity [31] (Appendix B).

As the magnetic unit cell doubles the lattice unit cell, the Bravais lattice changes from face-centered cubic to body-centered tetragonal. A reduction of symmetry in general introduces extra degree of freedom or uncertainties to specify the lattice space group of the magnetic unit cell. Here, the loss of the translational vector $(a/2,0,a/2)$, the face-center vector in the parent cubic structure, leads to possibilities of both symmorphic ($I\bar{4}2m$, #121) and non-symmorphic ($I\bar{4}2d$, #122) types of body-centered tetragonal space groups to describe the ionic lattice of ZnFe₂O₄. The difference is that the cubic face-center vector $(a/2,0,a/2)$ is reincorporated as the gliding and screw axis vector $(a/2,0,3a/2)$ in $I\bar{4}2d$.

From these lattice space groups $I\bar{4}2m$ and $I\bar{4}2d$, only two magnetic space groups of the black-white type of the second kind, $I_{c}\bar{4}2m$ (#121.332) and $I_{c}\bar{4}2d$ (#122.338), are compatible with the specific magnetic wavevector $(1,0,1/2)$ in ZnFe₂O₄. Both magnetic space groups include lattice space group operations in combination with anti-translations made of $(a/2,a/2,0)$ or $(0,0,a)$ and the time reversal operator $1^{\prime}$. Together with normal operations at the origin $(0,0,0)$ and body-center translation vector $(a/2,a/2,a)$, spins in these space groups acquire the parallel/antiparallel relationship as Fig. 4.

The combination of eight $\bar{4}2m$ point group operations, the body-center vector $(a/2,a/2,a)$, and the time reversal operation $1^{\prime}$ symmetrically connects spins in the magnetic unit cell. It leaves only one independent spin in the space group $I_{c}\bar{4}2d$ to describe all 32 spins. However, in the space group $I_{c}\bar{4}2m$, the lack of the gliding vector $(a/2,0,3a/2)$ leads to a degeneracy of two symmetry operations on each site under all scenarios, and there are always two independent sets of spins (colored vs. grey in Fig. 5) in the magnetic unit cell. This degeneracy of symmetry operations leads to strong constraints for the spin direction as we discuss below.

We further explore potential magnetic space groups of lower symmetry. While disordered ZnFe₂O₄ often demonstrates a ferromagnetic state [13], our clean samples do not have any perceivable FM behavior. We thus rule out magnetic space groups with ferromagnetic type of point groups based on Dzyaloshinsky’s principle [37]. We note that Ref. [10] suggested $I_{c}\bar{4}2d$ as the choice of the highest symmetry magnetic space group but ruled out $I_{c}\bar{4}2m$ based on molecular field considerations in conjunction with Mossbauer studies. Here, we work under the simple assumption of a body-centered, purely antiferromagnetic state. A survey of the symmetry operation reveals all suitable magnetic space groups are nonmagnetic type, including choices in $I_{c}\bar{4}$ (#82.42), which is suggested by JANA2020 and was implied by Ref. [11], and $I_{c}222$ (#23.52), which was suggested by Ref. [10]. Together with $I_{c}\bar{4}2m$ and $I_{c}\bar{4}2d$, these four magnetic space groups are the only choices for antiferromagnetism in ZnFe₂O₄ with the symmetry no lower than orthorhombic.

$I_{c}\bar{4}$ and $I_{c}222$ are both symmorphic space groups and both point groups $\bar{4}$ and $222$ have half of the symmetry operations removed from $\bar{4}2m$, including two mirror planes. The reduced total number of 16 symmetry operations from the point group, BCC-translation, and time reversal $1^{\prime}$ leaves both magnetic space groups $I_{c}\bar{4}$ and $I_{c}222$ with two independent sets of spins for the whole magnetic unit cell, and with no degeneracy of symmetry operations on each site.

The preceding analysis can be formally derived through representation analysis [38], which is incorporated in JANA2020 software [31, 32]. In the paramagnetic phase of ZnFe₂O₄, the $F\bar{4}3m$ space group of the parent lattice has 24 symmetry operations separated into five classes. With the non-zero wave vector $(1,0,1/2)$, a purely rotational point group $\bar{4}$ is derived from the original point group to form the wavevector group $G_{0}$, which leaves the wavevector invariant [38]. Here, $\bar{4}$ represents the kernel symmetry of the antiferromagnetic $(1,0,1/2)$ order, and the point group $\bar{4}2m$ represents the higher level epikernel symmetry because of the mirror planes. We can thus separate all four choice magnetic space groups into three levels of symmetry, with $I_{c}\bar{4}2d$ and $I_{c}\bar{4}2m$ at the epikernel level, $I_{c}\bar{4}$ at the kernel level, and $I_{c}222$, which cannot be directly derived through the epikernel/kernel relationship; $I_{c}222$ can only be derived from $I_{c}\bar{4}2m$ [10] based on Landau’s theory of continuous phase transition, but not from $I_{c}\bar{4}2d$ and $I_{c}\bar{4}$.

As an Abelian group of four elements, $\bar{4}$ has four one-dimensional irreducible representations (irreps); Ref. [11] listed the irreps together with associated basis functions for ZnFe₂O₄. Each of the four irreps ($mW1$-$mW4$) can be associated with the three kernel/epikernel space groups we discussed above, based on a different origin of symmetry operations. Using JANA2020, we refine spin structures with for all distinct combinations of magnetic space group and irrep.

For magnetic diffraction data taken from the clean-limit crystal at CORELLI at 6 K, the lattice structure is first refined including the extinction effect, generating the results in Table 1. The magnetic structure is subsequently refined using the updated lattice information. All parameters of atomic positions and thermal parameters are fixed during this stage. This is mainly because for diffraction data at 6 K, all refined lattice thermal parameters $U_{iso}$ are near zero. If they are allowed to float during the refinement of magnetism, the refined values could fluctuate to negative values which would appear unphysical. In the second stage of magnetism refinement, we constrain the extinction effect for both lattice and magnetism to be the same and refine it once again.

In addition to kernel and epikernel space groups, we also include $I_{c}222$ as a low-symmetry alternative. The inclusion of space groups of different (and lower) symmetries provides an understanding of certain symmetry elements’ stability. Several irreps have essentially identical refined spin structures in the same magnetic space group, and we can identify seven unique spin structures that are summarized in Fig. 5 and Table 2. All refinements generate similar $R$ and $GOF$ values; these are dominated by systematic issues of extinction correction at the lattice refinement stage discussed above.

Before evaluating results under the single-$Q$ scenario, we first explore whether ZnFe₂O₄ could also be in a multi-$Q$ state. For a triple-$Q$ state, the magnetic unit cell doubles along all three dimensions to $2a\times 2a\times 2a$. As two spins separated by $(a,0,0)$ cannot simultaneously satisfy the parallel and antiparallel conditions set by the $(1,0,0.5)$ and $(0.5,1,0)$ orders respectively, each $Q$-order in the triple-$Q$ state must come from separated spin components and there will be no combined reflection such as $(1.5,1.5,1.5)$; the diffraction pattern of a multi-$Q$ state would be identical to that of single-$Q$ [39].

A single-$Q$ state can be unambiguously identified if there is a reduction of lattice symmetry associated with the magnetic phase transition [40, 41]. However, the antiferromagnetic transition in ZnFe₂O₄ is of second order nature and demonstrates no observable structure modification. On the other hand, there exist indirect evidence against a multi-$Q$ state, which results from higher order spin-spin interaction terms in the Hamiltonian, and is more often seen in $4f$/$5f$ magnets such as Gd₂Ti₂O₇, CeB₆, UO₂, and UP [42, 40]. The higher order terms could have stronger influence at lower $T$, so multi-$Q$ states emerge at a second phase transition below $T_{N}$ in Gd₂Ti₂O₇ and UP. As $3d$ spins are of Heisenberg type with little anisotropy in the exchange interaction, a single-Q state is more likely to happen than a multi-$Q$ state [40].

A multi-$Q$ state can be verified through anisotropic external fields such as uniaxial pressure or magnetic field; for example, cubic CeB₆ transitions from a double-$Q$ to a single-$Q$ state under increasing magnetic field [40]. We characterized the antiferromagnetic field-temperature phase diagram of ZnFe₂O₄ using both DC magnetic susceptibility and heat capacity. With $H||(1,0,0)$ and $H||(1,1,0)$, $M/H$ vs. $T$ in Fig. 6 has a very weak field dependence. With $H||(1,0,0)$ and $H||(1,1,1)$, heat capacity $C_{p}$ measured to 9 T are always consistent with $M/H$ (Fig. 6). The antiferromagnetic phase boundary persists to 14 T with $T_{N}$ changing by $\sim 0.5$ K (Fig. 6). Both heat capacity and magnetic susceptibility indicate that the magnetic phase transition is always second order and there is no additional phase transition over the whole $H$-$T$ space. For all three crystalline directions, the phase diagrams $T_{N}(H)$ is isotropic, indicating isotropic (Heisenberg type) spin interactions in ZnFe₂O₄. $H$-$T$ phase diagrams along all three directions support a single-$Q$ antiferromagnetic state.

Spins separated by $(a,a,a)$ would be of opposite sign for all three vectors of the triple-$Q$ state. The magnetic cell is thus a primitive but not body-centered type for spins. Using JANA2020, we refine for both double-$Q$ and triple-$Q$ spin structures with various irreps of several allowed magnetic space groups such as $P_{c}\bar{4}21m$, $P_{c}\bar{4}2m$, $P_{I}\bar{4}3m$, $P_{I}\bar{4}3n$, and $P_{I}213$. Both $R$ and $GOF$ values do not improve with the additional degrees of freedom, and these refinements all generate moments ranging from zero to full sizes on independent spin sites. We conclude a multi-$Q$ state is not favored.

All of the possible refined magnetic structures of ZnFe₂O₄, in the different magnetic space groups discussed above (Fig. 5 and Table 2), are non-collinear and of three-dimensional type. We note here the common features of these spin structures, as they represent the definitive characteristics of antiferromagnetism in ZnFe₂O₄. First, spins lie predominantly within the $a$-$b$ plane, as the $c$-axis component in all kernel/epikernel models varies only from zero to  0.5 $\mu_{B}$. Second, all refined spin structures can be viewed as consisting of eight spin clusters, each with four nearly parallel (ferromagnetically aligned) spins (Fig. 5); these 8 FM tetrahedra are arranged antiferromagnetically within the magnetic unit cell.

In Fig. 3a, the measured intensities of lattice and magnetic reflections are plotted against $(h^{2}+k^{2}+l^{2})$ or equivalently $q_{hkl}^{2}$, with $q_{hkl}$ the wave vector of reflection $(h,k,l)$. While magnetic diffraction intensities $I_{M}$ evolve by the overall $q$-dependent magnetic form factor $I_{M}\sim e^{-q^{2}}$, they also demonstrate significant variations as a function of $(h,k,l)$, reflective of the magnetic structure factor of the order. The magnetic intensities $I_{M}$ (Fig. 3b) can be modeled by a simple expression:

where $h$, $k$, and $l$ are the even, odd, and half-integer indices respectively of the reflection $(h,k,l)$. This form was included in a complex expression for the diffraction intensity in Ref. [11]. It is possible to track the origin of Eq. 1 to four spins on a single tetrahedron of the pyrochlore lattice. While multiple models for the spin structure of individual tetrahedra can generate Eq. 1, one scenario that is consistent with all our refinement results is four parallel spins along any direction, so ${I_{M}\sim[\sum_{i=1}^{4}\exp{(i\mathbf{q}_{hkl}\mathbf{r}_{i}})]^{2}}$. As the simple expression of Eq. 1 captures the main features of our experimental results (Fig. 3b), the remaining differences are indicative of fine details of antiferromagnetically aligned components.

Alternatively, the same spin structures can be viewed as a stack of tetrahedra where each is composed of four antiferromagnetically arranged spins in the $a$-$b$ plane (Fig. 5). In some spin models, there exists a small ferromagnetic component along the $c$-axis in one type of antiferromagnetic tetrahedra while the other type has a net zero moment; this issue wll be addressed in detail below. We thus have two equivalent perspectives of the spin structure in ZnFe₂O₄, regarding it as either a set of antiferromagnetically arranged FM spin clusters, or as stacked zero-moment, locally antiferromagnetic tetrahedra. These two perspectives are not contradictory, as the pyrochlore lattice is made of corner sharing tetrahedra positioned in a diamond/zincblende structured network. The nearest neighbor tetrahedra of an AF type are always of FM types, and vice versa, forming a three-dimensional checkerboard pattern. Each individual spin is situated at the junction of two tetrahedra and is ferromagnetically aligned to three spins on one side, and antiferromagnetically aligned to another three spins on the other side.

In order to differentiate between the possibilities and identify the proper magnetic space group and spin structure, we next explore the evolution of the checkerboard spin structure in ZnFe₂O₄ as imposed symmetry constraints are reduced. The four space groups in Fig. 5 provide at least three layers of symmetry hierarchy. The epikernel groups $I_{c}\bar{4}2m$ and $I_{c}\bar{4}2d$ both possess the mirror planes, and between them, they exchange the degrees of freedom of two independent spins to the displacement vector $(a/2,0,a/2)$ of the parent FCC lattice and one spin. The kernel group $I_{c}\bar{4}$ loses the mirror planes but keeps the four-fold axis. The lowest symmetry group $I_{c}222$ lacks both mirror planes and the four-fold axis.

Refining in the symmorphic group $I_{c}\bar{4}2m$, the degeneracy of symmetry operations has spins on the mirror planes confined to directions perpendicular to the planes they are located. Half of the Fe spins in the unit cell would therefore align strictly along either $(1,1,0)$ or $(1,-1,0)$ directions within the $a$-$b$ plane with no $c$-axis component (Table 2), blue spins in Figs. 5c and 5d). In space group $I_{c}\bar{4}2m$, the other half of spins (grey ones in Figs. 5c and 5d) are not located on a mirror plane, and mirror operations restrict them to be parallel to the planes, again along either $(1,1,0)$ or $(1,-1,0)$ directions and also the $c$-axis. This mirror symmetry restriction persists through all irreps with different origins of group $I_{c}\bar{4}2m$, and is the main reason spins reside in the $a$-$b$ plane and become parallel on a tetrahedron. The dominant components in the $a$-$b$ plane and FM clusters in all refinement models in Fig. 5, despite mirror operations are no longer being required in both magnetic space groups $I_{c}\bar{4}$ and $I_{c}222$, can be regarded as a confirmation of the mirror symmetry’s presence in the spin structure.

In all three space groups $I_{c}\bar{4}2m$, $I_{c}\bar{4}$, and $I_{c}222$, we notice that the two independent sets of spins each form an antiferromagnetic tetrahedron, which are separated by the displacement vector $(a/2,0,a/2)$ of the parent FCC lattice (color vs. grey in Fig. 5). These two antiferromagnetic spin clusters are of nearly orthogonal relationship as one has spins pointing toward the center of the tetrahedron while the other has spins aligned tangential to the tetrahedron (Fig. 5). This degree of freedom of two independent spins provides the possibility for spins to remain aligned closely to the $(1,1,0)$ family of directions in both groups $I_{c}\bar{4}$ and $I_{c}222$. In contrast, when the number of independent spins is reduced to one in space group $I_{c}\bar{4}2d$, the spin alignment is dramatically changed to be mostly along $(1,0,0)$ family of directions, despite the presence of mirror operations. The resulting antiferromagnetic tetrahedron is similar to the average of two orthogonal antiferromagnetic configurations in all groups $I_{c}\bar{4}2m$, $I_{c}\bar{4}$, and $I_{c}222$. Two key components of the spin structure in ZnFe₂O₄ are thus the mirror symmetry and two independent spin sites. They lead to $(1,1,0)$-aligned spins and their configuration can only be destroyed by excessive symmetry constraints during refinement. We can thus identify $I_{c}\bar{4}2m$ as the sole suitable magnetic space group.

There remain two distinctive spin structures based on two irreps in $I_{c}\bar{4}2m$. Their difference lies in how each independent set of spins form the antiferromagnetic tetrahedra. In Fig. 6, we note that irrep $mW1$ for $I_{c}\bar{4}2d$, $I_{c}\bar{4}2m$, and $I_{c}\bar{4}$ always provides a distorted all-in-all-out type of antiferromagnetic configuration (Figs. 5a, c, e). By contrast, irrep $mW2$ of all three groups always leads to a two-in-two-out configuration (Figs. 5b, f). As spin moments are mostly in the $a$-$b$ plane, the all-in-all-out and two-in-two-out configurations in two dimensions are both antiferromagnetic. However, when spins can be three-dimensional, as in ZnFe₂O₄ with a small $c$-axis component, the two-in-two-out type would introduce a minute yet finite FM moment for each tetrahedron, which eventually averages out across the magnetic unit cell. The all-in-all-out type would always sum to a zero net-moment along all directions in all models.

For irreps such as $mW2$, which lead to a two-in-two-out type of antiferromagnetism in all three space groups, we notice individual $c$-axis components, at $\sim 0\mu_{B}$ and $0.40-0.55\mu_{B}$, are largely different, and are significantly larger than that of all-in-all-out spin configuration ($0-0.15\mu_{B}$) based on irreps $mW1$ and its equivalents ( Table 2). In space group $I_{c}222$, where these four irreps no longer apply, the all-in-all-out antiferromagnetic configuration persists with a zero net moment, yet the largest $c$-axis component in our refinements rises in the antiferromagnetic configuration of spins tangential to the tetrahedron (Table 2, grey spins of Fig. 5g). Spin components along the $c$-axis that are permitted by symmetry would exist because of Dzyaloshinsky’s principle [37], yet large $c$-axis components seem like mathematical artifacts. Irrep $mW1$ in the magnetic space group $I_{c}\bar{4}2m$ (Fig. 5c) thus leads to the physically most sensible spin structure for the antiferromagnetism in ZnFe₂O₄. We note the irrep choice of $mW1$ or $mW2$ does not affect the discussion from here on.

The phenomenon of two contrasting types of tetrahedra with only one species of spins is not often observed in long-range ordered pyrochlore magnets [6]. For Ising-type all-in-all-out antiferromagnets such as Cd₂Os₂O₇ [5], the spin structure on tetrahedra is uniform in the bulk, varying at ferromagnetic domain walls which would not be counted as a stable thermodynamic phase [43, 44]. With a Curie-Weiss temperature $T_{CW}\sim-25$ K and $T_{N}=9.95$ K, the spin structure in ZnFe₂O₄ is not expected to be greatly frustrated. Instead, the viability of coexisting FM and AF tetrahedra stems from a degree of freedom that is already present in the lattice: the space group $F\bar{4}3m$ of broken inversion symmetry.

Because of the breathing mode of the pyrochlore lattice, every spin is located at the joint point of one large and one small tetrahedron. Its coupling to neighbor spins depends on both Fe-O-Fe super-exchange and direct Fe-Fe exchange, in a delicate balance among many different orbit choices [23]. The lattice structure based on CORELLI data refinement (Table 1) is drawn in Fig. 7. The Fe-O and Fe-Fe distances both differ by  2% between two different-sized tetrahedra, suggesting stronger exchange couplings for spins on smaller tetrahedra. Furthermore, the Fe-O-Fe bond angle $95.11^{\circ}$ on the small tetrahedra is about 0.6^∘ larger than the equivalent bond angle on the large tetrahedra. In our previous refinement [23], the relationship between these two angles is opposite with the small tetrahedra hosting a narrower Fe-O-Fe angle. As bond angles are sensitively dependent on details of refined atomic positions, here our improved, full refinement clarifies this relationship. The Fe-O-Fe angle at $\sim 95^{\circ}$ is close to the boundary between ferromagnetic and antiferromagnetic types of super-exchange interaction [23]. For parallel spins on the large tetrahedra, the exchange interaction would be weak even when it is ferromagnetic, and the increasing Fe-O-Fe angle on small tetrahedra would support stronger antiferromagnetic interaction [23]. Both the negative $T_{CW}$ and the long-range ordered ground state indicate the antiferromagnetic interaction should be the type that dominates.