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Keller, Lukas, [Tanaka, Akihiro](https://orcid.org/0000-0001-8049-0860), [Hase, Masashi](https://orcid.org/0000-0003-2717-461X), Stuhr, Uwe, [Kohno, Masanori](https://orcid.org/0000-0001-5900-9139), Pomjakushin, Vladimir Yu., [Doenni, Andreas](https://orcid.org/0000-0002-7300-9175)

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[Evaluation of field-induced magnetic moments in the spin-1/2 antiferromagnetic trimerized chain compound Cu3(P2O6OD)2](https://mdr.nims.go.jp/datasets/f909d7b0-1974-4ee8-b741-463b8f3f16b4)

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PHYSICAL REVIEW B 102, 014403 (2020)Evaluation of field-induced magnetic moments in the spin-12 antiferromagnetictrimerized chain compound Cu3(P2O6OD)2Masashi Hase ,1,* Vladimir Yu. Pomjakushin,2 Lukas Keller,2 Uwe Stuhr,2Andreas Dönni ,3 Masanori Kohno ,3 and Akihiro Tanaka31Research Center for Advanced Measurement and Characterization, National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan2Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut (PSI), CH-5232 Villigen PSI, Switzerland3International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science (NIMS),1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan(Received 9 December 2019; revised 17 June 2020; accepted 18 June 2020; published 6 July 2020)We report on our evaluation of field-induced magnetic moments in the paramagnetic state of the spin- 12antiferromagnetic trimerized (J1 − J2 − J2) chain compound Cu3(P2O6OD)2 with J1 = 111 K and J2 = 30 K[M. Hase et al., Phys. Rev. B 76, 064431 (2007)]. Magnetic reflections with integer indices, generated by field-induced magnetic moments, were observed at neutron-diffraction experiments performed in an applied magneticfield, and we evaluated the magnitudes of the moments to be M1 = 0.43(2) μB/Cu and M2 = 0.013(10) μB/Cuon the two crystallographic Cu2+ (Cu1 and Cu2) sites, respectively, at 6 T and 1.6 K. The resulting ratioM2/M1 = 0.03(2) is in good agreement with the theoretical value for the ratio of the magnetization at thetwo sites. We thus conclude that for this well-understood spin system (which has the advantage of beingamenable to detailed theoretical calculations) the extracted field-induced magnetic moments reproduce thecorrect information on the magnetization. Our result leads us to believe that we can precisely evaluate exchangeinteractions in paramagnets with multiple exchange interactions and multiple crystallographic magnetic-ion sitesby using field-induced magnetic moments in combination with macroscopic physical quantities. This idea isexpected to be applicable to a wide variety of quantum and frustrated magnets without long-range order.DOI: 10.1103/PhysRevB.102.014403I. INTRODUCTIONThe precise knowledge on exchange interactions is clearlya prerequisite toward understanding the magnetic proper-ties of quantum spin systems. Exchange interactions areoften evaluated using macroscopic physical quantities suchas magnetization and specific heat. There are cases, however,in which the use of such measurements alone can lead toambiguity, where the results of experiments are seeminglyaccountable by several alternative sets of exchange interac-tions, as is seen in the spin- 12 diamond chain compoundCu3(CO3)2(OH)2 (azurite) [1–5].We can evaluate exchange interactions in a more precisemanner by combing neutron-scattering results and macro-scopic physical quantities. Exchange interactions can be reli-ably determined through the dispersion relations of magneticexcitations that are observed in inelastic neutron-scattering(INS) experiments, especially after having gained knowledgefrom the macroscopic physical quantities and the crystalstructure as to which exchange interactions are likely to bedominant. A major setback of this scheme, however, is thatINS experiments usually require the use of large single crys-tals, limiting its availability. Inputs from neutron-diffraction*HASE.Masashi@nims.go.jpexperiments on ordered magnetic moments are also useful, asthey can be used to clarify the signs of the exchange inter-actions and the ratios among the exchange-interaction values.This approach was applied, e.g., to the spin tetramer (four-spinsystem) compounds Cu2Fe2Ge4O13 [6,7] and Cu2CdB2O6[8]. The powerfulness of this strategy may be highlightedin particular through its application to the latter compound,where it enabled the authors to correct a previous assignment[9] of a wrong sign to one of the exchange constants, resultingin a totally new (and correct) set of exchange interactions [8].The method, however, is apparently inapplicable to paramag-nets as they have no ordered magnetic moments.In a similar vein, it would seem natural to expect that thefield-induced magnetic moments determined using neutron-diffraction experiments in magnetic fields should be usefulfor evaluating the exchange interactions of paramagnets. Letus take, for example, the trimerized (J1 − J2 − J2) spin chaindepicted in Fig. 1 and consider its response to a magnetic field.First consider the case where the antiferromagnetic (AFM) J1interaction is dominant. The Cu2 spins will in this case pair upinto AFM dimers. Upon applying a magnetic field, the field-induced magnetic moments on both the Cu1 and Cu2 siteswill align parallel to the magnetic fields, as also illustrated inFig. 1. In weak magnetic fields, the magnitude of the momenton Cu1 sites is greater than that on Cu2 sites. On the otherhand, if the AFM J2 interaction is dominant, AFM trimers2469-9950/2020/102(1)/014403(6) 014403-1 ©2020 American Physical SocietyMASASHI HASE et al. PHYSICAL REVIEW B 102, 014403 (2020)ab c Cu2 Cu1OJ1 J2Jeff = J22/2J1HM2M1FIG. 1. Spin system and expected field-induced magnetic mo-ments in Cu3(P2O6OD)2 [10,11]. Cu2+ ions (3d9) have a localizedspin (S = 12 ). Two crystallographic Cu (Cu1 and Cu2) sites exist[12]. The red and blue bars represent the shortest and second-shortestCu-Cu pairs, whose Cu-Cu lengths are 3.06 Å and 3.28 Å at roomtemperature, respectively. The other Cu-Cu lengths are 4.27 Å orlonger. The J1 and J2 exchange interactions in the shortest andsecond-shortest Cu-Cu pairs, respectively, form a spin- 12 trimerized(J1 − J2 − J2) chain whose Hamiltonian is expressed as follows:H = ∑i J1SiSi+1 + J2Si+1Si+2 + J2Si+2Si+3. The J1 and J2 interac-tions were evaluated to be 111 and 30 K, respectively [11]. The redellipse denotes a spin-singlet-like pair at an AFM dimer. The spins(black arrows) on the Cu1 sites are coupled to one another by the Jeffinteraction (Jeff = J222J1= 4.1 K). The arrows at the bottom depict thefield-induced magnetic moments, M1 and M2, on the Cu1 and Cu2sites, respectively.(three-spin systems) are formed. In weak magnetic fields, thefield-induced magnetic moments on Cu1 and Cu2 sites areantiparallel and parallel to the magnetic fields, respectively.There are several previous reports on the evaluation offield-induced magnetic moments in paramagnets. Paschenet al. showed that the Ce moments at the 4a site weresmaller by at least a factor of 2 than those at the 8c sitesin paramagnetic Ce3Pd20Si6 [13]. In the spin- 52 AFM trimercompound SrMn3P4O14 [14], we evaluated the field-inducedmagnetic moments at 6 T and 1.6 K, where a 13 quan-tum magnetization plateau appeared [15]. The magnetizationplateau indicated that the ground state (GS) was a sort offully polarized paramagnetic state. The Hamiltonian of a spintrimer is expressed as follows: H = J (S1S2 + S2S3). In theplateau GS, the quantum-mechanical values were calculatedto be S1z = S3z = 157 and S2z = − 2514 . We confirmed that theexperimentally evaluated field-induced magnetic moment oneach site was the same as gSjz, where g denoted the g factor.To demonstrate that the exchange interactions can beprecisely evaluated in paramagnets by using field-inducedmagnetic moments in combination with macroscopic physi-cal quantities such as magnetization and specific heat, it isnecessary to confirm whether the experimental values of field-induced magnetic moments are consistent with the calculatedones of magnetizations in well-understood spin systems. Tothis end we focus in this study on Cu3(P2O6OD)2. No sig-nature of magnetic long-range order (LRO) appears down to2 K in the specific-heat result [10]. As detailed in Fig. 1,the spin degrees of freedom of this material form a spin- 12AFM trimerized (J1 − J2 − J2) chain. The AFM exchangeinteractions were evaluated to be J1 = 111 K and J2 = 30 K[11]. The spins on the Cu2 sites are coupled by the mostdominant J1 interaction and form AFM spin dimers. Thespins on the neighboring Cu1 sites in a chain are weaklyand antiferromagnetically coupled to one another throughthe intermediate AFM spin dimer and form an AFM spinchain whose effective interaction (Jeff ) is expressed as J222J1.Therefore the magnetizations on the Cu2 and Cu1 sites aresmall and large, respectively. The calculated magnetizationsare shown in Fig. 4(b) of Ref. [10]. The magnetization of Cu2is approximately zero in weak magnetic fields, whereas thatof Cu1 appears even in significantly weak magnetic fields. Aswe now describe in detail, we performed neutron-diffractionexperiments on Cu3(P2O6OD)2 polycrystalline pellets, exper-imentally evaluated the field-induced magnetic moments (M1and M2 on the Cu1 and Cu2 sites, respectively), and comparedthe values with the calculated magnetizations on the Cu sites.II. METHODS OF EXPERIMENTS AND CALCULATIONWe synthesized a crystalline powder of Cu3(P2O6OD)2from a mixture of CuO (5 g) and D3PO4-D2O (200 mL) [12].The mixture was stirred continuously while heating it untilthe CuO was completely dissolved. Subsequently, the mixturewas placed in a furnace in air at 463 K for 48 h. Consequently,Cu3(P2O6OD)2 appeared as a light blue powder. We measuredthe x-ray powder diffraction pattern at room temperature(T ) using a diffractometer (Rigaku RINT-TTR III). Mostreflections originated from Cu3(P2O6OD)2. In addition, wedetected significantly weak reflections of other materials.We performed neutron-diffraction experiments at the Swissspallation neutron source (SINQ) in the Paul Scherrer Institut(PSI), where we used the high-resolution powder diffractome-ter for thermal neutrons (HRPT) [16] and the high-intensitycold neutron powder diffractometer (DMC). The wavelengthsof the neutrons were λ = 1.89 and 4.51 Å for the HRPT andDMC experiments, respectively. We applied magnetic fieldsin the range of μ0H = 0−6 T almost perpendicularly to thescattering vectors (Q) by using a superconducting magnet.We used pressed polycrystalline pellets of Cu3(P2O6OD)2with a diameter of 8 mm to minimize the problem of powderrealignment in the presence of strong magnetic fields.We performed the Rietveld refinements of the crystal struc-ture using the FULLPROF SUITE program package [17]. Wedescribe the calculation method for the integrated intensitiesof the magnetic reflections generated by the field-inducedmagnetic moments in Sec. III.We calculated the magnetic-field dependence of the mag-netization of the spin- 12 AFM trimerized chain by quantumMonte Carlo (QMC) techniques using the directed-loop algo-rithm in the path-integral formulation [18]. The number of Cusites in the QMC simulations was 120. We performed morethan one million updates.III. RESULTS AND DISCUSSIONThe circles in Fig. 2(a) depict the neutron-diffraction pat-tern of Cu3(P2O6OD)2 pellets at 0 T and 1.8 K measured usingthe HRPT diffractometer. The wavelength of the neutrons (λ)was 1.89 Å. We performed Rietveld refinements using P1 (no.2) to evaluate the crystal-structure parameters. The line on the014403-2EVALUATION OF FIELD-INDUCED MAGNETIC MOMENTS … PHYSICAL REVIEW B 102, 014403 (2020)FIG. 2. Neutron-diffraction patterns (circles) of Cu3(P2O6OD)2polycrystalline pellets (a) measured using the HRPT diffractometer(λ = 1.89 Å) at 0 T and 1.8 K and measured using the DMCdiffractometer (4.51 Å) at (b) 0 T and 1.6 K, and (c) 6 T and20 K. The line on the measured pattern portrays the Rietveld-refinedpattern obtained using the crystal structure with P1 (no. 2). Theline at the bottom portrays the difference between the measured andRietveld-refined patterns. The hash marks represent the positions ofnuclear reflections.experimental pattern indicates the result of the refinements.The line is in good agreement with the experimental pattern.The values of the crystal-structure parameters presented inTable I are consistent with those obtained in the previousrefinements of an x-ray diffraction pattern [12]. Therefore weconsider that no alignment of the powder occurs during theproduction of the pellets. In this study, we could determinethe deuterium position.The circles in Figs. 2(b) and 2(c) depict the neutron-diffraction patterns of Cu3(P2O6OD)2 pellets at 0 T and 1.6 Kand at 6 T and 20 K, respectively, measured using the DMCdiffractometer (λ = 4.51 Å). We performed Rietveld refine-ments using the values of the crystal-structure parameters pre-sented in Table I. The results of the refinements are also shownin Figs. 2(b) and 2(c). The refined pattern is in good agreementwith the experimental one in each figure. Consequently, wecould not detect the magnetic reflections at either 0 T and1.6 K or at 6 T and 20 K within experimental accuracy. Theagreement between the refined and experimental patterns inFig. 2(c) indicates that no alignment of the powder in thepellets occurred upon the application of magnetic fields.TABLE I. Values of crystal-structural parameters ofCu3(P2O6OD)2 derived from the Rietveld refinements of theHRPT neutron powder diffraction pattern at 0 T and 1.8 K.The term Biso denotes the isotropic atomic displacementparameter. We used monoclinic P1 (no. 2). The lattice constantsare a = 4.7735(1) Å, b = 7.0365(2) Å, c = 8.3315(2) Å,α = 66.570(1)◦, β = 77.063(1)◦, and γ = 72.001(1)◦. Theestimated standard deviations are shown in the parentheses. Thereliability indices of the refinements are Rp = 4.74%, Rwp = 6.07%,Rexp = 1.02%, and χ 2 = 35.7.Atom Site x y z Biso A2Cu1 1a 0 0 0 0.07(4)Cu2 2i 0.9389(6) 0.8461(4) 0.4291(4) 0.07(4)P1 2i 0.6339(9) 0.5084(7) 0.7072(6) 0.52(6)P2 2i 0.6498(9) 0.8405(7) 0.8133(6) 0.52(6)O1 2i 0.6842(9) 0.6617(6) 0.5177(4) 0.23(3)O2 2i 0.8186(9) 0.9445(6) 0.6332(4) 0.23(3)O3 2i 0.2930(8) 0.5142(6) 0.7427(5) 0.23(3)O4 2i 0.6921(8) 0.6003(5) 0.8364(5) 0.23(3)O5 2i 0.7779(9) 0.8280(6) 0.9679(5) 0.23(3)O6 2i 0.1759(8) 0.7199(6) 0.2529(4) 0.23(3)O7 2i 0.3171(8) 0.9452(6) 0.8119(5) 0.23(3)D1 2i 0.2499(9) 0.3899(6) 0.8460(5) 1.94(9)The circles in Fig. 3(a) depict the difference pattern at 6 Tobtained by subtracting the pattern at 20 K from that at 1.6 K.We can see several reflections that are not artifacts from thesubtraction of the data because of the following findings. Thediffraction pattern at 6 T and 20 K is depicted in Fig. 3(b).We can see the reflections of other materials betweenFIG. 3. (a) Difference pattern at 6 T obtained by subtractingthe pattern at 20 K from that at 1.6 K. We labeled the indices ofseveral magnetic reflections generated by field-induced magneticmoments. The blue lines show the calculated reflections. (b) Theneutron-diffraction pattern at 6 T and 20 K. It is a magnified versionof that in Fig. 2(c). The hash marks represent the positions of thenuclear reflections of Cu3(P2O6OD)2.014403-3MASASHI HASE et al. PHYSICAL REVIEW B 102, 014403 (2020)Q = 1.14 and 1.37 Å−1. However, the reflections are not seenin Fig. 3(a), indicating that the subtraction was appropriatelyperformed. In Fig. 3(a), the reflections at 001, 010, and 011are clearly prominent. The intensity of the 001 reflection ishighest, and those of the 010 and 011 reflections are approxi-mately equal to each other. In addition, the reflections at 100,111, 110, and 0−11 may exist.We deduce that the reflections are generated by field-induced magnetic moments because of the following findings.The magnetic susceptibility is finite at low T [10]. No mag-netic LRO, however, appears in weak magnetic fields at lowT , indicating that the system can be effectively regarded asalmost decoupled spin chains, as depicted in Fig. 1. Therefore,a magnetic LRO dose not occur at 6 T and low T , and it cannotbe the origin of the reflections.We calculated the integrated intensities [IM(Q)] of themagnetic reflections generated by field-induced magnetic mo-ments as follows:IM(Q) = A|FM(Q)|2nQ1sin 2θ sin θ, (1)whereFM(Q) = −12γ e2mec2∑jf (Q) jm j exp(iQ · r j )× exp(−Bjsin2 θλ2). (2)The coefficient A in Eq. (1) denotes a scaling factor that iscommon to the magnetic and nuclear reflections. We evaluatedA to be 1.008 × 1028 cm−2 using the Rietveld refinements forthe crystal structure at 6 T and 20 K depicted in Fig. 2(c).The term nQ denotes the number of reflections with the sameintensity and |Q|. The term 2θ denotes the scattering angle.The value of γ e2mec2 is 5.39 × 10−13 cm. The summation inEq. (2) is performed within the unit cell. The terms r j , f (Q) j ,mj , and Bj denote the position of the jth site ion, magneticform factor [19], field-induced magnetic moment, and atomicdisplacement parameter, respectively. We used the atomicpositions and values of Bj listed in Table I.Here we explain the reason for our use of the scalar mjinstead of the vector m j⊥Q, where m and m j⊥Q denote themagnetic-moment vector and the perpendicular componentof m to Q, respectively. Because the perpendicular compo-nents contribute to the magnetic reflections, m j⊥Q is used tocalculate the magnetic reflections generated by the magneticLRO. In this study we applied the magnetic fields almostperpendicular to Q. The field-induced magnetic moments arealways almost perpendicular to Q. Therefore we input thescalar mj in Eq. (2).The calculated intensity ratio of the magnetic reflectionsis depicted in Fig. 4. The intensity ratios strongly dependon M2/M1. From the experimental value of I (001)/I (010) =1.24(14), indicated by the horizontal line, we determinedM2/M1 to be 0.03(2). The blue lines in Fig. 3(a) depictthe magnetic reflections calculated for M1 = 0.43 μB/Cuand M2 = 0.013 μB/Cu (M2/M1 = 0.03), and they can ex-plain the experimental results. The calculated intensity ratioI (011)/I (010), indicated using the squares in Fig. 4, decreases0.60.811.21.40 0.05 0.1Relative intensityM2/M1I(001)/I(010)I(011)/I(010)Expt.FIG. 4. Calculated intensity ratio of the magnetic reflections vsM2/M1. The horizontal line indicates the experimental value ofI (001)/I (010) = 1.24.with increasing M2/M1, thereby indicating that the M2/M1value of 0.05 or more cannot explain the experimental inten-sity ratio of I (011)/I (010). We estimated the error bars ofM1 and M2 to be 0.02 μB/Cu and 0.01 μB/Cu, respectively,by using the error bar of M2/M1 (0.02). As expected forCu3(P2O6OD)2, the field-induced magnetic moments on theCu1 and Cu2 sites are large and small, respectively.We calculated the magnetizations of Cu1 and Cu2 to be0.525 μB/Cu and 0.016 μB/Cu, respectively, in the spin- 12AFM trimerized chain with J1 = 111 K and J2 = 30 K at6 T and 1.6 K by using the QMC techniques. The ratiobetween both the magnetizations is 0.03 and is consistentwith M2/M1. The value of M1 [0.43(2) μB/Cu], however, issmaller than that of the magnetization of Cu1 (0.525 μB/Cu).We consider the reason for the difference in the following.Similar results were obtained in SrMn3P4O14 [15]. As previ-ously described, the field-induced magnetic moment on eachMn site evaluated in the experiments is the same as thecorresponding calculated magnetization at 6 T and 1.6 K,where the 13 quantum magnetization plateau appears. Asdepicted in Fig. 4(c) in [15], on the other hand, the nor-malized intensity of the magnetic reflection at 011 generatedby the field-induced magnetic moments is smaller than thesquare of the magnetization above 7 K, thereby indicatingthat the magnetization comprises the field-induced magneticmoments and fluctuating components. In Cu3(P2O6OD)2, themagnetization increases with increasing μ0H at 6 T and1.6 K and therefore comprises the field-induced magneticmoments and fluctuating components. Accordingly, the valueof M1 is smaller than that of the magnetization of Cu1. A 13quantum magnetization plateau appears in the magnetic fieldsof μ0H > 12 T at 1.6 K in Cu3(P2O6OD)2. We expect thevalue of the field-induced magnetic moment on each site to bethe same as that of the calculated magnetization on the corre-sponding site in the plateau magnetic fields, as in SrMn3P4O14[15].Let us now mention some noteworthy characteristics of ourmethod. We could evaluate significantly small M2, althoughthe error bar was considerable. As previously described, in thecase wherein the magnetic fields are applied perpendicularlyto Q, the field-induced magnetic moments are also perpen-dicular to Q. Because the magnetic moments perpendicular to014403-4EVALUATION OF FIELD-INDUCED MAGNETIC MOMENTS … PHYSICAL REVIEW B 102, 014403 (2020)Q contribute to the magnetic reflections, small field-inducedmagnetic moments can generate large magnetic reflections. Ina polycrystalline sample, magnetic reflections of field-inducedmagnetic moments are roughly 4π times larger than those ofordered magnetic moments in a long-range order. Here, thefactor 4π is the surface area of the unit sphere originatingin the random direction of powder. For example, when thedetection limit of ordered magnetic moments is 0.2 μB, that offield-induced magnetic moments is 0.2/√4π ∼ 0.06 μB. Wecan evaluate field-induced magnetic moments of many para-magnets except for spin-gap systems having a nonmagneticground state. In ordinary diffraction measurements, reflec-tions are smaller in a polycrystalline sample than in a single-crystal one because of the random direction of powder in theformer. The direction of field-induced magnetic moments isparallel to the applied magnetic fields and does not dependon the crystal axes. Therefore the intensities of the magneticreflections generated by the field-induced magnetic momentsin a polycrystalline sample are comparable with those ina single crystal. Single crystals are not necessary for ourmethod.We believe that it is possible to precisely evaluate ex-change interactions in paramagnets with multiple exchangeinteractions and multiple crystallographic magnetic-ion sitesby using field-induced magnetic moments in combinationwith macroscopic physical quantities because we could pre-cisely evaluate the exchange interactions by using informa-tion regarding ordered magnetic moments in conjunctionwith the magnetic-susceptibility and magnetization resultsin Cu2CdB2O6 [8,9]. This idea is applicable to research ofseveral spin systems such as diamond chain [1–5], spin clus-ter [20], distorted kagome lattice [21], and three-leg ladder[22,23]. Evaluation of field-induced magnetic moments is alsouseful in research of magnets containing both transition-metaland rare-earth magnetic ions. It is difficult to evaluate the mag-netization of transition-metal ions because it is usually muchsmaller than the magnetization of rare-earth ions. We candirectly study the spin system of transition-metal ions becausewe can separately evaluate field-induced magnetic momentsof transition-metal ions. Many multiferroic materials containboth transition-metal and rare-earth magnetic ions [24]. Thisidea can contribute to studies of multiferroic materials.IV. CONCLUSIONWe carried out neutron-diffraction experiments in mag-netic fields on the paramagnetic state of the spin- 12 AFMtrimerized (J1 − J2 − J2) chain compound Cu3(P2O6OD)2with J1 = 111 K and J2 = 30 K. Magnetic reflections withinteger indices appeared at 6 T and 1.6 K, and they wereattributed to the field-induced magnetic moments. The mag-nitudes of the field-induced magnetic moments were M1 =0.43(2) μB/Cu and M2 = 0.013(10) μB/Cu on two crystal-lographic Cu2+ (Cu1 and Cu2) sites, respectively, at 6 Tand 1.6 K. The value of M2/M1 was 0.03(2), and it is thesame as the ratio of magnetizations of Cu2 and Cu1 sitescalculated for the chain with J1 = 111 K and J2 = 30 K. Weconfirmed that the information regarding the field-inducedmagnetic moments was consistent with that regarding thecalculated magnetizations in the well-understood spin system(spin chain). We believe that it is possible to precisely evaluateexchange interactions in paramagnets with multiple exchangeinteractions and multiple crystallographic magnetic-ion sitesby using field-induced magnetic moments in conjunction withmacroscopic physical quantities. Evaluation of field-inducedmagnetic moments is useful in research of a wide variety ofquantum and frustrated magnets without long-range order.ACKNOWLEDGMENTSThis work was supported by the Japan Society for the Pro-motion of Science (JSPS) KAKENHI Grant No. 18K03551,a grant for advanced measurement and characterization tech-nologies accelerating materials innovation (PF2050) from theNational Institute for Materials Science (NIMS), and JST-Mirai Program Grant No. JPMJMI18A3, Japan. The travelexpenses for the neutron-diffraction experiments were par-tially supported by the General User Program for NeutronScattering Experiments, Institute for Solid State Physics, theUniversity of Tokyo (Proposals No. 16801 and No. 18804),at JRR-3, Japan Atomic Energy Agency, Tokai, Japan. Theexperiments were transferred from 5G:PONTA at JRR-3. Weare grateful to Takashi Mochiku, Hiroaki Mamiya, MasamichiNishino, and Noriki Terada at NIMS for fruitful discussionsand to Seiko Matsumoto at NIMS for the sample synthesesand x-ray diffraction measurements.[1] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T.Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta,Experimental Observation of the 13 Magnetization Plateau in theDiamond-Chain Compound Cu3(CO3)2(OH)2, Phys. Rev. Lett.94, 227201 (2005).[2] B. Gu and G. Su, Comment on “Experimental Observa-tion of the 13 Magnetization Plateau in the Diamond-ChainCompound Cu3(CO3)2(OH)2),”Phys. Rev. Lett. 97, 089701(2006).[3] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T.Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta,Experimental Observation of the 13 Magnetization Plateau inthe Diamond-Chain Compound Cu3(CO3)2(OH)2–Reply, Phys.Rev. Lett. 97, 089702 (2006).[4] J. Kang, C. Lee, R. K. Kremer, and M.-H. Whangbo, Conse-quences of the intrachain dimer-monomer spin frustration andthe interchain dimer-monomer spin exchange in the diamond-chain compound azurite Cu3(CO3)2(OH)2, J. Phys.: Condens.Matter 21, 392201 (2009).[5] H. Jeschke, I. Opahle, H. Kandpal, R. Valentí, H. Das,T. Saha-Dasgupta, O. Janson, H. Rosner, A. Brühl, B.Wolf, M. Lang, J. Richter, S. Hu, X. Wang, R. Peters, T.Pruschke, and A. Honecker, Multistep Approach to Micro-scopic Models for Frustrated Quantum Magnets: The Caseof the Natural Mineral Azurite, Phys. Rev. Lett. 106, 217201(2011).[6] T. Masuda, A. Zheludev, B. Grenier, S. Imai, K. Uchinokura, E.Ressouche, and S. Park, Cooperative Ordering of Gapped and014403-5MASASHI HASE et al. PHYSICAL REVIEW B 102, 014403 (2020)Gapless Spin Networks in Cu2Fe2Ge4O13, Phys. Rev. Lett. 93,077202 (2004).[7] M. Matsumoto, H. Kuroe, T. Sekine, and T. Masuda, Transverseand longitudinal excitation modes in interacting multispin sys-tems, J. Phys. Soc. Jpn. 79, 084703 (2010).[8] M. Hase, A. Dönni, V. Yu. Pomjakushin, L. Keller, F. Gozzo, A.Cervellino, and M. Kohno, Magnetic structure of Cu2CdB2O6exhibiting a quantum-mechanical magnetization plateau andclassical antiferromagnetic long-range order, Phys. Rev. B 80,104405 (2009).[9] M. Hase, M. Kohno, H. Kitazawa, O. Suzuki, K. Ozawa, G.Kido, M. Imai, and X. Hu, Coexistence of a nearly spin-singletstate and antiferromagnetic long-range order in quantum spinsystem Cu2CdB2O6, Phys. Rev. B 72, 172412 (2005).[10] M. Hase, M. Kohno, H. Kitazawa, N. Tsujii, O. Suzuki,K. Ozawa, G. Kido, M. Imai, and X. Hu, 13 magnetiza-tion plateau observed in the spin-1/2 trimer chain compoundCu3(P2O6OH)2, Phys. Rev. B 73, 104419 (2006).[11] M. Hase, M. Matsuda, K. Kakurai, K. Ozawa, H. Kitazawa, N.Tsujii, A. Dönni, M. Kohno, and X. Hu, Direct observation ofthe energy gap generating the 13 magnetization plateau in thespin-1/2 trimer chain compound Cu3(P2O6OD)2 by inelasticneutron scattering measurements, Phys. Rev. B 76, 064431(2007).[12] R. Baies, V. Caignaert, V. Pralong, and B. Raveau, Copper hy-droxydiphosphate with a one-dimensional arrangement of cop-per polyhedra: Cu3[P2O6OH]2, Inorg. Chem. 44, 2376 (2005).[13] S. Paschen, S. Laumann, A. Prokofiev, A. M. Strydom, P. P.Deen, J. R. Stewart, K. Neumaier, A. Goukassov, and J.-M.Mignot, First neutron measurements on Ce3Pd20Si6, Physica B403, 1306 (2008).[14] M. Hase, T. Yang, R. Cong, J. Lin, A. Matsuo, K. Kindo,K. Ozawa, and H. Kitazawa, High-field magnetization ofSrMn3P4O14 exhibiting a quantum-mechanical magnetizationplateau and classical magnetic long-range order, Phys. Rev. B80, 054402 (2009).[15] M. Hase, V. Yu. Pomjakushin, A. Dönni, T. Yang, R. Cong, andJ. Lin, Direct observation of the ground state of a 13 quantummagnetization plateau in SrMn3P4O14 using neutron diffractionmeasurements, J. Phys. Soc. Jpn. 83, 104701 (2014).[16] P. Fischer, G. Frey, M. Koch, M. Koennecke, V. Pomjakushin, J.Schefer, R. Thut, N. Schlumpf, R. Buerge, U. Greuter, S. Bondt,and E. Berruyer, High-resolution powder diffractometer HRPTfor thermal neutrons at SINQ, Physica B 276, 146 (2000); http://sinq.web.psi.ch/hrpt.[17] J. Rodriguez-Carvajal, Recent advances in magnetic structuredetermination by neutron powder diffraction, Physica B 192,55 (1993); http://www.ill.eu/sites/fullprof/.[18] O. F. Syljuåsen and A. W. Sandvik, Quantum Monte Carlo withdirected loops, Phys. Rev. E 66, 046701 (2002).[19] S. W. Lovesey, Theory of Neutron Scattering from CondensedMatter (Oxford University Press, Oxford, 1984), Vol. 2.[20] B. Koteswararao, P. Khuntia, R. Kumar, A. V. Mahajan, A.Yogi, M. Baenitz, Y. Skourski, and F. C. Chou, Bose-Einsteincondensation of triplons in the S = 1 tetramer antiferromagnetK2Ni2(MoO4)3: A compound close to a quantum critical point,Phys. Rev. B 95, 180407 (2017).[21] K. Matan, T. Ono, Y. Fukumoto, T. J. Sato, J. Yamaura, M.Yano, K. Morita, and H. Tanaka, Pinwheel valence-bond solidand triplet excitations in the two-dimensional deformed kagomelattice, Nat. Phys. 6, 865 (2010).[22] S. Vilminot, M. Richard-Plouet, G. André, D. Swierczynski,M. Guillot, F. Bourée-Vigneron, and M. Drillon, Magneticstructure and properties of Cu3(OH)4SO4 made of triple chainsof spins s = 12 , J. Solid State Chem. 170, 255 (2003).[23] S. Vilminot, G. André, F. Bourée-Vigneron, M. Richard-Plouet, and M. Kurmoo, Magnetic properties and magneticstructures of Cu3(OD)4XO4, X = Se or S: Cycloidal versuscollinear antiferromagnetic structure, Inorg. Chem. 46, 10079(2007).[24] W. Eerenstein, N. D. Mathur, and J. F. Scott, Multiferroic andmagnetoelectric materials, Nature (London) 442, 759 (2006).014403-6