Magnetic structure of MgCr 2 O 4 below the spin-Peierls transition

Although not much is known for the magnetic structure of MgCr2O4, extensive work has been done for ZnCr₂O₄. There, four propagation vectorsq₁=(0.5 0.5 0), q₂ =(1 0 0.5), q₃=(1 0 0), andq₄=(0.5 0.5 0.5) have been revealed by powder neutron diffraction [207].

While the intensities of theq3andq4components are rather weak and depend on the sample stoichiometry, intensities of theq₁andq₂components are strong and do not vary between samples.

Our neutron diffraction experiments for MgCr₂O₄were performed on DMC at SINQ of PSI in an orange cryostat. For the measurements, about 2.5 g of MgCr₂O₄powder was filled into an aluminum cylinder can. The setup with 2.45 Å incoming neutron wavelength was employed.

Fig. 5.5 presents measurement results atT =1.6 K. Similar to ZnCr₂O₄, the majority of the magnetic peaks can be indexed by the propagation vectorsq₁=(0.5 0.5 0) andq₂=(1 0 0.5). As is indicated by the stars in Fig. 5.5, two extra magnetic peaks are observed at 2θ=24 and 38^◦, which can be indexed byq₃=(1 0 0). Due to the limited peak number, a reliable refinement of the magnetic structure forq3is not possible and in the following we will concentrate on the structures forq₁andq₂.

In order to solve the MgCr2O4magnetic structure, representation analysis is performed for theI4₁/amdandF dddspace groups. However, no satisfactory fits can be obtained with the irreducible representations from these two space groups. Therefore, a different approach is employed. Irrespective of the symmetry, we construct all the possible magnetic long-range ordered states and directly compare them with the diffraction pattern. Following the work on ZnCr₂O₄[214], two constraints are imposed on the magnetic structures: one is that each tetrahedron in the long-range ordered state has zero total moment. This means that the spin-lattice coupling only perturbatively relieves the ground state degeneracy of the high-temperature cubic phase. Another constraint is that spins lie in theab-plane and have equal components along theaandbaxes. This constraint is derived from the polarized neutron

diffraction data on ZnCr₂O₄[207] and its validity for MgCr₂O₄will be discussed at the end of this section.

2theta

10 15 20 25 30 35 40 45 50 55

Intensity (a. u.)

0 1 2 3 4 5 6

* *

Figure 5.5 – Refinement for the neutron diffraction data of MgCr₂O₄measured atT =1.6 K.

Data points are shown as red circles. The calculated pattern is shown as the black solid line.

The magnetic strutures of Fig. 5.6a forq1and Fig. 5.7a forq2are used for the calculation, and the refined moment sizes are 1.35 and 1.47µBfor theq₁andq₂structures, respectively.

The vertical bars, from top to bottom, show the positions of the nuclear peaks,q₁= (0.5 0.5 0) magnetic peaks, andq2= (1 0 0.5) magnetic peaks, respectively. And the blue line at the bottom shows the difference of data and calculated intensities. The Rietveld agreement factors areR_p=8.1 andR_{w p}=7.4. The magneticR-factors areR_f =11. forq₁andR_f =16.2 forq₂. The stars at 2θ=24 and 38^◦indicate the weak magnetic peaks withq3=(1 0 0).

a b a

a b

b c

observed intensity (a. u.)

0 50 100 150

calculated intensity (a. u.)

0 50 100 150

Figure 5.6 –(a)Collinear magnetic structure for theq1=(0.5 0.5 0) component. Spins form

↑ ↑ ↓ ↓chains along the [110] direction. The solid (dashed) lines show the shortened (elongated) bonds with stronger (weaker) couplings.(b)Coplanar magnetic structure for theq₁=(0.5 0.5 0) component. Spins are perpendicular along the [110] and [110] directions.(c)Refinement results for the single crystal neutron diffraction data measured at 1.6 K. Refinements using the collinear structure (panel a) and coplanar structure (panel b) lead to similar results.

5.2. Crystal and magnetic structures of MgCr2O4

Magnetic structures for theq1component

As a first step, we construct all the possible (0.5 0.5 0) long-range ordered states with spins along the [110] and [110] directions. With this starting point, the constraint of equalabcomponents is automatically satisfied. For all the 614656 structures constructed in this way, only 768 have zero total moments for all tetrahedra. Next, we compare the calculated diffraction pattern for all these 768 structures with the experimental data. For this purpose, intensities for the reflections of (0.5 0.5 0) at 2θ=12.0^◦, (0.5 0.5 1) at 20.8^◦, (0.5 1.5 0) at 20.8^◦, (0.5 1.5 1) at 32.0^◦, and (0.5 0.5 2)/(1.5 1.5 0) at 36.5^◦are extracted from the DMC data and then compared with those of the 768 candidates using a script implemented with MATLAB. In the end, 32 structures are found to be compatible with the DMC data, and the refinement results are presented in Fig.

5.5. The refined moment size is 1.35(1)µB for both of the collinear and coplanar structures, which is only half of the Cr³⁺total moment of 3µB.

Due to some redundancy in the structure construction, some of the 32 structures are equiva-lent. For example, flipping of all the spins produces an equivalent structure but is counted as a different solution in our approach. After eliminating such a redundancy, 4 collinear and 4 coplanar structures can be identified, and representative ones are shown in Fig. 5.6a and b. The strengthened/weakened antiferromagnetic bonds are also indicated as solid/dashed lines along with the magnetic structure, and both the collinear and coplanar structures are compatible with a compressedcaxis.

Of special interest is that for all the 4 collinear structures,↑ ↑ ↓ ↓spin chains can be observed along one of the diagonal directions. For example, in the structure shown in Fig. 5.6a, the spin chains along the [110] direction are aligned in the↑ ↑ ↓ ↓fashion, and its alignment is exactly the same for the neighbouring chains. Due to the zero-moment constraint, the remaining spins can only have fixed FM or AF chains along the [110] direction. The difference between the 4 collinear solutions lies in the two choices of the↑ ↑ ↓ ↓chain direction and the parallel/perpendicular relationship between the spin direction and theq₁vector.

To verify the obtained magnetic structures, single crystal neutron diffraction experiments were performed on TriCS at SINQ of PSI. The setup with 2.32 Å incoming neutron wavelength was employed, and altogether 167 and 114 reflections were collected for the (0.5 0.5 0) and (1 0 0.5) families, respectively. Fig. 5.6c presents the comparison between the calculated intensities of the obtained structures with the observed intensities for the (0.5 0.5 0) dataset (65 reflections averaged from 100 reflections of the (0.5 0.5 0) and (0.5−0.5 0) domains). Satisfactory fits can be achieved for both the obtained collinear and coplanar structures, with the agreement factors ofR_f =12.2,R_f2=16.5.

Magnetic structures for theq2component

Similar to the approach employed in theq₁structure determination, 614656 structures with spins along the [110] and [110] directions are constructed forq₂=(1 0 0.5). Among them,

a b a

observed intensity (a. u.)

0 50 100 150

calculated intensity (a. u.)

0 50 100 150

a b c

a b d b

Figure 5.7 –(a)Collinear magnetic structure for theq2=(1 0 0.5) component. The spin chains along the diagonal directions are aligned in a↑ ↑ ↓ ↓fashion. The solid (dashed) lines show the shortened (elongated) bonds with stronger (weaker) couplings. Notice that sites shifted by one unit cell along thecdirection will have opposite spins.bRefinement results for the single crystal neutron diffraction data measured at 1.6 K. Refinements using the collinear structure (panel a) and coplanar structure (panel c and d) lead to similar results.(c)The first coplanar magnetic structure for theq2=(1 0 0.5) component. Spin chains along the diagonal directions also have the↑ ↑ ↓ ↓alignment but spins of the neighbouring chains are perpendicular.(d) The second coplanar magnetic structure for theq₂=(1 0 0.5) component. Spin chains along the diagonal directions are aligned in a cycloidal fashion.

1112 solutions satisfy the zero-moment constraint, and are compared with the experimental data. For this purpose, intensities for the reflections of (1 0 0.5) at 2θ=19.0^◦, (0 0 1.5)/(1 1 0.5) at 25.6^◦, (1 0 1.5) at 30.8^◦, (1 2 0.5) at 39.5^◦,(3 0 0.5) at 53.3^◦are extracted from the DMC data. In the end, 56 structures are found to be compatible with the experimental data, and the refinement results are presented in Fig. 5.5. Similar to that of theq₁component, the refined moment size is 1.47(1)µB for both of the collinear and coplanar structures, which is only half of the Cr³⁺total moment of 3µB.

Not all of the 56 structures are in-equivalent. After removing the redundancy, only one collinear and two coplanar solutions are obtained and their structures are shown in Fig. 5.7a, c, and d. In the collinear structure, spins chains along the diagonal directions are aligned in the

↑ ↑ ↓ ↓fashion. And the alignment of the neighbouring chains are shifted by one spin site.

For the coplanar structure shown in Fig. 5.7c, the spins chains along the diagonal directions also have a↑ ↑ ↓ ↓alignment, but the neighbouring chains have perpendicular spin directions.

5.2. Crystal and magnetic structures of MgCr2O4

The other coplanar structure has cycloidal alignment along the diagonal directions and is shown in Fig. 5.7d. The chirality of the neighbouring cycloids are opposite for the chains along the [110] direction and are the same for the chains along the [110] direction. Fig. 5.7b presents the comparison between the calculated intensities of the obtained structures with the observed intensities for the (1 0 0.5) dataset measured on TriCS (42 reflections averaged from 114 reflections of the (1 0 0.5) and (1 0 -0.5) domains). Satisfactory fits can be achieved for all the obtained collinear and coplanar structures, with the agreement factors ofR_f =6.1, R_f2=10.8.

Determination of the magnetic structures using spherical neutron polarimetry

Until now, the magnetic structures of MgCr₂O₄look very similar to those reported for ZnCr₂O₄ [207, 214]. There, no further discrimination among the candidate structures is available, and the possibility of an equal-momentq1-q2double-qstructure prefers collinear solutions for bothq₁andq₂[214]. Here we present our spherical neutron polarimetry experimental results on MgCr₂O₄, which reveal additional spin components out of theabplane for bothq₁andq₂ structures and unambiguously determine theq₂structure to be canted coplanar.

Our neutron polarimetry experiments were performed on IN12 with the CryoPAD at ILL using an orange cryostat. The setup with 3.70 Å incoming neutron wavelength were employed. Two configurations were used in our measurement: one with the (hk0) plane as the horizontal scattering plane; another with the (hhl) plane as the horizontal scattering plane.

Tab. 5.2 presents the measured polarimetry matrices for theq₁reflections. For all the measured matrices, only the diagonal elements are non-zero. With the (hk0) plane horizontal, they areP_xx∼ −1,P_{y y}∼1, andP_zz∼ −1; With the (hhl) plane horizontal, they becomeP_xx∼ −1, P_{y y}∼ −1, andP_zz∼1. Such values immediately remind us of the MnSc₂S₄collinear structure with spins along the [110] directions (see section 4.3.2). However, careful comparison of the P_{y y} elements for the (0.5 0.5 0) and (0.5 0.5 1) reflections reveals a small but reproducible difference of 0.09, which suggests finite spin components out of theabplane.

To fit the measured SNP matrices, a uniform rotation is applied to the collinear structure shown in Fig. 5.6a, and the Euler angles (ZXZ-convention) ofφ,θ,ψare used as the fitting parameters. The summation of the absolute difference between the measured and calculated P_{y y}elements of the six measured reflections is used as the fitting criterion. A satisfactory fit can be achieved withφ= −0.0250,θ=0.3818,ψ=0.0361. The nearly zero value of the fitted φandψparameters indicates that the spin components in theabplane should stay the same after the rotation. This allows us to refine the SNP results with a different approach: instead of using the Euler angles to rotate the spins, we add uniform-sizedccomponents to all the spins. The refined structure hasS_c/S_b= ±0.376(5) and the calculated SNP matrices assuming a bender efficiency of 98 % are shown in Tab. 5.2. The refinement of the neutron diffraction dataset is equally good as the original results shown in Fig. 5.6c, and the agreement factors are R_f =11.3 andR_{w p}=16.7.

Surprisingly, adding the sameS_ccomponents to the coplanar structure shown in Fig. 5.6b produce exactly the same SNP matrices, meaning that the SNP measurements are not able to differentiate the collinear and coplanar structures. However, they do allow us to constrain the spins to the [1 1±0.376] directions if the structure is collinear and to the [0±0.376 1] planes if the structure is coplanar. For the collinear structure, it is separate domains that contribute to the (0.5 0.5 0) and (0.5 0.5 0) reflections; while for the coplanar structures, each domain contributes equally to the (0.5 0.5 0) and (0.5 0.5 0) reflections. Thus we propose that a neutron diffraction experiment with a magnetic field along the [110] direction while cooling might

Table 5.2 – Spherical neutron polarimetry matrices for theq1=(0.5 0.5 0) reflections. The calculations are performed using the collinear structure shown in Fig. 5.6 with additional spin components along thecaxis (see text). The combination of domains withS_c/S_b=0.376 and Sc/S_b= −0.376 is necessary to eliminate the off-diagonal elements.

Reflections experiments calculations

(hk0) as the horizontal scattering plane (0.5 0.5 0) (hhl) as the horizontal scattering plane

(0.5 0.5 0)

5.2. Crystal and magnetic structures of MgCr2O4

help differentiate the collinear and coplanar structures [225, 226].

SNP measurements for theq₂reflections are even more informative. Tab. 5.3 lists the measured matrices with (hk0) as the horizontal scattering plane. Different from theq₁ results, the diagonal elements vary a lot for the q2 reflections. Specifically, thePy y element changes non-monotonously from−0.27 for (1 0.5 0) to 0.01 for (1 1.5 0) and then to−0.16 for (1 2.5 0), which is very different from the monotonous variation observed in ZnCr₂O₄[207] and can be fitted by none of the obtained structures whose spins are in theabplane. Similar to the fits of theq₁components, a uniform rotation is applied to the three obtained structures listed in Fig.

5.7 to fit thePy y elements of the four measured reflections. After averaging over equivalent domains, only the coplanar structure with diagonal cycloidal chains (shown in Fig. 5.7d) can fit the measured matrices, and the fitted Euler angles areφ=0.2097,θ=0.3934,ψ= −0.1871 in rad.

Compared to the structure of Fig. 5.7d, the rotated structure keeps a similar spin configuration in theabplane but has smallccomponents that are opposite for the antiparallel pairs. This again enables us to refine the magnetic structure by adding uniformccomponents to all the spins. Satisfactory fits can be achieved withS_c/S_b= −0.378(5), and the corresponding SNP matrices are listed in Tab. 5.3. The fit to the diffraction dataset is as good as that of Fig.

5.7b, with similar agreement factors ofR_f =5.6,R_f2=9.7. In this way, we unambiguously determine theq2structure to be canted coplanar for MgCr2O4, which is different from the collinear structure suggested for ZnCr₂O₄[214].

Table 5.3 – Spherical neutron polarimetry matrices for theq₂=(1 0 0.5) reflections. The calculations are performed using a coplanar structure that is be obtained from Fig. 5.7d with a uniform rotation (see text). For each reflection, contributions from four domains due to the two possible chirality of the [110] spin chains and the two reflections ofq₂and−q₂are averaged together, which is necessary to eliminate the off-diagonal elements.

Reflections experiments calculations

(hk0) as the horizontal scattering plane (1 0.5 0)

E (meV)

Figure 5.8 – 2D map for the spin dynamics in MgCr₂O₄measured on EIGER along the(a) (k−1.5k k−3)(b)and (0.5k1−k) directions. The dotted lines indicate the measuredq-scans andE-scans. Log-scale is employed for the intensities.(c)and(d)shows theq-scan at 4.5 meV along the (k−1.5k k−3) and (0.5k1−k) directions, respectively. The red solid lines are the calculation results for the hexagon modes.