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Yuka Kusanose, Yasuyuki Shimura, Kazunori Umeo, Naomi Kawata, Toshiro Takabatake, [Taichi Terashima](https://orcid.org/0000-0001-9239-0621), [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), [Takako Konoike](https://orcid.org/0000-0002-6037-5782), [Yuya Hattori](https://orcid.org/0000-0002-3805-4659), Kazuhiro Nawa, Hung-Cheng Wu, Taku J. Sato, Takahiro Onimaru

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[Multipolar Phase Transition in a 4f 2 fcc Lattice Compound PrCdNi4](https://mdr.nims.go.jp/datasets/9323d987-de8b-4169-adbf-4923d8b830c9)

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Multipolar Phase Transition in the 4f 2 fcc Lattice Compound PrCdNi4Yuka Kusanose,1, 2, ∗ Yasuyuki Shimura,1 Kazunori Umeo,3 Naomi Kawata,3 ToshiroTakabatake,1 Taichi Terashima,4 Naoki Kikugawa,5 Takako Konoike,4 Yuya Hattori,5Kazuhiro Nawa,6 Hung-Cheng Wu,6, † Taku J. Sato,6 and Takahiro Onimaru11Department of Quantum Matter, Graduate School of Advanced Science and Engineering,Hiroshima University, Higashi-Hiroshima 739-8530, Japan2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan3Natural Science Center for Basic Research and Development (N-BARD),Hiroshima University, Higashi-Hiroshima 739-8526, Japan4Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0003, Japan5Center for Basic Research on Materials (CBRM),National Institute for Materials Science (NIMS), Tsukuba 305-0003, Japan6Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan(Dated: March 26, 2025)Transport and magnetic properties of a 4f2 fcc lattice compound, PrCdNi4, were studied. Themagnetic susceptibility, χ(T ), follows the Curie–Weiss law from 300 K to 20 K, as expected for afree Pr3+ ion. As the temperature decreases below 5 K, χ(T ) approaches a constant, indicatingvan-Vleck paramagnetic behavior. The specific heat, C(T ), displays a broad shoulder at around 4K, which can be reproduced by a doublet-triplet two-level model with an energy gap of 12 K. Theseresults suggest a non-magnetic Γ3 doublet ground state of the Pr3+ ion in the cubic crystallineelectric field. C(T ) exhibits a peak at TO = 1.0 K and this peak remains robust against magneticfields up to 5 T. In powder neutron diffraction measurements, no magnetic reflection was observedat 0.32 K < TO. Two anomalies at B = 2.1 and 5.3 T in magnetoresistance ρ(B) at 0.05 K likelyoriginate from switching in the order parameter. These results suggest that the phase transition atTO is ascribed to an antiferro-type order of the electric quadrupole or magnetic octupole of the Γ3doublet in the 4f2 fcc lattice.I. INTRODUCTIONPraseodymium-based cubic compounds with a 4f2configuration have attracted significant interest [1–3]. Inthese compounds, a strong interaction between their ac-tive multipolar degrees of freedom in the (quasi-) degen-erated crystalline electric field (CEF) ground states andconduction electrons leads to various phenomena such asmetal-insulator transition [5, 6], heavy-fermion supercon-ductivity [7–9], and non-Fermi liquid (NFL) behavior dueto the two-channel (quadrupole) Kondo effect [10–12].When the Pr3+ ion site has a cubic point group, theCEF ground state can be a non-Kramers doublet. Inthis state, the magnetic dipole is quenched, and instead,the electric quadrupole and magnetic octupole becomeactive. As a result, the quadrupole or octupole or-der may manifest itself [13–16], while the two-channelKondo effect is expected to occur if the quadrupoles areover-screened by equivalent conduction bands [12, 17–19]. An example is PrPb3, which has a non-KramersΓ3 doublet ground state, showing an antiferroquadrupole(AFQ) order at TQ = 0.4 K [20–22]. The order param-eter is the O02-type quadrupole [23]. Neutron diffractionmeasurements in a magnetic field along the [100] direc-tion revealed incommensurate sinusoidal modulation of∗ kusanose.yuka.p7@f.mail.nagoya-u.ac.jp† Present address: National Sun Yat-sen Universitythe quadrupoles [24]. This modulated structure sug-gests not only long-range quadrupole interaction medi-ated by conduction electrons but also the quenching ofthe quadrupoles due to hybridization with the conduc-tion electrons. On the other hand, no long-range orderof quadrupoles has been found in PrAg2In and PrMg3with the Γ3 doublet ground state. In these compounds,quadrupoles are prevented from ordering by either thequadrupole Kondo effect or atomic disorder inherent intheir Heusler-type structures which lowers the site sym-metry of Pr3+ [25–27].The discovery of the coexistence of the supercon-ductivity and AFQ order in PrIr2Zn20 has drawn sig-nificant attention to Pr 1-2-20 systems crystallizing ina cubic CeCr2Al20-type structure [28]. PrIr2Zn20 ex-hibits the AFQ order at TQ = 0.11 K below which thesuperconducting transition was observed at Tc = 0.05K [29–37]. The coexistence of the superconductivityand quadrupole order was also observed in isostructuralPrRh2Zn20 [38, 39], PrTi2Al20 [40–44], and PrV2Al20[40, 45]. These findings imply that the superconductiv-ity is related to the interaction between conduction elec-trons and the multipoles in the non-Kramers doublet [46].Moreover, the NFL behaviors were observed not onlyin PrIr2Zn20 [10] and PrRh2Zn20 [39] but also in dilutePr systems Y(Pr)Ir2Zn20 [11, 47–49] and Y(Pr)Co2Zn20[50]. These NFL behaviors probably result from thequadrupole Kondo effect to form a composite electronicorder involving the local quadrupole and the itinerantmailto:kusanose.yuka.p7@f.mail.nagoya-u.ac.jp2bands as the ground state [12, 51–54].Face-centered cubic (fcc) compounds PrTNi4 (T = Mgand In) are another family of Pr-based intermetallic com-pounds having the Γ3 doublet ground state [55–58]. Theycrystallize in a cubic MgSnCu4-type structure [59]. Asillustrated in Fig. 1(a), since the Pr sublattice is equiv-alent to an fcc lattice, we anticipate an unusual groundstate due to the geometrical frustration between activemultipoles of the nearest neighbor Pr ions. Addition-ally, anisotropic terms in the quadrupole interactions onthe fcc lattice become effective alongside isotropic ones[61, 62]. These potentials are quite distinct from thePr-based compounds mentioned above in terms of theanisotropic and competitive multipolar interaction. Infact, though PrInNi4 exhibits ferromagnetic order at TC= 0.75 K due to exchange interaction induced by cou-pling between the Γ3 doublet and the first-excited triplet[57, 58], PrMgNi4 exhibits no phase transition down to0.1 K [55]. This hindered quadrupole order in PrMgNi4was attributed to a symmetry lowering caused by excessMg atoms substituting for the Pr atoms or strong hy-bridizations between the 4f2 and conduction bands [55].Otherwise, if the isotropic and anisotropic interactions inthe fcc lattice are relatively weak compared to the energysplitting from the Γ3 doubet to the excited Γ1 singlet, thequadrupole order may be suppressed [61, 62].In our current study, we have focused on PrCdNi4 crys-tallizing in the cubic MgSnCu4-type structure [63]. Asshown in Fig. 1(b), Pr is surrounded by Cd atoms at4c site and Ni atoms at 16e. If the atomic disorder isreduced compared to the sister compound PrMgNi4 [55],the electric quadrupolar and/or magnetic octupolar orderof the non-Kramers doublet may not be hindered. Withbearing this in mind, we conducted measurements of thetransport and magnetic properties of PrCdNi4 to deter-mine the CEF ground state and understand the possibleinvolvement of quadrupole and/or octupole in the forma-tion of the ground state. Powder neutron diffraction mea-surements were performed to judge whether the phasetransition results from a magnetic or non-magnetic ori-gin. Recently, physical properties of the series of RCdNi4have been reported for R = Ce, Nd, Sm, and Gd-Tm [64]except for R = Pr.II. EXPERIMENTAL PROCEDUREA. Preparation and characterization of samplesWe synthesized samples of PrCdNi4 using the Cd self-flux method. First, we prepared the binary PrNi4 alloyby arc melting. Substantially, we sealed the PrNi4 ingotand Cd shots in double quartz amples in an argon atmo-sphere. The ampoule was then heated up to 1100◦C inan electric furnace and slowly cooled down. At 500◦C,the ampoule was quickly removed from the furnace andcentrifuged to remove the molten Cd flux. The PrCdNi4samples were cubes of typically 3.0 mm3 and found toFIG. 1. (a) Cubic MgSnCu4-type crystal structure of PrTNi4(T = Mg, Cd, and In) with the space group of F 4̄3m [59].In the structure, large (red) spheres represent Pr atoms, thesmall (orange) spheres represent T atoms, and the tetrahe-dra are formed by four Ni atoms shown with the (green)spheres. The Pr sublattice forms an fcc lattice. (b) A Pr-centered atomic cage made of four T atoms at 4c and twelveNi atoms at 16e. The crystal structure images were drawnusing VESTA [60].consist of grains smaller than 0.3 mm, as characterizedby metallographic examination. The grain size was toosmall to select single crystalline samples for our transportand magnetic measurements.The samples were analyzed using powder x-ray diffrac-tion and electron-probe microanalysis (EPMA). Back-scattered electron images and x-ray diffraction patternsare shown in Supplementary Material [65]. The atomiccompositions were determined by averaging over 10 dif-ferent regions for each sample with a JEOL JXA-8200analyzer. Assuming that the sum of the compositionsof the Pr and Cd atoms were 2, the atomic ratio ofthe sample batch for measurements was determined asPr1.00(1)Cd1.00(1)Ni3.89(4), where the numbers in paren-theses are the standard deviations. It is noted thatthe composition obtained for Pr and Cd closely matchesthe stoichiometric ratio, which remains consistent acrosssamples within the standard deviations. Small amountsof impurity phases of PrNi2Cd20 and PrNi5 were detectednot only in the backscattered electron images but alsoin the powder x-ray diffraction patterns. The powderx-ray diffraction pattern confirms the cubic MgSnCu4-type structure for the main phase. The single-crystalx-ray structural analysis was performed at 293 K with acrystal smaller than 0.2 mm using the Mo Kα radiationwith the wavelength of λ = 0.71073 Å, monochromatedby a multilayered confocal mirror using a Rigaku Xta-LAB Synergy-DW area-detector diffractometer. Detailsof the measurement, data collection, and refinement aredescribed in Supplemental Materials [65]. Figure 1(a)shows the cubic MgSnCu4-type structure of PrCdNi4[63]. In this structure, the Pr atoms occupy the fcc posi-tion of the unit cell, and the point group of the Pr site isthe cubic Td. The structural refinement did not show anyevidence for the site exchange between Pr and Cd sites.The lattice constant was evaluated as a = 7.12932(8) Åfor PrCdNi4. This value is 0.15% larger than 7.11832(16)Å for PrMgNi4, determined by refining the powder x-ray3diffraction pattern at room temperature.B. Physical property measurementsThe electrical resistance was measured using a stan-dard four-probe AC method. The measurements weredone with a Gifford-McMahon-type refrigerator for 3–300 K and with a commercial Cambridge Magnetic Re-frigerator mFridge mF-ADR50 for 0.1–4 K. The magneticfield dependences of ρ(B) were measured up to 17.5 T attemperatures down to 0.05 K by the AC method using a3He-4He dilution refrigerator equipped with a 20 T mag-net at NIMS. Magnetization was measured from 1.8 to300 K in magnetic fields for B ≤ 5 T using a commercialsuperconducting quantum interference device (SQUID)magnetometer (Quantum Design, MPMS). For 0.3 < T< 4.2 K in B ≤ 8.5 T, a capacitive Faraday method wasadopted. Thereby, we used a high-resolution capacitiveforce-sensing device installed in a 3He single-shot refrig-erator (Heliox, Oxford Instruments) [66]. The specificheat was measured using the thermal relaxation methodin the temperature range of 0.4 < T < 300 K for B ≤ 7 Twith a Quantum Design physical property measurementsystem (PPMS).In order to observe possible magnetic reflections in anordered phase, powder neutron diffraction experimentswere conducted using the ISSP triple-axis spectrometerGPTAS at JRR-3M in JAEA at Tokai, Japan [67]. Neu-tron beams with a wavelength of λ = 2.4563 Å (∼13.7meV) were obtained by the 002 reflection of a pyrolyticgraphite (PG) monochromator. Horizontal collimationof 40′-Monochromator-40′-Sample-40′-Analyzer-80′ for atriple-axis mode was utilized. A 3He refrigerator achievedthe base temperature at 0.32 K. In order to preventstrong neutron absorption of Cd atoms, the powderedsample of 0.35 g was thinly spread (approximately 0.2mm thick) on a single-crystalline silicon wafer [68, 69].The scattering plane was tilted by 1 degree away fromthe [100] direction, which is perpendicular to the surfaceof the silicon wafer.III. RESULTS AND DISCUSSIONA. Electrical resistivityThe temperature-dependent electrical resistivity, ρ(T ),of PrCdNi4 is compared with that of PrMgNi4 [55] inFig. 2. The residual resistivity ratio (RRR), evaluatedas ρ(300 K)/ρ(0.1 K), is 6.3 for PrCdNi4. This RRRvalue is more than two times higher than 2.8 for a sin-gle crystalline PrMgNi4 [55]. The higher RRR value forPrCdNi4 is consistent with the EPMA result. The com-position of Pr:Cd:Ni = 1.00(1):1.00(1):3.89(4) is closeto the stoichiometric ratio by assuming that the totalcomposition of Pr and Cd is 2. For the PrMgNi4 sam-ple, on the other hand, the composition was Pr:Mg:Ni !"!#!$!%!&!'!! ()*+(,-.%!!&#!&!!'#!'!!#!!!()/.012345$666(7("8%019:45$(;##<666(7(&8='!>= %&'!012345$FIG. 2. Temperature dependence of the electrical resistivityρ(T ) of PrCdNi4 (red) and PrMgNi4 (black) [55]. The insetdisplays the ρ(T ) data for T ≤ 3 K. The open arrow indicatesa broad shoulder at 1.2 K.= 0.94(1):1.06(1):3.86(2), where excess Mg atoms substi-tute for the Pr atoms, leading atomic disorder [55].As shown with the arrow in the main panel of Fig. 2,ρ(T ) for PrCdNi4 exhibits a shoulder at around 15 K,possibly due to the increase in scattering of conductionelectrons by thermal excitations of the 4f2 electrons be-tween the CEF levels [70]. This energy scale is consistentwith the 12 K energy gap between the ground state Γ3doublet and the first excited triplet, as discussed later.Focusing on the lower temperature range in the inset ofFig. 2, there is a broad shoulder centered at around 1.2K. This shoulder results from a phase transition observedin the specific heat measurements shown later.B. Magnetic susceptibility and isothermalmagnetizationFigure 3 shows the temperature dependence of themagnetic susceptibility χ(T ) and the inverse χ−1(T ) ofPrCdNi4 measured in the magnetic field of B = 1 T.χ−1(T ) follows a modified Curie–Weiss equation: χ(T )= C/(T − θp) + χ0, where C, θp, and χ0 representthe Curie constant, paramagnetic Curie temperature,and temperature independent susceptibility, respectively.The (red) solid curve represents the fit to the χ(T ) databetween 20 and 300 K using the above equation withθp = −8.3(4) K and χ0 = 5.7(2) ×10−4 emu/mol. Thenegative value of θp indicates antiferromagnetic intersiteinteraction between the Pr moments. The effective mag-netic moment was evaluated to be µeff = 3.70(1) µB/f.u.,which is moderately close to the value of 3.58 µB for afree trivalent Pr ion. On cooling below 4 K, χ(T ) doesnot diverge but approaches a constant, as shown in theinset. This is a van-Vleck paramagnetic behavior of anonmagnetic CEF ground state of the Pr ion. As shown4 !" !# !$   %&'()*)'+,-"  #  $    !%.-#/ #  $/ $  /  $*  %'+,)*)&'(-")0)$)1)#&22)0)"!3 %$-)45)*)67)$8)))0)9:!"%;-).67<=>?; !$@ !$# ! :%&")%&'(*'+,-$ $ ")0)$)1"/:)1 !#FIG. 3. Temperature dependence of the magnetic suscepti-bility, χ(T ), and the inverse χ−1(T ) of PrCdNi4 measured ina magnetic field of B = 1 T. The χ−1 data can be fitted witha modified Curie–Weiss equation. See text in detail. Theinset shows the magnetization divided by the magnetic field,M(T )/B, at magnetic fields of B = 1, 3, 5, and 8 T, withoutany offset.in the inset of Fig. 3, the M(T )/B data at B = 1, 3, 5,and 8 T do not exhibit a clear anomaly for T < 2 K.The isothermal magnetization M(B) data of PrCdNi4at temperatures of 0.3, 0.8, 1.5, 4.2, and 10 K are shownin Fig. 4. The data for temperatures below 4.2 K wereobtained by the capacitive Faraday method, while thedata at 10 K were measured by the SQUID magnetome-ter. The M(B) data for T ≤ 4.2 K are vertically offset forclarity. All M(B) data show a monotonous increase withincreasing magnetic field up to 8.5 T. These magnetiza-tion curves are reproduced reasonably well using a CEF !"#!$#!""!$" %&'(%)%*+,-./"!"&0, !"#$ !%#$&!'#$ #(#& #$)!*#$ !"#$%&FIG. 4. Isothermal magnetization M(B) of PrCdNi4 at T =0.3, 0.8, 1.5, 4.2, and 10 K. The data at T ≤ 4.2 K are ver-tically offset for clarity. The dashed curves are calculationswith the CEF parameters of W = −3.3 K and x = 0.8 andantiferromagnetic exchange interaction of K1 = −0.6 K be-tween the Pr moments. See text in detail.level scheme and antiferromagnetic exchange interaction,which will be discussed later.C. Specific heat and magnetic entropyThe specific heat C(T ) data of PrCdNi4 for T ≤ 15 Kare shown in Fig. 5(a). Since a nonmagnetic counterpartLaCdNi4 could not be synthesized, the phonon contribu-tion Cph(T ) was evaluated using the Debye model withθD = 270.7(6) K as described in the Supplemental Mate-rials [65, 71]. By subtracting Cph(T ) from the total C(T )data, we estimated the magnetic contribution Cmag(T ).Cmag exhibits a shoulder at around 4 K, which is at-tributable to the Schottky specific heat due to thermalexcitations from the CEF ground state to excited levels.A two-level Schottky model provides the expression ofC(T ) with the equation asC =nm∆2kBT 2e−∆/kBT(n+me−∆/kBT)2 , (1)where n and m represent the degeneracy of the groundand excited multiplets, respectively, and ∆ is the energygap between the two levels. Considering the van-Vleckparamagnetic behavior shown in the inset of Fig. 3, theCEF ground state of the Pr ion for the cubic Td pointgroup can be either the nonmagnetic Γ1 singlet or Γ3doublet. Calculations with a singlet-triplet (Γ1) two- !"#$%! &&'()'*&+,-.. / !/!!&'*.%! / !/!"+01& &'()'*&+,-..234567$#&8&!'0. %&* !  "  !"!#$-9/$-9%' )%.:$-9% +01 ;<  !"#$%! &&'()'*&+,-..=% !!&'*.#&8&!=/>&? &?$$@/'A.FIG. 5. (a) Temperature variations of the specific heat C(T )(open circles) and the magnetic contribution Cmag (close cir-cles), as well as the magnetic entropy Smag (right-hand scale)of PrCdNi4. Cph is the phonon contribution evaluated us-ing a Debye model as explained in Supplemental Materials[65]. The (red) dashed and (blue) dotted curves representthe Schottky specific heat calculated by a two-level model ofdoublet-triplet (Γ3) and singlet-triplet (Γ1) with an energygap of 12 K. The (orange) dot–dashed line represents thecalculation with W = −3.3 K and x = 0.8 determined forPrMgNi4 [56]. (b) The specific heat C(T ) at magnetic fieldsof B ≤ 7 T. The data for B ≥ 1 T are offset for clarity. Thearrows indicate the peak temperature, which remains robustagainst magnetic fields up to 4.5 T.5level model and a doublet-triplet (Γ3) one, with an en-ergy split of 12 K, are shown with the (red) dashed and(blue) dotted curves, respectively. It is assumed that thefirst excited state is a triplet, either the Γ4 or Γ5 triplet[72]. The respective two-level schemes are depicted inthe lower part of Fig. 5(a). The Γ3 model better repro-duces the shoulder of Cmag(T ) at around 4 K than theΓ1 model. Therefore, the CEF scheme likely consists ofthe Γ3 doublet ground state and the first excited triplet.Note that the absolute value of Cmag at around 4 K issmaller than that expected by the Γ3 model. This dis-crepancy may be ascribed to a relatively small fractionof the pristine PrCdNi4 in the measured sample, whichcould contain some impurities of PrNi5 and PrNi2Cd20.On the other hand, the larger value of Cmag observedabove 7 K may result from the contribution of the CEFlevels at higher energy. Otherwise, it is attributable toan underestimation of Cph.In order to verify this CEF scheme, we calculated theisothermal magnetization. The cubic CEF Hamiltonianis described by the equation asHCEF = W[x60(O04 + 5O44)+1− |x|1260(O06 − 21O46)],where W and x represent the CEF parameters, and Omn ’sstand for the Stevens operators [72]. The energy gap of12 K between the Γ3 doublet and the first excited tripletis nearly the same as that observed in PrMgNi4 [55], al-though it is anticipated that the energy splitting wouldbe smaller in PrCdNi4 due to its larger lattice parameter.Therefore, we use the parameters of W = −3.3 K and x= 0.8 for PrMgNi4 as determined by the inelastic neu-tron scattering experiments; Γ3(0)–Γ4(13.2 K)–Γ1(31.7K)–Γ5(134.6 K) [56]. The isothermal magnetization iscalculated with the above CEF parameters and an in-tersite magnetic interaction using the following Hamilto-nian.H = HCEF + gJµBJB −K1⟨J ⟩J ,where gJ = 4/5 is the Landé g-factor for a Pr3+ ion,J a total angular momentum, and K1 a coefficient ofthe magnetic inter-site interaction between the Pr ions.The isothermal magnetization calculated for B || [111]with antiferromagnetic interaction of K1 = −0.6 K isshown with the dashed lines in Fig. 4. The M(B) calcu-lations at T = 1.5, 4.2, and 10 K match the data of thepolycrystalline sample. It is noted that the specific heatcalculated using the above CEF parameters moderatelyreproduces the shoulder of Cmag around 4 K, as depictedby the (orange) dot-dashed line in Fig. 5. These resultsgive further support that the CEF ground state is the Γ3doublet carrying the electric quadrupole and magneticoctupole.With decreasing temperature below 2 K, Cmag(T ) ex-hibits a peak at TO = 1.0 K. This peak is the manifesta-tion of a phase transition of the Γ3 doublet. It is crucial toexclude the contribution of the impurity phases; neitherPrNi5 nor PrNi2Cd20 exhibit any phase transition near 1K [73–76]. The magnetic entropy Smag, estimated by in-tegrating the Cmag/T data with respect to temperature,is plotted on the right-hand axis of Fig. 5(a). Here, weassumed that the Cmag/T would approach zero linearlyas T decreases from 0.4 K to 0 K. Then, we estimated themagnetic entropy Smag at 0.4 K to be 0.21 J/K, whichwas added to Smag(T ) for T > 0.4 K. At TO, Smag isonly 40% of Rln2, and at around 6 K it reaches Rln2expected for the full order of the doublet. This suggeststhat the phase transition is due to the multipolar degreesof freedom in the Γ3 doublet, such as Γ3-type electricquadrupoles or Γ2-type magnetic octupole. Smag reachesRln5 at 18 K, which is consistent with the aforementionedCEF level scheme with the triplet state separated by 12K from the Γ3 doublet ground state.Figure 5(b) shows the C(T ) data in magnetic fields ofB ≤ 7 T. The peak temperature TO remains unchangeduntil the magnetic field is increased up to 4.5 T. Therobustness of TO to the magnetic field is reasonable whenthe phase transition does not arise from the magneticdipole but from the quadrupole or the octupole in theΓ3 doublet ground state. With further increase in themagnetic field above 5 T, the peak is suppressed andbecomes vague.D. Electrical resistivity in magnetic fieldsFigure 6(a) shows the electrical resistivity ρ(T ) in mag-netic fields of B = 0, 1, 3, 5, and 6 T. Here, the directionof the magnetic field is parallel to that of the electricalcurrent. In the inset of Fig. 6(a), the derivative of theelectrical resistivity, dρ(T )/dT , is shown. At B = 0, abroad maximum is observed at 0.88 K, which is compa-rable to TO, the peak of Cmag(T ). This maximum shiftsslightly to lower temperatures with increasing magneticfields, reaching 0.86 K at B = 6 T. The magnetic-fieldvariation of the maximum is consistent with that of TOin C(T ) shown in Fig. 5(b).Figure 6(b) shows the magnetic-field dependence of thenormalized magnetoresistance {ρ−ρ0}/ρ0 at various con-stant temperatures. The data were obtained in magneticfields up to 17.5 T at temperatures of 0.05, 0.2, 0.75, and1.2 K. At 0.05 K, there are a shallow minimum at B1= 2.8 T and a hump at B2 = 4.7 T. At B > 5 T, themagnetoresistance decreases and then remains constantfor B ≥ 10 T. The anomalies at B1 and B2 exist at 0.2and 0.75 K. However, at 1.2 K above TO, the minimumat B1 disappears, and the hump changes to a broad max-imum. These anomalies are probably attributed to theswitching of the order parameter in the ordered state, asdiscussed later.6FIG. 6. (a) Temperature dependence of the electrical resistiv-ity ρ(T ) of PrCdNi4 for T < 3 K in various constant magneticfields B up to 6 T. The inset shows the derivative dρ/dT ,where the arrows indicate the peak temperatures. The datain B are offset for clarity; for example, the bold arrow out-side the vertical left-axis indicates the offset for the data at B= 1 T. (b) Normalized magnetoresistance at various constanttemperatures 0.05, 0.2, 0.75, and 1.2 K in B up to 17.5 T. Thearrows highlight a shallow minimum at 2.8 T and a hump at4.7 T.E. Powder neutron diffractionNeutron diffraction patterns of the powdered sampleof PrCdNi4 at T = 2.0 K (red) > TO and 0.32 K (blue)< TO are represented in Fig. 7. Most of the observedpeaks are identified as nuclear peaks of PrCdNi4 andaluminum from the sample container. A peak at 2θ =33.2◦, indicated by a star, is ascribed to a small amountof impurity PrNi5. In the differential pattern (green), nopeak appears. Therefore, it is concluded that the phasetransition at TO does not result from a magnetic origin.This finding is consistent with the manifestation of themultipole order in the Γ3 doublet ground state, as willbe discussed with the B–T phase diagram.The Rietveld profile fitting was conducted to deter-mine the structural parameters of PrCdNi4. The fits areshown with the (black) solid lines in Fig. 7. The latticeparameter of a = 7.1112(3) Å at 0.32 K was estimatedby RIETAN-FP [77]. Details are provided in the Supple-mental Materials [65]. !!!"!!!#!!!$!!!%!!!!&'()'*+(,-./'(*0%1!*2-1!3!4! !"!#!$!%!!$5-.6)782   !"##$%  ###!"&9:;6<+"9=>6):-*?@AB)&CC9D"E +-F-%#83-@)G-!"F-!8#$-H-!"F-$8!-HFIG. 7. Powder neutron diffraction patterns of PrCdNi4 at T= 2.0 K (red) and 0.32 K (blue), measured above and belowTO = 1.0 K, respectively. The (green) line shows the differencebetween the data at 2.0 K and 0.32 K. The black lines repre-sent the fits with the Rietveld analysis. The bars indicate thescattering angles for the nuclear Bragg peaks of PrCdNi4 Apeak indicated by a star signifies an impurity phase of PrNi5.F. Magnetic field vs temperature phase diagramFigure 8 shows the magnetic-field vs temperature (B–T ) phase diagram of PrCdNi4. The phase transition tem-perature TO is determined from the C(T ) and ρ(T ) data.TO remains almost unchanged up to B = 6.5 T. The ro-bustness of TO to the magnetic field is a characteristicof the multipole order, such as the electric quadrupoleor magnetic octupole order in the Γ3 doublet groundstate. This behavior is consistent with the absence ofmagnetic reflection at 0.32 K. Moreover, the anomaliesfound in the ρ(B) data are plotted in the B–T phase di-agram. Two horizontal lines at 2 T and 5 T look likephase boundaries within the ordered phase. Since M(B)shows no anomaly at 0.3 K, as shown in Fig. 4, theseboundaries must relate to the multipolar order param-eter rather than the realignment of magnetic dipoles ina conventional antiferromagnetic order. Thereby, theseboundaries are likely attributed to the switching of theorder parameters as observed in the AFQ order in thePr-based compounds with the Γ3 doublet ground state[24, 30, 36]. When ferroquadrupole (FQ) order arisesfrom isotropic quadrupole interactions, its uniform or-der parameter remains unchanged in a magnetic field.Therefore, these results suggest that an antiferro-typemultipole order manifests itself below TO.Let us discuss why the multipolar transition manifestsitself in PrCdNi4 even though the long-range order ofthe Γ3 doublet is hindered in the isostructural PrMgNi4[55]. One possible reason is that the composition ofPr1.00(1)Cd1.00(1)Ni3.89(4) is close to the stoichiometric ra-tio, while excess Mg atoms substitute for the Pr atomsin the sample of Pr0.94(1)Mg1.06(1)Ni3.86(2) [55]. With-7out the atomic disorder in PrCdNi4, the non-KramersΓ3 doublet’s degeneracy could be conserved to give riseto the phase transition due to intersite exchange interac-tions. It has been noted that the AFQ order easily col-lapses due to a small amount of atomic exchange [78, 79].This may also apply to the absence of quadrupole orderin PrMgNi4, which further supports the AFQ order inPrCdNi4.On the other hand, the competitive isotropic J andanisotropicK nearest-neighbor interactions in the fcc lat-tice may affect quadrupolar order, combined with theCEF effect, as theoretically investigated using a four-site mean-field approximation [61]. According to thiscalculation, the order parameter is strongly dependenton the values of J and K scaled by the excitation energyfrom the Γ3 doublet to the excited Γ1 singlet, denoted asE1. Regarding the finite temperature properties of thequadrupole orders, there appear not only the FQ phase,as well as O02- and O22-type AFQ phases, but also triple-qorder with partially ordered sites. Moreover, the transi-tion temperature is significantly suppressed over a widerange of small values of J and K [61]. In PrCdNi4, E1 isestimated as 31.7 K for W = −3.3 K and x = 0.8, lead-ing to TO/E1 = 0.03. This value is roughly comparableto that obtained where the absolute values of K/E1 andJ/E1 are less than 0.01 in the J-K phase diagram. Fur-thermore, since the phase transition is of second orderand no successive transition was observed, the groundstate is likely characterized by the O22-type AFQ order.To reveal the impact of the competitive anisotropic andisotropic interactions between the multipoles inherent inthe fcc lattice, it is essential to observe the anisotropic de-pendence of physical properties with respect to the crys-tal axes and external field axes using single-crystallinesamples.IV. SUMMARYWe have conducted various measurements on a cubiccompound PrCdNi4, including magnetic susceptibility χ,isothermal magnetization M , specific heat C, electri-cal resistivity ρ, magnetoresistance, and powder neutrondiffraction. χ(T ) approaches a constant value on coolingbelow 10 K. This van-Vleck paramagnetic behavior isconsistent with the expected non-magnetic CEF groundstate for Pr3+. The magnetic specific heat Cmag exhibitsa shoulder at around 4 K, ascribed to CEF excitationsfrom the Γ3 doublet ground state to an excited triplet.Furthermore, C(T ) shows a peak at TO = 1.0 K, whichtemperature does not change significantly in magneticfields up to 4.5 T. This peak is likely attributed to multi-polar degrees of freedom in the Γ3 doublet. No magneticreflections were observed below TO in the powder neu-tron diffraction pattern. It is noted that Smag reachesonly 40% of Rln2 at TO. This reduction in the entropyis probably attributed to competitive anisotropic mul-tipole interactions inherent in the 4f2 fcc lattice [61]. !"! #$%&' !!#$(&!)*)+,-,#.$!&#"#!&#"# &FIG. 8. Magnetic field (B) vs temperature (T ) phase diagramof PrCdNi4 determined by the measurements of the specificheat C(T ), electrical resistivity ρ(T ), and magnetoresistanceρ(B). “MO” represents a multipole order, To is the orderingtemperature, and “Para” indicates the state where long-rangemultipole order is absent.Moreover, the magnetoresistance ρ(B) at T = 0.05 K(< TO) showed two anomalies at B = 2.8 and 4.7 Twhich are attributable to switching in the order parame-ter. If this is the case, the phase transition at TO resultsfrom an antiferro-type multipole order in the Γ3 doubleton the 4f2 fcc lattice. To further understand the or-der parameter, it is necessary to measure the anisotropyof the magnetic and transport properties using single-crystalline samples of PrCdNi4. In addition, neutrondiffraction and nuclear magnetic/quadrupole resonancemeasurements should be conducted in different orienta-tions of magnetic fields. Resonant x-ray diffraction andultrasonic measurements are promising methods for di-rectly detecting quadrupoles and/or octupoles in a zeromagnetic field.ACKNOWLEDGMENTSThe authors would like to thank Y. Yamane, H.Kusunose, T. Ishitobi, K. Hattori, M. Nohara, T. Mat-sumura, H. Funashima, and H. Harima for helpful dis-cussion. The authors also would like to thank R. Ya-mamoto and S. Mizutani for their measurements of thelow-temperature magnetoresistance at NIMS. The au-thors thank Y. Shibata for the electron-probe micro-analysis carried out at N-BARD, Hiroshima University.The measurements of the magnetization with MPMS andthe capacitance Faraday method with the 3He Helioxrefrigerator, the specific heat with the PPMS and theCambridge Magnetic Refrigerator mFridge mF-ADR50were performed at N-BARD, Hiroshima University. Thesingle-crystal x-ray structural analysis was performed us-8ing a Rigaku XtaLAB Synergy-DW area-detector diffrac-tometer at N-BARD, Hiroshima University. 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