# Fileset

[PhysRevB.109.094434.pdf](https://mdr.nims.go.jp/filesets/9d9db154-73c6-4689-ba5a-ca79a066c4ab/download)

## Creator

[Masashi Hase](https://orcid.org/0000-0003-2717-461X), [Ryo Tamura](https://orcid.org/0000-0002-0349-358X), Koji Hukushima, Shinichiro Asai, Takatsugu Masuda, Shinichi Itoh, [Andreas Dönni](https://orcid.org/0000-0002-7300-9175)

## Rights

[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Inelastic neutron scattering studies on the eight-spin zigzag-chain compound <math>  <mrow>    <msub>      <mi>KCu</mi>      <mn>4</mn>    </msub>    <msub>      <mi>P</mi>      <mn>3</mn>    </msub>    <msub>      <mi>O</mi>      <mn>12</mn>    </msub>  </mrow></math>: Confirmation of the validity of a data-driven technique based on machine learning](https://mdr.nims.go.jp/datasets/3f003f55-4b7f-422c-b111-fb08dc043703)

## Fulltext

Inelastic neutron scattering studies on the eight-spin zigzag-chain compound ${\rm KCu}_4{\rm P}_3{\rm O}_{12}$: Confirmation of the validity of a data-driven technique based on machine learningPHYSICAL REVIEW B 109, 094434 (2024)Inelastic neutron scattering studies on the eight-spin zigzag-chain compound KCu4P3O12:Confirmation of the validity of a data-driven technique based on machine learningMasashi Hase ,1,* Ryo Tamura ,2,3 Koji Hukushima ,4,5 Shinichiro Asai,6Takatsugu Masuda,6 Shinichi Itoh,7 and Andreas Dönni 11Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan2Center for Basic Research on Materials (CBRM), National Institute for Materials Science (NIMS),1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan3Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan4Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan5Department of Basic Science, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan6The Institute for Solid State Physics (ISSP), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan7Institute of Materials Structure Science (IMSS), High Energy Accelerator Research Organization (KEK),1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan(Received 16 October 2023; accepted 8 March 2024; published 25 March 2024)We performed inelastic neutron scattering (INS) experiments on KCu4P3O12 powder and compared theexperimental results with those calculated for the spin model (an eight-spin zigzag chain with S = 12 ) usingthe data-driven technique based on machine learning. We observed magnetic excitations at approximately 3.0,4.1, 5.9, and 8.8 meV at 5.5 K and at approximately 3.8 and 5.9 meV at 49 K. The excitations correspondingto 3.0, 4.1, and 8.8 meV were magnetic excitations from the ground state to the first, second, and fourth excitedstates (2.87, 4.23, and 8.53 meV from the calculations), respectively. The excitations corresponding to 3.8and 5.9 meV were magnetic excitations from the first excited state to the third and fourth excited states (3.78and 5.67 meV from the calculations), respectively. An excitation was likely to exist between the first and secondexcited states at approximately 1.35 meV in the experimental results. The excitation energies obtained fromthe INS experiments were almost consistent with those calculated from the exchange interaction values via thedata-driven technique (data-driven values). The experimental I (Q) curves could not be reproduced. We found thatI (Q) curves could be changed largely by small changes of exchange-interaction values. Therefore, we expect thatexchange-interaction values, which can explain not only the magnetic susceptibility, magnetization curves, andexcitation energies but also INS intensity, are in the vicinity of the data-driven values.DOI: 10.1103/PhysRevB.109.094434I. INTRODUCTIONThe precise knowledge of exchange interactions is aprerequisite for understanding the magnetic properties ofquantum spin systems. Typically, exchange interactions areevaluated by analyzing the magnetic susceptibility and mag-netization curves. However, it is often difficult to uniquelyevaluate multiple exchange interactions. For example, severalsets of exchange interaction values have been reported forthe diamond chain compound Cu3(CO3)2(OH)2 [1–8] and thedimer-monomer compound Ni2V2O7 [9–12].Tamura and Hukushima have developed a data-driventechnique based on machine learning to evaluate multipleexchange interactions and their uncertainty from multiplephysical quantities [13,14]. This data-driven technique wasbased on Bayesian statistics, and the exchange interactionswere determined such that the posterior distribution was max-imized. The posterior distribution constitutes the differencebetween the experimental and calculated results obtained*HASE.Masashi@nims.go.jpusing an effective model and appropriate prior distribution ofexchange interactions. The prior distribution corresponds toprior knowledge of the target material.Recently, this data-driven technique based on machinelearning was applied to KCu4P3O12 [15]. In this com-pound, the Cu2+ ions possess S = 12 spins. Figure 1 showsthe spin model (an eight-spin zigzag chain) based on thecrystal structure [16]. The spin Hamiltonian is definedas −∑7i=1 Ji,i+1SiSi+1, where J1 = J1,2 = J7,8, J2 = J2,3 =J6,7, J3 = J3,4 = J5,6, and J4 = J4,5. Using the magneticsusceptibility and magnetization curves at various temper-atures, we evaluated the exchange interactions as follows:J1 = −8.54 ± 0.51 meV [antiferromagnetic (AFM)], J2 =−2.67 ± 1.13 meV (AFM), J3 = −3.90 ± 0.15 meV (AFM),and J4 = 6.24 ± 0.95 meV [ferromagnetic (FM)] [15]. Weused the values of the exchange interactions to calculatethe excitation energies of the magnetic excitations. The ex-citation energies of the low-lying eigenstates are listed inTable I. Figure 2 shows the histogram of the excitationenergies.To demonstrate that the data-driven technique based onmachine learning is an important tool for scientific research,2469-9950/2024/109(9)/094434(8) 094434-1 ©2024 American Physical Societyhttps://orcid.org/0000-0003-2717-461Xhttps://orcid.org/0000-0002-0349-358Xhttps://orcid.org/0000-0003-1153-1758https://orcid.org/0000-0002-7300-9175https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.109.094434&domain=pdf&date_stamp=2024-03-25https://doi.org/10.1103/PhysRevB.109.094434MASASHI HASE et al. PHYSICAL REVIEW B 109, 094434 (2024)TABLE I. Low-lying eigenstates in the spin model of KCu4P3O12. S denotes the total spin of the eigenstates. Excitation energieswere evaluated from J1 = −8.54 ± 0.51, J2 = −2.67 ± 1.13, J3 = −3.90 ± 0.15, and J4 = 6.24 ± 0.95 meV determined by the data-driventechnique based on machine learning (Cal1) and J1 = −8.71, J2 = −3.89, J3 = −4.00, and J4 = 4.27 meV determined by a steepest descentmethod (Cal2) described in the Sec. III. We evaluated uncertainties of the Cal1 results from the uncertainties of the exchange-interaction valuesas is shown in Fig. 2. We also show the excitation energies estimated from the INS results. The magnetic excitations indicated by the symbol“O” are experimentally observed. The excitation between GS and 3ES is forbidden (“F”). We estimated the excitation energy between GS and3ES from that between GS and 1ES and that between 1ES and 3ES.Excitation energy (meV) ObservationS Cal1 Cal2 Exp From GS From 1ESGS 0 0 0 01ES 1 2.87 ± 0.11 3.02 3.0 ± 0.2 O2ES 1 4.23 ± 0.13 4.17 4.1 ± 0.5 O O3ES 2 6.65 ± 0.21 6.75 6.8 ± 0.3 F O4ES 1 8.53 ± 0.30 8.80 8.8 ± 0.5 O Oit is necessary to confirm that other physical quantities (par-ticularly the microscopic quantities) of KCu4P3O12 can bereproduced using the spin model. We can determine the ex-citation energies of the magnetic excitations using inelasticneutron scattering (INS) measurements and compare the ex-perimental excitation energies with the calculated excitationenergies. Accordingly, we performed INS measurements onKCu4P3O12 powder.II. EXPERIMENTAL METHODSWe synthesized a crystalline powder of KCu4P3O12 viasolid-state reaction. The starting materials were K2CO3, CuO,and (NH4)2HPO4 powders with purities of 99.9%, 99.99%,and 99%, respectively. A stoichiometric mixture of the pow-ders was calcined at 523 K for 48 h in air. The calcinedpowders were sintered at 973 K for 168 h in air. In Ref. [16],fabrication of single crystals with diameters up to 0.3 mmusing hydrothermal synthesis was reported. Many samples(typically 10 g) are necessary for neutron scattering exper-iments. Therefore, we used the solid-state reaction. X-raypowder diffraction patterns were recorded at room tempera-ture using an x-ray diffractometer (RINT-TTR III, Rigaku).We performed Rietveld refinements of the crystal struc-ture using the FULLPROF Suite software package containingFIG. 1. Schematic of the spin model (an eight-spin zigzag chain)in KCu4P3O12 [15,16] drawn using VESTA [17]. Using the data-driventechnique based on machine learning, the exchange interactionswere evaluated as J1 = −8.54 ± 0.51 meV (AFM), J2 = −2.67 ±1.13 meV (AFM), J3 = −3.90 ± 0.15 meV (AFM), and J4 = 6.24 ±0.95 meV (FM).internal tables for the scattering lengths [18]. We useda physical property measurement system (Quantum De-sign) for specific heat measurements. We performed INSmeasurements using the High Resolution Chopper (HRC)spectrometer at BL 12 at the Japan Proton Accelerator Re-search Complex (J-PARC) [19–21].III. RESULTS AND DISCUSSIONSThe blue circles in Fig. 3 represent the x-ray diffractionpattern of KCu4P3O12 at room temperature. The line on theexperimental pattern shows the results of the Rietveld refine-ments using the crystal structure reported in Ref. [16]. Thisline is consistent with the experimental results.The red circles in Fig. 4 show the temperature T depen-dence of the specific heat [C(T )] of KCu4P3O12. No phasetransition appears between 1.9 and 300 K. The blue circles inFIG. 2. The histogram of the excitation energy evaluated fromJ1 = −8.54 ± 0.51, J2 = −2.67 ± 1.13, J3 = −3.90 ± 0.15, andJ4 = 6.24 ± 0.95 meV determined by the data-driven techniquebased on machine learning. We divided the uncertainty range into20 equal parts for each exchange-interaction value and calculatedeigenenergies for 194 481 (= 214) sets of the exchange interactions.Vertical and horizontal lines show the excitation energies and theiruncertainties, respectively, estimated from the INS results.094434-2INELASTIC NEUTRON SCATTERING STUDIES ON THE … PHYSICAL REVIEW B 109, 094434 (2024)FIG. 3. X-ray diffraction pattern (circles) of KCu4P3O12 atroom temperature. The line on the measured pattern portrays theRietveld-refined pattern obtained using the crystal structure with P1̄(No. 2) [16]. The line at the bottom portrays the difference be-tween the measured and Rietveld-refined patterns. The hash marksrepresent the positions of reflections. The lattice constants are a =7.4273(2) Å, b = 7.8327(2) Å, c = 9.4573(2) Å, α = 108.285(1)◦,β = 112.671(1)◦, and γ = 92.743(1)◦. The reliability indices of therefinements are Rp = 1.82%, Rwp = 2.43%, and Rexp = 0.82%.Fig. 4(b) represent C(T ) of the spin model with the exchange-interaction values described in Fig. 1. The experimental andcalculated specific heats are consistent with each other below10 K where the lattice specific heat is probably small. Thecalculated specific heat has a broad maximum at around 30 K.The broad maximum is not seen in the experimental C(T )probably due to overlap of the lattice specific heat. Accord-ingly, we cannot prove that the exchange-interaction valuesare correct for KCu4P3O12 by the C(T ) result.Figure 5 shows the INS intensity I (Q, ω) maps ofKCu4P3O12 powder below 7.0 meV at various temperatures.FIG. 4. Temperature T dependence of the specific heat C(T ) ofKCu4P3O12 (red circles). The blue circles in panel (b) indicate C(T )calculated for the spin model with J1 = −8.54, J2 = −2.67, J3 =−3.90, and J4 = 6.24 meV.Here, Q and ω denote the magnitudes of the scattering vectorand the energy transfer, respectively. The energy of the inci-dent neutrons (Ei) is 15.3 meV. At 5.5 K, strong excitations atapproximately 3.0 meV and weak excitations up to 6.0 meVare observed. The 3.0-meV excitation exists over a wide Qrange, indicating a cluster excitation. The intensities of theexcitations decrease with increasing T ; therefore, they aremagnetic in nature. Figure 6 shows the I (Q, ω) maps up to15.0 meV at 5.5 K. The value of Ei is 25.4 meV. Anothermagnetic excitation exists at approximately 8.8 meV.Figure 7 shows the ω dependence of the INS intensityI (ω) at Ei = 15.3 meV. Magnetic excitations are observed atapproximately 3.0, 4.1, 5.9, and 8.8 meV at 5.5 K as shownin Figs. 7(a) and 7(b) and at approximately 3.8 and 5.9 meVat 49 K as shown in Fig. 7(e). The horizontal bars indicatethe energy resolution. The value at ω = 0 meV (�ω0) is0.855 meV. We assumed that the energy resolution at a finite ωwas �ω0(1 − ω/Ei )3/2. The width of the 3.0-meV excitationis slightly greater than the energy resolution. The widths ofthe other excitations seem greater than the energy resolutions.Considering the excitation energies from the ground andfirst excited states (represented by the pink and blue triangles,respectively) calculated for the eight-spin zigzag chain withJ1 = −8.54, J2 = −2.67, J3 = −3.90, and J4 = 6.24 meV,the 3.0, 4.1, and 8.8 meV excitations were magnetic excita-tions from the ground state (GS) to the first, second, and fourthexcited states (1ES, 2ES, and 4ES), respectively; we refer tothem as the 0-1, 0-2, and 0-4 excitations, respectively. Asshown in Table I, the excitation between the GS and the thirdexcited state (3ES) is forbidden. As the temperature increases,the intensity of the 3.0-meV excitation decreases rapidly andthe excitation is not observed at 49 K. However, the 5.9-meVexcitation is observed at 49 K, as shown in Fig. 7(e). Becausethe T dependence of the excitation intensity is determinedby the thermal population factor of the initial state [22],excitations from the same initial state exhibit the same Tdependence of the intensity. Therefore, the initial state of the5.9-meV excitation is not the GS. It is likely that the 5.9-meVexcitation is the excitation from 1ES to 4ES (1-4 excitation).Similarly, the initial state of the 3.8-meV excitation at 49 Kshown in Fig. 7(e) is not the GS. We consider that the 3.8-meVexcitation is the excitation from 1ES to 3ES (1-3 excitation),and the excitations at around 4.0 meV at 5.5 K compriseboth the 0-2 and the 1-3 excitations. Four excitations (0-1,0-2, 1-3, and 1-4 excitations) exist between 2.0 and 6.0 meV.We could not separate them and evaluate the T dependenceof the intensity of each excitation. The intensity between 1.0and 2.0 meV increases with T . Therefore, excitations exist inthis range and probably include the 1-2 excitation, althoughwe cannot separate the 1-2 excitation. As shown in Table Iand Fig. 2, we observed all the low-lying excitations fromthe GS and 1ES. The excitation energies evaluated by thedata-driven technique based on machine learning were almostconsistent with those obtained using the INS experiments onKCu4P3O12.The circles in Fig. 8 represent the Q dependence of the INSintensity I (Q) for the 0-1, 0-2, 0-4, 1-3, and 1-4 excitations atEi = 15.3 meV. We observed an apparent Q dependence of theexperimental I (Q) except for the I (Q) for the 0-4 excitationdepicted in Fig. 8(c). The pink lines in Fig. 8 denote I (Q)094434-3MASASHI HASE et al. PHYSICAL REVIEW B 109, 094434 (2024)FIG. 5. The INS intensity I (Q, ω) maps of KCu4P3O12 powder below 7.0 meV at various temperatures. The energy of incident neutronsEi is 15.3 meV. The vertical key on the right shows the INS intensity in arbitrary units.calculated using J1 = −8.54, J2 = −2.67, J3 = −3.90, andJ4 = 6.24 meV. The formulas for I (Q) are described in theAppendix [23–29]. The two peak positions are slightly lowerin the experimental I (Q) curves than in the calculated ones(pink) for the 0-1 and 1-3 excitations. For the 0-4 excitation,FIG. 6. (a) The INS intensity I (Q, ω) map of KCu4P3O12 powderbelow 15.0 meV at 5.5 K. The energy of incident neutrons Ei is25.4 meV. The vertical key on the right shows the INS intensity inarbitrary units. (b) Map between 7 and 11 meV.the Q dependence of the experimental I (Q) is weaker than thatof the calculated I (Q). The experimental and calculated I (Q)curves (pink) for the 0-2 and 1-4 excitations do not agree witheach other.FIG. 7. ω dependence of the INS intensity I (ω) of KCu4P3O12powder. The energy of the incident neutrons Ei is 15.3 meV. Thepink and blue triangles indicate the excitation energies from theGS and 1ES, respectively. The horizontal bars represent the energyresolution.094434-4INELASTIC NEUTRON SCATTERING STUDIES ON THE … PHYSICAL REVIEW B 109, 094434 (2024)FIG. 8. Q dependence of the INS intensity I (Q) of KCu4P3O12powder (circles). The energy of incident neutrons Ei is 15.3 meV.The lines indicate I (Q) calculated for (a) the 0-1 and 1-3 exci-tations, (b) the 0-2 excitation, (c) the 0-4 excitation, and (d) the1-4 excitation. The pink lines were obtained from J1 = −8.54,J2 = −2.67, J3 = −3.90, and J4 = 6.24 meV evaluated using thedata-driven technique based on machine learning [15]. The bluelines were obtained from J1 = −8.71, J2 = −3.89, J3 = −4.00, andJ4 = 4.27 meV evaluated using a steepest descent method.Unlike the case of AFM dimers, we could not utilize thepeak positions of I (Q) to directly determine the Cu-Cu pairsthat exhibited strong exchange interactions in KCu4P3O12.This is because I (Q) consists of multiple terms as describedin the Appendix. In an AFM dimer with a spin-spin dis-tance R, I (Q) is expressed as f (Q)2A[1 − sin(QR)QR ]. We canuse the value of R to determine the Cu-Cu pair that formsan AFM dimer. For example, when the positions of the firstpeak in the I (Q) for the 0-1 and 0-2 excitations (Q = 0.7and 0.9 Å−1, respectively) were matched to those in theI (Q) of an AFM dimer, we obtained R = 6.1 and 4.8 Å,respectively. In contrast, the spin-spin distances in the fourexchange interactions in KCu4P3O12 are located between 2.94and 3.19 Å.The difference in the ω dependence of the INS intensitybetween Figs. 7(a) and 7(b) is mainly caused by the structuralfactor of the magnetic excitations. As shown in Fig. 8(a),the experimental intensity of the 3.0 meV (0-1) excitationis larger at around Q = 0.7 Å−1 corresponding to Fig. 7(b)than at around Q = 1.6 Å−1 corresponding to Fig. 7(a). Thus,the 4.1-meV (0-2) excitation is less apparent in Fig. 7(b). Asshown in Fig. 8(b), the experimental intensity of the 4.1-meV(0-2) excitation is larger at around Q = 1.6 Å−1 than at aroundQ = 0.7 Å−1. Thus, the 5.9-meV (1-4) excitation is less ap-parent in Fig. 7(a).The inconsistency between the experimental and calculatedI (Q) curves (pink) indicates that the present exchange-interaction values determined by the data-driven techniquebased on machine learning (data-driven values) are not ap-propriate. However, to obtain exchange-interaction valuesthat can reproduce quantitatively the INS intensity, we needthe INS intensity of magnetic excitation alone. As is seenin Figs. 8(a) and 8(c), the 0-1 and 0-4 excitations showweak Q dependencies above Q = 1 Å−1, suggesting that thecontributions of the phonon excitation and background cannotbe ignored. However, we could not subtract the contributionsbecause of the lack of information. Accordingly, it is impos-sible at present to evaluate the exchange interactions from theINS intensity.Next, we considered evaluations of the exchange interac-tions from the excitation energies. There may be differentsets of exchange-interaction values that can reproduce themagnetic susceptibility, magnetization curves, and excitationenergies. However, we cannot determine which set is cor-rect because the experimental results can be also explainedby the data-driven values. Therefore, we searched exchange-interaction values, which could reproduce more correctly theexcitation energies, using the data-driven values as the startingpoint. Here, we utilized a steepest descent method to finda local minimum. The results are J1 = −8.71, J2 = −3.89,J3 = −4.00, and J4 = 4.27 meV of which excitation energiesare 3.02, 4.17, 6.75, and 8.80 meV from GS to 1ES, 2ES, 3ES,and 4ES, respectively, as described in Table I. The values areclose to the experimental values (3.0, 4.1, 6.8, and 8.8 meV,respectively). The new J1, J2, and J3 values are almost withinthe uncertainties of the data-driven values, whereas the newJ4 value is slightly smaller than the data-driven value. Asshown in Fig. 9, the curves calculated using the new valuesare close to those calculated using the data-driven values. Weevaluated differences between the experimental and calculatedresults. The agreements at the magnetic susceptibility andthe magnetization curve at 50 K are slightly better in theresults using the new values and those at the magnetizationcurves between 1.3 and 30 K are slightly better in the resultsusing the data-driven values. On the other hand, as shown inFigs. 8(b) and 8(d), the I (Q) curves calculated using the newvalues (blue) are clearly different from those calculated usingthe data-driven values (pink) in spite of the small differencesbetween the data-driven values and new values. This resultsuggests that I (Q) curves can be changed largely by smallchanges of exchange-interaction values.As shown in Fig. 8(b), the peak positions and the ra-tio of intensities between the two peaks of the blue curvecalculated using the new values are close to those of theexperimental I (Q) curve, although there are apparent differ-ences between the calculated and experimental I (Q) curvesat Q < 0.6 and Q > 2.0 Å−1 probably due to the contribu-tions of the phonon excitation and background. Therefore, weexpect that exchange-interaction values, which can explainnot only the magnetic susceptibility, magnetization curves,and excitation energies but also the INS intensity of magneticexcitation alone, are in the vicinity of the data-driven values.In future, we will perform further neutron scattering exper-iments and obtain the INS intensity of magnetic excitationalone after subtracting the contributions of phonon excitationsand the background. Our current simulation code based onthe Bayesian statistics does not support INS data. We willdevelopan algorithm including an investigation of prior distri-bution for interactions to obtain exchange-interaction valuesthat can reproduce the magnetic susceptibility, magnetizationcurves, and INS intensities. If we cannot obtain the exchange-interaction values that can reproduce all the experimentalresults, we will have to consider a model containing otherexchange interactions.094434-5MASASHI HASE et al. PHYSICAL REVIEW B 109, 094434 (2024)FIG. 9. Comparison plots of the magnetic susceptibility at 0.01 T and the magnetization curves at various temperatures between the ex-perimental (blue circles) and calculated results. The red circles were obtained from J1 = −8.54, J2 = −2.67, J3 = −3.90, and J4 = 6.24 meVevaluated using the data-driven technique based on machine learning [15]. The green triangles were obtained from J1 = −8.71, J2 = −3.89,J3 = −4.00, and J4 = 4.27 meV evaluated using a steepest descent method.IV. CONCLUSIONWe performed INS experiments on KCu4P3O12 powderand compared the experimental results with those calculatedfor the spin model (an eight-spin zigzag chain with S = 12 )using the data-driven technique based on machine learning.We observed magnetic excitations at approximately 3.0, 4.1,5.9, and 8.8 meV at 5.5 K and at approximately 3.8 and5.9 meV at 49 K. The 3.0-, 4.1-, and 8.8-meV excitationswere magnetic excitations from the ground state to the first,second, and fourth excited states (2.87, 4.23, and 8.53 meVfrom the calculations), respectively. The 3.8- and 5.9-meVexcitations were magnetic excitations from the first excitedstate to the third and fourth excited states (3.78 and 5.67 meVfrom the calculations), respectively. We considered that, inthe experimental results, an excitation between the first andsecond excited states existed at approximately 1.35 meV. Theexcitation energies obtained from INS experiments almostagree with those calculated from the exchange interactionvalues evaluated using the data-driven technique. We observedall the low-lying excitations from the ground and first excitedstates. The experimental I (Q) curves cannot be reproduced,suggesting that the data-driven values are not appropriate.Therefore, we searched exchange-interaction values, whichcould reproduce more correctly the excitation energies, us-ing the data-driven values as the starting point. The newlyobtained J1, J2, and J3 values are almost within the uncer-tainties of the data-driven values, whereas the newly obtainedJ4 value is slightly smaller than the data-driven value. Wecompared the I (Q) curves calculated using the data-drivenvalues with those calculated using the new values. We foundthat I (Q) curves could be changed largely by small changesof exchange-interaction values. Therefore, we expect thatexchange-interaction values, which can explain not only themagnetic susceptibility, magnetization curves, and excitationenergies but also the INS intensity of magnetic excitationalone, are in the vicinity of the data-driven values.ACKNOWLEDGMENTSThis study was supported by Japanese Society for the Pro-motion of Science (JSPS) KAKENHI Grant No. 18K03551and the World Premier International Research Center Initia-tive (WPI), Ministry of Education, Culture, Sports, Scienceand Technology (MEXT), Japan. The neutron scattering094434-6INELASTIC NEUTRON SCATTERING STUDIES ON THE … PHYSICAL REVIEW B 109, 094434 (2024)experiments on KCu4P3O12 were approved by the NeutronScience Proposal Review Committee of J-PARC/MLF (Pro-posal No. 2020B0026) and supported by the Inter-UniversityResearch Program on Neutron Scattering of IMSS, KEK. Wethank Seiko Matsumoto at the National Institute for MaterialsScience (NIMS) for the sample synthesis and x-ray diffrac-tion measurements. We are grateful to Masamichi Nishino,Kazunari Yamaura, Alexei Belik, and Yoshihiro Tsujimoto atNIMS for fruitful discussions.APPENDIX: CALCULATION OF I(Q)I (Q) for the transition from the |I〉 to the |F〉 states isproportional to the following term:k′k〈I|Q̂+⊥(Q)|F〉〈F|Q̂⊥(Q)|I〉 (A1)[23–26]. The parameters k and k′ are the initial and finalneutron wave numbers, respectively. The operator Q̂(Q) isexpressed by the following equation:Q̂(Q) = f (Q)8∑i=1exp(iQ · Ri )Si. (A2)Here, f (Q) denotes the magnetic form factor of an isolatedCu2+ ion [27–29]. The position of the spin Si is indicatedby Ri. The center of two Cu4 sites connected by the J4interaction, as shown in Fig. 1, is set as the origin of Ri. Thesubscript ⊥ in Eq. (A1) indicates a projection onto a plane per-pendicular to the scattering vector Q. Since no magnetic orderexists in KCu4P3O12, we can consider that Q̂⊥(Q) = Q̂(Q).Here, we express the matrix elements as follows:〈F|Q̂⊥(Q)|I〉 j =8∑i=1exp(iQ · Ri )a ji, j = x, y, z. (A3)Note that the values of aji also depend on eigenstates (|I〉 and|F〉 states). We obtained that aji = a j(9−i) (i = 1 ∼ 4) for the0-1, 0-4, 1-3, and 1-4 excitations and that a ji = −a j(9−i) (i =1 ∼ 4) for the 0-2 and 1-2 excitations. Therefore,〈I|Q̂+⊥(Q)|F〉〈F|Q̂⊥(Q)|I〉= 4∑jii′cos(Q · Ri ) cos(Q · Ri′ )a jia ji′ (A4)for the 0-1, 0-4, 1-3, and 1-4 excitations and〈I|Q̂+⊥(Q)|F〉〈F|Q̂⊥(Q)|I〉= 4∑jii′sin(Q · Ri ) sin(Q · Ri′ )a jia ji′ (A5)for the 0-2 and 1-2 excitations. After a spherical average, I (Q)is expressed as follows:I (Q) = f (Q)2⎡⎣∑i �=i′Aii′,IF(sin Q|Ri − Ri′ |Q|Ri − Ri′ | + sin Q|Ri + Ri′ |Q|Ri + Ri′ |)+∑iAi,IF(1 + sin 2Q|Ri|2Q|Ri|)⎤⎦ (A6)for the 0-1, 0-4, 1-3, and 1-4 excitations andI (Q) = f (Q)2⎡⎣∑i �=i′Aii′,IF(sin Q|Ri − Ri′ |Q|Ri − Ri′ | − sin Q|Ri + Ri′ |Q|Ri + Ri′ |)+∑iAi,IF(1 − sin 2Q|Ri|2Q|Ri|)⎤⎦ (A7)for the 0-2 and 1-2 excitations. Ai,IF is always positive becauseit consists of a2ji. Aii′,IF can be both positive and negativebecause it consists of a jia ji′ . The dominant term in the 0-1and 1-3 excitations is A3,IF(1 + sin 2Q|R3|2Q|R3| ). Therefore, the cal-culated I (Q) curves for the 0-1 and 1-3 excitations are similarto each other as shown in Fig. 8(a).[1] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T.Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta, Ex-perimental 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 observation ofthe 1/3 magnetization plateau in the diamond-chain compoundCu3(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,Kikuchi et al. Reply:, Phys. Rev. Lett. 97, 089702 (2006).[4] B. Gu and G. Su, Magnetism and thermodynamics of spin-1/2Heisenberg diamond chains in a magnetic field, Phys. Rev. B75, 174437 (2007).[5] K. C. Rule, A. U. B. Wolter, S. Süllow, D. A. Tennant, A. Brühl,S. Köhler, B. Wolf, M. Lang, and J. Schreuer, Nature of the spindynamics and 1/3 magnetization plateau in azurite, Phys. Rev.Lett. 100, 117202 (2008).[6] 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).[7] 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, andA. Honecker, Multistep approach to microscopic models forfrustrated quantum magnets: The case of the natural mineralazurite, Phys. Rev. Lett. 106, 217201 (2011).[8] K. C. Rule, M. Reehuis, M. C. R. Gibson, B. Ouladdiaf, M. J.Gutmann, J.-U. Hoffmann, S. Gerischer, D. A. Tennant, S.Süllow, and M. Lang, Magnetic and crystal structure of azurite094434-7https://doi.org/10.1103/PhysRevLett.94.227201https://doi.org/10.1103/PhysRevLett.97.089701https://doi.org/10.1103/PhysRevLett.97.089702https://doi.org/10.1103/PhysRevB.75.174437https://doi.org/10.1103/PhysRevLett.100.117202https://doi.org/10.1088/0953-8984/21/39/392201https://doi.org/10.1103/PhysRevLett.106.217201MASASHI HASE et al. PHYSICAL REVIEW B 109, 094434 (2024)Cu3(CO3)2(OH)2 as determined by neutron diffraction, Phys.Rev. B 83, 104401 (2011).[9] Y. C. Sun, Z. W. Ouyang, J. F. Wang, Z. X. Wang, Z. C. Xia,and G. H. Rao, Breaking of 1D magnetism in a spin-1 chainantiferromagnet Ni2V2O7: ESR and first-principles studies, Eur.Phys. J. Plus 131, 343 (2016).[10] Z. W. Ouyang, Y. C. Sun, J. F. Wang, X. Y. Yue, R. Chen,Z. X. Wang, Z. Z. He, Z. C. Xia, Y. Liu, and G. H. Rao,Novel half-magnetization plateau and nematiclike transitionin the S = 1 skew chain Ni2V2O7, Phys. Rev. B 97, 144406(2018).[11] J. J. Cao, Z. W. Ouyang, X. C. Liu, T. T. Xiao, Y. R. Song,J. F. Wang, Y. Ishii, X. G. Zhou, and Y. H. Matsuda, Un-usual dimerization and magnetization plateaus in S = 1 skewchain Ni2V2O7 observed at 120 T, Phys. Rev. B 106, 184409(2022).[12] M. Hase, A. Dönni, N. Terada, V. Yu. Pomjakushin, J. R. Hester,K. C. Rule, and Y. Matsuo, Neutron diffraction studies underzero and finite magnetic fields of the 12 quantum magnetiza-tion plateau compound Ni2V2O7, Phys. Rev. B 107, 224415(2023).[13] R. Tamura and K. Hukushima, Method for estimating spin-spin interactions from magnetization curves, Phys. Rev. B 95,064407 (2017).[14] R. Tamura and K. Hukushima, Bayesian optimization for com-putationally extensive probability distributions, PLoS One 13,e0193785 (2018).[15] R. Tamura, K. Hukushima, A. Matsuo, K. Kindo, and M. Hase,Data-driven determination of the spin Hamiltonian parametersand their uncertainties: The case of the zigzag-chain compoundKCu4P3O12, Phys. Rev. B 101, 224435 (2020).[16] H. Effenberger, KCu4(PO4)3: A compound with two trigonaldipyramidal Cu(II)O5 coordination polyhedra, Z. Kristallogr.180, 43 (1987).[17] K. Momma and F. Izumi, VESTA for three-dimensional visu-alization of crystal, volumetric and morphology data, J. Appl.Crystallogr. 44, 1272 (2011).[18] J. Rodríguez-Carvajal, Recent advances in magnetic structuredetermination by neutron powder diffraction, Phys. B: Condens.Matter 192, 55 (1993); http://www.ill.eu/sites/fullprof/.[19] S. Itoh, T. Yokoo, S. Satoh, S. Yano, D. Kawana, J. Suzuki,and T. J. Sato, High resolution chopper spectrometer (HRC) atJ-PARC, Nucl. Instrum. Methods Phys. Res., Sect. A 631, 90(2011).[20] S. Yano, S. Itoh, T. Yokoo, S. Satoh, T. Yokoo, D. Kawana, andT. J. Sato, Data acquisition system for high resolution chopperspectrometer (HRC) at J-PARC, Nucl. Instrum. Methods Phys.Res., Sect. A 654, 421 (2011).[21] S. Itoh, K. Ueno, and T. Yokoo, Fermi chopper developed atKEK, Nucl. Instrum. Methods Phys. Res., Sect. A 661, 58(2012).[22] R. E. Watson and A. J. Freeman, Hartree-Fock atomic scatteringfactors for the iron transition series, Acta Crystallogr. 14, 27(1961).[23] A. Furrer and H. U. Güdel, Interference effects in neutron scat-tering from magnetic clusters, Phys. Rev. Lett. 39, 657 (1977).[24] S. W. Lovesey, Theory of Neutron Scattering from CondensedMatter (Clarendon, Oxford, 1984), Vol. 2.[25] A. Zheludev, G. Shirane, Y. Sasago, M. Hase, and K.Uchinokura, Dimerized ground state and magnetic excitationsin CaCuGe2O6, Phys. Rev. B 53, 11642 (1996).[26] M. Hase, K. M. S. Etheredge, S.-J. Hwu, K. Hirota, and G.Shirane, Spin-singlet ground state with energy gaps in Cu2PO4:Neutron-scattering, magnetic-susceptibility, and ESR measure-ments, Phys. Rev. B 56, 3231 (1997).[27] J. Akimitsu and Y. Ito, Magnetic form factor of Cu2+ inK2CuF4, J. Phys. Soc. Jpn. 40, 1621 (1976).[28] T. Freltoft, G. Shirane, S. Mitsuda, J. P. Remeika, and A. S.Cooper, Magnetic form factor of Cu in La2CuO4, Phys. Rev. B37, 137 (1988).[29] P. J. Brown, Magnetic form factors, in International Tables forCrystallography, Vol. C: Mathematical, physical and chemicaltables, edited by E. Prince, Sec. 4.4.5 (Wiley and Hoboken, NewJersey, USA, 2006), pp. 454–461.094434-8https://doi.org/10.1103/PhysRevB.83.104401https://doi.org/10.1140/epjp/i2016-16343-8https://doi.org/10.1103/PhysRevB.97.144406https://doi.org/10.1103/PhysRevB.106.184409https://doi.org/10.1103/PhysRevB.107.224415https://doi.org/10.1103/PhysRevB.95.064407https://doi.org/10.1371/journal.pone.0193785https://doi.org/10.1103/PhysRevB.101.224435https://doi.org/10.1524/zkri.1987.180.14.43https://doi.org/10.1107/S0021889811038970https://doi.org/10.1016/0921-4526(93)90108-Ihttp://www.ill.eu/sites/fullprof/https://doi.org/10.1016/j.nima.2010.11.107https://doi.org/10.1016/j.nima.2011.06.042https://doi.org/10.1016/j.nima.2011.09.048https://doi.org/10.1107/S0365110X61000048https://doi.org/10.1103/PhysRevLett.39.657https://doi.org/10.1103/PhysRevB.53.11642https://doi.org/10.1103/PhysRevB.56.3231https://doi.org/10.1143/JPSJ.40.1621https://doi.org/10.1103/PhysRevB.37.137