Spin dynamics of heterometallic Cr7M wheels (M =Mn, Zn, Ni) probedby inelastic neutron scattering

R. Caciuffo and T. GuidiDipartimento di Fisica ed Ingegneria dei Materiali e del Territorio, Università Politecnica delle Marche,

Via Brecce Bianche, I-60131 Ancona, Italy

G. Amoretti, S. Carretta, E. Liviotti, and P. SantiniDipartimento di Fisica, Università di Parma, I-43100 Parma, Italy

C. MondelliIstituto Nazionale per la Fisica della Materia, and Institut Laue Langevin, Boîte Postale 220 X, F-38042 Grenoble Cedex, France

G. Timco, C. A. Muryn, and R. E. P. WinpennyDepartment of Chemistry, The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

sReceived 29 September 2004; published 9 May 2005d

Inelastic neutron scattering has been applied to the study of the spin dynamics of Cr-based antiferromagneticoctanuclear rings where a finite total spin of the ground state is obtained by substituting one Cr3+ ion ss=3/2d with Zn ss=0d, Mn ss=5/2d, or Ni ss=1d dications. Energy and intensity measurements for severalintramultiplet and intermultiplet magnetic excitations allow us to determine the spin wave functions of theinvestigated clusters. Effects due to the mixing of different spin multiplets have been considered. Such effectsproved to be important to correctly reproduce the energy and intensity of magnetic excitations in the neutronspectra. On the contrary to what is observed for the parent homonuclear Cr8 ring, the symmetry of the firstexcited spin states is such that anticrossing conditions with the ground state can be realized in the presence ofan external magnetic field. Heterometallic Cr7M wheels are therefore good candidates for macroscopic obser-vations of quantum effects.

DOI: 10.1103/PhysRevB.71.174407 PACS numberssd: 75.50.Tt, 75.10.Jm, 75.40.Gb, 75.45.1j

I. INTRODUCTION

Magnetic wheels are polynuclear molecular clusters witha ring-shaped cyclic structure and a dominant antiferromag-netic sAFd coupling between nearest-neighbor ions. For aneven number of spin centers, the ground state is a singlet andthe excitation spectrum is characterized by rotational andspin-wave-like bands.1–3 Heterometallic rings withSÞ0 canbe obtained from anS=0 homonuclear ring by chemical sub-stitution of one or two magnetic centers.4,5 Theoretical cal-culations suggest that such systems could exhibit interestingquantum-coherence phenomena6,7 and are therefore of con-siderable interest. Indeed, the replacement of a magnetic ionin a cyclic structure allows one to modify the topology of theexchange interactions, that in turn plays a key role in deter-mining the macroscopic behavior of the system.8–10

Here we report the results of inelastic neutron scatteringsINSd experiments on heterometallic AF rings, and we showthat the spin level sequence and dynamics are substantiallymodified with respect to those of the parent compound. Theinvestigated wheels are derived from the spin-compensatedneutralfCr8F8sO2CCMe3d16g ring11 by substitution of one di-valent cationsM =Mn, Zn, Nid for a trivalent Cr ion.4 Due tothe difference between the Cr3+ spin ss=3/2d and the spin ofthe dication, the ground state of the so-obtained Cr7M wheelshas a nonzero total spinsS=1/2, 1, and 3/2 forM =Ni, Mn,and Zn, respectivelyd. Recent investigations have shown that,within this Cr7M family, S=3/2 rings exhibit magnetocaloriceffects that could be exploited belowT,2 K,12 while Cr7Ni

could be a suitable candidate for the physical implementationof qubits.13 The suggested opportunities are linked to prop-erties that are very sensitive to the spin energy spectrum andto the composition of the spin wave functions. A reliabledetermination of these quantities is therefore relevant, andINS is known to be the most appropriate technique to ad-dress this problem.14–17

Our INS experiments on Cr7M provide energy and inten-sity for several transitions between the anisotropy-split low-est spin multiplets. The analysis of the data leads to an ac-curate determination of the main features of the microscopicintracluster interactions, including nearest-neighbor isotropicexchange, zero-field splitting, and dipole-dipole interactionparameterssintercluster interactions are very weak, of theorder of 10 neVd. The results obtained indicate that anticross-ing conditions between the ground state and excited low-lying energy levels with the same symmetry can be met inCr7M, by applying a suitable magnetic field. This is at vari-ance with the situation in the parent Cr8 ring, where groundstate crossings involve states with different symmetry,15 asconfirmed by the vanishingly small level repulsion observedby heat capacity measurements.18 The occurrence of anti-crossing conditions suggests that Cr7M wheels are good can-didates for the macroscopic observation of quantum phenom-ena arising fromS mixing, such as the quantum fluctuationof the magnitude of the total spin.8,9 A correct interpretationof the experimental spectra requires a generalization of theorientation-averaged INS cross sections for polycrystals,19 asproposed by Waldmann.20

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II. EXPERIMENTAL DETAILS

Deuterated microcrystalline samples ofsC2D5d2NH2

Cr7MF8fO2sC5D9dg16 sM2+=Ni,Mn,Znd have been preparedaccording to a slightly modified literature procedure,4,5 bydissolving chromium fluoride tetrahydratesAldrichd in amixture of trimethyl-d9-acetic acid and diethyl-d10-amines98% deuterated, Aldrichd before adding an excess of thesecond metal saltsnickel carbonate hydroxide tetrahydrate,manganese chloride tetrahydrate, or basic zinc carbonated.The preparation of trimethyl-d9-acetic acid starting fromacetone-d6 was adapted from standard methods.21 All com-pounds were purified additionally on a silica gel columnusing toluene as the eluent4 and finally crystallized from amixture of pentane/acetone by evaporation of the mixtureof solvents at 313–318 K. X-ray-diffraction analyses forall three compounds showed that crystals don’t have anysolvent molecules in the crystal lattice. The observed Braggpeaks could be indexed in the tetragonalP4 space group,with lattice parameters at 100 K ofa=b=19.9598s2d Å,c=16.0609s2d Å for Cr7Ni; a=b=19.9375s2d Å,c=16.1079s2d Å for Cr7Mn; and a=b=19.8973s2d Å,c=16.0227s2d Å for Cr7Zn. The magnetic rings have a diam-eter of about 10 Å and the metal ions have the shape of analmost regular planar octagonsFig. 1d.

INS measurements have been performed at the InstituteLaue-Langevin in GrenoblesFranced, with the direct-geometry time-of-flight spectrometer IN5, using about 2 g ofeach compound. A flat disk geometry has been chosen for thealuminum sample holders1.1-mm thickness, 5-cm diameterd,in order to reduce multiple-scattering background. Neutron

spectra have been recorded with the sample kept at differenttemperatures, betweenT=2 K and T=12 K, inside a stan-dard liquid-4He cryostat. Solid angle and detector efficiencycalibrations have been performed using the spectrum of avanadium metal sample. Neutron incident wavelengthsl=4, 5, and 9 Å were used, corresponding to instrumentalresolutions at the elastic peak of 0.147, 0.091, and 0.019meV, respectively. The three sets of measurements allowedus to cover an energy-transfer range fromqv,0.04 to about4 meV. The angular interval spanned by the detector bankscorresponds to scattering vector amplitudes varying bet-ween Q,0.55 and 2.6 Å−1 at qv,1 meV, and betweenQ,0.85 and 2.2 Å−1 at qv,3.7 meVsfor l=4 Åd.

III. SPIN HAMILTONIAN AND INS MAGNETICCROSS SECTION

The investigated systems are ensembles of independentidentical magnetic units, each one described by the Hamil-tonian operator:

H = oi=1

6

JCr-Crsi ·si+1 + JCr-Mss7 ·s8 + s8 ·s1d

+ oi=1

8

si ·Di ·si + oi, j=1

8

si ·Dij ·sj . s1d

In the above equation we assume that sitesi =1−7 of theoctanuclear ring are occupied by Cr3+ ions, and sitei =8 isfilled by theM dication. The first two terms are the isotropic,nearest-neighbor Heisenberg-Dirac-Van Vleck exchange in-teraction, withJCr-Cr andJCr-M being the exchange integralsfor Cr-Cr and Cr-M pairs, respectively. The third term in Eq.s1d describes the local crystal-field interaction, whereas thefourth term gives the dipole-dipole intracluster interaction,evaluated within the point-dipole approximation. Assumingthe z axis along the perpendicular to the ring plane, thesecond-order local anisotropy is expected to be dominated bythe axial termsdi, with smaller rhombic termsei:

oi=1

8

si ·Di ·si = oi=1

8

diFsz,i2 −

1

3sissi + 1dG + o

i=1

8

eifsx,i2 − sy,i

2 g.

s2d

With si ø3/2, no fourth-order local anisotropy must be con-sidered.

As pointed out by Waldmann,20 the INS cross section de-scribed in Ref. 19 for polycrystalline samples of molecularnanomagnets is not of general validity. In particular, it mayfail to describe correctly the scattering from systems withlarge magnetic anisotropy and does not properly take intoaccount intramolecular interference effects. Here we use thefollowing expression, which is obtained by averaging themagnetic dipole INS cross section22 with respect to the pos-sible orientations of the scattering vectorQ,20

FIG. 1. The molecular structure of the Cr7M sM =Mn,Ni,Zndmagnetic ring, with hydrogen atoms omitted for clarity. TheA ionssA=7/8 Cr and 1/8M at each sited are represented by large darkcircles; fluorine positions are indicated by small dark circles, theoxygen and carbon atoms are large and small gray circles, respec-tively. The cation is found H bonded inside the cavity.

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]2s

]V ] v=

A

Nm

kf

k0e−2Wo

n,m

e−bEn

ZInmsQddsqv − Em + End,

s3d

whereA=0.29 barn,Nm is the number of magnetic ions,Z isthe partition function,k f andk0 are the wave vectors of thescattered and incident neutrons, exps−2Wd is the Debye-Waller factor,Q=k0−k f is the scattering vector,En is theenergy of the generic spin stateunl, and the function InmsQdis defined as

InmsQd = oi,j

Fi*sQdFjsQdh 2

3f j0sQRijd + C02j2sQRijdgszi

szju

+ 23f j0sQRijd − 1

2C02j2sQRijdgssxi

sxj+ syi

syjd

+ 12 j2sQRijdfC2

2ssxisxj

− syisyj

d + C−22 ssxi

syj+ syi

sxjdgu

+ j2sQRijdfC12sszi

sxj+ sxi

szjd + C−1

2 sszisyj

+ syiszj

dgj,

s4d

whereFsQd is the magnetic form factor,Rij gives the relativeposition of ionsi and j , j0,2sQRijd are spherical Bessel func-tions, and

C02 =

1

2F3SRijz

RijD2

− 1G ,

C22 =

Rijx2 − Rijy

2

Rij2 ,

C−22 = 2

RijxRijy

Rij2 ,

C12 =

RijxRijz

Rij2 ,

C−12 =

RijyRijz

Rij2 , s5d

and

saisg j

= knusaiumlkmusg j

unl sa,g = x,y,zd. s6d

Equations4d is the explicit form of the formula given byWaldmann in Ref. 20. In addition to being valid whatever thesymmetry and amplitude of the magnetic anisotropy, Eq.s4dcan be easily implemented in a numeric code.

The parameters of the spin HamiltonianfEqs.s1d ands2dghave been determined from a best fit of the neutron spectrabased on calculations of Eq.s4d. We recall that from the INSselection rules only transitions withDS=0,61 andDM =0,61 have nonzero intensity. In the fitting procedure, aGaussian line shape is associated with each allowed transi-tion, with a full width at half maximumsFWHMd equal tothe instrumental resolution and an area proportional to thecalculated transition strength. For high-energy transitions, abroadening of the final state reflecting lifetimes effects hasbeen assumed. The parameters defining the Hamiltonian

have then been varied until the best agreement between cal-culated and experimental spectra was obtained.

The anisotropic part of the spin Hamiltonian can be writ-ten in terms of irreducible tensor operatorssITOd of rank k=2, and therefore mixes states with differentS andM, or atleast states with differentS if the anisotropy is purely axial.Therefore the total spinS is not a good quantum number andthe total Hamiltonian cannot be diagonalized within eachs2S+1d-dimensional block. This difficulty can be overcomeby the procedure proposed in Ref. 23 and used to evaluatethe mixing between the lowest lyingS multiplets in the Cr8ring.15

First, the minimum ensemble of spin manifolds requiredto reproduce the INS cross section at high-energy transfer isdetermined assuming isotropic exchange only. Then the com-plete Hamiltonian is diagonalized in the corresponding sub-space. In this way one obtains the energy spectrum and thespin statesunl as linear combinations of basis vectors

usSdSMl labeled by the set of intermediate spin statessSd,with coefficientsksSdSMunl.

A stringent test for the spectroscopic assignment of theobserved transitions is provided by theQ dependence of theirintensity, which is essentially determined by the geometry ofthe cluster and the composition of the spin wave functions.This dependence can be easily measured with properly cali-brated detectors at different scattering angles, and the resultscan be compared with calculations using Eq.s4d.

IV. EXPERIMENTAL RESULTS

A. Cr 7Ni

Figure 2 shows the angle-integrated INS intensity re-corded atT=2 K for the Cr7Ni sample, with IN5 operated atl=5 Å. The two peaks emerging at 1.19 and 1.34 meV cor-respond to transitions between theuS=1/2,M = ±1/2l

FIG. 2. Low-energy transfer INS response for Cr7Ni at 2 Ksl=5 Åd. The displayed intensity is the sum for the whole detectorbank. Background from the sample holder has been subtracted.Solid line: intensity calculated for the model Hamiltonian describedin the text, taking into accountS mixing; broken line: calculatedresponse withS mixing set to zero. Inset: intensity calculated withthe approximate formula for the cross-section given in Ref. 19ssolid lined.

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ground state and the anisotropy split sublevels of theuS=3/2l first excited spin stateuS=3/2,M = ±1/2l anduS=3/2,M = ±3/2l.

The solid line represents the intensity corresponding tothe Hamiltonian given by Eqs.s1d and s2d, with parametersJCr-Cr=1.46 meV, JCr-Ni=1.69 meV, dCr=−0.03 meV, anddNi =−0.6 meV. ThedCr value is the one determined for theCr8 ring15 and was kept fixed in the fitting procedure. As theINS response appears to be quite insensitive to the in-planeanisotropy, the nonaxial part of the Hamiltonian has beenneglected. With the parameters determined in this work, theground state composition is dominated by theuS=1/2l com-ponent, with a smalluS=3/2l contaminationsabout 1%d; thefirst excited state is an almost pureuS=3/2l multiplet witheasy-plane zero-field splitting. The INS cross section hasbeen calculated using the expression given by Eq.s4d, inte-

grated over theQ interval corresponding to the experimentalconditions. Although quantitatively small,S mixing must betaken into account to reproduce correctly the intensity ratioof the observed doubletsthe intensity calculated withS mix-ing neglected is shown in Fig. 2 by the dashed lined.

In the inset of Fig. 2, the experimental data are comparedwith the curve calculated by the approximate INS cross sec-tion reported in Ref. 19, and the Hamiltonian parametersquoted above. Due to the large anisotropy at the Ni site, theapproximation of Borras-Almenaret al. is not adequate inthe present case, and its use instead of Eq.s4d would lead toa completely wrong estimate of the Hamiltonian parameters.This difference is more evident when comparing theQ de-pendence of the transition intensities with theoretical esti-mates provided by the two formulas. As shown in Fig. 3sad,the intensity of the transition involving theuS=3/2,M= ±1/2l excited level at 1.19 meV is underestimated over alargeQ interval by the expression given in Ref. 19, while theintensity of the transition towards theuS=3/2,M = ±3/2lstate at 1.34 meV is overestimated by more than a factor 2.In comparison, an excellent agreement between calculatedcurves and observed data is obtained when using Eq.s4dfFig. 3sbdg.

Excitations involving spin manifolds with higher energycan be observed by reducing the incident wavelength. Re-sults obtained withl=4 Å at T=2 and 12 K are shown inFig. 4. At T=2 K, weak excitations are observed between 3and 4 meV. They correspond to transitions from the groundstate to uS=1/2l and uS=3/2l excited states. When thesample is warmed at 12 K, a strong peak appears at 2.08meV. This excitation is attributed to the transition from thefirst exciteduS=3/2l state to auS=5/2l state lying at 3.31meV. The calculated spectra, represented by the smooth linesin Fig. 4, compare very favorably with the experiment andconfirm the validity of the proposed set of parameters. Theratio JCr-Ni /JCr-Cr=1.16 is somewhat larger than the recentestimate based on the fit of heat capacity and torque magne-

FIG. 3. Cr7Ni INS intensity of the peaks at 1.19 meVscirclesdand 1.34 meVstrianglesd as a function of the scattering vector am-plitude, Q. Data have been obtained with incident wavelengthl=5 Å and sample temperatureT=2 K. Calculated curves are repre-sented by solid linessto be compared with filled circlesd and dashedlines sto be compared with trianglesd. Panelsad shows the resultobtained with the INS cross section as reported in Ref. 19, panelsbdshows the output of Eq.s4d. In both casesJCr-Cr=1.46 meV,JCr-Ni=1.69 meV,dCr=−0.03 meV, anddNi =−0.6 meV. The insetin panel sbd shows the INS response integrated from 1.1 and 1.4meV as a function ofQ, compared with the calculated curve.

FIG. 4. sColor onlined Cr7Ni INS spectra collected withl=4 Å at T=2 K scirclesd and T=12 K strianglesd. Smooth linesrepresent the spectra calculated from eigenvalues and eigenvectorsof the Hamiltonian given in Eqs.s1d ands2d, by associating to eachtransition a Gaussian line shape with a height proportional to thecalculated probability and a width of 147meV, corresponding to theexperimental resolution.

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tometry data.13 In addition the weighted average axial aniso-tropy parameterd=−0.045 meVsRef. 24d is about 1.7 timeslarger than previously reported, because of the very largeanisotropy of the Ni ion that was not estimated in previousworks.

B. Cr7Zn

Substitution of a diamagnetic Zn2+ ion for one Cr3+ in theAF Cr8 ring results in auS=3/2l spin ground state, whoseanisotropy splitting can be observed by high-resolution INSexperiments. Figure 5 shows the spectra recorded on IN5with incident wavelength 9 Åsenergy resolution at the elas-tic peak of 19meVd, with the sample kept at 2 and 10 K.

Assuming crystal-field parameters similar to those deter-mined in the Cr8 parent compound, we expect an easy-axisanisotropy for theS=3/2 ground manifold and an easy-planesplitting for the lowestS=5/2 multiplet. In this hypothesis,the peak observed at about 0.11 meV must be attributed tothe uS=3/2,M = ±3/2l→ uS=3/2,M = ±1/2l transition, asonly the lowest manifold is thermally populated atT=2 K.The solid and broken lines give the calculated intensity forT=2 and 10 K assuming a purely axial local crystal field forthe Cr ions, withdCr=−0.028 meVsdZn=0d. As for the Nicase, the INS spectra are not sensitive to the in-plane com-ponent of the anisotropy and again we consider only theaxial part ofH.

As shown in Fig. 6, intermultiplet transitions at higher-energy transfer are observed with incident wavelengths of 4and 5 Å. The peaks appearing in theT=2 K spectrum aredue to transitions from the splituS=3/2l ground state to-wards uS=1/2l s0.83 meVd, uS=5/2l s1.91 meVd, uS=1/2land uS=3/2l s2.25 meVd, uS=1/2l s3.0 meVd, and unre-solveduS=5/2l and uS=3/2l statess3.6 meVd.

The spurious peak at 0.4 meV in panelsbd is due to neu-trons diffused incoherently by the sample and diffracted bythe cryostat walls. The instrumental resolution in this con-figuration is not high enough to resolve the small anisotropysplitting of the excited levels, so that the energy separation ofthe doublets around 0.83 and 1.91 meV reflects the splittingof the ground state. Hot peaks associated to excitations fromthe first excited levels appear atT=12 K around 1.39 and 2.6meV. The solid and broken lines in Fig. 6 are intensitiescalculated at the two temperatures assuming an exchangeparameter very similar to the one reported for the Cr8 ring inRef. 15, JCr-Cr=1.43 meV for all nearest-neighbor Cr-Crpairs, JCr-Zn=0 sas Zn is diamagneticd and dCr=−0.028 meV. The agreement with experimental data is verygood and justifies the use of just one value for theJCr-Crexchange integrals, although the use of a less symmetric

FIG. 5. Cr7Zn high-resolution INS spectra collected withl=9 Å at T=2 K scirclesd and T=10 K strianglesd. The peak at0.11 meV is an intramultiplet transition involving the anisotropysplit components of theuS=3/2l ground state. Solids2Kd anddasheds10 Kd lines are intensities calculated for a purely axial, easyplane magnetic anisotropy of the Cr ionssdCr=−0.028 meVd. Theelastic peak and a quasielastic contribution have been included inthe best fit procedure.

FIG. 6. sColor onlined INS spectra for Cr7Zn at 2 Kscirclesd and12 K strianglesd, measured with incident wavelength ofsad 5 Å andsbd 4 Å. The 0.4 meV peak in panelsbd is spurious. Smooth linesare best fits of the experimental data to a superpositionof Gaussian line shapes associated to allowed transitions. EachGaussian contribution has an area proportional to the transitionstrength calculated with Eq.s4d assuming isotropic exchangesJCr-Cr=1.43 meV,JCr-Zn=0d and axial local anisotropysdCr=−0.028 meVd.

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magnetic model would improve the fitting at high energytransfer. The spectroscopic assignments are confirmed by theQ dependence of the transition intensities, as shown in Fig. 7where experimental data for some of the excitations observed

are compared with curves calculated with Eq.s4d. It is inter-esting to notice that the general behavior predicted for cyclicspin clusters by Waldmann20 is also followed in the presentcase where the ring symmetry is broken by the dication.

C. Cr7Mn

Substitution of one high spin,s=5/2, Mn2+ ion for a Cr3+

in the Cr8 ring results in an uncompensateduS=1l groundstate. In the presence of purely axial anisotropy, the zero-field splitting would lead to a quasitriplet structure, with anuS=1,M = ±1l doublet and anuS=1,M =0l singlet separatedby an energy proportional to the axial crystal field parameter.As shown in Fig. 8, thel=9 Å high-resolution spectrumrecorded atT=2 K shows a broad excitation that can befitted by the superposition of two Gaussians, each with aFWHM equal to the instrumental resolutions19 meVd andcentered at 0.095 and 0.11 meV, respectively. This is a clearindication of a sizeable rhombic term in the zero-field split-ting Hamiltonian. By fixing thedCr parameters to the valuesdetermined by INS for the parent Cr8 compound sdCr

=−0.03 meVd,15 the best fit of the data is obtained fordMn=−3 meV sa factor of 200 smaller than the Ni cased andeCr/dCr=eMn/dMn=0.14. If the S mixing is neglected, theZFS parameters describing the anisotropic splitting of theS=1 ground multiplet can be obtained by projecting the com-plete Hamiltonian onto the corresponding subspace. We ob-tain HS=1=DfSz

2−SsS+1d /3g+EsSx2−Sy

2d with a rhombicityE/D=0.13. The latter is reduced byS mixing to E/D=0.1ssee Fig. 8d.

Intermultiplet excitations measured with incident wave-length l=4 Å are shown in Fig. 9. AtT=2 K only onestrong magnetic peak is observed at 1.6 meV. A second mag-netic transition appears at 2.3 meV if the sample is heated at12 K. The two peaks correspond to the transitions involving

FIG. 8. sColor onlined Cr7Mn high-resolution INS spectra col-lected withl=9 Å at T=2 K. Two intramultiplet transitions withinthe anisotropy split components of theuS=1l ground state are ob-served. The solid line is the intensity calculated assuming both axialand in-plane anisotropy. The dash-dot lines represent the individualcontributions to the broad, unresolved excitation from the transi-tions between the splituS=1,M = ±1l quasidoublet and theuS=1,M =0l singlet. The dot line gives the intensity calculated withS mixing set to zero. The elastic peak and a quasielastic contribu-tion have been included.

FIG. 9. sColor onlined Inelastic spectra atT=2 K scirclesd andT=12 K strianglesd obtained for Cr7Mn with an incident wave-length l=4 Å. The peak at 0.4 meV is due to spurious effects.Smooth lines are intensities calculated at 2ssolidd and 12 Ksdashedd. The elastic peak and a quasielastic contribution have beenincluded.

FIG. 7. sColor onlined Q dependence of the peaks appearing inthe Cr7Zn inelastic spectra shown in Fig. 6, compared with theoret-ical estimatesssolid linesd. Triangles:uS=3/2l→ uS=1/2l at 0.83meV; circles: uS=1/2l→ uS=3/2l at 1.38 meV; squares:uS=3/2l→ uS=5/2l at 1.91 meV; open circles:uS=3/2l→ uS=5/2l at 3.6meV.

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the ground stateuS=1l and theuS=2l and uS=3l multipletswith minimal energy.

Both peaks therefore correspond to excitations within thelowest rotational-like band and their energy follows theLandé rule very closely. In these conditions, only an averageexchange parameter can be estimated. AssumingJCr-Cr=JCr-Mn=1.43 meV, calculations provide the spectra shownin Fig. 9 by smooth lines. The agreement is satisfactory anda model with more parameters would not be justified by thedata. Alternatively, by fixingJCr-Cr=1.46 meV as in Cr7Ni,the same results are obtained withJCr-Mn=1.37 meV. Theassignment of the two transitions to the Landé band is cor-roborated by theQ dependence of their intensity, shown inFig. 10. For both peaks a pronounced oscillatory behavior isobserved, while an almost flatQ dependence is expected fortransitions to states not belonging to the rotational band.3

V. DISCUSSION

The energy spectra of the heterometallic Cr7M magneticwheels resulting from our INS investigation are shown inFig. 11 as functions of the total spinS. A parabolic band,formed by states with minimal energy for eachS value, iseasily identified. The levels belonging to this parabolic bandhave energies that closely follow the Landé interval ruleES=D10fSsS+1d−S0sS0+1dg / fS1sS1+1d−S0sS0+1dg, where S0

is the spin of the ground state andD10 is the energy of thefirst excited level, with spinS1. Excitations involving adja-cent levels of the parabolic band have intensities with similarQ dependencies, with an oscillatory behavior and a pro-nounced maximum at aQ value related to the radius of thewheel.3 As discussed in Ref. 2 and 3, these excitations arerelated to the combined quantum rotation of the oppositelyoriented total spin on each Néel sublattice of the AF wheel.25

Effects due to the mixing of different spin multiplets havebeen considered. Such effects proved to be important to cor-rectly reproduce the energy and intensity of magnetic exci-tations in the neutron spectra. An interesting difference be-tween the spin wave functions of the parent Cr8 ring andthose of the heterometallic derivatives concerns their sym-metry. In Cr8 the ground state and the first excited state be-long to different irreducible representations of the moleculepoint group, whereas they have the same symmetry in thesubstituted wheels. This has important consequences on thesystem behavior in the presence of an external magneticfield. In Cr8 the Zeeman splitting of theuS=1, ±1l doubletleads to a crossing with the ground state atB=6.9 T with nolevel repulsion and a vanishingly small Schottky anomaly inthe heat capacity.18 On the other hand, according to the re-sults of the diagonalization of the Hamiltonian Eqs.s1d ands2d, anticrossings between the lowest lying levels are ex-pected in the Cr7M compounds. At the anticrossing condi-tions, a finite Schottky anomaly will occur, with an ampli-tude that depends on the angle between the applied magneticfield and the easy-axis. Moreover, the enhancement ofSmix-ing at the level anticrossings will produce maxima in thetorque signal corresponding to oscillations of the total spinamplitude, as observed for the Mnf333g grid.8,9

In the investigated series, Cr7Ni is particularly interesting.The amount ofS mixing in this ring is quite small, corre-sponding to about 1% admixture ofSÞ1/2 components inthe ground state. Cr7Ni can therefore be considered as aneffective S=1/2 system suitable for the implementation ofthe qubit.13

VI. SUMMARY

Intramultiplet and intermultiplet excitations involvingspin manifolds with energy smaller than 4 meV have been

FIG. 10. Q dependence of the peaks appearing in the Cr7Mninelastic spectrum at 12 K, compared with theoretical estimatesuS=1l→ uS=2l at 1.6 meVstriangles and solid lined; uS=2l→ uS=3l at 2.3 meVscircles and dashed lined. TheQ dependence of theintramultiplet transition at 0.1 meVsT=2 Kd is compared with cal-culations in the inset.

FIG. 11. Energy of the lowest spin eigenstates as a function ofthe total spinScalculated for Cr7M magnetic wheels;sad M =Cr, sbdM =Ni, scdM =Mn, sddM =Zn. Isotropic exchange interactions onlyare assumed, with parameterssad JCr-Cr=1.46 meV Ref. 15,sbdJCr-Cr=1.46 meV andJCr-Ni=1.69 meV, scd JCr-Cr=1.43 meV andJCr-Mn=1.43 meV, andsdd JCr-Cr=1.43 meV andJCr-Zn=0.

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measured with inelastic neutron scattering in polycrystallinesamples of the heterometallic wheels Cr7M sM=Ni,Mn,Znd. The minimum set of exchange and local crys-tal field parameters necessary to describe the physics of eachinvestigated compound has been determined by comparingthe experimental spectra with theoretical cross sections. Theresults obtained show that chemical substitution of magneticions in a cyclic structure can be used to tailor the magneticproperties of the wheels, by controlling the microscopic ex-change interaction. Finally, the explicit form of the generalpowder-averaged INS cross section given in Ref. 20 and suit-

able to describe molecular nanomagnets of any symmetryhas been presented.

ACKNOWLEDGEMENT

This work was partly supported by Ministerodell’Università e della Ricerca Scientifica e Tecnologica,FIRB Project No. RBNE01YLKN and by EPSRCsUKd. Wethank the Institut Laue Langevin, Grenoble, France for ac-cess to the neutron beam facility.

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