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## Creator
[Masashi Hase](https://orcid.org/0000-0003-2717-461X), [Andreas Dönni](https://orcid.org/0000-0002-7300-9175), Vladimir Yu. Pomjakushin, Martin Rotter
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[Magnetic structure of TbNbO determined using neutron diffraction experiments and magnetic anisotropy calculations](https://mdr.nims.go.jp/datasets/d6f2c15a-f620-4ebf-b8c8-e67fca45c0c5)
## Fulltext
Magnetic structure of Tb3NbO7 determined usingneutron diffraction experiments and magneticanisotropy calculationsMasashi Hasea , Andreas Dönnia , Vladimir Yu. Pomjakushinb , MartinRottercaResearch Center for Materials Nanoarchitectonics (MANA), National Institute forMaterials Science (NIMS), Tsukuba, 305-0047, Ibaraki, JapanbLaboratory for Neutron Scattering and Imaging, Paul Scherrer Institute (PSI), VilligenPSI, CH-5232, SwitzerlandcMcPhase Project, Venice, 30121, ItalyAbstractWe determined the magnetic structure of Tb3NbO7 based on powder neutrondiffraction experiments and magnetic anisotropy calculations. We calculatedthe energy levels split by crystalline electric fields, and the magnetic momentsusing McPhase. The c axis is the easy axis for Tb1 moments, and magneticanisotropy is small for Tb2 moments. Magnetic reflections were observedat low temperatures. The propagation vector is k1 = (0, 0, 0) at 1.8 K andk2 = (1/2, 1/2, 0) at 2.5 K between transition temperatures TN1 = 2.0 Kand TN2 = 3.2 K. We selected candidates for an irreducible representation(IR) of the magnetic structures using magnetic anisotropy. From Rietveldrefinements, we determined the following: The IRs are mΓ4 and mS2 at 1.8K and 2.5 K, respectively. The main component of Tb1 ordered moments isw at 1.8 K. The order of the main components in each chain along the c axisis antiferromagnetic. Tb1 moments are disordered at 2.5 K. Tb2 momentsform noncollinear magnetic structures in the ab plane perpendicular to Tb1chains with a small w component at both the 1.8 K and 2.5 K. The magneticstructure at 1.8 K is consistent with the magnetic anisotropies obtained usingMcPhase calculations. Only Tb2 moments are ordered at 2.5 K, indicatingthe appearance of a partially disordered state. We also considered the originof the two magnetic transitions.Keywords:Magnetic structures of Tb3NbO7, Magnetic anisotropy calculations usingPreprint submitted to J. Magn. Magn. Mater. April 25, 2024Manuscript File Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 https://www2.cloud.editorialmanager.com/magma/viewRCResults.aspx?pdf=1&docID=27297&rev=1&fileID=510182&msid=7560ad80-9a12-4522-968b-bb6c4c889534https://www2.cloud.editorialmanager.com/magma/viewRCResults.aspx?pdf=1&docID=27297&rev=1&fileID=510182&msid=7560ad80-9a12-4522-968b-bb6c4c889534McPhase, Two propagation vectors, Partially disordered state,1. IntroductionSelection of the propagation vector and irreducible representation (IR) isimportant to determine magnetic structures using neutron diffraction datasets[1, 2]. However, it is difficult to select a correct IR among many candidatesof IR. Therefore, we narrow down IRs considering the other results. Forexample, when we know the signs of exchange interactions or appearance ofspontaneous magnetization, IRs consistent with the results remain as candi-dates.Furthermore, magnetic anisotropy is crucial in selecting a correct IR. Forexample, when the c axis is the easy axis of the magnetic moments, IRswith finite c (w) components are candidates. The magnetic moments andanisotropy can be calculated using McPhase, which is an open source programpackage for the calculation of magnetic properties [3, 4, 5]. In addition,McPhase plays an important role in the analysis of magnetic structures [6, 7,8, 9, 10]. Therefore, we focus on McPhase to efficiently determine magneticstructures.We focus on Tb3NbO7 [11] because of the following two results. One resultis the appearance of two magnetic transitions in R3AO7 (R = Tb or Nd, A= nonmagnetic ion) [17]. In R3MO7 (R = Tb or Nd, M = magnetic ion),on the other hand, one magnetic transition occurs. We confirmed that bothR and Ru magnetic moments ordered at the same antiferromagnetic (AFM)transition temperature TN = 17 K [14] and 19 K [16] in R3RuO7 (R = Tband Nd), respectively, indicating existence of interactions between R and Rumagnetic moments [12]. Therefore, it is expected that the magnetic modelis simpler in R3AO7 than in R3MO7, whereas mechanism of appearance ofthe magnetic order may be more complicated in R3AO7 because of the twomagnetic transitions.The other result is the unique magnetic structure of Tb3RuO7 as shown inFig. 1(a) [12, 13]. Tb3NbO7 and Tb3RuO7 are nearly isostructural. There aresix crystallographic Tb sites, and two crystallographic Ru sites in Tb3RuO7.The Tb1 (blue) and Tb2 (light-blue) ordered moments are parallel to the baxis along which Tb1-Tb2 chains are formed. The order of the Tb1 and Tb2moments is AFM in each Tb1-Tb2 chain. The Tbi (i = 3 − 6) moments(red) form a noncollinear magnetic structure in the ac plane perpendicular2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 to Tb1-Tb2 chains. The Ru1 and Ru2 moments are ordered and disordered(paramagnetic), respectively, indicating the presence of a partially disorderedstate, although the two Ru sites are very similar. In the family of R3RuO7(R = rare earth elements), magnetic structures of Tb3RuO7 and Nd3RuO7have been determined [12]. They are significantly different from each other,although the two compounds are nearly isostructural [14, 15, 16, 17].Here, we describe the magnetism and crystal structure of Tb3NbO7 [11].The magnetic susceptibility and specific heat of a powder sample were mea-sured. The first and second order phase transitions occurred at TN1 = 2.0and TN2 = 3.2 K, respectively. It was speculated that Tb1 and Tb2 mo-ments were ordered at TN1 = 2.0 and TN2 = 3.2 K, respectively [11]. Nospontaneous magnetization appeared down to 1.8 K. The magnetic suscepti-bility shows a broad maximum at around 4 K, probably indicating existenceof short-range magnetic correlations. Figure 1(b) shows the Tb positions.The space groups of Tb3NbO7 and Tb3RuO7 in the low-temperature (T )phase are orthorhombic C2221 (No. 20) [11] and Pna21 (No. 33) [14], re-spectively. The a, b, and c axes in Tb3NbO7 correspond to the a, c, and baxes in Tb3RuO7, respectively. There are two crystallographic Tb sites. Tb1(blue) and Tb2 (red) sites in Tb3NbO7 correspond to the Tbi (i = 1 − 2)and Tbi (i = 3− 6) sites in Tb3RuO7, respectively. Thus, a Tb1 chain alongthe c axis in Tb3NbO7 corresponds to a Tb1-Tb2 chain along the b axisin Tb3RuO7. These two compounds are isostructural in the high-T phase[14, 19, 20]. The space group is orthorhombic Cmcm (No. 63). Figure 1(c)shows oxygen ions around Tb ions. Oxygen ions form cubic-like square prismand pentagonal bipyramid around Tb1 and Tb2 ions, respectively.It is necessary to determine the magnetic structure of Tb3NbO7 to in-vestigate mechanism of appearance of the magnetic order and to understandthe magnetic structure of Tb3RuO7. Consequently, we performed powderneutron diffraction experiments on the Tb3NbO7, determined the magneticstructure of Tb3NbO7 using supports of McPhase, and then compared withthe magnetic structure of Tb3RuO7.2. Methods of experiments and calculationsWe synthesized crystalline powders of Tb3NbO7 via a solid-state reac-tion. The starting materials were Tb4O7 and Nb2O5 powders with puritiesof 99.99% and 99.9%, respectively. Stoichiometric mixtures of the powderswere sintered at 1673 K for 24 h in air. Powder X-ray diffraction patterns3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 (a) Tb3RuO7(b) Tb3NbO7(c) Tb3NbO7Figure 1: (a) Magnetic structure of Tb3RuO7 at 1.5 K [12] drawn using VESTA [18]. Thespace group of the low-T phase is orthorhombic Pna21 (No. 33) [14]. The lattice constantsare a = 14.588 Å, b = 7.345 Å, and c = 10.560 Å at room temperature. The rectangularrepresents a unit cell. Blue, light-blue, red, green, and light-green circles denote Tb1, Tb2,Tbi (i = 3−6), Ru1, and Ru2 sites, respectively. Tb1-Tb2 chains indicated by the dashedline are formed along the b axis. (b) Tb positions in the low-T phase of Tb3NbO7. Thespace group is orthorhombic C2221 (No. 20) [11]. The lattice constants are a = 7.465 Å,b = 10.552 Å, and c = 7.514 Å at room temperature. The rectangular represents a unitcell. The a, b, and c axes in Tb3NbO7 correspond to the a, c, and b axes in Tb3RuO7,respectively. The blue and red circles denote Tb1 and Tb2 sites, respectively. Tb1 chainsindicated by the dashed line are formed along the c axis. (c) Oxygen ions around Tb ionsin Tb3NbO7.4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 were recorded at room temperature using an X-ray diffractometer (RINT-TTR III, Rigaku). Within the experimental accuracy range the obtainedsample was in a single phase.Neutron diffraction experiments were performed at the Swiss SpallationNeutron Source of the Paul Scherrer Institut, where the high-resolution pow-der diffractometer for thermal neutrons (HRPT diffractometer) [21] was used.We performed group-theory analysis of magnetic structures using the pro-grams ISODISTORT [1] and BasIreps in the FullProf Suite program package[2]. We performed Rietveld refinements of crystal and magnetic structuresusing the FullProf Suite program package [2] containing the internal tablesfor scattering lengths, and magnetic form factors. We calculated energy lev-els split by crystalline electric fields (CEFs) and magnetic moments usingMcPhase [3, 4, 5].3. Results and discussion3.1. Crystal structure at 12.0 KThe blue circles in Fig. 2 represent the neutron diffraction patterns ofTb3NbO7 at 12.0 K. We performed Rietveld refinements in the orthorhombicspace group C2221 (No. 20) to evaluate the crystal structure parameters.The line on the experimental pattern shows the results of these refinements.This is consistent with the experimental results. The values of the crystalstructure parameters listed in Table 1 are consistent with those obtained inprevious refinements of an X-ray diffraction pattern at room temperature[11].3.2. McPhase calculationsWe evaluated energy levels split by CEFs and magnetic moments usingMcPhase to investigate the magnetic anisotropy of Tb moments in Tb3NbO7[3, 4, 5]. We calculated CEFs based on the point-charge model using thevalues of the structural parameters at 12.0 K. We considered all neighboringTb3+, Nb5+, and O2− ions up to a distance of 20 Å from each Tb3+ ion. Thenumbers of ions are 2532 and 2522 from Tb1 and Tb2 ions, respectively. Weobtained the splitting into thirteen (2J + 1 = 13) CEF levels (all singlets)owing to the surrounding ions. Here, the magnitude of the total angularmomentum, Landé g factor, and magnitude of the moment are J = 6, gJ =3/2, and gJJ = 9 µB, respectively, for the ground-state multiplet of Tb3+ions (4f 8). The results are presented in Table 2. The ground and first-excited5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 01x1042x1043x1041 2 3 4 502x1044x1046x1048x1043.2 3.3 3.4 3.5� Å��Intensity (arb. units)12.0 KFigure 2: Powder neutron-diffraction pattern (circles) of Tb3NbO7 at 12.0 K. The wave-length of the neutrons was 2.450 Å. The line on the measured pattern portrays the Rietveld-refined pattern obtained using the crystal structure with C2221 (No. 20) [11]. The line atthe bottom portrays the difference between the measured and Rietveld-refined patterns.The hash marks represent the positions of nuclear reflections. The reliability factors areχ2 = 2.32, Rwp = 2.97 %, Rexp = 1.95 %, and RBragg = 3.48 %. The inset shows thepattern between 3.2 and 3.5 Å−1. The large reflection consists of the three reflections at0 0 4, 2 4 2, and 4 0 0.6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 1: Values of crystal-structural parameters of Tb3NbO7 derived from the Rietveldrefinements of the neutron-diffraction pattern at 12.0 K (upper line of each site). Thespace group is orthorhombic C2221 (No. 20) [11]. The lattice constants are a = 7.4660(1)Å, b = 10.5517(2) Å, and c = 7.5023(2) Å. The estimated standard deviations are shown inthe parentheses. The term Biso denotes the isotropic atomic displacement parameter. Thevalues of Biso of Tb and Nb became negative when they were treated as free parameters.Thus, we set the values to zero. To reduce the number of fitting parameters, we used onevalue of Biso for O. The reliability factors are χ2 = 2.32, Rwp = 2.97 %, Rexp = 1.95 %,and RBragg = 3.48 %. We also show the values derived from the Rietveld refinements ofthe X ray diffraction pattern at room temperature (lower line of each site) [11].Atom Site x y z Biso (Å2)Tb1 4a 0.4882(5) 0 0 00.4895(7) 0 0 0.36(2)Tb2 8c 0.2321(2) 0.2346(1) 0.2550(8) 00.2356(3) 0.2346(1) 0.2502(7) 0.36(2)Nb1 4a 0.9907(6) 0 0 00.9987(9) 0 0 0.03(6)O1 8c 0.2930(4) 0.3787(4) 0.0215(4) 0.44(2)0.295(2) 0.376(2) 0.014(3) 1.1(2)O2 8c 0.3145(4) 0.3745(3) 0.4637(5) 0.44(2)0.309(2) 0.383(2) 0.453(3) 1.1(2)O3 4b 0 0.6284(3) 0.25 0.44(2)0 0.627(2) 0.25 1.1(2)O4 4b 0 0.3648(3) 0.25 0.44(2)0 0.368(2) 0.25 1.1(2)O5 4b 0 0.0676(3) 0.25 0.44(2)0 0.065(2) 0.25 1.1(2)7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 2: Energy levels (second and fourth columns) of the thirteen states split by CEFs inTb3NbO7 calculated using McPhase based on the point charge model [3, 4, 5]. The thirdand fifth columns show the energies from the lowest-energy level named 1. The energiesof levels 1 and 2 for Tb1 are almost the same but different. Uncertainties due to theexperimental errors in the crystal-structural parameters were evaluated using gauss law oferror propagation and were shown in parentheses.Tb1 (4a) energy (meV) Tb2 (8c) energy (meV)13 117.0 233.8(2.6) 115.6 213.6(1.7)12 117.0 233.8(2.6) 115.6 213.5(1.7)11 59.5 176.4(2.2) 63.6 161.6(1.3)10 58.6 175.5(2.2) 63.2 161.1(1.3)9 22.3 139.2(2.0) 19.1 117.0(1.1)8 16.3 133.2(1.9) 16.1 114.0(1.0)7 0.2 117.1(2.0) -14.0 83.9(1.3)6 -17.3 99.5(1.6) -23.6 74.4(0.9)5 -20.6 96.3(1.6) -36.8 61.2(1.3)4 -59.6 57.2(0.8) -60.2 37.7(0.6)3 -59.7 57.1(0.8) -62.8 35.2(0.7)2 -116.8 0.0(0.0) -97.8 0.2(0.0)1 -116.8 0 -98.0 0states form a quasi doublet that is well separated from the other excited CEFlevels.We calculated magnetic moments at 1 K, where higher CEF levels werenot populated, under an external magnetic field of 10 T applied parallel to thethree axes. The magnetic moments are presented in Table 3. An externalmagnetic field induces a magnetic moment that is not oriented parallel tothe applied field owing to the CEF anisotropy. The values of ⟨Mc⟩ at Tb1sites are large, indicating that the c axis is the easy axis for Tb1 moments.The anisotropy of Tb2 moments is small. As described later, we used themagnetic anisotropy to select the candidates of IR.3.3. Magnetic structures at 1.8 and 2.5 KFigure 3(a) shows the powder neutron diffraction patterns of Tb3NbO7at 1.8 K, 2.5 K (between TN1 = 2.0 K and TN2 = 3.2 K), and 12.0 K.At 1.8 K and 2.5 K below TN2, there are numerous reflections that do notexist at 12.0 K. Therefore, they are magnetic reflections. We can index allmagnetic reflections using the propagation vector k1 = (0, 0, 0) at 1.8 K and8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 3: Magnetic moments of Tb3NbO7 at 1 K in the external magnetic field of 10 Tapplied parallel to the three axes calculated from the point charge model for the CEFs usingMcPhase [3, 4, 5]. The values of ⟨Mj⟩ (j = a, b, c) are the same at Tb1i and Tb1(i + 2)(i = 1, 2) and at Tb2i and Tb2(i + 4) (i = 1 − −4). The positions of Tb1(i + 2) andTb2(i + 4) are those of Tb1i plus ( 12 ,12 , 0) and those of Tb2i plus ( 12 ,12 , 0), respectively.Uncertainties due to the experimental errors in the crystal-structural parameters wereevaluated using gauss law of error propagation and were shown in parentheses only forTb11 and Tb21 for simplicity.H ∥ a position ⟨Ma⟩ ⟨Mb⟩ ⟨Mc⟩ ⟨M⟩Tb11 (x, 0, 0) 0.129(2) 0.000(0) 0.000(0) 0.129(2)Tb12 (x̄, 0, 12) 0.129 0.000 0.000 0.129Tb21 (x, y, z) 4.801(27) -6.209(91) 3.863(139) 8.748(168)Tb22 (x̄, ȳ, z + 12) 4.801 -6.209 -3.863 8.748Tb23 (x̄, y, z̄ + 12) 4.801 6.209 3.863 8.748Tb24 (x, ȳ, z̄) 4.801 6.209 -3.863 8.748H ∥ b position ⟨Ma⟩ ⟨Mb⟩ ⟨Mc⟩ ⟨M⟩Tb11 (x, 0, 0) 0.000(0) 1.394(45) -8.886(8) 8.994(46)Tb12 (x̄, 0, 12) 0.000 1.394 8.886 8.994Tb21 (x, y, z) -4.686(27) 6.394(87) -3.696(140) 8.747(167)Tb22 (x̄, ȳ, z + 12) -4.686 6.394 3.696 8.747Tb23 (x̄, y, z̄ + 12) 4.686 6.394 3.696 8.747Tb24 (x, ȳ, z̄) 4.686 6.394 -3.696 8.747H ∥ c position ⟨Ma⟩ ⟨Mb⟩ ⟨Mc⟩ ⟨M⟩Tb11 (x, 0, 0) 0.000(0) -1.234(43) 8.911(6) 8.996(43)Tb12 (x̄, 0, 12) 0.000 1.234 8.911 8.996Tb21 (x, y, z) 4.720(26) -6.075(95) 4.180(135) 8.755(167)Tb22 (x̄, ȳ, z + 12) -4.720 6.075 4.180 8.755Tb23 (x̄, y, z̄ + 12) 4.720 6.075 4.180 8.755Tb24 (x, ȳ, z̄) -4.720 -6.075 4.180 8.7559 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 k2 = (1/2, 1/2, 0) at 2.5 K. Figure 3(b) shows the patterns at 1.8 K, 2.2 K,and 2.5 K. We can see magnetic reflections belonging to k1 = (0, 0, 0) andthose belonging to k2 = (1/2, 1/2, 0) at 2.2 K, indicating coexistence of twophases. This result is consistent with the specific heat result (the first-orderphase transition at TN1 = 2.0 K). Figure 3(c) shows the patterns obtained atvarious temperatures. Diffuse magnetic scattering is observed at 3.5 K and5.0 K in the paramagnetic phase.Symmetric analyses were performed to derive the possible magnetic con-figurations of 4a and 8c sites (Tb1 and Tb2 sites, respectively) using pro-grams ISODISTORT [1] and BasIreps [2]. As shown in Table 4, four one-dimensional real IRs are possible for k1 = (0, 0, 0) at 1.8 K. Each 4a and 8csite splits into two independent orbits (without symmetry coupling betweenthem). As shown in Table 5, two IRs are possible for k2 = (1/2, 1/2, 0) atthe 2.5 K. Each 4a and 8c site splits into two and four independent orbits,respectively. All Tb1 and Tb2 ions were located at the same crystallographicsites, 4a and 8c sites, respectively. Therefore, in the refinements, we assumedthat magnitudes of components of magnetic moments were the same for allTb1 (as well Tb2) ions, for example, |v11| = |v12| = |v1|.Rietveld refinements were performed to determine the magnetic struc-tures. In the refinements, we used the crystal structural parameters deter-mined at 12.0 K. According to the results of the McPhase calculations, thec axis is the easy axis for Tb1 moments at 1 K. Therefore, candidates for IRat 1.8 K are mΓ2 and mΓ4. We found that only mΓ4 explains the patternobserved at 1.8 K. At 2.5 K, only mS2 reproduced the observed pattern. Asshown in Table 6, we succeeded in refinements at 2.5 K without Tb1 mo-ments. A Tb22 site has [u21, v21, w̄21], [u21, v21, w̄21], and [ū21, v̄21, w21] formΓ4, mS2, and mS1, respectively, indicating that mΓ4 and mS2 possess thesame symmetry. Figure 4 shows the results of the Rietveld refinement at (a)1.8 K and (b) 2.5 K. The lines on the measured patterns (circles) representthe Rietveld refined patterns including both nuclear and magnetic contribu-tions, and they could reproduce the measured patterns.Figure 5 shows the magnetic structure. The magnetic moments are pre-sented in Table 6. As expected from the results of the McPhase calculations,the main component of the Tb1 ordered moments is w at a 1.8 K. The orderof the main components in each chain along the c axis is AFM. The smallferromagnetic (FM) v components in each chain indicate canting of Tb1 or-dered moments. Tb1 moments are disordered at 2.5 K. Tb2 moments formnoncollinear magnetic structures in the ab plane perpendicular to Tb1 chains10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 02x1044x1046x1040.5 1 1.5 2 2.5 31.8 K2.512.0Intensity (arb. units) (a)02x1044x1046x1040.5 1 1.5 21.8 K2.22.5Intensity (arb. units) (b)050001x1040.5 1 1.5 21.8 K2.5� Å��Intensity (arb. units) 3.5 K5.012.0(c)(1)(1)(2)(3)(4)(5)(2)(3)Figure 3: Powder neutron-diffraction patterns of Tb3NbO7. The wavelength of the neu-trons was 2.450 Å. Indices of the major magnetic reflections are (1) 0 ± 1 1, (2) ±1 0 1,(3) ±1 ± 2 1, (4) ±2 ± 1 1, and (5) ±2 ± 3 1 for k1 = (0, 0, 0) indicated by red numbers.Indices of the major magnetic reflections are (1) ±0.5 ± 0.5 1, (2) ±0.5 ± 1.5 1, and (3)±1.5 ± 0.5 1 for k2 = (1/2, 1/2, 0) indicated by blue numbers.11 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 4: Group theory analysis for the magnetic structure of Tb3NbO7 at 1.8 K calculatedusing the programs ISODISTORT [1] and BasIreps [2]. The crystallographic space groupis C2221 (No. 20) [11]. The magnetic propagation vector is k1 = (0, 0, 0). The characterset corresponds to the following four symmetry elements [2]: Symm(1): 1; Symm(2):2 (0, 0, 1/2) 0, 0, z; Symm(3): 2 0, y, 1/4; Symm(4): 2 x, 0, 0. IR denotes irreduciblerepresentation. Tb1 on 4a site splits into two independent orbits. Tb2 on 8c site splitsinto two independent orbits. The components of the magnetic moments are expressedusing uij , vij , and wij for site i and orbit j.Character set (1, 1, 1, 1) (1, 1, -1, -1) (1, -1, 1, -1) (1, -1, -1, 1)IR (ISODISTORT) mΓ1 mΓ2 mΓ4 mΓ3IR (BasIreps) IRrep(1) IRrep(2) IRrep(3) IRrep(4)Orbit 1 Tb11 (x, 0, 0) [u11, 0, 0] [0, v11, w11] [0, v11, w11] [u11, 0, 0]Tb12 (x̄, 0, 12) [ū11, 0, 0] [0, v̄11, w11] [0, v11, w̄11] [u11, 0, 0]Orbit 2 Tb13 (x+ 12, 12, 0) [u12, 0, 0] [0, v12, w12] [0, v12, w12] [u12, 0, 0]Tb14 (x̄− 12,−12, 12) [ū12, 0, 0] [0, v̄12, w12] [0, v12, w̄12] [u12, 0, 0]Orbit 1 Tb21 (x, y, z) [u21, v21, w21] [u21, v21, w21] [u21, v21, w21] [u21, v21, w21]Tb22 (x̄, ȳ, z + 12) [ū21, v̄21, w21] [ū21, v̄21, w21] [u21, v21, w̄21] [u21, v21, w̄21]Tb23 (x̄, y, z̄ + 12) [ū21, v21, w̄21] [u21, v̄21, w21] [ū21, v21, w̄21] [u21, v̄21, w21]Tb24 (x, ȳ, z̄) [u21, v̄21, w̄21] [ū21, v21, w21] [ū21, v21, w21] [u21, v̄21, w̄21]Orbit 2 Tb25 (x+ 12, y + 12, z) [u22, v22, w22] [u22, v22, w22] [u22, v22, w22] [u22, v22, w22]Tb26 (x̄− 12, ȳ − 12, z + 12) [ū22, v̄22, w22] [ū22, v̄22, w22] [u22, v22, w̄22] [u22, v22, w̄22]Tb27 (x̄− 12, y + 12, z̄ + 12) [ū22, v22, w̄22] [u22, v̄22, w22] [ū22, v22, w̄22] [u22, v̄22, w22]Tb28 (x+ 12, ȳ − 12, z̄) [u22, v̄22, w̄22] [ū22, v22, w22] [ū22, v22, w22] [u22, v̄22, w̄22]12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 5: Group theory analysis for the magnetic structure of Tb3NbO7 at 2.5 K calculatedusing the programs ISODISTORT [1] and BasIreps [2]. The crystallographic space group isC2221 (No. 20) [11]. The magnetic propagation vector is k2 = (1/2, 1/2, 0). The characterset corresponds to the following two symmetry elements [2]: Symm(1): 1; Symm(2):2 (0, 0, 1/2) 0, 0, z. IR denotes irreducible representation. Tb1 on 4a site splits into twoindependent orbits. Tb2 on 8c site splits into four independent orbits. The componentsof the magnetic moments are expressed using uij , vij , and wij for site i and orbit j.Character set (1, 1) (1, -1)IR (ISODISTORT) mS1 mS2IR (BasIreps) IRrep(1) IRrep(2)Orbit 1 Tb11 (x, 0, 0) [u11, v11, w11] [u11, v11, w11]Tb12 (x̄, 0, 12) [ū11, v̄11, w11] [u11, v11, w̄11]Orbit 2 Tb13 (x+ 12, 12, 0) [u12, v12, w12] [u12, v12, w12]Tb14 (x̄− 12,−12, 12) [ū12, v̄12, w12] [u12, v12, w̄12]Orbit 1 Tb21 (x, y, z) [u21, v21, w21] [u21, v21, w21]Tb22 (x̄, ȳ, z + 12) [ū21, v̄21, w21] [u21, v21, w̄21]Orbit 2 Tb23 (x̄, y, z̄ + 12) [u22, v22, w22] [u22, v22, w22]Tb24 (x, ȳ, z̄) [ū22, v̄22, w22] [u22, v22, w̄22]Orbit 3 Tb25 (x+ 12, y + 12, z) [u23, v23, w23] [u23, v23, w23]Tb26 (x̄− 12, ȳ − 12, z + 12) [ū23, v̄23, w23] [u23, v23, w̄23]Orbit 4 Tb27 (x̄− 12, y + 12, z̄ + 12) [u24, v24, w24] [u24, v24, w24]Tb28 (x+ 12, ȳ − 12, z̄) [ū24, v̄24, w24] [u24, v24, w̄24]13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 6: Magnetic moments of Tb3NbO7. All Tb1 and Tb2 ions were located at thesame crystallographic sites, 4a and 8c sites, respectively. Therefore, we assumed thatmagnitudes of components of magnetic moments were the same for all Tb1 (as well Tb2)ions, for example, |v11| = |v12| = |v1|. Parameters m1 and m2 denote the magnitude ofthe magnetic moments of Tb1 and Tb2.Temperature 1.8 K 2.5 KPropagation vector k1 = (0, 0, 0) k2 = (1/2, 1/2, 0)IR mΓ4 mS2Tb11 (0.488, 0, 0) [0, v1, w1]Tb12 (−0.488, 0, 0.5) [0, v1, w̄1]Tb13 (0.988, 0.5, 0) [0, v̄1, w̄1]Tb14 (−0.988,−0.5, 0.5) [0, v̄1, w1]Tb21 (0.232, 0.235, 0.255) [u2, v2, w2] [u2, v2, w2]Tb22 (−0.232,−0.235, 0.755) [u2, v2, w̄2] [u2, v2, w̄2]Tb23 (−0.232, 0.235, 0.245) [ū2, v2, w̄2] [ū2, v2, w̄2]Tb24 (0.232,−0.235,−0.255) [ū2, v2, w2] [ū2, v2, w2]Tb25 (0.732, 0.735, 0.255) [ū2, v̄2, w̄2] [ū2, v̄2, w̄2]Tb26 (−0.732,−0.735, 0.755) [ū2, v̄2, w2] [ū2, v̄2, w2]Tb27 (−0.732, 0.735, 0.245) [u2, v̄2, w2] [u2, v̄2, w2]Tb28 (0.732,−0.735,−0.255) [u2, v̄2, w̄2] [u2, v̄2, w̄2]v1 (µB) -1.16(5)w1 (µB) 7.22(10)m1 (µB) 7.31(10)u2 (µB) -3.66(6) 3.64(8)v2 (µB) 6.48(4) 5.31(6)w2 (µB) 1.57(3) -1.76(4)m2 (µB) 7.60(6) 6.68(3)14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Intensity (arb. units)Intensity (arb. units)� Å��1.8 K (a)2.5 K (b)-1x10401x1042x1043x1040.5 1 1.5 2 2.5 3-2x10402x1044x1046x1040.5 1 1.5 2 2.5 3Figure 4: Powder neutron-diffraction pattern (circles) of Tb3NbO7. The wavelength of theneutrons was 2.450 Å. The magnetic propagation vector and IR are k1 = (0, 0, 0) and mΓ4at 1.8 K (a) and k2 = (1/2, 1/2, 0) and mS2 at 2.5 K (b). The line on the measured patternportrays the Rietveld-refined pattern including both nuclear and magnetic contributions.We used atomic parameters determined by Rietveld refinements for the pattern at 12.0K using orthorhombic C2221 (No. 20). The line at the bottom portrays the differencebetween the measured and Rietveld-refined patterns. The upper and lower hash marksrepresent the positions of the nuclear and magnetic reflections, respectively. The reliabilityfactors are χ2 = 6.81 and 12.1, Rwp = 5.01 and 6.75 %, Rexp = 1.92 and 1.94 %,RBragg = 2.02 and 3.55 %, and Rmag = 3.04 and 7.22 % at 1.8 (a) and 2.5 K (b),respectively.15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 with a small w component at both the 1.8 K and 2.5 K. The magnetic struc-ture at 1.8 K is consistent with the magnetic anisotropies obtained using theMcPhase calculations. Only Tb2 moments are ordered at 2.5 K, indicatingthe appearance of the partially disordered state observed in several frustratedantiferromagnets [12, 24, 26, 29, 22, 23, 28, 25, 27, 30]. The magnitudes ofthe Tb1 and Tb2 moments at 1.8 K are smaller than the theoretical values(gJJ = 9 µB) probably because of frustration among exchange interactions.Our results are consistent with the following results reported in [11]. Thefirst and second order phase transitions occur at TN1 = 2.0 and TN2 = 3.2 K,respectively. Tb1 and Tb2 moments are ordered at TN1 = 2.0 and TN2 = 3.2K, respectively. No spontaneous magnetization appears. Both the broadmaximum at around 4 K in the magnetic susceptibility and the diffuse mag-netic scattering at around 3.5 K and 5.0 K indicate existence of short-rangemagnetic correlations.3.4. Comparison of magnetic structures of Tb3RuO7 and Tb3NbO7Magnetic structures of Tb moments in Tb3RuO7 at 1.5 K [Fig. 1(a)] [12]and Tb3NbO7 at 1.8 K (Fig. 5) were found to be similar. This similarityindicates that the influence of Ru moments on the magnetic structure ofTb moments is weak except for the high transition temperature (TN = 17 K)because of the Tb-Ru exchange interactions in Tb3RuO7. The T dependencesof the magnitude of Tb1 moments were also found to be similar. In Tb3RuO7,the magnitude is large [8.58(2) µB] and small [1.78(12) µB] at 1.5 and 15K, respectively. In Tb3NbO7, the magnitude is large [7.97(8) µB] and zeroat 1.8 and 2.5 K, respectively (TN2 = 3.2 K). Here, we refer to Tbi withi = 1 − 2 (Tb1) and Tbi with i = 3 − 6 (Tb2) moments for Tb3RuO7(Tb3NbO7) as the chain and plane moments, respectively. The easy axisof the chain moments is parallel to the chains, whereas the plane momentsare mainly ordered perpendicular to the chains. Therefore, the exchangeinteractions between the chain and plane moments force chain moments to beperpendicular to the chains, indicating that competition between anisotropyand exchange interactions occurs. Consequently, we propose the followinghypothesis: chain moments cannot develop at 15 K and 2.5 K in Tb3RuO7and Tb3NbO7, respectively.We considered the reasons why two transitions and one transition appearin Tb3NbO7 and Tb3RuO7, respectively. In Tb3NbO7, the order of the Tb1moments is the origin of transition at TN1. In addition, the propagationvector was changed from k2 = (1/2, 1/2, 0) to k1 = (0, 0, 0). Chain and16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 (c) T = 1.8 K T = 2.5 K(b) T = 1.8 K T = 2.5 K(a) Tb3NbO7T = 1.8 K T = 2.5 KFigure 5: Magnetic structures of Tb3NbO7 shown as three dimensional (a) and as projec-tions onto the bc (b) and ab planes (c) drawn using VESTA [18]. The magnetic propagationvector and IR are k1 = (0, 0, 0) and mΓ4, respectively, at 1.8 K and k2 = (1/2, 1/2, 0) andmS2, respectively, at 2.5 K. The blue and red circles denote Tb1 and Tb2 sites, respec-tively. The vertical dashed line in (a) indicates a Tb1 chain.17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 plane moments prefer the FM and AFM arrangements, respectively, in thea and b directions. We speculate that the transition from k2 = (1/2, 1/2, 0)to k1 = (0, 0, 0) occurs because chain moments become dominant in thearrangement of moments below TN1. In Tb3RuO7, all Tb moments prefer theFM arrangements along the a and c directions (chains ∥ b) and no furthertransition occurs.4. SummaryWe determined the magnetic structures of Tb3NbO7 based on powderneutron diffraction experiments and magnetic anisotropy calculations. Wecalculated the energy levels split by CEFs and the magnetic moments us-ing McPhase. The c axis is the easy axis for Tb1 moments, and magneticanisotropy is small for Tb2 moments. Magnetic reflections were observedat low temperatures. The propagation vector is k1 = (0, 0, 0) at 1.8 K andk2 = (1/2, 1/2, 0) at 2.5 K between transition temperatures TN1 = 2.0 K andTN2 = 3.2 K. Magnetic anisotropy was used to select candidates for IRs of themagnetic structures. We determined that the IRs are mΓ4 and mS2 at 1.8 Kand 2.5 K, respectively, from Rietveld refinements. The main component ofTb1 ordered moments (chain moments) is w at 1.8 K. The order of the maincomponents in each chain along the c axis is AFM. Small FM v componentsin each chain indicate the canting of Tb1 ordered moments. Tb1 momentsare disordered at 2.5 K. Tb2 moments (plane moments) form noncollinearmagnetic structures in the ab plane perpendicular to Tb1 chains with a smallw component at both the 1.8 K and 2.5 K. The magnetic structure at 1.8 K isconsistent with the the magnetic anisotropies obtained using McPhase calcu-lations. Only Tb2 moments are ordered at 2.5 K, indicating the appearanceof the partially disordered state observed in several frustrated antiferromag-nets. Magnetic structures of Tb moments in Tb3RuO7 at 1.5 K and Tb3NbO7at 1.8 K are similar, indicating that the influence of Ru moments on magneticstructure of Tb moments is weak. We speculate that competition betweenmagnetic anisotropy and exchange interactions between chain and plane mo-ments generates the small and zero chain moments in Tb3RuO7 at 15 K andTb3NbO7 at 2.5 K, respectively. We also speculate that the transition fromk2 = (1/2, 1/2, 0) to k1 = (0, 0, 0) at TN1 in Tb3NbO7 occurs because thechain moments become dominant for the arrangement of moments below TN1.18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 AcknowledgmentsThis study was supported by Japan Society for the Promotion of Science(JSPS) KAKENHI Grant No. 18K03551. JST-Mirai Program Grant No.JPMJMI18A3, Japan, and the World Premier International Research CenterInitiative (WPI), Ministry of Education, Culture, Sports, Science and Tech-nology (MEXT), Japan. This study is partially based on the experimentsperformed on the HRPT diffractometer (Proposal No. 20212393) at the SwissSpallation Neutron Source SINQ, the Paul Scherrer Institute, Switzerland.We thank Seiko Matsumoto at the National Institute for Materials Science(NIMS) for the sample synthesis and X-ray diffraction measurements. Wealso thank Masamichi Nishino, Kazunari Yamaura Alexei Belik, and Yoshi-hiro Tsujimoto at NIMS for their insightful discussions.References[1] B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, ISODIS-PLACE: An internet tool for exploring structural distortions, J. Appl.Cryst. 39 (2006) 607; [https://stokes.byu.edu/iso/isodistort.php].[2] J. Rodriguez-Carvajal, Recent advances in magnetic structure deter-mination by neutron powder diffraction, Physica B 192 (1993) 55;[http://www.ill.eu/sites/fullprof/].[3] M. Rotter, Using McPhase to calculate magnetic phase diagrams of rareearth compounds, J. Magn. Magn. Mater. 272 (2004) e481.[4] M. Rotter, M. 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