Transverse magnetic anisotropy in Mn12 acetate: Direct determinationby inelastic neutron scattering

Roland Bircher, Grégory Chaboussant, Andreas Sieber, and Hans U. GüdelDepartment of Chemistry and Biochemistry, University of Berne, Freiestrasse 3, CH-3000 Berne 9, Switzerland

Hannu MutkaInstitut Laue-Langevin, rue Jules Horowitz 6, Boîte Postale 156, 38042 Grenoble Cedex 9, France

(Received 18 October 2004; published 30 December 2004)

A high-resolution inelastic neutron scattering(INS) study of fully deuterated Mn12 acetate provides accuratespin-Hamiltonian parameters for this prototype single-molecule magnet. The Mn12 clusters deviate from axialsymmetry, a nonzero rhombic term in the model Hamiltonian leading to excellent agreement with observedpositions and intensities of the INS peaks. The following parameter set provides the best agreement with theexperimental data:D=−0.0570s1d meV, B4

0=−2.78s7d310−6 meV, B44=−3.2s6d310−6 meV, and uEu

=6.8s15d310−4 meV. Crystal dislocations are not the likely cause of the symmetry lowering. Rather, thisstudy lends strong support to a recently proposed model, which is based on the presence of several molecularisomers with distinct spin-Hamiltonian parameters.

DOI: 10.1103/PhysRevB.70.212413 PACS number(s): 75.50.Xx, 75.30.Gw, 75.45.1j, 78.70.Nx

Single-molecule magnets (SMMs) are crystallinecompounds consisting of nominally identical poly-nuclear molecular complexes of transition metal ions. Theyexhibit slow magnetization relaxation phenomena at cryo-genic temperatures. Discrete steps in the magnetization are atypical feature directly related to quantum tunneling pro-cesses of the magnetization(QTM). Mn12 acetate,fMn12O12sOAcd16sH2Od4g ·2HOAc·4H2O, was the first re-ported SMM and remains the best studied up till now. Sincethe pioneering work of Sessoliet al.1,2 Mn12 acetate has beenextensively studied by inelastic neutron scattering(INS),3,4

EPR,5,6 NMR,7 Raman and infrared spectroscopy,8 specificheat,9,10 micro-Hall probe techniques, micro superconductingquantum interference device techniques,11–13 magnetizationrelaxation,1,9,14,15 and numerous other bulk measurements.Identifying and quantifying the interactions leading to slowrelaxation and QTM has been the focus of several studies invery recent years.16–19 In this Brief Report we report a high-resolution inelastic neutron scattering study on fully deuter-ated Mn12 acetate. In contrast to most other studies no exter-nal magnetic field is involved. This leads to very accuratevalues of the relevant interaction parameters responsible forQTM and allows us to unambiguously discriminate betweenthe various microscopic models which have been proposedto account for the observed macroscopic behavior.16,17

fMn12O12sOAcd16sH2Od4g ·2HOAc·4H2O crystallizes in

space groupI4, and thefMn12O12sOAcd16sH2Od4g moleculeoccupies a position withS4 point symmetry.20 The four H2Oand two HOAc(acetic acid) solvent molecules are incorpo-rated between the Mn12 acetate complexes, with the twoHOAc molecules disordered on a fourfold position. Ex-change coupling between the Mn3+ and Mn4+ ions within thecomplexes leads to aS=10 ground state. A strong axial an-isotropy with theS4 axis as the easy axis splits the groundstate into sublevels fromMS= ±10 toMS=0 and thus createsthe energy barrier which is responsible for the slow magne-tization relaxation phenomena at cryogenic temperatures.

The appropriate spin Hamiltonian to account for this zero-field splitting in theS4 point group is given by Ref. 21:

Haxial = DFSz2 −

1

3SsS+ 1dG + B4

0O40 + B4

4O44 s1d

where O40=35Sz

4−f30SsS+1d−25gSz2−6SsS+1d+3S2sS+1d2

and O44= 1

2sS+4+S−

4d. The first term of Eq.(1) is the leadingterm. Numerous parameter sets have been proposed, withthose based on EPR and INS studies, which are very similar,usually considered the most reliable.3,5 Equation(1) cannotaccount for all the observed QTM phenomena, in particulartunneling throughMS= ±10 in zero field below 2 K. Addi-tional terms to the ones in Eq.(1) are therefore needed. AHamiltonian including a rhombic term of the form

Haniso= Haxial + EsSx2 − Sy

2d s2d

requires a deviation from the crystallographically determinedS4 molecular symmetry. This can result from the disorder inthe solvent structure, and this possibility was examined indetail in Refs. 17 and 22. The presence of six different geo-metrical molecular isomers with slightly different environ-ments and thus differentD andE parameters was postulated.An alternative model, in which the fourfold molecular sym-metry is broken by crystal dislocations, was proposed in Ref.16. This model corresponds to a broad distribution of sitegeometries withE values broadly distributed arounduEu=0.Very recent EPR and QTM studies strongly favor a discretedistribution,18,19 but exact parameter values to test the spe-cific predictions of Refs. 17 and 22 are still missing.

In an earlier INS study of partially deuterated Mn12 ac-etate the data were of sufficient quality to determine the threeparametersD, B4

0, andB44 in Eq. (1).3 The important question

of an E term remained open. By an upgrade, the time-of-flight instrument IN5 at the ILL in Grenoble has gained anorder of magnitude in speed, and data of significantly higher

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quality than in Ref. 3 are obtained. They allow a deeperanalysis in terms of Eq.(2) and lead to a clear distinctionbetween the proposed models. 6.5 g of a fully deuteratedsample of Mn12 acetate were used in the present study. Thematerial was prepared as described in Ref. 20 using deuter-ated precursors and solvents. A cylindrical aluminum con-tainer of 14 mm diameter and 55 mm length was used for theINS experiments. Data were corrected for the backgroundand detector efficiency using standard procedures. Figure 1shows an overview INS spectrumsl=5.9 Åd for a tempera-ture T=24 K, at which all theMS levels of the zero-field-split S=10 ground state have some population. The well de-veloped and almost regular pattern of inelastic peaks on boththe energy loss and energy gain sides(positive and negativeenergy transfer, respectively) are assigned toDMS= ±1 tran-sitions between adjacent ground state levels. This assignmentis straightforward,3 since onlyDMS= ±1 transitions are al-lowed in an axially zero field split state; see Fig. 3(a). Theirrelative intensities are calculated using Ref. 23. The higher-orderB4

4 andE terms in Eqs.(1) and (2) will mix MS func-tions, and this becomes relevant in the following. The innerpart of the INS pattern was measured at the same tempera-ture with increased experimental resolution(l=8 Å, fullwidth at half maximum of the elastic line 23meV). The re-sult, displayed in Fig. 2(a) shows a sharpening of both theelastic and inelastic features and, in particular, some wellresolved structure below 0.3 meV on both the gain and theloss sides. The features in this spectral range are broadenedessentially, while at higher energy transfers the peak widthsare resolution limited. Below 0.3 meV the energy intervalsand the relative intensities significantly deviate from theregular pattern observed in Fig. 1. These deviations reflectthe higher order terms in the spin Hamiltonian, and they willnow be analyzed. The data in Fig. 2 are reproduced severaltimes without error bars to allow comparisons with differenttheoretical models. For all the calculated spectra in Figs.2(b)–2(d) instrumental bandwidths are used. The curve inFig. 2(b) corresponds to a calculation using Eq.(1) with theD, B4

0, andB44 parameter values in set I of Table I. This is the

best parameter set within the axial approximation, and it isthe same within experimental error to the one derived previ-

ously using INS.3 As for all the other models considered herethe energy transfer range above 0.2 meV is very well repro-duced by these parameters. But below 0.2 meV there is pooragreement with the experimental data of the present study.Maxima of the calculation coincide with minima of the dataand vice versa. The upper part of the energy splitting patternfor this calculation is shown in Fig. 3(c). TheB4

4 term splitsand mixes theMS= ±2 states in first order. Fitting our ob-served peak positions with the eigenvalues of Eq.(2), i.e.,including anE term, leads to the calculated curve in Fig.2(c). The best parameters are listed in set I of Table I. Thecalculated energy pattern is shown in Fig. 3(d). The E termsplits and mixes theMS= ±1 states in first order, and thissplitting is significant. This calculation shows a remarkablygood agreement with the experiment, both in terms of peakpositions and intensities. Each of the broadened peaks below0.3 meV contains two or three unresolved transitions. Analmost equally good agreement is obtained with the param-eter set II of Table I. The two solutions I and II correspond tonegative and positive values of the parameterB4

4, respec-tively. In a purely axial model, eigenvalues for equal positive

FIG. 1. INS spectrum measured on IN5 with an incident wave-length of l=5.9 Å at 24 K for 15 h, summed over all scatteringangles. The peaks are assigned toDMS= ±1 transitions; see Fig.3(a).

FIG. 2. (a) INS spectrum measured with an incident wavelengthof l=8 Å at 24 K for 12.5 h, summed over all scattering angles. In(b)–(d) the same data are plotted without error bars including the-oretical simulations. The same background, shown as dash-dottedlines in(b), was used in all the calculations.(b) Equation(1), D, B4

0,andB4

4 values in set I of Table I.(c) Equation(2), parameter set I inTable I. (d) Equation(2), parameter set III in Table I.

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and negativeB44 terms are degenerate. Figure 4 shows a con-

tour plot, in which the goodness of the least-squares fit isplotted in the parameter space spanned byB4

4 and uEu. Thetwo minima corresponding to the solutions I and II in TableI are very well defined. We confidently conclude that anEterm is necessary to account for the observed INS data. Andin contrast to earlier studies we can quantify theuEu param-eter:uEu=6.8s15d310−4 and 5.3s9d310−4 meV for solutionsI and II, respectively. Of the two parameter sets we givepreference to set I, i.e., a negativeB4

4 parameter, because ourvalue B4

4=−3.2s6d310−6 meV lies close to the most recentand very accurate determination by EPR ofuB4

4u=3.8s1d310−6.19

Of the two models proposed in the literature to accountfor a deviation from axial molecular symmetry we definitelyrule out the dislocation model,16 because it corresponds to abroad distribution of sites withE terms centered arounduEu=0. This is the energy level pattern in Fig. 3(c) and wouldlead to a broadened version of the calculated spectrum inFig. 2(b). The alternative model with a set of six discretesites for the Mn12 acetate molecule17 has recently received

qualitative support from both EPR and tunneling studies.18,19

In Ref. 17 theD and E parameters of the six isomers wereestimated on the basis of the angular overlap model. Basedon the same isomer distribution modelD and E parametersobtained by a density functional theory calculation were re-cently reported.22 The distribution ofD andE values is verysimilar to that of Ref. 17, but the absolute values are morereliable. In order to reproduce the high-energy-transfer partof our data it was necessary to scale theD parameters ob-tained from Ref. 22 with our bestB4

0 value by a factor of1.014(3). We calculated the energy splitting patterns for thesix isomers and the corresponding INS spectrum at 24 K,using theuEu and the scaledD values from Ref. 22 and ourbestB4

0 and B44 values; see set III in Table I. The result is

shown in Fig. 2(d). The agreement with the experimentaldata is remarkably good, considering that no adjustment ofthe uEu parameters was attempted. It is nota priori clear whythe distribution of six species with populations and param-eter values as given in set III of Table I ives a discrete INSspectrum with well defined peaks. Inspection of the param-eter set III in Table I reveals that the three isomers 1, 2 cis,and 3 make up 75% of all complexes in the crystal. Theisomers 1 and 3 have very similarD and, more importantly

FIG. 4. Contour plot of the goodness of the fit of Eq.(2) to theexperimental peak positions as a function ofB4

4 andE with constantvalues forD and B4

0. The two least-squares minima correspond tothe parameter sets I and II in Table I.

TABLE I. Parameter values obtained from fits to the experimental INS peak positions. I and II correspondto the minima in the goodness of fit plot in Fig. 4. Set III corresponds to the model proposed in Ref. 17, butwith uEu and scaledD parameter values from Ref. 22. The numbering of isomers is the same as in Refs. 17and 22. TheB4

0 andB44 parameters are the same as in set I.

Set Isomer OccupancyD

(meV)B4

0

s10−6 meVdB4

4

s10−6 meVduEu

s10−4 meVd

I −0.0570s1d −2.78s7d −3.2s6d 6.8(15)

II −0.0570s1d −2.78s7d 5.1(7) 5.3(9)

III 0 0.0625 −0.0563 −2.78 −3.2 0

4 0.0625 −0.0580 −2.78 −3.2 0

1 0.25 −0.0563 −2.78 −3.2 6.9

2 cis 0.25 −0.0572 −2.78 −3.2 0.02

2 trans 0.125 −0.0572 −2.78 −3.2 13.8

3 0.25 −0.0572 −2.78 −3.2 6.9

FIG. 3. Energy level splitting of theS=10 ground state andallowed (energy loss) INS transitions.(a) Total splitting, purelyaxial; (b) upper part of(a); (c) splitting of Eq.(1); and(d) splittingof Eq. (2) with parameters in set I of Table I.

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in this context, practically identicaluEu parameters to our bestparameter set I. This, together with some near coincidencesof transition energies for the other isomers, leads to a dis-crete spectrum which is in good agreement with experiment.For the peaks above 0.3 meV the calculated broadening dueto the presence of the six isomers lies within the experimen-tal accuracy of the data and is thus not observable. Our studythus provides support for the model proposed in Ref. 17 andrefined in Ref. 22.

Our spectroscopic study thus clearly reveals that the Mn12acetate clusters infMn12O12sOAcd16sH2Od4g ·2HOAc·4H2Odeviate from axial symmetry. A rhombic term in the spinHamiltonian is essential, and all three parameter sets in TableI reproduce the INS spectra very well. Earlier studies basedon Landau-Zener(LZ) magnetization relaxation18 and EPR(Ref. 19) measurements provided upper limits of 8.7310−4

and 12.4310−4 meV, respectively, for theuEu values in themost strongly rhombically distorted complexes in Mn12 ac-etate. Our study significantly narrows down the range of pos-sible uEu values. In contrast to LZ and EPR measurements,we are not primarily probing the complexes with the fastestrelaxation or the strongest rhombic distortion but the total ofall complexes. TheuEu=6.8s15d310−4 meV value of set I inTable I represents this. If we adopt the isomer distributionmodel, this value accounts for 50% of all the complexes; for37.5% theuEu value is at least an order of magnitude smaller,and for 12.5% it is 13.8310−4 meV. This latter value is inreasonable agreement with upper limit estimates of 8.7

310−4 and 12.4310−4 meV from LZ and EPR, respectively.Interestingly, it was recently suggested, based on EPR ex-periments, that in deuterated Mn12 acetate the upper limit ofuEu was 24.8310−4 meV, twice as high as for the undeuter-ated material.24 This value is high and outside the range ofour parameter values. Our experiment was performed on acompletely deuterated sample, and we took great care tohandle it in a water free atmosphere. In a sample which is notfully deuterated there is additional disorder in the acetic acidand water structure, and this could possibly lead to a broaderdistribution of rhombic distortions and thussEd values.Rhombic distortions of individual complexes, likely causedby disorder in the solvent structure, are responsible for someof the observations in QTM measurements which are notcompatible with a purely axial model. The present study pro-vides quantitative information about the size of the rhombicparameters. This model is not able to account for the obser-vation in Mn12 acetate of QTM between levels with oddDMvalues. Possible mechanisms have been proposed in Refs.16, 17, and 25. These effects must be small and our experi-ment provides no information to quantify them.

The authors acknowledge useful discussions with OliverWaldmann. This work was supported by the Swiss NationalScience Foundation and the TMR programs Molnanomagand Quemolna of the European Union(Grants No. HPRN-CT-1999-00012 and No. MRTN-CT-2003-504880).

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