Neutron scattering study of magnetic ordering and excitations in the ternary rare-earth diborocarbide Ce^{11}B_2C_2
Isao Nakanowatari, Rei Morinaga, Takahiro Onimaru, Taku J. Sato
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Neutron scattering study of magnetic ordering and excitations in the ternaryrare-earth diborocarbide Ce B C Isao Nakanowatari, Rei Morinaga, Takahiro Onimaru,
2, 1, ∗ and Taku J Sato
1, 2, † Neutron Science Laboratory, Institute for Solid State Physics,University of Tokyo, 106-1 Shirakata, Tokai, Ibaraki 319-1106, Japan SORST, Japan Science and Technology Agency, Kawaguchi, Saitama, Japan (Dated: November 1, 2018)Neutron scattering experiments have been performed on the ternary rare-earth diborocarbideCe B C . The powder diffraction experiment confirms formation of a long-range magnetic orderat T N = 7 . q = [0 . , . , . ~ ω ≈ , f spins and conduction electrons in the paramagnetic phase. A prominent featureis suppression of the quasielastic fluctuations, and concomitant growth of a sharp inelastic peak in alow energy region below T N . This suggests dissociation of the conduction and localized 4 f electronson ordering, and contrasts the presently observed incommensurate phase with spin-density-waveorder frequently seen in heavy fermion compounds, such as Ce(Ru − x La x ) Si . I. INTRODUCTION
The ternary rare-earth diborocarbides REB C (RE:rare-earths) with heavy RE elements attract special at-tentions recently, as they exhibit intriguing successive or-dering at low temperatures. Exemplified by DyB C ,the specific heat measurement detects two anomalies at T c2 = 15 . T c1 = 24 . The higher temperatureanomaly at T c1 is attributed to ordering of quadrupolemoments by various measurements such as the ultra-sonic vibration measurement and resonant X-ray diffrac-tion. It has been suggested that the quadrupolar mo-ments originate from nearly degenerated two Kramers’doublets realized by pseudo-cubic local symmetry aroundthe Dy ions. On the other hand, the lower tempera-ture anomaly at T c2 is due to magnetic (dipole) order-ing. The magnetic structure established below T c2 isnon-trivial; a complicated multi- q ferrimagnetic struc-ture is realized with the four modulation vectors q =(1 , , , (0 , , / , (0 , ,
0) and (0 , , / Asexemplified above, interplay of dipolar and quadrupolardegrees of freedom is essential for the complex succes-sive ordering in the REB C compounds with heavy REelements.In contrast, REB C with the light rare-earth elementRE = Ce does not have quadrupolar degree of freedom;absence of the ground state degeneracy has been reportedin the ultrasonic vibration measurement. Nonetheless,CeB C is of particular interest because influence ofmagnetic interactions to the complex successive orderingmay be separately deduced from that of the quadrupo-lar interactions by comparing ordering behavior of non-quadrupolar CeB C with those in the heavy RE sys- tems. Also, strong electron correlations originating fromlarger hybridization between 4 f and conduction electronsmay bring about intriguing low-temperature properties.Earlier studies on CeB C may be summarized asfollows. Electronic specific heat coefficient γ was esti-mated in the paramagnetic temperature range as γ =98 . . Thus, CeB C may be classified asa heavy-fermion system with moderate mass enhance-ment, a typical consequence of strong electron correla-tions in the 4 f electron systems. A sharp anomaly at T N = 7 . indicating that a certain mag-netic order is established below this temperature. Thespecific heat anomaly is followed by a broad shoulderaround T = 6 . Q = ( δ, δ, δ ′ )with δ ≈ .
161 and δ ′ ≈ . On the basis of thisobservation, an incommensurately modulated structurewas inferred in the ordered phase. However, because ofinsufficient quality of the crystal, it was the single peakthat the diffraction experiment could detect, and con-sequently, the magnetic structure of the ordered phasecould not be determined. Inelastic neutron scattering hasalso been carried out to determine the crystalline-electric-field (CEF) splitting of the Ce
4f levels; the inelasticspectrum was measured at 7 K for − < ~ ω <
45 meV.However, because of the limited energy range and tem-perature point, it hardly provides details of the spin fluc-tuations and excitations in CeB C .Knowledge on the low-temperature magnetic structureis essential for the understanding of the magnetic order-ing in the CeB C , and its relation to REB C withheavy RE elements. The CEF splitting and spin fluc-tuation spectrum are also crucial information to eluci-date its magnetic properties. Hence, to address theseissues we have undertaken low-temperature neutron pow-der diffraction and inelastic scattering experiments inthe present work. In the powder diffraction we havesucceeded in obtaining intensity data for a number ofmagnetic Bragg reflections, whereas inelastic spectra ina wide energy range up to 80 meV were collected atseveral temperatures below 100 K. These experimentsenable us to unambiguously determine the magneticstructure in the ordered phase, as well as peak ener-gies in the inelastic spectrum; it will be shown that thelowest-temperature magnetic structure is a sinusoidallymodulated structure with the modulation vector q =[0 . , . , . ~ ω ≈ , T N , and a sharp inelastic peak emerges in the lowenergy region. II. EXPERIMENTAL DETAILS
Polycrystalline samples of Ce B C (about 10 gramsin total) were prepared using an arc furnace under an Ar gas atmosphere. Purity of the starting elements was99.9 % for Ce and C. To avoid strong neutron absorp-tion of natural boron, we used the isotope enriched B(99.53 % enrichment). Structural quality of the resultingpolycrystalline samples was checked by the X-ray powderdiffraction as well as the neutron diffraction.The polycrystalline samples were crushed into powderand loaded in a thin Al sample cell for the neutron scat-tering experiments. The cell was then set to a He closed-cycle refrigerator. Neutron powder diffraction was per-formed using a powder-diffraction detector bank newlyinstalled to the LAM-80ET inverted-geometry time-of-flight (TOF) spectrometer at the KENS spallation neu-tron source, KEK, Japan. The diffraction detector bankhas 34 detectors covering the scattering angle range of138 ◦ < θ < ◦ . We used the incident neutrons ina wave-length range of 1 . < λ < .
40 ˚A, which cor-responds to the d -range of 0 . < d < . MSAS-TOF . To perform profile fitting of highly asymmetric peaks atlarge time-of-flights, a special profile function is used: f prof ( t ′ ) = ( a α [(1 − η ) f G ( t ′ )+ ηf L ( t ′ )] a α +2 α for t ′ < αα { [2 a +(1 − a − a ) α t ′ ] exp( − α t ′ )+8 a α t ′ exp( − α t ′ ) } a α +2 α for t ′ >
0. (1)In the above expression, t ′ = t TOF − t , where t is atime-of-flight at which a Bragg reflection appears. Thetwo functions f G ( t ′ ) and f L ( t ′ ) are the Gaussian andLorentzian functions defined by: f G ( t ′ ) = exp( − t ′ /γ ) √ πγ G ,f L ( t ′ ) = γ L π ( γ + t ′ ) . (2)The profile for negative t ′ is a modified version of thepseudo-Voigt function, allowing independent valuesfor γ G and γ L . On the other hand, to simulate a longtail due to the solid methane moderator at the KENSspallation source, a set of exponentially decaying func-tions is assumed for t ′ > The parameters are assumed tobe linearly t dependent in the present TOF range: η = η + η t , γ G = γ G1 + γ G2 t , γ L = γ L1 + γ L2 t , a = a + a t , a = a + a t , α = α + α t ,and α = (1 − η ) / ( √ πγ G ) + η/ ( πγ L ). Details of the newdiffraction bank at LAM-80ET and the analysis code willbe published elsewhere. Neutron inelastic scattering experiment was performed using the LAM-D inverted-geometry time-of-flight spec-trometer, also installed at the KENS spallation neu-tron source. Final energy was fixed to 4.59 meV us-ing the pyrolytic graphite (PG) 002 reflections, whereashigher harmonic neutrons were eliminated by cooled Befilters. There are four analyzers at the scattering angles2 θ = ± ◦ and ± ◦ ; we mainly show data taken withthe lower angle detectors in this report, and higher an-gle data are used only for phonon subtraction purpose.Energy resolution was estimated as ∆ E = 0 .
42 meV [fullwidth at half maximum (FWHM)] at the elastic positionusing a vanadium standard. Background was subtractedfrom the raw data using a proper combination of empty-cell and absorber runs, and absorption effect was cor-rected using numerically calculated absorption factors.
III. RESULTS AND DISCUSSIONA. Crystal structure
First of all, the validity of the newly installed detec-tor bank and the home-made Rietveld analysis code is
Parameters T = RT a (˚A) 5.3948(3) c (˚A) 3.8646(3) V (˚A ) 112.47(1) x B x C B isoCe B isoB B isoC R wp B C at the roomtemperature. The space group was assumed to be P /mbm .The Ce atoms occupy the 2( a ) sites at (0,0,0), whereas the Band C atoms occupy the 4( h ) sites at ( x B , x B + 1 / , /
2) and( x C , x C + 1 / , / Z = 2). confirmed by solving the room temperature structure ofCe B C . Figure 1 shows the powder diffraction pat-tern at the room temperature recorded without usingthe refrigerator. The pattern is analyzed assuming twophases; one is the tetragonal CeB C structure with thespace group P /mbm determined by Onimaru et al. , whereas the other is polycrystalline Al used as the sam-ple cell in the present experiment. For both the phases,the lattice parameters, atom positions (except for the Alphase), isotropic atom displacement parameters ( B iso ),and preferred orientation parameters were optimized, inaddition to the profile and background parameters. Re-sulting calculated diffraction profile is shown in the figureby the solid line. Difference between the observation andthe calculation is also presented in the figure. Coinci-dence between the calculated and observed intensities isquite satisfactory. Obtained parameters are listed in Ta-ble I, which are in perfect agreement with the previousresult. This confirms the validity of the newly installeddetector bank and the Rietveld analysis code.Figure 1(b) shows a powder diffractogram measured at T = 4 . < T N . In this low temperature experiment, aserious background due to vacuum chamber walls of therefrigerator appears in a TOF region of 22 . < t TOF <
31 ms. Therefore, the data in the range were removedin the figure. It can be seen that the diffraction patternbelow T N is essentially the same as that at the roomtemperature. Hence, no change in the crystal structureis concluded in the present diffraction experiment. B. Magnetic structure
Figure 2(a) shows the powder diffraction patterns atlarge TOF (30 < t
TOF <
45 ms) in the low tempera-ture range (
T < . T N . A huge Braggpeak at t TOF = 33 . -1000 0 1000 2000 3000 4000 5000 15 20 25 30 35 40 I n t en s i t y ( a r b . un i t s ) Time of Flight (ms)(a) Ce B C Powder LAM-80D Iobs (RT)Icaldiff-1000 0 1000 2000 3000 4000 5000 15 20 25 30 35 40 I n t en s i t y ( a r b . un i t s ) Time of Flight (ms)(b) Ce B C Powder LAM-80DIobs (T = 4.7K)
FIG. 1: (a) Powder diffraction patterns of Ce B C atthe room temperature. The calculated diffraction pattern isshown in the figure by the dashed line, whereas the differencebetween the calculated and observed intensities is depicted bythe solid line. (b) Powder diffraction patterns of Ce B C at T = 4 . can clearly see evolution of several new Bragg peaks be-low 7 . t TOF ≈ . T ≈ T N = 7 . q = ( δ, δ, δ ′ ), andits symmetrically equivalents. ( δ and δ ′ are refined inthe present study as described later.) Thus, the presentpowder diffraction result is consistent with the previoussingle-crystal study, and provides intensity informationfor more Bragg reflections that is mandatory for the spinstructure analysis.Because of the limited TOF range and arbitrariness formagnetic reflection indexing due to powder averaging, itis not straightforward to find a spin structure model di-rectly from the observed diffraction pattern. Thus, tofind possible spin-structure candidates, we use the mag-netic representation analysis introduced by Izyumov etal. . In this method, structure candidates are given by -100-50 0 50 100 150 200 250 300 350 30 32 34 36 38 40 42 44 46 I n t en s i t y ( a r b . un i t s ) Time of Flight (ms) Ce B C Powder LAM-80D
Iobs - Icalc (a) 0 1000 2000 3000 4000 5000 6000 7000 8000 4.5 5 5.5 6 6.5 7 7.5 8 8.5 I n t en s i t y ( a r b . un i t s ) Temperature (K)(b) Ce B C Powder LAM−80D (2− d , d , d ’) reflection FIG. 2: (a) Neutron powder diffraction patterns at severaltemperatures between 8.0 K and 4.7 K. Magnetic Bragg re-flections clearly appear at the low temperatures. Dashed linestands for the result of the Rietveld fitting for the patternat 4.7 K, whereas the solid line denotes the difference be-tween the observed and calculated intensities. Vertical linesshown in the lower part of the figure are the magnetic Braggpeak positions and intensities for magnetic structures givenby the linear combinations of the irreducible representationsIR n : m . Note that only IR2:1+IR2:2 can reproduce the ob-served diffraction pattern. (b) Temperature dependence ofthe integrated intensity for the (2 − δ, δ, δ ′ ) reflections, ap-pearing at t TOF ≈ . linear combinations of magnetic basis vectors of the irre-ducible representations in the paramagnetic phase. In theLandau theory of second order phase transition, a singleirreducible representation may be selected as a symmetryof the ordered phase. In reality, it frequently happensthat two or more irreducible representations are neces- sary to reproduce the symmetry of the ordered phase.Nonetheless, the number of the necessary representationsis usually small. Thus, we may expect that the orderedphase in CeB C may be given by a combination of afew irreducible representations. Here, we try to find themagnetic structure model using the smallest number ofthe magnetic basis vectors. For this purpose, a repre-sentation analysis code, named MBASE , has been newlydeveloped, which can calculate magnetic representationbasis vectors for arbitrary k -group and magnetic-ion po-sitions. Assuming a single- q structure with multiple domainsof the equivalent modulation vectors, direction of a spin(or total angular momentum) at the d -th site in the l -thunit cell may be generally written as: h J l,d i = J a d exp( − i q · R l ) + a ∗ d exp(i q · R l )] , (3)where R l denotes the position of the l -th unit cell. Thepolarization vector a d is given as a linear combination ofthe magnetic basis vectors: a d = X n,m c n,m a n,m,d , (4)where a n,m,d denotes the basis vector of the irreduciblerepresentation IR n : m for the spin at the d -th site. Thebasis vectors for the q -domain and for the domains withsymmetrically equivalent modulations (arms) are listedin Table II. In the present Rietveld analysis, the do-mains with the equivalent modulations are assumed tobe equally populated. For the selection of the basis vec-tors, we note that there is very large anisotropy in themagnetic susceptibility; χ c is considerably smaller than χ a and χ . This indicates that spins most likely lie inthe basal plane. Hence, we may use only the in-plane ba-sis vectors. In the bottom part of Fig. 2(a), reflectionpositions and intensities from the magnetic structuresgiven by single or linear combinations of IR n : m areshown by the vertical thick solid lines. We find that a sin-gle irreducible representation cannot reproduce the peakpositions; for instance IR1:1 definitely gives a peak at t TOF ≈ . c n,m = ±√
2, we find that the combina-tion IR2:1 + IR2:2 [ i.e. , a d = ( a , ,d + a , ,d ) / √ c -plane isschematically shown in Fig. 3.The profile fitting has been performed assuming theIR2:1+IR2:2 structure; the fitting parameters were themodulation vector q and amplitude J . Dashed line inthe Fig. 2(a) shows the result of the profile fitting to thediffraction pattern taken at 4.7 K. The difference betweenthe calculation and the observation is also shown in thefigure by the solid line. Reasonable coincidence can be IR n : m a n,m, a n,m, q = ( δ, δ, δ ′ )1:1 (1 , ,
0) (0 , − ǫ ∗ , , ,
0) ( − ǫ ∗ , , , ,
1) (0 , , − ǫ ∗ )2:1 (1 , ,
0) (0 , ǫ ∗ , , ,
0) ( ǫ ∗ , , , ,
1) (0 , , ǫ ∗ ) q = ( − δ, δ, δ ′ )1:1 (1 , ,
0) (0 , , , ,
0) (1 , , , ,
1) (0 , , − , ,
0) (0 , − , , ,
0) ( − , , , ,
1) (0 , , q = ( δ, − δ, δ ′ )1:1 (1 , ,
0) (0 , , , ,
0) (1 , , , ,
1) (0 , , − , ,
0) (0 , − , , ,
0) ( − , , , ,
1) (0 , , q = ( − δ, − δ, δ ′ )1:1 (1 , ,
0) (0 , − ǫ, , ,
0) ( − ǫ, , , ,
1) (0 , , − ǫ )2:1 (1 , ,
0) (0 , ǫ, , ,
0) ( ǫ, , , ,
1) (0 , , ǫ )TABLE II: Magnetic representation basis vectors a n,m,d forthe irreducible representations IR n : m . They are calculatedfor the four magnetic domains with the modulation vectors q = ( δ, δ, δ ′ ) , ( − δ, δ, δ ′ ) , ( δ, − δ, δ ′ ), and ( − δ, − δ, δ ′ ). ǫ is de-fined as ǫ = exp(2 π i δ ). The basis vectors for − q are given bycomplex conjugates of the above vectors. The vector a n,m, is for the (0,0,0) site, whereas a n,m, for the ( , ,
0) site. found in the figure between the calculated and observedintensities, despite the rather deficient statistics of the ex-perimental data. From the profile fitting, J is estimatedas J = 1 . g J Jµ B = 1 . µ B at 4.7 K. The q -vectoris refined as q = [0 . , . , . q -vector is quite close to the com-mensurate value of (1 / , / , / ( d , d , d ’) domainab FIG. 3: Schematic drawing of the spin structure in the basal c -plane at T = 4 . q = ( δ, δ, δ ′ ) is shown in the figure. S ( - h w ) ( a r b . un i t s ) - h w (meV)Ce B C Powder LAM−DMagnetic partPhonon partTotal
FIG. 4: Neutron inelastic scattering spectrum at T = 15 . T = 100 K data is shown by the solid line. C. Inelastic scattering
1. Phonon subtraction procedure
Next, to investigate the spin fluctuations and excita-tions, we have performed the inelastic scattering experi-ment using the same powder sample. To reliably obtainmagnetic scattering intensity from the raw inelastic spec-trum, a phonon contribution has to be carefully removed.Hence, we first make an estimation of the phonon con-tribution using high temperature data at T = 100 K.For the rare-earth compounds in a paramagnetic tem-perature range, magnetic scattering may originate fromlocal (single-site) transitions, and thus, Q -dependence ofthe magnetic scattering may be dominantly given by the S ( - h w ) ( a r b . un i t s ) - h w (meV) T=15.5KT=23.25KT=32.3KT=52.05KCe B C Powder LAM−D
FIG. 5: Magnetic scattering spectra at T = 52 . , . , . magnetic form factor. On the other hand, the phononscattering may approximately exhibit Q dependence.Thus, we assume the following Q dependence for the total( i.e. , magnetic + phonon) scattering function: S tot ( Q, ~ ω ) = A mag ( ~ ω )[ f mag ( Q )] + A ph ( ~ ω ) Q , (5)where A mag ( ~ ω ) and A ph ( ~ ω ) stand for the Q inde-pendent parts of the magnetic and phonon scattering,whereas f mag ( Q ) for the magnetic form factor of theCe ions. By comparing the inelastic spectra at T =100 K measured with the two different scattering angles(2 θ = 35 ◦ and 85 ◦ ), we estimate the phonon contribution A ph ( ~ ω ) Q . Then, the phonon contribution at low tem-peratures is obtained using the [1 − exp( − ~ ω/k B T )] − dependence. Figure 4 exemplifies this phonon subtrac-tion procedure for the representative data at T = 15 . S ( - h w ) ( a r b . un i t s ) - h w (meV) T=15.5KT=23.25KT=32.3KT=52.05KCe B C Powder LAM−D
FIG. 6: Magnetic scattering spectra at T = 52 . , . , . ( ~ ω >
30 meV). This large phonon contamination re-duces statistical accuracy of the estimated magnetic in-tensity, however, one finds that there definitely exists fi-nite magnetic intensity for ~ ω >
50 meV.
2. Inelastic spectrum in the paramagnetic phase (
T > T N ) The phonon-subtracted magnetic scattering spectra atfour representative temperatures T = 52 . , . , . δ -function like elastic peak: (i) quasielastic component cen-tered at ~ ω = 0 (Fig. 6); (ii) inelastic peak at ~ ω ≈ ~ ω ≈
65 meV(Fig. 5). Since those spectra were measured in the param-agnetic phase, the peaks are most likely due to transitionsbetween the CEF splitting levels. Hence, the observedspectra are analyzed using the following CEF Hamilto-nian derived for Ce in the C h site symmetry of the2( a ) site in CeB C : H CEF = B O + B O + B O + B − O − , (6)where O nm stands for the Stevens operators. Underabove CEF, J = 5 / splits into threeKramers’ doublets. The transition strengths between theCEF splitting levels are given as follows: b αnm = 2e − E n /k B T Z |h n | J α | m i| E m − E n , ( m = n ) b αnn = e − E n /k B T Z |h n | J α | n i| k B T , (otherwise) (7) where | n i and | m i are wave functions for the initial andfinal states of the CEF splitting levels (see Fig. 7 fornumbering of the states), and Z is the partition function.The scattering function from a powder sample may begiven by a sum of spectral weights of the CEF transitions: S ( Q, ~ ω ) inel = 23 (cid:20) g J f mag ( Q ) (cid:21) N ~ ω − exp( − ~ ω/k B T ) X nmα b αnm P nm ( ~ ω ; ~ ω nm , Γ nm ) . (8)In the above equation, we assume a Lorentzian-type peak profile for the inelastic CEF excitations: P nm ( ~ ω ; ~ ω nm , Γ nm ) = Γ nm π (cid:20) ~ ω − ~ ω nm ) + Γ nm + 14( ~ ω + ~ ω nm ) + Γ nm (cid:21) . (9)For the quasielastic peak shape, we assume a pseudo-Voigt function, which is a reasonable approximation ofLorentzian function convoluted by a Gaussian-shaped in-strumental resolution function. For the pseudo-Voigtfunction, the width of the unconvoluted Lorentzian isdenoted by Γ nn (FWHM). We note that the lowest or-der coefficient B can be determined precisely using asingle crystal magnetization measurement; B is esti-mated as 6.34 K in the earlier work. Thus, we fix B ,and try to find the optimum values for B , B and B − that reproduce all the inelastic spectra in the para-magnetic phase simultaneously. In the fitting, we as-sume that temperature dependence of the Hamiltonianparameters B , B , B , and B − is negligible in thepresent temperature range. On the other hand, most ofthe width parameters Γ nm are assumed as temperaturedependent; only Γ and Γ have to be fixed to the em-pirical value 12 meV because of the insufficient statisticsin the high energy regions. The resulting optimum pa-rameters for the CEF Hamiltonian are B = − . B = 10 . B − = − . B − could not be estimated reliably; its uncertaintyrange is considerably larger than those of other param-eters. This parameter exists in the CEF Hamiltonianbecause of the broken four fold symmetry due to the B- C ordering. Since B and C have relatively similar elec-tronegativity, the symmetry breaking may possibly bemoderate, and thus we might infer that B − may beirrelevant.Earlier neutron inelastic scattering experiment pro-vides considerably different Hamiltonian parameters; B = 0 . .
28 K) and B = 0 . .
13 K). The discrepancy is due to the assignmentof the highest inelastic peak; the previous study assignedthe highest energy peak to a very broad hump found at ~ ω ≈
23 meV in their spectrum. The hump is completelyabsent in the presently observed inelastic spectrum. Itmay be noteworthy that a weak peak in phonon densityof states of the elemental aluminum, which is commonlyused for sample cells in inelastic neutron scattering, ex-ists in this energy range, and thus this may become asource of uncertainty. On the other hand, we clearly seethe highest energy peak at ~ ω ≈
65 meV, which is out ofthe observation energy range of the previous study. Wealso note that earlier macroscopic study provides roughestimate of the CEF levels at 102 K and 1110 K, sup-porting our result.Among the three peaks, the quasielastic peak is of par-ticular interest in the Ce compounds, since it reflects thelow-energy scattering process between the ground statedoublets and the conduction electrons. Hence, we pa-rameterize the quasielastic-peak width as a function oftemperature. To obtain the width precisely, the fitting isperformed in the limited energy range − < ~ ω < qel as fitting parameter. The fitting J = 5/2 n=1n=2n=3free Ce C E = 71KE = 744K FIG. 7: Schematic illustration of the CEF splitting scheme inthe CeB C . The energy separations of the Kramers’ doubletsare determined in the present study. See text for details. P ea k w i d t h ( m e V ) Temperature (K)T N FIG. 8: Temperature dependence of the quasielastic peakwidth Γ qel (FWHM) in the paramagnetic temperature range.The solid line is a guide to the eyes. result is depicted by the solid lines in Fig. 6. Obtainedtemperature variation of the quasielastic peak width Γ qel is shown in Fig. 8. The quasielastic peak becomes nar-rower on cooling, as generally seen in heavy-fermion sys-tems. It may be noteworthy that the quasielastic widthis noticeably large even at T ≈
12 K, just above T N .This indicates that quasielastic fluctuations due to thescattering by the conduction electrons are still dominantin vicinity of the ordering temperature. S ( - h w ) ( a r b . un i t s ) - h w (meV) T=4.77KT=12.27K(a) Ce B C Powder LAM−D 0 0.005 0.01 0.015 0.02−4 −2 0 2 4 6 S ( - h w ) ( a r b . un i t s ) - h w (meV) T=4.77KT=12.27K(b) Ce B C Powder LAM−D
FIG. 9: (a) Neutron inelastic spectra at T = 12 .
27 K > T N and T = 4 .
77 K < T N . Solid line is the calculated spectrumassuming the CEF Hamiltonian Eq. (7), whereas the dashedline for T = 4 .
77 K is the result of the scattering-functioncalculation assuming an additional Zeeman term in the CEFHamiltonian. (b) Excitation spectra in the low energy region.Solid lines in this plot are guides to the eyes.
3. Inelastic spectrum in the ordered phase (
T < T N ) As seen in the previous section, the temperature de-pendence of inelastic spectrum is found to be moderatein the paramagnetic phase. However, it shows drasticchange across T N . Representative inelastic spectrum be-low T N is shown in Fig. 9, in comparison with the param-agnetic spectrum at T = 12 .
27 K. The broad quasielasticsignal around ~ ω = 0 disappears below T N , and a newinelastic peak develops at ~ ω = 2 . T > T N . Thus,the low-energy spin fluctuations in the ground state dou-blet are strongly suppressed on ordering. This stronglysuggests dissociation of 4 f electrons from the conductionelectrons, i.e. formation of the localized moment in theordered phase.On the origin of the sharp inelastic peak, several pos-sibility may be anticipated. The simplest possibility maybe splitting of the ground state doublet by the inter-nal (exchange) molecular field appearing in the orderedphase. To check this possibility, we calculated the single-site CEF excitation spectrum under the internal field byintroducing the Zeeman term H Zeeman = g J µ B J · H int in the CEF Hamiltonian Eq. (7). Direction of the inter-nal field H int is parallel to the spin direction [110]. Thecalculated inelastic spectrum assuming H int = 17 T isshown in Fig. 9(a) by the dotted line. The peak positionis reproduced by introducing the internal field, however,relative spectral weights of the peaks are apparently in-consistent with the observed spectrum. Since in the si-nusoidally modulated structure the molecular field is notuniform, another possibility may be the doublet split-ting due to the distributed (non-uniform) internal fields.However, the distribution will apparently introduce peakbroadening, and thus will not reproduce the sharpnessof the 2.1 meV peak. As above, the single-site CEF ori-gin is unlikely for the sharp inelastic peak. As anotherpossibility, we may thus speculate that the inelastic peakplausibly stems from a certain collective excitation of theinteracting Ising-like spins formed by the ground statedoublets. To pursue the origin of the 2.1 meV inelasticpeak, further study on its Q -dependence using a singlecrystal is necessary. Such a single-crystal inelastic scat-tering experiment is in progress.It should be reminded that the long-period sinusoidallymodulated structure has a number of spins that havestrongly reduced average spin magnitudes h S i << S (seeFig. 3). In a localized spin system, such reduced spins arerealized by thermal fluctuations, and thus becomes un-stable for further lowering temperature. On the otherhand, for an incommensurate phase in heavy-fermion sys-tems, such reduced spins are formed by quantum fluctu-ations; coupling with conduction electrons enables a for-mation of spin-density-wave-type (SDW) order, as typi-cally seen in Ce x La − x Ru Si or Ce(Ru − x Rh x ) Si . In both the cases, either thermal or quantum spin fluc-tuations should remain in the sinusoidally modulatedphase. In contrast to the above understanding of the si-nusoidally modulated phase, quasielastic fluctuations arecompletely suppressed in CeB C . This is a very uniquecharacteristic of the sinusoidally modulated phase in theCeB C , and further experimental as well as theoreticalstudy is highly desired to clarify this issue.Finally, we compare the ordered spin structure ofCeB C with those in the heavy RE systems. Incom-mensurately modulate structures have been frequentlyobserved in the heavy RE systems, such as HoB C , TbB C , and ErB C . In HoB C , incommensu-rately modulated spin structure is realized blow T c1 =5 . q = (1 , ,
0) and(1 ± δ, δ, δ ′ ) where δ = 0 .
112 and δ ′ = 0 .
04. Quadrupolarorder is established at lower temperatures
T < T C2 =5 . q = (1 , , , , / , ,
0) and (0 , , / C . TbB C also shows similar multi- q in-commensurate structure. This compound does not showquadrupolar ordering in the zero external field, however,quadrupolar ordering is known to take place in a verylow external magnetic field of 1 T, indicating that a likelysituation is realized for the formation of the quadrupolarmoment in the CEF ground state. In contrast, ErB C exhibits the single- q sinusoidally modulated structure be-low T N = 15 . q = (1 + δ, δ,
0) ( δ = 0 . q = (1 , , T t = 13 . q sinusoidal phase in the CeB C withno quadrupolar degree of freedom. Therefore, this re-sult, in addition to the above observations for the otherRE systems, supports the claim that the magnetic sec-tor of the REB C compounds has a tendency to form along-period single- q structure, and it is the effect of thequadrupolar degree of freedom that realizes the complexmulti- q structures. IV. CONCLUSIONS
We have performed neutron powder diffraction and in-elastic scattering experiments on the cerium diborocar-bide Ce B C . In the powder diffraction study, we haveclearly detected the magnetic reflections below the or-dering temperature T N = 7 . q = [0 . , . , . ~ ω = 0 and broad in-elastic peaks at ~ ω ≈ T < T N , drastic change has been seen in the low energyspin fluctuation spectrum; a sharp inelastic peak developsat ~ ω = 2 . f and conductionelectrons in the paramagnetic temperatures are stronglysuppressed in the ordered phase. Acknowledgments
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