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Matsuo, Akira, Matsumoto, Masashige, [Hase, Masashi](https://orcid.org/0000-0003-2717-461X), Kindo, Koichi

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[Magnetism of the spin-1 tetramer compound A2Ni2Mo3O12 (A = Rb or K)](https://mdr.nims.go.jp/datasets/ce712807-3f49-4150-adda-d4168f879d8a)

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PHYSICAL REVIEW B 96, 214424 (2017)Magnetism of the spin-1 tetramer compound A2Ni2Mo3O12 (A = Rb or K)Masashi Hase,1,* Akira Matsuo,2 Koichi Kindo,2 and Masashige Matsumoto31Research Center for Advanced Measurement and Characterization, National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba-shi, Ibaraki 305-0047, Japan2The Institute for Solid State Physics (ISSP), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8581, Japan3Department of Physics, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka-shi, Shizuoka 422-8529, Japan(Received 11 October 2017; revised manuscript received 30 November 2017; published 19 December 2017)We measured the temperature dependence of the magnetic susceptibility χ (T ) and the specific heat C(T ) andthe magnetic-field dependence of the magnetization M(H ) of A2Ni2Mo3O12 (A = Rb or K) powder. We considerthat the probable spin model is an interacting spin-1 antiferromagnetic tetramer model. We evaluated values ofthe intratetramer interactions as J1 = 9 K and J2 = 18 K, and the effective intertetramer interaction as Jeff = 4 Kfor Rb2Ni2Mo3O12. The susceptibility and magnetization at 1.3 K of K2Ni2Mo3O12 are very close to those ofRb2Ni2Mo3O12. We observed a phase transition to a magnetically ordered state in C(T )/T in magnetic fieldsabove 3 T. The transition temperature increases with magnetic field. Probably, the ordered state appears around1.8 K even in 0 T. The ordered state in 0 T, however, is not stable enough like an order in the vicinity of a quantumcritical point. Longitudinal-mode magnetic excitations may be observable in single crystalline A2Ni2Mo3O12(A = Rb or K).DOI: 10.1103/PhysRevB.96.214424I. INTRODUCTIONTwo types of magnetic excitations exist in a magneticallyordered state. They are gapless transverse-mode (Nambu-Goldstone mode) [1] and gapped longitudinal-mode (ampli-tude Higgs mode) [2–6] excitations corresponding to fluctua-tions in directions perpendicular and parallel to ordered mo-ments, respectively. The transverse-mode (T-mode) excitationsare well known as spin wave excitations. Investigations of thelongitudinal-mode (L-mode) excitations are now in progress.The L-mode excitation has weak intensity and spontaneouslydecays into a pair of T-mode excitations [7,8]. However, itcan be well-defined in the ordered state in the vicinity of thequantum critical point [9]. In interacting antiferromagnetic(AF) spin- 12 dimer compounds TlCuCl3 and KCuCl3, theground state (GS) is a spin-singlet state at atmosphericpressure and zero magnetic field. The L-mode excitations wereactually observed in a pressure-induced magnetically orderedstate of TlCuCl3 and KCuCl3 by inelastic neutron scattering(INS) experiments [10–12] and Raman scattering experiments[13,14], respectively, and in a magnetic-field-induced orderedstate of TlCuCl3 by Raman scattering experiments [15].According to results of theoretical investigations, the L-mode excitations may be observed in an antiferromagneticallyordered state appearing on cooling at atmospheric pressureand zero magnetic field in interacting spin-cluster compounds[16]. A shrinkage of ordered magnetic moments by quantumfluctuations leads to a large intensity of the L-mode excitations.If the GS of the corresponding isolated spin cluster is aspin-singlet state, the shrinkage of ordered moments can beexpected in an ordered state of the interacting spin-clustercompounds. In interacting spin clusters, the ordered state canappear under the condition that the value of � is comparableto or less than that of an effective intercluster interaction [16].Here � is the energy difference (spin gap) between the singlet*HASE.Masashi@nims.go.jpGS and lowest triplet states in the isolated cluster. The effectiveintercluster interaction is given by the sum of the productsof the absolute value of each intercluster interaction (|Jint,i |)and the corresponding number of interactions per spin (zi)as Jeff = ∑i zi |Jint,i |. The effective intercluster interactionis usually smaller than dominant intracluster interactions.Therefore it is advantageous to the appearance of the orderedstate that � is smaller than dominant intracluster interactions.In the AF spin dimer given by JS1 · S2, the value of �/Jis 1 irrespective of the spin value. It is rare that spin dimercompounds show a magnetically ordered state at atmosphericpressure and zero magnetic field. Examples are NH4CuCl3(spin 12 ) [17,18] and CrVMoO7 (spin 32 ) [19].In a spin tetramer expressed by the following Hamiltonianwith J1 > 0 or J2 > 0, the GS is a spin-singlet state [20]. TheHamiltonian isH = J1S2 · S3 + J2(S1 · S2 + S3 · S4). (1)�/J1 can be sufficiently small [21–24]. Interacting spintetramers are advantageous to the appearance of the orderedstate. Several spin tetramer compounds having a magneticallyordered state have been reported. Table I shows values ofJ1, J2,�, and TN of the spin tetramer compounds. Magneticexcitations in Cu1142 Cd11B2O6 were studied by INS experi-ments on its powder [23]. The results suggest the existenceof the L-mode excitations. It is important to investigate the L-mode excitations in the known spin tetramer compounds usingsingle crystals. It is also important to find more spin tetramercompounds having an antiferromagnetically ordered state. Wecan expect spin-1 tetramers in A2Ni2Mo3O12 (A = Rb or K)from its crystal structure [27,28]. We report magnetism ofthese compounds.II. EXPECTED SPIN SYSTEMThe two compounds A2Ni2Mo3O12 (A = Rb or K)are isostructural. The space group is P 21/c (No. 14). Thelattice constants are a = 6.996, b = 9.186, c = 19.895 Å, and2469-9950/2017/96(21)/214424(8) 214424-1 ©2017 American Physical Societyhttps://doi.org/10.1103/PhysRevB.96.214424HASE, MATSUO, KINDO, AND MATSUMOTO PHYSICAL REVIEW B 96, 214424 (2017)TABLE I. Values of the exchange interaction parameters, the spingap of the isolated tetramers, and the AF transition temperature inthe spin tetramer compounds. In Cu2Fe2Ge4O13, Cu2+ and Fe3+ ionshave spins 1/2 and 5/2, respectively. The spin tetramers Fe-Cu-Cu-Feare formed. In Rb2Ni2Mo3O12, TN in 0 T was estimated from theextrapolation of TN(H ).spin J1 (K) J2 (K) � (K) TN Ref.Cu2CdB2O612 317 −162 19 9.8 [23]CuInVO512 240 −142 17 2.7 [24]SeCuO312 225 160 84 8 [25]Cu2Fe2Ge4O1312 , 52 255 26.7 1.3 39 [16,26]Rb2Ni2Mo3O12 1 9 18 12 1.8(2) this workβ = 108.71◦ in Rb2Ni2Mo3O12 [27]. They are a = 6.952, b =8.910, c = 19.733 Å, and β = 108.06◦ in K2Ni2Mo3O12 [28].The Ni2+ ions (3d8) have localized spin-1. The positions ofthe Ni ions and the O ions connected to the Ni ions are shownschematically in Fig. 1(a). Two crystallographic Ni sites (Ni1and Ni2) exist. Red and blue bars indicate two types of shortNi-Ni pairs. We show the Ni-Ni distances and Ni-O-Ni anglesin Table II. If dominant exchange interactions exist in the Ni-Nipairs, spin tetramers given by Eq. (1) are formed. The otherNi-Ni lengths are 4.999 (4.961) Å or greater in Rb2Ni2Mo3O12(K2Ni2Mo3O12). Crystal structures depend on A and M incompounds expressed as A2M2Mo3O12 (A = alkali metaland M = 3d metal). There are several types of spin systems.For example, the spin system is a frustrated spin- 12 chain inA2Cu2Mo3O12 (A = Rb or Cs) [29,30].Figure 1(b) shows NiO6 octahedra. Table III shows Ni-Olengths and O-Ni-O angles in the octahedra. The Ni1O6 andNi2O6 octahedra are similar to each other. The symmetries ofcrystal fields affecting the Ni2+ ions are not so far from cubic.We expect that the single-ion anisotropy is small.III. EXPERIMENTAL AND CALCULATION METHODSCrystalline A2Ni2Mo3O12 (A = Rb or K) powder wassynthesized by a solid-state reaction. Starting materials areRb2CO3 (purity 99 %), K2CO3 (purity 99.9 %), NiO (purity99.97 %), and MoO3 (purity 99.99 %) powder. Stoichiometricmixtures of powder were sintered at 853 and 923 K in airfor 150 h with intermediate grindings for the Rb and K com-pounds, respectively. We measured x-ray powder diffractionpatterns at room temperature using an x-ray diffractometer(RINT-TTR III; Rigaku). We confirmed that each sample wasa nearly single phase of A2Ni2Mo3O12 (A = Rb or K).We measured the magnetization in magnetic fields ofup to 5 T using a superconducting quantum interferencedevice magnetometer magnetic property measurement system(Quantum Design). High-field magnetization measurementswere conducted using an induction method with a multilayerpulsed field magnet installed at the Institute for Solid StatePhysics (ISSP), the University of Tokyo. We used a physicalproperty measurement system (Quantum Design) for thespecific heat measurements. We measured the specific heat inmagnetic fields of not only Rb2Ni2Mo3O12 but also CuInVO5and CrVMoO7 for comparison.FIG. 1. (a) Schematic drawing of positions of Ni2+ ions havingspin-1 and O2− ions connected to Ni2+ ions in A2Ni2Mo3O12 (A =Rb or K) [27,28]. Blue, red, and white circles indicate Ni1, Ni2,and O sites, respectively. Red and blue bars indicate two types ofshort Ni-Ni pairs. We define J1 and J2 as the exchange interactionparameters for the Ni2-Ni2 and Ni1-Ni2 pairs, respectively. The J1and J2 interactions form a spin-1 tetramer. (b) NiO6 octahedra ofNi1 (left) and Ni2 (right). (c) Interacting spin-1 tetramer model usedto calculate magnetization using a mean-field theory based on thetetramer unit (tetramer mean-field theory). We adopted the interactionin the third shortest Ni-Ni pair (4.999 Å) as the effective intertetramerinteraction (Jeff ).We obtained eigenenergies of isolated spin-1 tetramersusing an exact diagonalization method. We calculated thetemperature dependence of the magnetic susceptibility χ (T )and the specific heat C(T ), and the magnetic-field dependenceof the magnetization M(H ) using the eigenenergies.We calculated M(H ) for the model shown in Fig. 1(c)using a mean-field theory based on the tetramer unit (tetramerTABLE II. Ni-Ni lengths and Ni-O-Ni angles in the two types ofshort Ni-Ni pairs in A2Ni2Mo3O12 (A = Rb or K).Rb KNi-Ni Ni-O-Ni Ni-Ni Ni-O-Nilength (Å) angle (deg.) length (Å) angle (deg.)J1 Ni2-Ni2 3.174 97.48 (O10) 3.128 96.24 (O10)J2 Ni1-Ni2 3.081 96.31 (O8) 3.089 97.89 (O8)93.64 (O11) 93.96 (O11)214424-2MAGNETISM OF THE SPIN-1 TETRAMER COMPOUND . . . PHYSICAL REVIEW B 96, 214424 (2017)TABLE III. Ni-O lengths and O-Ni-O angles in NiO octahedra in Rb2Ni2Mo3O12. The deviation is defined as the difference between eachNi-O length and the average Ni-O length [Lave,i(i = 1,2)] divided by Lave,i . The values of Lave,i are 2.080 and 2.062 Å for Ni1-O and Ni2-O,respectively. The maximum deviation in the Ni-O lengths from the average is 0.045. The minimum and maximum O-Ni-O angles are 81.88◦and 101.18◦, respectively.Å deviation Å deviationNi1-O4 2.024 0.027 Ni2-O7 2.023 0.019O1 2.035 0.022 O3 2.032 0.014O5 2.058 0.011 O8 2.044 0.009O8 2.092 0.006 O11 2.050 0.006O12 2.098 0.009 O10(2) 2.077 0.007O11 2.174 0.045 O10(4) 2.145 0.040degrees degreesO11-Ni1-O1 90.58 O10(4)-Ni2-O7 84.43O5 92.02 O8 88.76O8 82.48 O10(2) 82.52O12 81.88 O11 83.03O4-Ni1-O1 92.04 O3-Ni2-O7 95.17O5 93.38 O8 91.67O8 91.92 O10(2) 101.18O12 95.31 O11 93.33O12-Ni1-O5 88.43 O10(2)-Ni2-O7 92.56O8 86.99 O8 85.83O1-Ni1-O5 93.49 O11-Ni2-O7 93.09O8 90.42 O8 86.78O11-Ni1-O4 173.84 O10(4)-Ni2-O3 176.30O12-Ni1-O1 172.28 O10(2)-Ni2-O11 163.89O8-Ni1-O5 173.29 O8-Ni2-O7 173.16mean-field theory). Finite magnetic moments were initiallyassumed on the Ni sites in the tetramer. The mean-fieldHamiltonian was then expressed by a 81 × 81 matrix formunder consideration of the external magnetic field and themolecular field from neighboring tetramers. The eigenstatesof the mean-field Hamiltonian were used to calculate theexpectation value of the ordered moments on the Ni sites.We continued this procedure until the values of the magneticmoments converged. We finally obtained a self-consistentlydetermined solution for M(H ).IV. RESULTS AND DISCUSSIONFigure 2 shows the temperature T dependence of themagnetic susceptibility χ (T ) of A2Ni2Mo3O12 (A = Rb or K)in a magnetic field of H = 0.01 T. The susceptibilities of thetwo compounds are very close to each other. A broad maximumcan be seen around 16 K, indicating a low-dimensional AFspin system. The susceptibility decreases rapidly at low T .However, χ (T ) does not seem to approach a small value at 0 Kexpected for a spin singlet GS with a spin gap [31–33]. Figure 3shows the magnetic field H dependence of the magnetizationM(H ) of Rb2Ni2Mo3O12 at 2 K. The magnetization hasthe finite slope even around 0 T, indicating that the GS ismagnetic. The red circles in the inset of Fig. 4(a) show thespecific heat divided by T [C(T )/T ] of Rb2Ni2Mo3O12 in 0T. A broad maximum can be seen around 5 K, indicating alow-dimensional AF spin system. C(T )/T seems to approachzero at 0 K, indicating no T -linear term. A spin liquid statecannot be expected [34]. We will describe later the results infinite magnetic fields.FIG. 2. Temperature T dependence of the magnetic susceptibilityχ (T ) of Rb2Ni2Mo3O12 (red circles) and K2Ni2Mo3O12 (green line)in a magnetic field of H = 0.01 T below 300 K (a) and 30 K (b).The light-blue line indicates χ (T ) calculated for the isolated spin-1tetramer with J1 = 9 K and J2 = 18 K.214424-3HASE, MATSUO, KINDO, AND MATSUMOTO PHYSICAL REVIEW B 96, 214424 (2017)00.050.10 1 2 3 4 5H (T)M(H) (µ B/Ni)(b)05x10-41x10-30 0.02 0.04 0.06 0.08 0.1M(H) (µ B/Ni)H (T)(a)FIG. 3. Magnetic field H dependence of the magnetization M(H )of Rb2Ni2Mo3O12 at 2 K below 0.1 T (a) and 5 T (b).The red lines in Fig. 5 show M(H ) of Rb2Ni2Mo3O12.The magnetization at 1.3 K increases monotonically and issaturated around 45 T. The g value was evaluated to be 2.24from the saturated magnetization. The saturation is also seenat 4.2 K and is smeared above 10 K. As shown in the inset ofFig. 5, the magnetizations of the two compounds at 1.3 K arevery close to each other.We compared experimental χ (T ) and M(H ) at 1.3 Kwith those calculated for the isolated spin-1 tetramer modelexpressed in Eq. (1). The experimental results are close to thecalculated ones with J1 = 9 K and J2 = 18 K indicated bythe light-blue lines in Figs. 2 and 5. The light-blue line in theinset of Fig. 4(a) indicates C(T )/T calculated for the samemodel. We cannot evaluate precisely the magnetic specificheat of Rb2Ni2Mo3O12. Although the red circles in the insetof Fig. 4(a) show the total C(T )/T including lattice specificheat, the experimental and calculated results are similar to eachother.The experimental M(H ) at 1.3 K increases monotonicallyup to the saturation, whereas the calculated M(H ) at 1.3 Kshows quantum magnetization plateaus. According to theresults in CuInVO5 [24] and CrVMoO7 [19], the discrepancybetween the experimental and calculated M(H ) is probablycaused by intertetramer interactions. It is difficult, however, todetermine which intertetramer interactions are effective. Weadopted the interaction in the third shortest Ni-Ni pair (4.999Å) as the effective intertetramer interaction (Jeff). Figure 1(c)shows schematically the interacting spin-1 tetramer model.The blue lines in Fig. 5 indicate M(H ) calculated forthe interacting spin-1 tetramer model using the tetramermean-field theory. The values of the exchange interactions areJ1 = 9 K, J2 = 18 K, and Jeff = 4 K. The experimental andcalculated M(H ) at 1.3 K are in agreement with each other.FIG. 4. (a) Temperature T dependence of the specific heat dividedby T [C(T )/T ] of Rb2Ni2Mo3O12 in various magnetic fields. Theinset shows C(T )/T in zero magnetic field below 10 K. The light-blueline indicates C(T )/T calculated for the isolated spin-1 tetramer withJ1 = 9 K and J2 = 18 K. (b) C(T )/T of Rb2Ni2Mo3O12 in 3, 6,and 9 T indicated by pink circles, red squares, and green squares,respectively. Lines indicate calculated Schottky specific heat of g =2. The inset shows C(T ) of Rb2Ni2Mo3O12 (squares) and the Schottkyspecific heat (line) in 9 T.The calculated M(H ), however, is larger than the experimentalM(H ) in high fields at 10 K and higher T s. As J1 or J2increases, the calculated M(H ) at high T approaches theexperimental M(H ), whereas the discrepancy between theexperimental and calculated results of M(H ) at low T andχ (T ) becomes apparent. We also calculated M(H ) for theinteracting spin-1 tetramer model including the D term of thesingle-ion anisotropy. Here, the Hamiltonian of the single-ionanisotropy is expressed asHs = D(Sz)2. The calculated M(H )214424-4MAGNETISM OF THE SPIN-1 TETRAMER COMPOUND . . . PHYSICAL REVIEW B 96, 214424 (2017)FIG. 5. Magnetic field H dependence of the magnetization M(H )of Rb2Ni2Mo3O12 (red lines) at various temperatures. Light-blue andblue lines indicate M(H ) calculated for the isolated spin-1 tetramermodel labeled by “cal 1” and for the interacting spin-1 tetramermodel in Fig. 1(c) labeled by “cal 2,” respectively. The values ofthe parameters are J1 = 9 K, J2 = 18 K, Jeff = 4 K, and g = 2.44.Vertical positions are shifted by 1 μB/Ni per line. The inset showsM(H ) at 1.3 K of Rb2Ni2Mo3O12 (red) and K2Ni2Mo3O12 (green).depends on the D value strongly and weakly at low andhigh T , respectively. The present discrepancy between theexperimental and calculated M(H ) could not be reduced bythe introduction of the D term.We can see 0, 14 , 12 , and 34 magnetization plateaus inthe calculated line of the isolated spin-1 tetramer model at1.3 K. The 14 , 12 , and 34 magnetization-plateau phases arepolarized paramagnetic phases in which STz = −1,−2, and −3,respectively. Here, STz represents the z value of the total spin ofthe four S = 1 spins. In interacting spin tetramers, an orderedphase can appear in a magnetic-field range where M(H )increases. As described later, the specific-heat results indicatea magnetically ordered phase at low T in Rb2Ni2Mo3O12. Nomagnetization plateau in experimental lines indicates a singlemagnetically ordered phase is formed until the saturation ofthe magnetization [19].Superexchange interactions are probably ferromagnetic inthe two types of short Ni-Ni pairs because of the Ni-O-Ni angles. Antiferromagnetic direct exchange interactionsare also expected because of the short Ni-Ni lengths. Weinfer that the summation of the ferromagnetic superexchangeinteractions and AF direct exchange interaction generates thesmall AF J1 and J2 interactions.Figure 4(a) shows C(T )/T of Rb2Ni2Mo3O12 in variousmagnetic fields. We can see a peak above 3 T. The peakis not caused by isolated Ni2+ spins of impurities showingthe Schottky specific heat. Figure 4(b) shows C(T )/T ofRb2Ni2Mo3O12 and that of the Schottky specific heat of g = 2.The Schottky specific heat has a stronger H dependence than01230 2 4 6 8 10T N (K)H (T)(a)Rb2Ni2Mo3O120.811.21.41.60 2 4 6 8 10T N(H)/TN(0)H (T)(b)Rb2Ni2Mo3O12CrVMoO7Cu2CdB2O6CuInVO5FIG. 6. (a) Magnetic field H dependence of the AF transitiontemperature TN(H ) of Rb2Ni2Mo3O12. The line is a guide to eyes.(b) Magnetic field H dependence of TN(H )/TN(0) of the spin tetramercompounds Rb2Ni2Mo3O12, Cu2CdB2O6, and CuInVO5. We used 2 Kas TN(0) of Rb2Ni2Mo3O12 [the upper limit of estimated TN(0)]. Theresults of the spin-3/2 dimer compound CrVMoO7 are also shown asreference.Rb2Ni2Mo3O12. The H dependence of the Schottky specificheat is stronger when g is larger. The inset of Fig. 4(b) showsC(T ) in 9 T of Rb2Ni2Mo3O12 and the Schottky specific heatof g = 2. The peak positions are different from each other. Thedifference is larger when g is larger. Consequently, the peaksin C(T )/T of Rb2Ni2Mo3O12 indicate a phase transition to amagnetically ordered state.Figure 6(a) shows the H dependence of the peak tem-perature TN(H ) in C(T )/T of Rb2Ni2Mo3O12. The peaktemperature increases with H . As suggested by the blue line,we infer that the ordered state appears around 1.8 K even in0 T. This inference is consistent with the susceptibility andmagnetization results indicating that the GS is magnetic.Figure 6(b) shows the H dependence of TN(H )/TN(0) of thespin tetramer compounds Rb2Ni2Mo3O12, Cu2CdB2O6 [35],and CuInVO5, and the spin-3/2 dimer compound CrVMoO7.As H is increased, TN(H )/TN(0) increases in Rb2Ni2Mo3O12,whereas TN(H )/TN(0) is almost constant or decreases slightlyin the other three compounds. The phase transition temperatureincreases with H in the spin-tetrahedra system Cu2Te2O5Br2[36]. Investigations using chemical and hydrostatic pressuresuggest the closeness of the system to a quantum critical point[37–40]. In Rb2Ni2Mo3O12, we speculate that the order in 0 Tis not stable enough like an order in the vicinity of a quantumcritical point [10–12] and that the order is stabilized by theapplication of magnetic fields.214424-5HASE, MATSUO, KINDO, AND MATSUMOTO PHYSICAL REVIEW B 96, 214424 (2017)FIG. 7. Eigenenergies measured from the spin-singlet groundstate in the isolated spin-1 tetramer with J1 = 9 K and J2 = 18 K.Figure 7 shows the eigenenergies measured from the spin-singlet GS in the isolated spin-1 tetramer with J1 = 9 K andJ2 = 18 K. The spin gap value (12 K) is smaller than thedominant J2 value and is larger than the Jeff value (4 K). Weconsider that the order in 0 T is not stable enough because ofthe small Jeff value.In future, we will make single crystals of A2Ni2Mo3O12(A = Rb or K) and evaluate the single-ion anisotropy. Wewill determine the magnetic structure by neutron diffractionexperiments. We will confirm the signs of J1 and J2 fromthe magnetic structure. We will consider which intertetramerinteractions are effective to stabilize the magnetic structure.We will calculate χ (T ) and M(H ) of a more realistic modelusing quantum Monte Carlo techniques and reproduce theexperimental results.As described, the order of Rb2Ni2Mo3O12 in 0 T may besimilar to an order in the vicinity of a quantum critical point.The L-mode magnetic excitations may be observable like inthe pressure-induced or magnetic-field-induced magneticallyordered state of the interacting AF spin- 12 dimer compoundsTlCuCl3 and KCuCl3 [10–15]. We intend to perform inelasticneutron scattering and Raman scattering experiments to inves-tigate L-mode magnetic excitations. In the weakly orderedspin- 12 chain antiferromagnet Sr2CuO3, unusual magneticexcitations were recently observed by ESR experiments [41].It is reported that the excitations can be attributed to theNambu-Goldstone mode renormalized due to its interactionwith the high-energy L-mode. We also pursue such unusualexcitations in A2Ni2Mo3O12 (A = Rb or K).V. CONCLUSIONWe measured the temperature T dependence of the mag-netic susceptibility χ (T ) and the specific heat C(T ) and themagnetic-field H dependence of the magnetization M(H ) ofRb2Ni2Mo3O12 powder. A broad maximum appears around16 K in χ (T ). Although χ (T ) decreases rapidly at lowT , χ (T ) does not seem to approach a small value at 0 Kexpected for a spin singlet GS with a spin gap. The low-fieldmagnetization at 2 K has the finite slope even around 0 T.The high-field magnetization at 1.3 K increases monotonicallywithout magnetization plateaus and is saturated around 45 T.The susceptibility and magnetization at 1.3 K of K2Ni2Mo3O12are very close to those of Rb2Ni2Mo3O12. The isolated spin-1antiferromagnetic tetramer model with J1 = 9 K and J2 = 18K can closely reproduce the experimental susceptibility. Wewere able to explain the magnetization curves using the inter-acting spin-1 tetramer model with the effective intertetramerinteraction Jeff = 4 K. In C(T )/T , we can see a peak above3 T, indicating a phase transition to a magnetically orderedstate. The transition temperature TN(H ) increases with H .From the H dependence of TN(H ), probably, the ordered stateappears around 1.8 K even in 0 T. The ordered state in 0 T,however, is not stable enough like an order in the vicinity ofa quantum critical point. Longitudinal-mode magnetic excita-tions may be observable in single crystalline A2Ni2Mo3O12(A = Rb or K).ACKNOWLEDGMENTSThis work was financially supported by Japan Societyfor the Promotion of Science (JSPS) KAKENHI (Grant No.15K05150) and grants from National Institute for MaterialsScience (NIMS). M.M. was supported by JSPS KAKENHI(Grant No. 17K05516). 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