role of single-ion excitations in the mixed-spin quasi-one-dimensional quantum antiferromagnet

PHYSICAL REVIEW B 1 MAY 2000-IVOLUME 61, NUMBER 17

Role of single-ion excitations in the mixed-spin quasi-one-dimensional quantumantiferromagnet Nd2BaNiO5

A. Zheludev and S. MaslovDepartment of Physics, Brookhaven National Laboratory, Upton, New York 11973

T. Yokoo and J. AkimitsuDepartment of Physics, Aoyama-Gakuin University, 6-16-1 Chitosedai, Setagaya-ku, Tokyo 157 Japan

S. RaymondUniversitede Geneve DPMC 24, quai Ernest Ansermet 1211, Geneve 4, Switzerland

S. E. NaglerOak Ridge National Laboratory, Building 7692, MS 6393, P.O. Box 2008, Oak Ridge, Tennessee 37831

K. HirotaDepartment of Physics, Tohoku University, Aramaki, Aoba-ku, Sendai 980-8578, Japan

~Received 21 October 1999!

Inelastic neutron scattering experiments on Nd2BaNiO5 single crystals and powder samples are used tostudy the dynamic coupling of one-dimensional Haldane-gap excitations in theS51 Ni21-chains to localcrystal-field transitions, associated with the rare earth ions. Substantial interference between the two types ofexcitations is observed even in the one-dimensional paramagnetic phase. Despite that, the results provide solidjustification for the previously proposed ‘‘static staggered field’’ model forR2BaNiO5 nickelates. The ob-served behavior is qualitatively explained by a simple chain-random-phase-approximation model.

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I. INTRODUCTION

Linear-chain rare earth nickelates with the general fmula R2BaNiO5 were recently recognized as unique modcompounds for experimental studies of Haldane-gap qutum spin chains in strongstaggeredapplied fields.1 In thesematerials the Haldane subsystem is formed by chains oS51 Ni21 ions with relatively strong, almost isotropicnearest-neighbor antiferromagnetic~AF! exchange interactions (J'27 meV). The corresponding Haldane energy gin the magnetic excitation spectrum isD'11 meV.2–7 Aneffective staggered exchange fieldHp acting on these onedimensional~1D! objects is generated by three-dimension~3D! magnetic long-range ordering in the ‘‘auxiliary’’ sublattice of magnetic rare earth ions, which typically occursNeel temperatures of several tens of kelvin.8 The magnitudeof the staggered field is proportional to the ordered momon the R31 sites, and can be varied in an experiment,changing the temperature. Static magnetic propertiesR2BaNiO5 compounds are rather well described by tchain-mean-field~MF! model.9,6 Using this theory to analyzethe measured temperature dependences of magnetic orderameters for a number ofR2BaNiO5 species previously enabled us to extract thestaggered magnetization functionfor asingle Haldane spin chain.9 These experimental results wefound to be in excellent agreement with theoretical calcutions for isolated chains.10,11The view ofR2BaNiO5 systemsas Haldane spin chains immersed in a static staggered~SSF! model turned out to be successful in describingbehavior of Haldane-gap excitations as well.10,6 In agreementwith analytical10,12 and numerical11 predictions, the energy

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gap was shown to increase quadratically withHp inPr2BaNiO5 , Nd2BaNiO5 , and (NdxY12x)2BaNiO5 systemsin their magnetically ordered phases.

The focus of the present paper is on a different, pottially very important, aspect in the magnetism ofR2BaNiO5materials, namely, on the strong spin-orbit interactions tycally associated with rare earth ions. Of particular interestlocal ~single-ion! crystal-field ~CF! excitations of the rareearths and their possible interactions with the Haldane-modes propagating on the Ni chains. Until now this effehas not been investigated in sufficient detail. The modused forR31 in R2BaNiO5 were in most cases restrictedconsidering the ion’s lowest-energy electronic configutions: effectiveS51/2 doublets for Kramers ions like Nd31

or Er31 ~Ref. 9! or singlets for Pr31 ~Ref. 3!. Higher-energyelectronic states were entirely ignored. Yet, in a numbermaterials, particularly in Nd2BaNiO5,4 several CF transitionsoccur very close to the Haldane-gap energy. In this situaa strong coupling between the two types of excitationsbe expected. Does this interaction limit the applicabilitythe SSF model? Does it give rise to peculiar ‘‘mixed’’ Ni-Rexcitations? If so, do these ‘‘mixed’’ modes retain a prdominantly 1D character of Haldane-gap excitations? Inpresent paper we address these important and interequestions in a series of inelastic neutron scattering measments on Nd2BaNiO5.

II. EXPERIMENT

The crystal structure ofR2BaNiO5 compounds is dis-cussed in detail elsewhere.13 Nd2BaNiO5 crystallizes in an

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orthorhombic unit cell, space groupImmm, a'3.8 Å, b'5.8 Å, andc'11.7 Å.5 The S51 Ni chains are formedby orthorhombically distorted NiO6 octahedra that are lineup in the (100) crystallographic direction, sharing their acal oxygen atoms. The Nd sites are positioned in betwthese chains, linking nearest-neighbor chains along the ctallographicb axis ~Fig. 1!. We shall denote the corresponing exchange constant asJ1. In addition, one can expecsubstantial coupling of each Nd31 ion to two Ni21 sites in athird nearby chain~exchange constantJ2). The site symme-try for the rare earth ions is sufficiently low, and the eletronic ground state for Nd31 is a Kramers doublet. The Ne´eltemperature for Nd2BaNiO5 is TN548 K. The magneticstructure is as described in Ref. 8 and 14.

The simplest technique used to locate CF excitationinelastic neutron scattering on powder samples. Such msurements were previously performed on Nd2BaNiO5 for en-ergy transfers up to 30 meV using a three-axis spectrome4

To learn about higher-energy CF transitions we carriedpreliminary experiments on the time-of-flight~TOF! spec-trometer LAM-D, installed at the spallation neutron sourKENS in the High Energy Accelerator Research~KEK!.LAM-D implements the so-called inversion geometry andequipped with four fixed PG crystal analyzers. The measuspectra are automaticallyQ integrated in a certain momentum range that roughly corresponds to 1 –2 Å21 at \v50. Energy transfers from22 meV to 200 meV are accessible. The energy resolution at the elastic scattering posiis about 300meV and the relative resolutiond(\v)/\v isfairly constant, around 4%, between\v52 and 60 meV. Inthe experiment we used roughly 10 g of Nd2BaNiO5 powderprepared by the solid-state reaction method. The samplevironment was a standard closed-cycle refrigerator.

Most of the data presented below were obtainedsingle-crystal samples. These experiments were carried othe High Flux Isotope Reactor~Oak Ridge National Laboratory! on the HB-1 triple-axis spectrometer. We used the saassembly of three coaligned single crystals as in previstudies.6 The sample was in all cases mounted with t(h,k,0) reciprocal-space plane parallel to the scattering plof the spectrometer, to compliment previous (h,0,l ) mea-surements. A neutron beam of 14.7 meV or 30.5 meV fixfinal energy of 30.5 meV fixed incident energy was us

FIG. 1. A schematic representation of the magnetic interacgeometry in the structure ofR2BaNiO5.

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with 608-408-408-1208 collimations and a pyrolitic graphite~PG! filter positioned after the sample. PG~002! reflectionswere used for monochromator and analyzer. The measments were done with energy transfers up to 35 meV. Tsample environment was a standard Displex refrigerator,the temperature range 35–65 K was explored. The grodirections of the crystals roughly coincide with the (10crystallographic axis. In the experiment the crystal rods wtherefore positioned horizontally in the scattering plane.this geometry absorption effects, primarily due to the prence of Nd nuclei, may be substantial, as will be discussedetail below.

III. RESULTS

A. Powder data

Inelastic powder experiments, while obviously less infomative than single-crystal measurements, provide a connient view of the excitation spectrum in a wide energy ranFigures 2~a!–2~d! show the measured inelastic spectraNd2BaNiO5 powder atT516.5, 55, 100, 300 K, respectively. The background originating from the sample hold

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FIG. 2. LAM-D inelastic spectra measured in Nd2BaNiO5 pow-der samples at different temperatures. The contribution ofsample holder has been subtracted.

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was measured separately and subtracted from the datashown. Each scan corresponds to 16–20 h of counting tAt T555 K, above the Ne´el temperatureTN548 K, oneclearly sees at least three features that are likely to be Ndexcitations @Fig. 2~b!#. In addition to the two previouslyidentified low-energy modes atE1518 meV and E2524 meV energy transfer,4 a broad yet intense peak is seat 40 meV. As in previous three-axis powder measuremeat 11 meV one can also see a shoulder correspondinHaldane excitations. The latter are known to have a stdispersion along the chain axis. As a result, compared todispersionless CF excitations, the Ni-chain modes appsubstantially weakened in a powder experiment, as oppoto single-crystal measurements. No substantial scatterinseen within the Haldane gap. AtT5100 K the 11 meVshoulder is much less pronounced, while the CF peaksmain almost unchanged@Fig. 2~c!#. As could be expected, ahigh temperatures the entire spectrum becomes smeare@Fig. 2~d!#.

A drastic change in the spectrum occurs upon coolthrough the Ne´el temperature@Fig. 2~a!, T516.5 K#. Anintense resolution-limited peak emerges inside the gap ancentered at 4 meV energy transfer. This feature correspoto a dispersionless Ising-like spin wave associated with lo

FIG. 3. Typical constant-Q scans measured in Nd2BaNiO5 atT555 K at the 1D AF zone centerh52.5, at different momentumtransfers perpendicular to the chain axis~symbols!. The solid linesare a global fit to the model cross section, as described in theThe hatched, light-gray, and gray peaks represent partial conttions from the Haldane-gap and two crystal-field excitations,spectively.

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range magnetic ordering.6 Below T520 K the Haldane gapwas previously shown to exceed 15 meV.6 In apparent con-tradiction with this single-crystal result, the 11 meV shouldis weakened but still visible in our new 16.5 K powder daOne possible explanation of this is that the shoulder is ptially due to a weak, previously unobserved CF excitationlow temperatures the Haldane gap increases, but the newmode stays at 11 meV energy transfer. Some of the sincrystal results presented below support this interpretation

A more complete, quantitative, analysis of the powddata is not presently possible, as the phonon backgrounnot known. In the future we plan to supplement these dwith ‘‘background’’ measurements on Y2BaNiO5, to isolatethe magnetic contribution of the rare earth ionsNd2BaNiO5. For the purpose of the present paper, howevwe can draw one useful conclusion: in trying to understathe coupling of the Ni and Nd subsystems on energy sccomparable toD, we can safely restrict ourselves to consiering the two strong CF excitations at 18 meV and 24 monly.

B. Transverse intensity modulation

1. Constant-Q scans

Having verified the location of major CF excitationspowder measurements, we proceeded with more detasingle-crystal experiments. As a first step, we investigathe b-axis dependence of intensities and energies for thechain and Nd-CF excitations near the 1D AF zone cenTypical constant-Q scans collected atQ5(0,k,2.5), for sev-eral values ofk, are shown in Fig. 3. A total of nine scans othis type were measured, withk ranging from 0 to 1, atT555 K ~just above the temperature of magnetic orderinTN548 K). These data are summarized in the intenscontour plot in Fig. 4~a!. Peaks corresponding to the Nchain excitations and to the two CF modes are clearly seThe central result of this work is thatthe relative intensitiesof these excitations appear to vary significantly with tranverse momentum transfer.

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FIG. 4. ~a! Contour plot of inelastic scattering intensity mesured in Nd2BaNiO5 at T555 K at the 1D AF zone centerh52.5 in a series of constant-Q scans~20 counts/3 min contoursteps!. The symbols show the energies of Haldane~circles! and twocrystal-field excitations~triangles!, as determined from fitting amodel cross section to individual scans.~b! The result of a global fitto the data shown in~a!.

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To better understand this intensity variation and to lofor a transverse dispersion in any of the three modes,analyzed each of the measured const-Q scans using a semiempirical parameterized model cross section:

S~Q,v!5SH~Q,v!1S1~Q,v!1S2~Q,v!, ~1!

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S2~Q,v!5S2(0)d~\v2E2!@ f Nd~Q!#2, ~4!

~\vQ!25D21v2sin2~Qia!. ~5!

In these equationsSH(Q,v), S1(Q,v), andS2(Q,v) are thepartial contributions of the Haldane-gap and two CF modrespectively. The coefficientsSH

(0) , S1(0) , andS2

(0) representthe intensities of each component. The spin wave velov56.07D ~Ref. 15! and intrinsic energy widthG characterizethe Haldane-gap excitations. In our analysisG was fixed to avalue of 1 meV, as in previous studies.6 Qi is the projectionof the scattering vector onto the chain axis. Finally,f Ni(Q)and f Nd(Q) are magnetic form factors for Ni21 and Nd31

ions. The data analysis was performed assuming seven ipendent parameters: the three intensity prefactors, a conbackground, and the three characteristic energiesD, E1, andE2. The cross section was convoluted with the spectromresolution function and fit separately to each scan. Typfits are shown in solid lines in Fig. 3. This analysis yielded

FIG. 5. ~a! k dependence of excitation intensities measuredNd2BaNiO5 at T555 K, as deduced from model cross section fito individual const-Q scans@see Fig. 4~a!#. ~b! k dependence ofHaldane-gap energy, determined in the same fashion. Solid lineguides for the eye.

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measurement of theb-axis dispersion relation in each brancplotted with symbols in Fig. 4~a!. To within experimentalerror the CF branches appear flat, and only a very smdispersion is seen in the Haldane modes@Fig. 5~b!#. TheHaldane-gap energy is a maximum at the transverse zboundaryk50.5. The intensity variation for each modeshown in Fig. 5~a!. The intensity of the 18 meV excitationappears almostk independent. At the same time, intensitiof both the Haldane and the 24 meV CF modes are stronk dependent, and are seemingly in ‘‘antiphase’’ one toother.

Data sets similar to those described above were alsolected atT535 K, i.e., below the Ne´el temperature. Theseresults are summarized in Figs. 6 and 7. Compared tohigh-temperature measurements, the Haldane-gap enerincreased to about 14 meV. The Haldane branch thus occloser to the 18 meV CF mode. It is also almost twiceweak as atT555 K. As a result, the scattering of dapoints in the measured transverse dispersion curve showFig. 7~b! is rather large, and the bandwidth cannot be reliadetermined. Compared to the high-temperature regime,magnitude of intensity variations in the both the Haldane a24 meV excitations are visibly reduced@see Fig. 7~a!#.

2. Global fits to data

The final analysis of the measured constant-Q scans wasperformed by fitting a model cross section simultaneouslyall data collected at each temperature@Figs. 4~a! and 6~a!#.Formulas 1–5 used to analyze individual scans were mfied to allow for a transverse dispersion and intensity molation in each branch. The appropriate forms for thesek de-pendences are derived in the theory section below.corresponding expression for the magnetic dynamic strucfactor near the 1D AF zone center is as follows:

S~Q,v!5SH~Q,v!1S1~Q,v!1S1~Q,v!, ~6!

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FIG. 6. ~a! Contour plot of inelastic scattering intensity mesured in Nd2BaNiO5 at T535 K at the 1D AF zone centerh51.5 in a series of constant-Q scans~20 counts/3 min contoursteps!. The symbols show the energies of Haldane~circles! and twocrystal-field excitations~triangles!, as determined from fitting amodel cross section to individual scans.~b! The result of a global fitto the data shown in~a!.

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1A2„f Ni~Q! f Nd~Q!…cos~Q'b/2!%, ~9!

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E2~Q'!5E21B2cos2~Q'b/2!. ~12!

In these equationsQ' is the component of momentum tranfer parallel to theb axis. As explained in the theory sectiothe dimensionless coefficientsAH , A1, andA2 quantify theadmixture of Nd-CF spin fluctuations to the chain modes avice versa.BH , B1, and B2 are respective magnitudes otransverse dispersion. To minimize the number of adjustaparameters the valuesB1[0, A2[0 ~no dispersion ob-served!, andA1[0 ~the mode appears not to be modulate!were fixed, which hardly affected the finalx2. The results ofthe global least-squares fitting to theT555 K and T535 K data are shown in Figs. 4~b! and 6~b!, respectively.The refined values of all parameters are summarized in TI. In both cases ax251.7 was achieved, which suggests th

FIG. 7. ~a! k dependence of excitation intensities measuredNd2BaNiO5 at T535 K, as deduced from model cross section fito individual const-Q scans@see Fig. 6~a!#. ~b! k dependence ofHaldane-gap energy, determined in the same fashion. Solid lineguides for the eye.

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the model cross section describes the observed behaviorsonably well. As discussed above, the coefficients for intsity modulations in all three modes, as well as the dispersbandwidth for the Haldane branch atT555 K, determinedin this analysis, can be considered rather reliable. Oncontrary, the large value forBH at T535 K is somewhatsuspect. In the refinement this parameter has a strong clation with E1; i.e., the Haldane gap is not very well resolved. This also accounts for the large scattering inpoints obtained in fits to individual scans in Fig. 7~b!. Thereis little doubt, however, that the bandwidth of transverse dpersion remains very small even in the magnetically ordestate.

3. Wide-range constant-E scans

It is implied in Eqs.~7!–~9! that the intensity modulationin all modes is periodic along theb axis. To verify this pe-riodicity we performed constant-E scans at the Haldane gaenergy and at\v525 meV, covering a broad range otransverse momentum transfers. Typical raw data collecteT555 K are shown in Figs. 8~a! and 9~a!. A serious tech-nical problem with such measurements is that theystrongly influenced by neutron absorbtion in the sampleeach scank varies significantly at a constant value ofh. As aresult, the crystal rods rotate relative to the incident and ogoing beams by a large angle. Absorbtion corrections cobe calculated, but additional undesirable effects arising frthe size ~length! of the sample, being comparable to thwidth of the incident neutron beam, are difficult to accoufor analytically. To correct the data for any transmissionfects, we measured the actual transmission coefficientin situ.This was done separately for each point of the constanEscans performed. The monochromator, sample, and scaing angles of the spectrometer were set to values calculfor a given point in the inelastic scan. The analyzer angwere then set to the elastic position to measure the intenof incoherent scattering. Incoherent scattering being isopic, the measured incoherent intensity can be assumed t

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TABLE I. Parameters characterizing the transverse dispersand intensity modulation of the Haldane-gap and crystal-field etations in Nd2BaNiO5, as obtained from global fits of a model crossection to series of constant-Q scans.

T555 K,Q5(2.5,k,0)

T535 K,Q5(1.52,k,0)

D ~meV! 10.97~0.09! 13.26~0.07!AH 20.17 ~0.01! 20.12 ~0.02!BH ~meV! 20.19 ~0.14! 0.64 ~0.15!SH

(0) ~arb. units! 1.17 ~0.03! 0.66 ~0.02!

E1 ~meV! 18.33~0.09! 19.7 ~0.1!A1 0 ~fixed! 0 ~fixed!

B1 ~meV! 0 ~fixed! 0 ~fixed!

S1(0) ~arb. units! 0.20 ~0.01! 0.22 ~0.02!

E2 ~meV! 23.8 ~0.06! 24.6 ~0.09!A2 0.72 ~0.06! 0.37 ~0.05!B2 ~meV! 0 ~fixed! 0 ~fixed!

S2(0) ~arb. units! 0.34 ~0.01! 0.30 ~0.01!

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proportional to the effective transmission coefficient. Typicmeasured transmission curves are shown in Figs. 8~b! and9~b!. A few Bragg reflections were accidentally picked upthese elastic scans, and the corresponding points weremoved from the data sets shown. In Figs. 8~b! and 9~b! ar-rows indicate geometries of minimum transmission, realizwhen the sample rods are parallel to the incident or the stered beams. The measured transmission coefficientsused to scale the measured const-E scans point by point. Theresulting corrected data are plotted in Figs. 10 and 11.

Figure 10 shows transmission-correctedk scans taken athe Haldane-gap energy, near two 1D AF zone centerh51.5 andh52.5. The intensity falloff at largeuku is mostlydue to focusing and to the effect of magnetic form factoOn top of this falloff, a periodic intensity modulation iclearly seen. The data can be rather well fit to Eq.~7! con-voluted with the spectrometer resolution function. Theare shown in solid lines in Fig. 10 and correspond toAH520.19(0.02). This value is in good agreement withAH520.17, obtained from the analysis of constant-Q scans de-scribed above.

Figure 11~b! shows that near the 1D AF zone center tintensity of the 25 meV CF mode is modulated as well,

FIG. 8. Measuring the transverse modulation of the Haldane-excitations intensity:~a! constant-E scan measured in Nd2BaNiO5

at T555 K near the 1D AF zone centerh51.5 at \v511.5 meV~open circles!. The background is measured off the 1AF zone center~solid circles!. ~b! Transimission curve for the scashown in panel~a!, measured as explained in the text.

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antiphase with that of the Haldane branch. The data can bto Eq. ~9!, assumingA250.28(0.05) @solid line in Fig.11~b!#. Note that this value is less than a half of that deducfrom the analysis of constant-Q scans at the same temperture (A250.72). This discrepancy is easily understood. Evat 25 meV energy transfer the Haldane-gap excitations ctribute to inelastic scattering, thanks to their steep dispersalong the chain axis that produces long ‘‘tails’’ on the higenergy side of the peak in constant-Q scans. The contribu-tion of these tails was not taken into account in the analyof the wide-range const-E scan shown in Fig. 11~b!, but wasincluded in the global fit to constant-Q scans. Our previousestimate forA2 should thus be considered more accurate

An important result is that away from the 1D AF zoncenter, ath52.25, the 25 meV mode shows no oscillatioof intensity @Fig. 11~a!#. To within experimental error themeasuredk dependence can be explained by the Nd fofactor alone~solid line!. This observation supports the notiothat intensity oscillations result from a mixing between Cand Haldane-gap excitations. Away from the 1D AF zoncenter Ni-chain modes occur at very-high-energy transand therefore do not interact with the dispersionlessbranch.

p FIG. 9. Measuring the transverse modulation of the 25 mexcitation intensity:~a! constant-E scan measured in Nd2BaNiO5 atT555 K near the 1D AF zone centerh52.5 at\v525 meV. ~b!Transmission curve for the scan shown in panel~a!, measured asexplained in the text.

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4. Previous data analysis procedures revised

The discovery of a substantial variation of the intensitythe CF excitations raises concerns about the procedureto analyze the inelastic data for Nd2BaNiO5 in Refs. 6 and 7.In these studies the CF ‘‘background’’ was measured awfrom the 1D AF zone-centers, where the Haldane modesnot seen. The background was then subtracted point by pfrom the ‘‘signal’’ scans measured at the 1D AF zone cenIt was assumed that such background-subtracted scansgood measure of inelastic scattering originating from thechains alone. We now understand that this approach isprinciple, not valid. Although CF excitations are almost dpersionless, their intensity in the ‘‘background’’ scans wnot necessarily match that in the ‘‘signal’’ scans. Accordito the theoretical model described below, the procedureonly be applied atk5(2n11)/2, h5(2m11)/2 (m, n inte-ger!, where there is no mixing between Ni-chain and Nsingle-ion excitations. The data analysis employed in Refwhere measurements were performed atQ5(1.5,0.5,0), isthus fully appropriate. In Ref. 6, on the other hand, the dwere taken atk50, and the background subtraction may,principle, have produced systematic errors. It appears, hever, that point-by-point background subtraction is a fai

FIG. 10. Transverse modulation of the Haldane-gap excitatiintensity: constant-E scans measured in Nd2BaNiO5 at T555 Knear the 1D AF zone centersh51.5 ~a! and h52.5 ~b! at \v511.5 meV~symbols!. The measured intensity has been correcfor absorbtion and the measured background has been subtraSolid lines are fits to a model cross section, as described in the

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reliable way to determine the Haldane-gapenergy at anytransverse wave vector. Indeed, in Ref. 6 all scans wtaken below 25 meV energy transfer, and thus include othe 18 meV CF excitation. This mode, as we now know, isfact flat. The main results of Ref. 6 pertain to the temperatdependence of the Haldane gap, which occurs even belowmeV in the studied temperature range, and are totally va

C. Dispersion along the chain axis

The mixing between the Haldane-gap and 24 meVmodes should be most prominent at points of reciprospace where the two excitations have similar energies. Wthe CF excitation is practically dispersionless, the Haldabranch has a steep dispersion along the chain axis, andbe expected to anticross with the 24 meV branch rather cto the 1D AF zone center. To investigate this phenomenwe measured the intensity of inelastic neutron scatterfrom Nd2BaNiO5 on a grid of points inh-E space atk50.5, h51.5–1.8, atT555 K. The result of this measurement is summarized in the grayscale-contour plot in Fig.The circles and squares show the positions of peaks swhen the data are broken up into a collection of constanQand constant-E scans, respectively. The most prominent fe

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FIG. 11. Transverse modulation of the 25 meV excitation intesity: constant-E scans measured in Nd2BaNiO5 at T555 K awayfrom ~a! and close to~b! the 1D AF zone centerh52.5 at \v525 meV ~symbols!. The measured intensity has been correcfor absorbtion and the measured background has been subtraSolid lines are fits to a model cross section, as described in the

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ture of the spectrum is an anticrossing at the point of insection of the flat 24 meV CF mode and the paraboHaldane branch. The solid lines are guides for the eyeemphasize this effect. The anticrossing gap 2dE can be es-timated to be roughly 4 meV. Note that there is no evidenof anticrossing of the Haldane and 18 meV CF branches

A much less significant feature that is barely seen witthe statistics of the measurement is what appears to be adispersionless mode at\v'12 meV ~dash-dotted line inFig. 12!. In the single-crystal sample it appears much weathan the the Haldane mode, but nevertheless may accounthe ‘‘shoulder’’ seen in powder samples atT516.75 K, asdiscussed previously. This feature is substantially weathan the other two CF excitations, and we shall disregarin the following analysis.

IV. THEORY

It is rather interesting that in Nd2BaNiO5 the pronouncedtransverse modulation of intensity in the Ni-chain andexcitations isnot accompanied by any significant transverdispersion. If we were dealing with a one-component syste.g., directly coupledchains, a narrow bandwidth alonggiven direction would automatically result in only a weaperiodicity in the structure factor. For magnetic excitationsa Bravais crystal, this follows from the sum rule for the firmoment of the magnetic dynamic structure factorS(q,v).16

The strong modulation and lack of dispersion in Nd2BaNiO5must therefore be an interference effect that involves both

FIG. 12. Logarithmic contour plot of inelastic neutron scatteriintensity measured in Nd2BaNiO5 at T555 K near the 1D AF zonecenter h51.5. Circles and squares show peak positions as tappear in constant-E and constant-Q scans, respectively. The baare peak widths at half height. Solids lines are guides for theDashed lines represent the dispersion relation expected forHaldane excitations~parabola! and 25 meV CF mode~straight line!in the absence of mode coupling. The dash-dotted line showsposition of what seems to be a weak, previously unobservedexcitation.

r-cat

e

new

rfor

erit

,

t

i

and Nd magnetic degrees of freedom. In the following stions we shall demonstrate that this mixing can be qualtively accounted for by a simple random-phasapproximation~RPA! model for Ni-R interactions.

A. Bare susceptibilities

1. Ni chains

The first step in the RPA analysis is to determine the b~noninteracting! dynamic susceptibilities for the Ni and Nsublattices. For the Haldane spin chains in the vicinity of1D AF zone centerqi5p we can safely use the single-modapproximation17–19

xNi(0)~Q,v!5

12cosQia

2

Zv

D21v2sin2~Qia!2~\v1 i e!2.

~13!

In this formulaQi is the projection of the momentum tranfer Q onto the chain axis,v is the spin wave velocity in thechains,v'6.07D, andZ'1.26~Ref. 15!. It is easily verifiedthat for scattering vectors close to the 1D AF zone center~2! used to fit the the inelastic scans coincides withimaginary part of Eq.~13! in the limit e→0. In the magneti-cally ordered state, i.e., in the presence of an effective mfield He f f , Eq. ~13! can still be utilized. However, the corresponding gap energies for the three components ofHaldane triplet will be modified byHe f f , as discussed inRefs. 9, 10, 20, and 6.

2. Rare earth ions

For the rare earth subsystem the bare single-ion suscbility cannot be written down so easily, as the exact eltronic configuration ofR31 in R2BaNiO5 has not been cal-culated to date. When we were concerned withmechanism of magnetic ordering in Nd2BaNiO5 in Refs.9and 6, we were dealing with thestatic susceptibility of therare earths. For Nd31 at low temperatures the static suscetibility can be, to a good approximation, attributed to tlowest-energy Kramers doublet and written as aS51/2 Bril-louin function. For our present purpose of studying thedy-namiccoupling of Nd and Ni subsystems we are interestedthe dynamicsusceptibility of Nd31 at frequencies comparable to the Haldane-gap energy. In the paramagnetic phthe contribution toxNd from the ground state doublet ipurely elasticand can thus be totally ignored in the RPcalculation. AtT.TN we therefore only need to consider thcontribution of transitions between the ground-state ofion and higher-energy crystal-field levels. In the orderphase, we also have to take into account single-ion trations within the ground-state doublet, which becomes sby the mean exchange field.6 As a first step, we shall consider only one excited level of Nd31, at energyE. Transi-tions from the ground state to the excited state producefollowing term in Nd single-ion susceptibility:

xNd(0)~v!5M2

E

E22~\v1 i e!2R~T!. ~14!

In this formula M is the matrix element of the magnetmoment operator between the ground state and the exc

y

e.he

heF

edn

ecic

gedw

ms

it

be-

s-tion

his


state.R(T) is the temperature renormalization factor definas the difference in populations of the excited and groustates. To simplify our qualitative analysis, in Eq.~14! wehave ignored any terms that are off diagonal in spin projtion indexes, and the above formula thus applies to a partlar selected channel of spin polarization.

B. Interaction geometry

Let us now consider the geometry of Ni-Nd exchaninteractions~Fig. 1!. The antiferromagnetic coupling denoteby J1 stabilizes the magnetic structure seen in the lotemperature phase, and is responsible for the staggeredfield at T,TN . As is often the case in body-centered cry

e

ulur

dh

uaden

d

-u-

-ean-

tals, the couplingJ2 is geometrically frustrated. Indeed,links a singleR31 moment to two subsequent Ni21 spins inthe chains that are strongly coupled antiferromagneticallytween themselves. As a result, at the MF levelJ2 plays norole in long-range ordering. Below we shall arbitrarily asume that the exchange matrix is diagonal in spin projecindexes. In the body-centered structure, with twoR sites perevery Ni, we have to consider six magnetic ions: Ni~1! at~0.5,0.5,0!, Ni~2! at ~0,0,0.5!, Nd~3! at ~0.5,0,d!, Nd~4! at(0.5,0,2d), Nd~5! at (0,0.5,d20.5), and Nd~6! at (0,0.5,[email protected]#), where d'0.2. For the RPA calculation weneed the Fourier transform of the exchange matrix in tbasis set:

J'~Q!5S 0 0 J1~Q! J1* ~Q! J2~Q! J2* ~Q!

0 0 J2~Q! J2* ~Q! J1~Q! J1* ~Q!

J1* ~Q! J2* ~Q! 0 0 0 0

J1~Q! J2~Q! 0 0 0 0

J2* ~Q! J1* ~Q! 0 0 0 0

J2~Q! J1~Q! 0 0 0 0

D , ~15!

J1~Q!52J1cos~pk!exp~2p i ld !, ~16!

J2~Q!52J2cos~ph!exp@2p i l ~d20.5!#. ~17!

ainla-CF

d-

One readily sees that, in the reciprocal-space planes„h,2n11/2,l … (n integer!, J1 interactions cancel out and can bignored. A similar cancellation occurs forJ2 in the „(2n11)/2,k,l … (n integer! planes. These cancellations resfrom interaction topology, and do not rely on any of oassumptions regarding the bare susceptibilities.

C. RPA susceptibility

The assumptions made above bring us to a simple mocoupling problem in each channel of spin polarization. TRPA susceptibility matrix is written asxRPA(Q,v)5@11x0(Q,v)J(Q)#21x0(Q,v).

1. Dispersion

The dispersion relation is obtained by solving the eqtion det(@xRPA#21)50. In our case of only one excited Nstate this can easily be done analytically. With six indepdent atoms we obtain six modes~two of them degenerate!:

2~\v1,2!25DQi

2 1E26A~DQi

2 2E2!218EM2ZvJ12 ,

2~\v3,4!25DQi

2 1E26A~DQi

2 2E2!218EM2ZvJ22 ,

\v5,65E, ~18!

where we have introduced the notationDQi

2 5D2

1v2sin2(Qia), and

t

e-e

-

-

J 62 5uJ1~Q!6J2~Q!u254J1

2cos2~pk!14J22cos2~ph!

616J1J2cos~pk!cos~ph!cos~p l !. ~19!

We have implicitly dropped the (12cosQia)/2 factor inxNi(0)

and the temperature factorR in xNd(0) .

Particularly useful is the linearized version of Eq.~18!which becomes valid in the weak-coupling limit:

\v1~Q!5DQi2J1~Q!2

ZvD

EM2

E22DQi

2,

\v2~Q!51J1~Q!2ZvD

EM2

E22DQi

2,

\v3~Q!5DQi2J2~Q!2

ZvD

EM12

E22DQi

2,

\v4~Q!5E1J2~Q!2ZvD

EM12

E22DQi

2,

\v5,6~Q!5E. ~20!

The first and third branches can be described as Ni-chHaldane-gap modes with an admixture of Nd-spin corretions. The second and fourth branches, conversely, areexcitations with a small component of Ni spins. If the N

ti

ono

dertin

inqs

es,try

d toheties


single-ion excitation lies above the Haldane branch agiven wave vector, the energy of the Ni-chain modepushed down by Ni-Nd interactions, while the CF excitatienergy is increased. The reverse is expected in the casesingle-ion excitation inside the Haldane gap. Modes 5 anappear as totally unperturbed single-ion transitions. A vimportant result is that the transverse dispersion resulfrom Ni-R interactions isquadraticin J and is thus expectedto be very small in the weak-coupling limit.

Also useful for our purposes are values of anticrossgaps at the point of mode intersection. According to E~18! these are given by

2dE1↔25J1MA2ZvE

,

te

4,thca

si

as

as

f a6yg

g.

2dE3↔45J2MA2ZvE

, ~21!

for the two pairs of intersecting same-symmetry branchrespectively. Intersection of branches of different symme~1,4! and ~2,3! does not lead to an anticrossing effect.

2. Intensity modulation

The inelastic neutron scattering cross section is relatethe imaginary part of dynamic susceptibility through tfluctuation-dissipation theorem. To get the actual intensiwe also have to take into account the Ni andR magneticform factorsf Ni(Q) and f Nd(Q):

ds

dV}@ f Ni~Q! f Ni~Q! f Nd~Q! f Nd~Q! f Nd~Q! f Nd~Q!#x9~Q,v!S f Ni~Q!

f Ni~Q!

f Nd~Q!

f Nd~Q!

f Nd~Q!

f Nd~Q!

D . ~22!

be

eu-

: thes

e

The task of calculating the totalx9(Q,v) is greatly simpli-fied in the weak-coupling limit, where one can wrixRPA(Q,v)'@12x0(Q,v)J(Q)#x0(Q,v). At this level ofapproximation the two Haldane excitations~modes 1 and 3!are not distinguished. The same applies to CF modes 2,and 6. For the sum of the partial cross sections ofHaldane modes a straightforward yet somewhat tediousculation gives

ds113

dV}

ZvDQi

S f Ni~Q!224 f Ni~Q! f Nd~Q!EM2

E22DQi

2

3Re@J1~Q!1J2~Q!# D . ~23!

Similarly, for the single-ion modes,

ds214

dV}2M2S f Nd~Q!214 f Ni~Q! f Nd~Q!

Zv

E22DQi

2

3Re@J1~Q!1J2~Q!# D . ~24!

One readily sees that, unlike the corrections to the disperrelation, corrections to mode intensities arelinear in J1 andJ2 and can be substantial. The linear form of Eqs.~20!, ~23!,and~24! enables a straightforward generalization to the cof several Nd-single-ion transitions.

5,el-

on

e

3. Results for the 1D AF zone center

The expressions for the dynamic structure factor canfurther simplified for scattering vectors in„(2n11)/2,k,l …(n integer! reciprocal-space planes, where most of the ntron data were collected. Here the effect ofJ2 is totally can-celed, and there are only three branches in the spectrumHaldane branch~labeled ‘‘H’’ !, a CF branch that interactwith the Ni-chain excitations~‘‘CF’’ !, and the unperturbedCF transition(‘‘CF0’’). Using Eq.~22! on the exact~ratherthan linearized! RPA susceptibility matrix and truncating thresult to the first order inJ1, we obtain

dsH

dV}

ZvDQi

S f Ni~Q!228J1f Ni~Q! f Nd~Q!

3EM2

E22DQi

2cos~pk!cos~2p ld !D , ~25!

dsCF

dV}2M2S f Nd~Q!2@11cos~4p ld !#

18J1f Ni~Q! f Nd~Q!Zv

E22DQi

2cos~pk!cos~2p ld !D ,

~26!

dsCF0

dV}2M2f Nd~Q!2@12cos~4p ld !#. ~27!

Th

e

o

io

ua

tra

jesncareo

teVaparuexe

have

oa.n

a

n

u.

nt

edhes

he

neqs.

th–

d-ve

neareh-neri-

en-re-

he

anet inthe

24

tedhein

arthntededns,is

on-n-

bece

the

rsefield


The intensity of all three modes becomes modulated.period of modulation along theb axis is exactly 2b* , whilethat along c is incommensurate with the lattice. For threciprocal-space rods„(2n11)/2,k,0… (n integer! the modeCF0 vanishes and only two branches remain. For the purpof convenience we shall also rewrite Eqs.~20! for the „(2n11)/2,k,l … (n integer! planes:

\vH~Q!5DQi24J1~Q!2

ZvD

EM2

E22DQi

2cos2~pk!, ~28!

\vCF~Q!5D114J1~Q!2ZvD

EM2

E22DQi

2cos2~pk!, ~29!

\vCF0~Q!5E. ~30!

As noted previously, the magnitude of transverse dispersis proportional toJ1

2.We cannot expect the model described above to be q

titatively valid for any realR2BaNiO5 system. Its main limi-tation is that it assumes the exchange matrix and the maelements of momentum between the rare earth groundexcited states to be simultaneously diagonal in spin protion indexes. In theR2BaNiO5 structure, the rare earth sitehave a very low symmetry and the bare susceptibility aexchange matrixes are bound to be rather complex. Wehowever, expect the model to make qualitatively correct pdictions regarding the periods and signs of modulationsexcitation intensities and energies.

4. Paramagnetic phase

For Nd2BaNiO5 we should distinguish two regimes. AT.TN , at energy transfers below 30 meV, only the 18 mand the 24 meV Nd modes will mix with the Haldane-gexcitations. Experimentally, the 18 meV transition appetotally decoupled from the Ni chains. This is most likely dto its particular polarization and the actual form of the echange tensors. For example, if the 18 meV excitation wpolarized along thex axis andJ1

xx is zero, then no mixingwould occur. The 24 meV excitation is thus the only one tmixes with the Haldane branches, producing the obsermodulations.

The RPA model, oversimplified as it is, justifies the useEqs. ~6!–~12! in fitting the inelastic neutron scattering datFrom this analysis we can even extract some useful quatative information. According to Eqs.~28! and~25!, the ratioBH /AH @see Eqs.~7! and~10!# is independentof any param-eters of the rare earth ions:

BH /AH5Zv2D

J1 . ~31!

This is a robust result that allows us to obtain at leastorder-of-magnitude estimate forJ1, without making any as-sumptions regarding Nd31. For an isolated Haldane spichainZv'7.6D, soBH /AH'3.8J1. From theT555 K ex-perimental valuesA520.17 meV andB520.19 we getJ1'0.3 meV. We can compare this value to the MF copling a for Nd2BaNiO5 that we obtained in Refs. 9 and 6

e

se

n

n-

ixndc-

dn,-f

s

-re

td

f

ti-

n

-

Indeed,J' should be related toa through J1M (Ni)M (Nd)

5a/2, which givesJ1'0.5 meV, in reasonable agreemewith our new estimate.

A more crude estimate can also be obtained forJ2, whichin our model is the only possible source of the observanticrossing between the Haldane and 24 meV CF branc~Fig. 12!. Indeed, atk50.5 where these data were taken teffect of J1 is canceled, according to Eq.~16!. Experimen-tally, the anticrossing is about 4 meV. From Eq.~21! we getJ2M1'0.8 meV. The 24 meV CF mode and the Haldaexcitation are of comparable intensity, and based on E~25! and ~26! we can assume thatM2;Zv/D, which givesM;3. This gives us an estimate forJ2: of the order of 0.2meV. This is a rather reasonable value for a rare-eartransition-metal superexchange bond.

5. Ordered state

In the magnetically ordered phase an additional Ncentered excitation, namely, the 4 meV Ising-like spin wathat appears belowTN ~seeT516.5 K data in Fig. 2!, willalso mix with the Ni-chain modes. Being inside the Haldagap, it will produce intensity and energy modulations thatopposite to those resulting from the mixing with the higenergy 24 meV branch. Intensity oscillations in the Haldabranch will thus be reduced, in agreement with our expemental findings. As observed in previous studies, the intsity of the peak corresponding to Haldane excitations isduced by almost a factor of 2. This results from tsuppression of theb-axis-polarized Haldane excitation~lon-gitudinal mode! in the ordered state.7 In the framework of theRPA model, a decrease of spectral weight in the Haldbranches inevitably leads to a smaller interference effecthe CF mode. Such a reduction of intensity modulation inordered state is indeed observed experimentally for themeV excitation~see Table I!.

V. SUMMARY

We have demonstrated that local excitations associawith magnetic rare earth ions play an important role in tmagnetism ofR2BaNiO5 quantum antiferromagnets. Eventhe 1D ~paramagnetic! phase there are substantialdynamicinteractions between quantum spin chains and the rare esubsystem. This phenomenon can be qualitatively accoufor by a simple chain-RPA model. Not having more detailinformation on the electronic states of the rare earth iousing this model for a quantitative analysis of the dataobviously a leap of faith. It does, however, produce reasable order-of-magnitude estimates for Ni-Nd coupling costants.

At any temperature the Haldane-gap modes cannotconsidered as purely Ni excitations, and their interferenwith Nd-single-ion modes is substantial. Despite that,Haldane-gapenergy is practically the same as inisolatedchains in astatic mean exchange field, and the transvedispersion is very small. The static staggered exchangemodel is thus fully justified for Nd2BaNiO5, as are all ourprevious results for the temperature~staggered field! depen-dence of the Haldane gap.

Bk2-

ntfor

po-


ACKNOWLEDGMENTS

The authors would like to thank P. Dai, J. Zarestky,Taylor, and R. Rothe for expert technical assistance. WorBNL was carried out under Contract No. DE-AC0

J.

ro

ys

s.

,v

nd

.at

98CH10886, Division of Material Science, U.S. Departmeof Energy. Oak Ridge National Laboratory is managed

the U.S. DOE by Lockheed Martin Energy Research Corration under Contract No. DE-AC05-96OR22464.

hys.

ett.

in,

1For a recent review, see A. Zheludev, Neutron News10, 16~1999!.

2J. Darriet and L.P. Regnault, Solid State Commun.86, 409~1993!; T. Yokoo, T. Sakaguchi, K. Kakurai, and J. Akimitsu,Phys. Soc. Jpn.64, 3651~1995!; G. Xu et al., Phys. Rev. B54,R6827~1996!.

3A. Zheludev, J.M. Tranquada, T. Vogt, and D.J. Buttrey, Euphys. Lett.35, 385 ~1996!; Phys. Rev. B54, 6437~1996!.

4A. Zheludev, J.M. Tranquada, T. Vogt, and D.J. Buttrey, PhRev. B54, 7216~1996!.

5T. Yokoo, A. Zheludev, M. Nakamura, and J. Akimitsu, PhyRev. B55, 11 516~1997!.

6T. Yokoo, S. Raymond, A. Zheludev, S. Maslov, E. RessoucheZaliznyak, R. Erwin, M. Nakamura, and J. Akimitsu, Phys. ReB 58, 14 424~1998!.

7S. Raymond, T. Yokoo, A. Zheludev, S.E. Nagler, A. Wildes, aJ. Akimitsu, Phys. Rev. Lett.82, 2382~1999!.

8E. Garcı´a-Matreset al., J. Solid State Chem.103, 322 ~1993!;149, 363 ~1995!.

-

.

I..

9A. Zheludevet al., Phys. Rev. Lett.80, 3630~1998!.10S. Maslov and A. Zheludev, Phys. Rev. Lett.80, 5786~1998!.11J. Lou, X. Dai, S. Quin, Z. Su, and L. Yu, Phys. Rev. B60, 52

~1999!.12I. Bose and E. Chattopadhyay, cond-mat/9904311~unpublished!.13E. Garcı´a-Matreset al., J. Solid State Chem.103, 322 ~1993!.14V. Sachan, D.J. Buttrey, J.M. Tranquada, and G. Shirane, P

Rev. B49, 9658~1994!.15E.S. Sorensen and I. Affleck, Phys. Rev. B49, 15 771~1994!.16G. Muller, H. Thomas, M.W. Puga, and H. Beck, J. Phys. C14,

3399 ~1981!.17D.P. Arovas, A. Auerbach, and F.D.M. Haldane, Phys. Rev. L

60, 531 ~1988!.18G. Muller, H. Thomas, M.W. Puga, and H. Beck, J. Phys. C14,

3399 ~1981!.19S. Ma, C. Broholm, D.H. Reich, B.J. Sternlieb, and R.W. Erw

Phys. Rev. Lett.69, 3571~1992!.20S. Maslov and A. Zheludev, Phys. Rev. B57, 68 ~1998!.

role of single-ion excitations in the mixed-spin quasi-one-dimensional quantum antiferromagnet

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