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Neutron Spectroscopy for the Magnetic Anisotropy of Molecular Clusters

Roberto CaciuffoIstituto Nazionale per la Fisica della Materia, Dipartimento di Scienze dei Materiali e della Terra,

Università di Ancona, Via Brecce Bianche, I-60131 Ancona, Italy

Giuseppe AmorettiIstituto Nazionale per la Fisica della Materia, Dipartimento di Fisica,

Università di Parma, Viale delle Scienze, I-43100 Parma, Italy

Amir MuraniInstitut Laue Langevin, Avenue des Martyrs, B.P. 156 X, F-38042 Grenoble, France

Roberta Sessoli, Andrea Caneschi, and Dante GatteschiDipartimento di Chimica, Università di Firenze,

Via Maragliano 72, I-50144 Firenze, Italy(Received 15 July 1998)

Neutron spectroscopy has been used to observe transitions between the energy levels for thdifferent orientations of theS ­ 10 spin ground state of an octanuclear iron molecular clustersFe8dexhibiting quantum tunneling of the magnetization. The results obtained on nondeuterated samprovide the first direct measurement of the zero-field splitting in a large cluster nanomagnet. A reliaand accurate determination of the composition of the cluster’s spin wave functions and the spHamiltonian parameters are deduced from the experimental observations, giving crucial informatfor the study of the macroscopic quantum tunneling process. [S0031-9007(98)07673-X]

PACS numbers: 75.50.Tt, 75.10.Jm, 75.40.Gb, 75.45.+ j

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Molecular clusters composed by a large numberstrongly interacting metal ions have been of growing scietific interest as models of nanometer-sized single-domamagnetic particles with high spin ground state [1]. Compared to conventional nanomagnets obtained by sputting, wet precipitation synthesis, or ball milling of bulkmagnetic materials, the molecular systems offer the advatage of being formed by identical particles, whose structucan be precisely determined by diffraction measuremenMoreover, diluted systems characterized by a very wecluster-cluster interaction (typically of the order of a fewmeV) can be easily prepared [2]. Ensembles of weakly iteracting magnetic clusters with large Ising-type anisotropappeared to be very promising for the observation of messcopic quantum coherence. It has recently been realizthat the cluster magnetization can tunnel coherently btween the two opposite directions corresponding to the pof degenerate potential wells created by the anisotropy brier [3,4]. Such a phenomenon provides the signaturequantum behavior in mesoscopic systems and is of grrelevance not only from a fundamental point of view, bualso because it may prove to have important technicapplications [5,6].

Knowledge of the magnetic anisotropy of the clusterscentral to our understanding of the conditions under whicthe quantum tunneling of the magnetization (QTM) caoccur. In the presence of uniaxial anisotropy, the spectenergy of a cluster depends on the orientation of the spS relative to its symmetry axis,z. In the simplest case,

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with zero external magnetic field, the spin Hamiltoniandefined by a single anisotropy parameterD and dependson the spin projection alongz, H ­ DS2

z . However, forQTM to be allowed, the rotational symmetry about thezaxis must be broken by fourth-order spin terms, which dnot commute withSz [7–9]. In general, theories of QTMdepend critically on the knowledge of the anisotropy prameters, whose precise determination requires a suitaexperimental technique.

Recently, high-frequency electron paramagnetic resnance (EPR) was used to determine the spin Hamiltonfor Mn12ac [10], a molecular cluster for which resonanQTM was observed below 2 K [3,4]. The analysis of EPspectra taken in magnetic fields up to 25 T proved the prence of fourth-order spin terms, which in Mn12ac are rsponsible for the transverse magnetic anisotropy andoccurrence of QTM. In EPR measurements, however,zero-field-splitting terms in the Hamiltonian are deducefrom their effect on the level scheme in the presence ostrong magnetic field, and the data interpretation requicomplementary assumptions on the parameters definingZeeman term. Inelastic neutron scattering (INS) is particlarly appealing under this respect, because it can in prciple give a detailed picture of the low lying energy levefrom a straightforward analysis of spectra taken with zemagnetic field.

In this Letter we report the results of high resolutioINS experiments which give directly the anisotropy spliting of theS ­ 10 spin ground state in an octanuclear iro

© 1998 The American Physical Society

VOLUME 81, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 23 NOVEMBER 1998

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molecular clustersFe8d where the existence of QTM hasbeen clearly established [11]. The system investigatedfFe8O2sOHd12stacnd6g81, where tacn is the organic ligandtriazacyclononane [1]. The cluster of 8 Fe(III ) ions hasthe two-dimensional structure shown in Fig. 1 and is chaacterized by an overall symmetryD2. The four ions inthe middle of the molecule are in the so-called butterflarrangement, which can be considered as the first stepward the formation of a triangular planar lattice. Hydroxobridges connect the central motif to the four peripheraFe(III ) ions, while the organic ligands prevent the growthof the iron hydroxide.

The inelastic neutron scattering experiments were peformed with the high energy resolution multichopper timeof-flight spectrometer IN5, at the Institute Laue Langevinin Grenoble, France. Because of the large scattering crosection of hydrogen and the associated unwanted incohent scattering and multiple scattering, it is often found to bconvenient for experimental reasons to replace hydrogby deuterium, which has a much lower cross section. Whave, instead, used a thin sample consisting of two gramof a nondeuterated powder which was encapsulated inaluminum can and put into a standard liquid He cryostaBecause of the enormous amount of hydrogen presentthe investigated system, the quality of the magnetic specprovided by the experiment is quite astonishing.

Data were collected at temperatures between 1 a10 K, using an incident neutron energy of 1.01 meVcorresponding to a resolution of19 meV at zero energytransfers1 eV ­ 1.6 3 10219 Jd. The scattered neutronsare counted by an array of 10683He detectors spanning

FIG. 1. The structure offFe8O2sOHd12stacnd6g81. The largeempty circles represent iron atoms, full circles are oxygeatoms, hatched circles are nitrogen, and small empty circlare carbon atoms. The spin structure of theS ­ 10 groundstate is illustrated by the arrows.

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the scattering angle range between 11± and 128.7±. Thesignals from the individual detectors are sorted accordinto time of flight and added up. The maximum value for thescattering vector wasQmax ­ 1.28 Å21, and the neutronflux at the sample was about2 3 105 s21.

Magnetization, susceptibility, and high frequency EPRmeasurements show that the spin ground state of the F8cluster isS ­ 10, arising from competing antiferromag-netic interactions between theSFe ­ 5y2 spins of theFe(III ) centers [12]. The EPR spectra have been interpretby a zero-field spin HamiltonianH ­ DS2

z 1 EsS2x 2

S2y d, with an axial termD ­ 223.7 meV and a rhom-

bic parameterE ­ 24 meV [12]. With this Hamiltonian,the components of the ground manifold at the bottom othe anisotropy potential wells would mainly beM ­ 610states, followed by theM ­ 69 levels at about443 meV.On approaching the top of the anisotropy barrier, the mixing of states with differentM becomes more and more ef-fective due to the nondiagonal terms in the Hamiltonian.

The neutron experiments confirm that the above picture is essentially correct, except for several modificationas discussed below. As shown in Fig. 2, the scatterincross section measured for the sample at 1.3 K exhiba single excitation at465 meV. At such a low tempera-ture the excited states are not populated, and the onallowed dipolar transition (with selection ruleDM ­ 0or 61) is that between thejS ­ 10; M ­ 610l and thejS ­ 10; M ­ 69l levels. However, on increasing thetemperature the excited levels become populated progresively and transitions between them can be observed, ofering the possibility of studying the details of the spinHamiltonian. The results are shown in Fig. 3 where the intensity distribution measured at 4.8 and 9.6 K is reportePeaks on the positive energy-transfer side are due to trantions from lower to higher energy states; the correspondindeexcitation peaks appear on the negative energy-transside of the spectrum. We find that the simple Hamiltonian used for the interpretation of the EPR response [12

FIG. 2. Inelastic neutron scattering spectrum measuredT ­ 1.3 K, with an incident energy of 1.01 meV. The energytransfer resolution is19 meV at the elastic line.

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FIG. 3. Inelastic neutron scattering spectrum measuredT ­ 4.8 K and T ­ 9.6 K. The empty can signal has beesubtracted.

does not give a sufficiently accurate reproduction of the oserved spectrum and further symmetry-allowed terms mbe added. The width of the observed excitations is relution limited, and no information on the relaxation rateobtained from the present experiment.

In the dipole approximation, the magnetic scatterinfunctionSsQ, hvd for unpolarized neutrons can be writteas [13]

SsQ, hvd ­N4

sgNred2f2sQdg2J

3Xi,f

pijk fjS'jilj2Pshv 2 Dfi , Gfid , (1)

wheresgN red2 ­ 0.29 barnysr, hv is the neutron energytransfer, jnl are spin eigenstates with energiesEn andthermal occupation probabilitiespn ­ exps2EnykBT dyP

n exps2EnykBT d, Dfi ­ Ef 2 Ei , S' is the spin com-ponent perpendicular to the neutron scattering vectorQ,andPshv 2 Dfi , Gfid is the line shape for a peak of fulwidth at half maximumGfi and energy transfer centereat Dfi. For a polycrystalline sample, it is

jk fjS'jilj2 ­13

s2jk fjSzjilj2 1 jk fjS1jilj2

1 jk fjS2jilj2d . (2)

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The matrix elements and the energies can be obtaineddiagonalization of the spin Hamiltonian [14]

HS ­ DfS2z 2 SsS 1 1dy3g 1 EsS2

x 2 S2yd

1 D0O04sSd 1 E0O2

4sSd 1 CO44sSd , (3)

where the fourth-order spin operators are defined as

O04sSd ­ 35S4

z 2 f30SsS 1 1d 2 25gS2z

2 6SsS 1 1d 1 3S2sS 1 1d2,

O24sSd ­

14

hf7S2z 2 SsS 1 1d 2 5g sS2

1 1 S22d (4)

1 sS21 1 S2

2d f7S2z 2 SsS 1 1d 2 5gj ,

O44sSd ­

12

sS41 1 S4

2d .

The position of the peaks in the INS cross section givea direct measurement of the eigenvalues of the spin Hamtonian, and their intensities provide information about thwave functions through the matrix elements ofS'. In prin-ciple, the parameters of the Hamiltonian can be determinfrom the experimental data by a least-squares fit procedu

In the first step of this procedure only the second-ordeterms were considered. We found that the best fit valuethe axial parameterD ­ 225.2s2d meV is 6% larger thanthe EPR value quoted in [12], while the rhombic parameter E ­ 24.02s3d meV remains unchanged. With theseparameters, however, it was impossible to account at tsame time for the exact position of the high energy shapeaks and the low energy part of the experimental spetrum, which is characterized by broad unresolved peakBy allowing for a simultaneous variation of all the parameters in (3), we found that the best-fit values ofD andE did not change within the errors, while for the fourth-order parameters we obtainedD0 ­ 0.87s6d 3 1024 meV,E0 ­ 0.1s1d 3 1024 meV, andC ­ 7.4s6d 3 1024 meV.With these parameters, the height of the barrier for the rorientation of the magnetization isAyk ­ 32.8s1d K, to becompared with the value of 24.5 K obtained from magnetiac susceptibility measurements [11].

Figure 4 shows that the agreement between calculatiand experiment is very good, both for the position and thintensity of the peaks. In the excitation part, the calculated intensity of the peak at 0.17 meV is lower than thexperimental one, whereas the agreement is good for tcorresponding peak at negative energy transfer. Thisalmost certainly due to a spurious signal originated by thdetector electronics. Also, the excitation at 0.074 meV appears to be more intense than calculated, probably becaof the overlap with the tail of the very strong elastic peakas confirmed by a preliminary scan with even higher energy resolution.

The spectral energy region below about 0.3 meV (anin particular the interval 0.2–0.3 meV) presents morcomplex features than the higher energy one. This

VOLUME 81, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 23 NOVEMBER 1998

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FIG. 4. Inelastic neutron scattering spectrum measured9.6 K. The smooth line is the theoretical spectrum for thHamiltonian given by Eq. (3), with the parameters quotedthe text. The individual excitations are represented by Gaussline shapes.

due to the mixing of thejSMl states by the nondiagonaterms in the spin Hamiltonian, which is very effective fojMj # 6. As appears from Fig. 5, the scattered intensiin this energy range is due to the superposition of sevevery close transitions between the mixed wave functionhaving nonzero matrix elements of comparable magnituAs a consequence, the shape of the peaks is very sensto small variations of the spin Hamiltonian coefficients.

In conclusion, the high resolution INS spectrum of thnondeuterated Fe8 nanomagnet shows how accurate thtechnique can be for the determination of the zero-fiesplitting and the magnetic anisotropy of molecular clusteThe results obtained open exciting perspectives for thevestigation of this class of materials and offer the possibity of a very precise determination of the spin Hamiltoniaparameters, which critically influences the compositionthe wave functions taking part in the macroscopic tunneing process. Important information has been obtaineddevelop a quantitative understanding of the mechanismQTM in Fe8. In particular, the height of the barrier for thereorientation of the magnetization has been determinedbe Ayk ­ 32.8s1d K. This value is significantly higherthan the 24.5 K deduced from measurements of thelaxation time [11]. A similar situation has been observefor Mn12ac, where the spin Hamiltonian provided by EPgives a barrier height of 67.1 K, whereas a fitting of threlaxation data leads toAyk ­ 61 K [10]. A relaxationbarrier lower than the energy difference between the loest M ­ 610 and the highestM ­ 0 level could be anindication of QTM acting between excitedM levels andspeeding up the relaxation of the magnetization alsothe thermally activated regime. A high resolution neutroscattering experiment in Mn12ac would be interestingverify the actual strength of the zero-field splitting poten

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FIG. 5. Zero-field splitting of theS ­ 10 ground state mul-tiplet of Fe8. Thicker lines correspond to doubly degenerastates (within 0.01 meV); the main transitions allowed in thdipole approximation are indicated by arrows.

tial and confirm if the above difference is a characteristof systems exhibiting QTM. Moreover, it has been showthat the INS spectrum is sensitive to the higher order prameters, which in this way can be accurately determinin zero field. This is particularly important for systemwith axial anisotropy, where the off-diagonal fourth-ordespin operatorsS4

1 1 S42d is essential to induce tunneling

being the only term in the spin Hamiltonian which contributes to the in-plane magnetic anisotropy.

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