structural transition in c clusters - max planck societystructural transition in —c60–n clusters...

PHYSICAL REVIEW B 66, 094107 ~2002!

Structural transition in „C60…n clusters

W. Branz, N. Malinowski,* A. Enders, and T. P. MartinMax-Planck-Institut fu¨r Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany

~Received 21 March 2002; published 17 September 2002!

The structure of (C60)n clusters (n51– 150) was investigated by heating them to definite temperatures in ahelium bath~heating cell! with subsequent characterization in a time-of-flight mass spectrometer. The intensityanomalies in the resulting mass spectra reflect particularly stable configurations that resisted evaporation.However, the set of enhanced peaks~magic numbers! turned out to be temperature dependent. Clusters heatedto a moderate temperature of 490 K yielded mass spectra indicating the presence of icosahedral structures thatcan be characterized by a set of unusually strong peaks, namelyn513, 19, 23, 26, 35, 39, 43, 46, 49, 55, 5513m, 116, 125, 131, 137, and 147 (m51– 14). Based on constant-temperature molecular dynamics simula-tions it could be concluded that these icosahedra represent metastable structures that had not yet attainedthermal equilibrium on the experimental time scale (ts50.5 ms). Only when the (C60)n clusters were heatedto appreciably higher temperatures they could reach thermodynamical equilibrium, manifested by a thermallyinduced structural transition into close-packed structures, decahedral structures and structures based on the98-Leary tetrahedron. Thus, mass spectra of clusters heated to 600 K demonstrate the new set of magicnumbers,n538, 48, 58, 61, 64, 68, 71, 75, 77, 84, 89, 91, 96, and 98. Additionally, the experimental arrange-ment used allows both charged and neutral (C60)n clusters to be studied. Thereby, it was possible to show thatthe structure of large (C60)n clusters is insensitive to their charge state (2/n/1), disproving a hypothesis thathas been discussed in the literature for several years. The heating cell was also used to study the thermallyinduced dissociation in the fullerene dimers (C60)2 , (C70)2, and (C60)2

1 . The dissociation energies weredetermined to be 0.275 eV, 0.313 eV, and 0.37 eV, respectively (60.08 eV).

DOI: 10.1103/PhysRevB.66.094107 PACS number~s!: 61.46.1w, 36.40.2c, 61.48.1c

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I. INTRODUCTION

After the discovery of the large scale C60 synthesis,1 the

properties of fullerene solids have been intensively invegated. As the intermolecular C60 potential is very shortranged compared to the large fullerene diameter, C60 mol-ecules might be regarded as ‘‘nanoscopic hard spherTherefore it is not very surprising that in a C60 crystal themolecules are placed on the fcc close-packed sites expefor hard spheres.2 However, some bulk properties are alvery sensitive to the exact shape of the unusually narpotential well. So it is, for example, not yet clear if C60 has aslightly stable liquid phase3–6 or not.7,8

Several years ago the observation of clusters of C60 mol-ecules stimulated considerable interest in their structu9

The intensity anomalies in the photofragmentation mspectra of positively charged (C60)n clusters clearly indicatedthe presence of structures based on Mackay icosahed10

The especially high stability of the (C60)13 icosahedron wasalso demonstrated by experiments of Hansenet al.11,12 In allthese experiments the heating of the clusters was accplished by a short excimer laser pulse (Dt'15 ns). How-ever, the experimental results contradicted subsequent tretical studies on the structure of (C60)n clusters. Althoughthere is still controversy about the most suitable intermolelar C60 potential,

13–23 to date all global optimization calculations predict close-packed or decahedral structures for (C60)nclusters above a certain crossover cluster size in the rabetweenn513 and 16.22–28 For example, when the sphercally averaged Girifalco~GF! potential is used, icosahedrastructures represent only the global energy minima up tn

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513.22,24 Even with a full 60-site atom-atom potential, thcrossover shifts only slightly to larger sizes (n516).23,25,26

Recently, Pacheco and Prates-Ramalho~PPR! published anew intermolecular C60 potential that was obtained fromabinitio calculations of solid C60.

14 When compared with theGirifalco potential, the PPR potential indeed gives a bedescription of some bulk properties, such as, for examthe volume-pressure dependence.14 However, structure cal-culations for PPR clusters also failed in explaining the apearance of icosahedral structures in (C60)n clusters, even ifthe additional three-body term of the PPR potential wincluded.27,28 The sizes where close-packed and decahestructures appear aren5 15 for the three-body andn516 inthe case of the two-body PPR potential.28 The general reasonfor this rapid disappearance of icosahedral global minimasmall cluster sizes is the short-ranged intermolecular potial. With the decreasing range of the potential, structuwith high internal strain, such as icosahedra or disordeliquidlike configurations, are energetically less favorable bcause they involve nearest-neighbor distances that devfrom the equilibrium pair separation.29,30

One possible explanation for the discrepancy betweenexperimental and theoretical findings that has been discufor several years in the literature is the suggestion thatcalculations were performed for neutral clusters, whereasmass spectra of Martin et al. allow conclusions only for potively charged clusters.22,25,26Indeed, due to the large polaizability of the C60 molecule,

31 it seems not unlikely that theintroduction of a long-range Coulomb term could fundametally influence the structure of fullerene clusters. In ordertest this hypothesis we have performed a more detailed

©2002 The American Physical Society07-1

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W. BRANZ, N. MALINOWSKI, A. ENDERS, AND T. P. MARTIN PHYSICAL REVIEW B66, 094107 ~2002!

perimental investigation of both charged and neutral (C60)nclusters.32 In the present work we show experimental daclearly indicating that the structure of (C60)n clusters is in-deed independent from their charge state.

Although the cluster charge has little influence on thstructure, another parameter does—the clustertemperature.In contrast to the former laser heating experiments, innew experiments the clusters were heated by a heliumas described in detail below. When annealing the clustercomparatively low temperatures, e.g., atTh5490 K for atime span of ca. 0.5 ms, in principle, the same icosahemagic numbers as in the spectra yielded by laser heawere obtained.32 At higher temperatures~e.g., Th5600 K)the set of magic numbers completely changes and hasinterpreted as a thermally induced structural transitionclose-packed and decahedral structures and also strucbased on the recently discovered 98-Leary tetrahedro33

These structural assignments were confirmed by structcalculations on (C60)n clusters modeled by the PPpotential.28

The experimental findings pose a challenge for a theoical description. A temperature-dependent set of magic nbers cannot be explained only by global optimization callations, but thermodynamics and kinetics have to playkey role for the understanding of the observed effects. Fthermore, for theoretically examined systems, icosahestructures were preferred athigh temperatures becausetheir larger vibrational entropy.34–39These results give a firshint that the icosahedral structures occurring for (C60)n clus-ters at low temperatures are most probably due to thermonamicalnonequilibriumconditions. Such a metastable icoshedral phase has been observed in growth simulationsilver clusters40 and the same method has very recently givevidence for kinetic trapping effects in icosahedral structufor (C60)n clusters.

41 In this paper we will discuss the themally induced structural transition in (C60)n clusters ingreater detail by the presentation of new experimentaltheoretical results.

II. EXPERIMENTAL METHOD

The fullerene clusters were studied by the techniquetime-of-flight ~TOF! mass spectrometry. In our new Refletron TOF ~Ref. 42! we applied high acceleration (Uacc518 kV) and post acceleration (Upost530 kV) voltages inorder to avoid detection problems for large, singly chargclusters (n.100). For such large clusters only a weak signcould be achieved in our old machine.9 As cluster source weused a low-pressure inert gas condensation cell. Fullevapor from a resistively heated oven was quenched in alium atmosphere~1–3 mbars! cooled by liquid nitrogen~Fig.1!. The steady helium stream transported the clusters cdensed out of the vapor from the condensation cell intheating cell with a length of 14 cm and 6 mm in diametSubsequently, the cluster beam passed through two diffetial pumping stages into a high-vacuum chamber containthe acceleration region of the TOF spectrometer.

Particularly stable clusters can be identified in a mspectrum if the clusters are heated to temperatures at w

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they evaporate molecules. Then the corresponding peaksappear enhanced in the spectrum. In our former investigaof fullerene clusters9 heating and ionization were carried oalmost simultaneously by laser pulses. Therefore no informtion about the stability ofneutral fullerene clusters could beobtained. In contrast, the experimental setup shown in Fiallows the properties of either charged or neutral clusterbe studied. In the first case, positive and negative clusterwere formed by a glow discharge inside the condensacell. After passing through the heating cell one charge scies is selected, depending on whether the TOF spectromis operated in positive- or negative-ion mode. For the invtigation of neutral clusters ionization was accomplishedthe light pulse of anF2 excimer laser~157 nm! after theclusters had passed through the heating cell. Becausephoton energy of 7.9 eV is just above the ionization threold of a C60 molecule ~ionization potential is 7.6 eV! theclusters can be ionized without additional heating. In a ophoton ionization process the small excess energy is cverted to kinetic energy of the electron. Consequently,intensity anomalies in the resulting mass spectra are indreflecting the stabilities ofneutral (C60)n clusters.

The heating cell is a further development of a designhave used for the investigation of the melting behaviorsodium clusters.43 As can be seen from Fig. 1, special attetion was paid to achieving a constant temperature pro

FIG. 1. Experimental setup consisting of the cluster condention cell, the heating cell, and the TOF spectrometer. For the iization of the clusters either a laser pulse in the ionization voluor a glow discharge in the condensation cell can be used. Theperature profile of the heating cell, measured with a thermocouinside the cell, is also shown.

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STRUCTURAL TRANSITION IN (C60)n CLUSTERS PHYSICAL REVIEW B 66, 094107 ~2002!

(65 K) over a long distance~stability region'9 cm). Theduration of stay,ts , of the (C60)n clusters in the stabilityregion was determined experimentally44 and can be variedbetweents50.5 ms and 1 ms, depending on the pressurethe condensation cell. From that it can be estimated thatclusters should be well thermalized by at least 105 collisionswith the helium buffer gas.45 Even the time the clusterspend in the initial rising region should be sufficient for ataining nearly the plateau temperature. In this manner ipossible to produce a cluster beam with a well-defined teperature in the range betweenTh5100 and 700 K.

III. EXPERIMENTAL RESULTS AND DISCUSSION

A. Decay behavior of fullerene dimers

In order to test the functionality of the heating cell, thdecay behavior of theneutral fullerene dimers (C60)2 ,(C70)2, and the singly positively charged dimer (C60)2

1 wasinvestigated. Because of thesingle induced dipole-dipolebond the dissociation energy of a dimer lies well below tof the larger fullerene clusters (n.2), where at leasttwo(n53) or three (n54) bonds have to be broken. Therefothe decay of the fullerene dimers occurs in a low-temperaregion well separated from that of the trimers and tetramwhich was also checked experimentally. Thus, the dimpopulationI 2(T,t) can be described by the simple rate eqution

d

dtI 2~T,t !52I 2~T,t !k2~T!, ~1!

wheret is the heating time andk2 the decay rate. Hence, it ipossible to investigate the temperature-dependent dimernal without additional preselection in the experiment. Tdissociation energy of a dimer can then be determinedrectly from the definition equation of the Arrhenius activtion energy:46

D2ª2kB] ln k2~T!

]~1/T!. ~2!

In our experiment the decay ratek2 was determinedfrom the normalized dimer decay curven2(T,ts)5I 2(T,ts)/I 2(100 K, ts):

k2~T!521

tsln@n2~T,ts!#. ~3!

As I 2(T,ts) corresponds to the peak signal of the dimerthe mass spectrum,n2(T,ts) represents the surviving probability when the dimer is heated for the time spants at thetemperatureT.46 Note that according to Eq.~2! the slope ofln k2(T) versus 1/T should be independent fromts as checkedexperimentally. The value of the activation energy obtainfor the neutral dimer (C60)2 from the evaluation of severadissociation measurements is

D2@~C60!2#50.27560.08 eV. ~4!

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This value is in good agreement with the one that canestimated from the cohesion energy of bulk C60 ~see Table I!.Using Eq.~4! the constantC2 of the Arrhenius law,

k2~D2 ,T!5C2expS 2 D2kBTD , ~5!was determined to be of the order ofC254310

10 s21.Combining the Arrhenius law~5! with the rate equation~1!one gets after time integration a theoretical expression forsurviving probability,

n2,theo~T,ts!5expF2tsC2expS 2 D2kBTD G , ~6!which can be compared directly with the experimentally dtermined valuesn2(T,ts). This is demonstrated in Fig. 2 containing experimental data pointsn2(T,ts50.5 ms) obtainedat different temperaturesT, as well as the fitted decay curven2,theo(T,ts50.5 ms), for the dimers (C60)2 , (C70)2, and(C60)2

1 . Thereby the dissociation energies of (C70)2 and(C60)2

1 were determined by fitting Eq.~6! to the experimen-tal data using the same value forC2 as in the case of (C60)2.The resulting dissociation energies areD2 @(C70)2] 50.31360.08 eV and D2 @(C60)2

1] 50.37260.08 eV. For thedimer (C70)2 the value is in good agreement with that calclated from the cohesion energy of bulk C70 listed in Table I.

In the case of the singly charged dimer (C60)21 the addi-

tional charge-dipole interaction energyDD2@(C60)21# in-

duced by the charge placed on one of the C60 molecules cansimply be estimated. Under the assumption that the chargin time average smeared out over the entire C60 surface, theelectrical fieldEW (r ) corresponds to that of a point chargThe interaction energy is then given by

DD252pW indEW 52aC60EW 2, ~7!

wherepW ind is the induced dipole moment andaC60 the polar-

izability of the C60 molecule. Using the value foraC60 re-

cently determined by Antoineet al.31,53 (aC60576.5 Å3)

TABLE I. Binding energies ~eV! of the dimers(C60)2„D2@(C60)2#… and (C70)2 „D2@(C70)2#… estimated from the co-hesion energies of the bulk fullerenes, independently measurefour different groups~Refs. 47–52!. As a fullerene molecule in thebulk has 12 nearest neighbors the corresponding dimer dissocienergyD2 is roughly given by the sixth part of the heat of submationDsubHm

0 per molecule. Including non-nearest-neighbor intactions up to the fourth coordination shell causes an additiobinding energy in the bulk of about 4% what was considered incalculated dimer-values.

D2@(C60)2# D2@(C70)2# Ref.

0.27960.009 0.360.015 500.26460.007 0.31360.007 510.27160.035 0.360.035 520.30160.004 0.3360.005 47–490.27960.014 0.31160.016 Mean value

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and the equilibrium distancer 5r 0 for two C60 moleculesfrom the PPR potential14 (r 0510.018 Å) one gets an addtional binding-energy contribution ofDD2,theo50.109 eV.This value is close to the experimentally observed shiftdissociation energy ofDD250.097 eV~see Fig. 2!.

The experimental data points in Fig. 2 are recorded adwell time ts in the heating cell of 0.5 ms. When the heatitime is increased the decay temperaturesTd shift as expectedto lower values. When, for example, the dimer (C60)2 isheated forts51.0 ms, according to Eq.~6! the decay tem-perature should shift fromTd@(C60)2#5189 K down toTd@(C60)2#5181 K. This behavior is indeed confirmed bthe experiment~not shown!. In principle, a decay occurs aany nonzero temperature, if the heating timets is longenough.

The overall picture of the results gained for the simpdimer systems indicates that the heating cell operates aspected. To the best of our knowledge no other gas phmeasurements of the dissociation energies of fullerdimers have been performed so far. The experimentallytermined values summarized in Table II are in reasonagood agreement with the bulk values of Table I, and withthe estimated values according to Eq.~7!.

B. The structure of large „C60…n clusters

One main result of the preceding section was thatbinding energy of the charged dimer (C60)2

1 is enhanced by

FIG. 2. Comparison between the experimentally determinedcay pointsn2(T,ts50.5 ms) of the dimers (C60)2 (L), (C70)2 (*),and (C60)2

1 (1) and the corresponding fitted decay curves accoing to Eq. ~6! @(C60)2, solid line; (C70)2, dashed line; (C60)2

1 ,pointed line#. At the top of the diagram the decay temperaturesTd atwhich the corresponding signal has decreased ton2(T,ts50.5 ms)51/2 are indicated.

TABLE II. Experimental gas phase dissociation energies~eV! ofthe fullerene dimers (C60)2 , (C70)2, and (C60)2

1 .

D2@(C60)2# D2@(C70)2# D2@(C60)21#

0.27560.08 0.31360.08 0.37260.08

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about 35% compared to the neutral value. This leads toinitial question as to whether the structure of large (C60)nclusters is also affected by the cluster charge. The answerbe seen in Fig. 3. The enhanced peaks in Fig. 3~b! are theresult of heating neutral (C60)n clusters atTh5490 K forts50.5 ms. A comparison with the upper and lower partFig. 3, containing the intensity anomalies of~a! negativelyand~c! positively charged (C60)n clusters heated at the samtemperature, shows that the prominent peaks in all thspectra are the same. This clearly indicates that the strucof (C60)n clusters atTh5490 K is insensitiveto their chargestate. Furthermore, mass spectra for negative, positive,neutral (C60)n clusters were recorded at more than 20 diffeent heating cell temperaturesTh . None of the enhancedpeaks at any temperature turned out to be dependent oncharge state (2/n/1) of the cluster. Therefore the result o

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FIG. 3. Mass spectra reflecting the stabilities of~a! negatively,~b! neutral, and~c! positively charged (C60)n clusters at 490 K (ts50.5 ms). The set of magic numbers of allthreespectra is identi-cal. Different intensities in individual peaks merely reflect differeinitial cluster distributions, but have no significance.

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charge insensitivity can be generalized to the whole tempture region investigated in this work.

Figure 4 shows mass spectra of (C60)n clusters obtained afour different heating cell temperatures at the same heatime ts50.5 ms. The size distribution of cold clusters,grown atTh5100 K in Fig. 4 ~a!, is smooth and thereforecontains no structural information.

In Fig. 4~b! again a mass spectrum recorded atTh5490 K is shown, but now in a somewhat larger marange. Summarizing the 490 K spectra of Figures 3 and 4~b!they can be characterized by a set of unusually strong mpeaks, namely, n513, 19, 23, 26, 28, 31, 33, 35, 39, 43, 455, 5513m, 116, 125, 131, 137, and 147 (m51– 14). These,in the cluster literature so-called ‘‘magic numbers,’’ anearly identical to those obtained in our earlier laser heaexperiment.9 Exceptions in the new experiment are fouonly for the enhanced peaks atn526, 28, 31, and 33.

When the temperature of the heating cell is furthercreased the icosahedral magic numbers begin to disapand other unusually strong mass peaks arise. Finally,spectrum is dominated by a completely new set of manumbers. This can be seen in Fig. 4~c! where a (C60)n spec-trum recorded atTh5585 K is shown. Thenewset of magicnumbers isn538, 48, 58, 61, 64, 68, 71, 75, 77, 84, 89, 91, 9and 98. The enhanced peaks which remained unchangelocated in the low-mass region, namely,n513, 19, 26,28, 31, 33, and 35.

Concerning the temperature dependence of the intenanomalies in the (C60)n spectra, two main questions arisFirst, which structures are represented by the different semagic numbers? Second, what is thereasonfor the observedtemperature dependence? Because the second questiohardly be answered without clarifying the first one we wfirst of all focus on the structure analysis.

The magic numbers in the 490 K spectra@Figs. 3 and4~b!# are very similar to those observed for argon and xenclusters54,55 — similar but not identical. However, thLennard-Jones~LJ! potential, by which the interatomic interaction and the occurrence of icosahedral structures in rgas clusters can be explained, is not suitable for the destion of (C60)n clusters. As already mentioned, these requirpotential that is ‘‘harder’’ and shorter in range than thepotential. Nevertheless, the appearance of icosahedral stures for (C60)n clusters can also be justified just by gemetrical arguments.

The strong mass peaks atn513, 55, and 147 are versuggestive, as they correspond to the number of C60 mol-ecules that are necessary to complete the first three shethe sequence of Mackay icosahedra. However, this consion is far from definite since there are two other typesstructures with major shell closings for these numberscuboctahedra and square-faced truncated decahedra. It iportant to make this distinction, since a cuboctahedronany size can be cut out of a C60 fcc-bulk crystal, while icosa-hedra and decahedra are not consistent with the translatsymmetry due to their fivefold symmetry axes. In this senthey are noncrystalline.

When, however, the subshell structure between the mshell closings is considered, cuboctahedra and decahedr

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FIG. 4. Mass spectra of (C60)n clusters recorded at heating cetemperatures of~a! 100 K, ~b! 490 K, ~c! 585 K, and~d! 610 K(ts50.5 ms). Representative structures are shown in~b! and ~c!,where each C60 molecule is represented by a sphere. The structuobtained after heating up to 490 K~b! have an icosahedral motifthose obtained after more intense heating~c!, ~d! are instead close-packed, decahedral, or tetrahedral~see text!.

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be ruled out. The removal of one triangular face from perficosahedral (C60)n clusters containing 55 and 147 moleculyields clusters with 49 and 137 molecules, respectively. Tremoval of two, three, and five faces yieldsn546, 43, 39,and 131, 125, 116, respectively. All of these eight numbare represented by strong peaks in the 490 K spectra.suggests plausible structures for clusters somewhat smthan perfect Mackay icosahedra. The magic numbersclusters withlow coverage of a complete icosahedral cocan be explained by the addition of triangular faces accoing to the findings of Fargeset al.56 and Northby.57 Hereafterthere are two mutually exclusive lattices for the coveragea closed-shell icosahedron, called IC and FC. The IC latconsists of all sites of the next larger Mackay icosahedand is preferred for high coverage, resulting in the manumbers explained above. For low coverage the IC latticenergetically disfavored, since the molecules of the new shave contact with only two molecules of the core. In thcase FC sites are preferred, where additional moleculessitting in a trough built of three core molecules. In particulfor the coverage of the 13-molecule icosahedron the clusizes n519, 23, and 26 are favorable fillings of the Flattice which are also reflected by enhanced peaks in theK spectra. Even more pronounced in the 490 K spectra isFC sequence based on the 55-molecule icosahedral corthis case each of the 20 triangular faces of the icosahehas three FC sites. Thus, the series of unusually strong pfor the sizesn55513m(m51 –14) can be explained by thfilling of FC sites on successive triangular faces. Further sport for this assumption is given by the mass peaks(C60)70 and (C60)79, which stand out within the (5513m)series and correspond to the completion of one@cf. Fig.3~b!#, two ‘‘umbrellas’’ respectively, around adjacent fivefoaxes of the 55-icosahedron.

Finishing the analysis of the 490 K spectra it is wormentioning that all observed magic numbers whichcannotbe explained by stable icosahedral structuresn528, 31, 33) are located in the intermediate region betwFC and IC growth. In this size regime the icosahedral symetry is not so distinct as in the vicinity of closed shelEven for LJ clusters the icosahedral structures of the cosponding cluster sizes are only slightly lower in energy ththe optimal decahedral structures.58

The interpretation of the magic numbers in the 585spectrum of Fig. 4~c! is more difficult because no inclusivsuperior structural motif, as in the case of the icosahecould be found. For this reason support from theoretistructural calculations is necessary in order to make reliaconclusions. In a recent work the structure of (C60)n clustershas been modeled by the PPR potential. By means ofthe appearance of nearly all magic numbers in the hitemperature spectra could be explained.28 The 38-moleculecuboctahedron, as well as the decahedral structure of (C60)48shown in Fig. 4~c! represent the global energy minima fPPR clusters and are also predicted to be especially stFurthermore, the enhanced peaks in the 585 K mass strum occurring atn564, 71, and 75 can be explained b

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structures based on the 75-Marks decahedron59 and the peaksat n531, 33, and 35 are most probably due to small dehedra.

A cluster with a quite unusual symmetry, the recently dcovered 98-Leary tetrahedron, represents almost certainlybasis for the series of strong peaks atn561, 68, 77, 84, 91, 96, and 98. Thereby it should be nothat these structures donot represent the global minima foPPR clusters. However, the energy gap to the optimal sttures is rather small ('0.1–1 eV) compared with the hugenergy difference to the icosahedral configuratio('1 –3.3 eV).28 When the structures are restricted to thoderived from the Leary tetrahedron, the above magic nubers are perfectly reflected in a theoretical energy differespectrum of the corresponding PPR clusters.28 Hence, ourinitial interpretation of these magic numbers, based onlygeometrical arguments,32 is confirmed. Figure 5 illustrateshow the especially stable structures can be cut out of98-Leary tetrahedron. The latter consists of a 56-molecstellated fcc-based tetrahedron~bright spheres!, whose sixfaces are each covered by hexagonal arrangements consof seven molecules~darker spheres!. If one of these7-patches is removed, the result is a stable structure wi

FIG. 5. Particularly stable clusters, based on the 98-Leary tehedron, whose peaks are observed as magic numbers in thetemperature (C60)n spectra. Each C60 molecule is represented bysphere with a diameter corresponding to the equilibrium pair seration. The structures are obtained by the removal of 7-patches f(C60)98 and/or truncation. The lower part shows in which way tstructures of (C60)58 and (C60)48 are contained within (C60)98.

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total of 91 molecules. Also the strong peaks atn589 andn584 can be explained in a similar way. Truncation at oedge of (C60)98 by removal of three bright and six darspheres leads to a quite compact cluster with a total ofmolecules. The structure of (C60)84 is obtained by the re-moval of one 7-patch and additional truncation. Continuthis procedure further reveals the configurations of (C60)68and (C60)61. Finally, the structure of (C60)77 can be derivedfrom (C60)91 by truncation at the two adjacent edges of tmissing 7-patch. Additionally, it is shown in Fig. 5, in whicway the structures of (C60)58 and (C60)48 are containedwithin the Leary tetrahedron. Thus, the structural familyclusters based on the 98-tetrahedron can be extended totwo configurations. Hence, the Leary tetrahedron appearrepresent the dominating structural motif in the higtemperature spectra.

The reason why the strong peak atn554 in Fig. 4~c! wasexcluded from the list of magic numbers is that the pebetweenn552 and 54 are remainders from the stable 5icosahedron. The molecules at the vertices of the icosadron are poorly bound to the layer underneath becauseare only in contact with one core molecule. Therefore,bundle of peaks atn552–54 corresponds to icosahedra wmissing vertex molecules. Evidence is given by the fact tthese peaks disappear when the temperature is slightlycreased. This is already the case in the spectrum showFig. 4~d!, which was recorded atTh5610 K. In contrast tothe other prominent features in the spectrum, the peaks aciated with the 55-icosahedron in the regionn552–55 van-ished. This finding is also important for the interpretationthe prominent peaks atn558 and 61. It indicates that theare no longer based on the 55-icosahedron, but represenstructures. In analogy, the peaks betweenn5141 and 146 inthe high mass region of the spectrum in Fig. 4~d! representmissing vertex molecules of the 147-icosahedron.

So far we have shown that (C60)n clusters heated up to610 K for ts50.5 ms adopt different structures. Howevewe have not discussed the reasons for the occurrence otwo structural phases. In principle, there are three possexplanations for the observed effect.

First, the spectra recorded at the different temperatucould reflect the thermodynamically most stable structuFor nonzero temperatures not the potential energyE, but thefree energyF5E2TS (T, temperature;S, entropy! is therelevant thermodynamical variable for the determinationthe preferred equilibrium structures. Since the icosahestructures definitely donot represent the global energminima ~at T50 K) and are also considerably higherpotential energy than, for example, the tetrahedstructures,28 they would only be preferred athigh tempera-tures because of their larger vibrational entropy.28,39 Conse-quently, this kind of equilibrium solid-solid transition, whicwas observed in some systems as a premelting effect,34–37 ismost probably not responsible for the structural change(C60)n clusters in the investigated temperature range.

A second possible explanation has recently been sgested by Balettoet al.41 They assume that besides prformed icosahedral configurations also a small amounclose-packed decahedral and tetrahedral structures cou

09410

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formed in the inert gas condensation cell. When heatinghigh temperatures the less stable icosahedral structwould decay while the more stable nonicosahedral cluswould survive and are observed. We cannot exclude this psibility directly from our experimental data, because the tocluster signal decreases continuously with increasing tperature due to diffusive losses at the walls and the exithe heating cell. Therefore, only the relative stability of larclusters can be investigated. However, the dissociation egies, i.e., the energy required to remove the most weabound molecule, are only slightly different for the tetrahedstructures and, for instance, the icosahedral structure(C60)55 (DE,0.09 eV, PPR potential!. Furthermore, in Fig.4~d! the strong peak of (C60)98 and residual peaks of th(C60)147-icosahedron are simultaneously present. But whthe temperature is further increased, the peak of (C60)98 dis-appears before those belonging to the 147-icosahedronalthough the dissociation energy of (C60)147 and its frag-ments is ca. 0.35 eV lower than that of (C60)98. Moreover,according to the growth simulations of Baletto and cworkers the formation of nonicosahedral structures wereserved only at relatively high growth temperatures~525–550K!. In our experiment, however, the (C60)n clusters grow in acondensation cell with a surrounding helium gas temperaof about 80 K~Fig. 1!. The binding energy per C60 moleculein a (C60)n cluster is rather weak and there exist a grenumber of internal degrees of freedom, (180n-6), due tointermolecular vibrations (3n-6), intramolecular vibrations(171n16), and rotations (6n-6) to which the internal en-ergy can be distributed. As the clusters stay, in general,several milliseconds in the condensation cell, there is enotime to redistribute the condensation energy to the interdegrees of freedom.60 Therefore only a moderate increasetemperature is expected during the growth process as a rof the heat of adsorption of newly bonded molecules.With-out cooling the maximum temperature of a cluster thatreached after growth can be estimated from the total bindenergy. The temperature of (C60)55, for instance, should beof the order of 400 K, when the contributions from the itramolecular vibrations are weighted corresponding to thEinstein functions. However, depending on how fast the hexchange with the atoms of the cold helium buffer gascurs, the real growth temperature in the condensation ceexpected to be considerably lower. Thus, the most likely snario is that depending on the growth temperature, the (C60)nclusters are either completely disordered or icosahedrasome disorder left. The amount of disorder will increase wdecreasing growth temperature.41 Subsequent moderate heaing then would lead to kinetical trapping in the perfect icoshedral configurations observed in the 490 K spectra.markably, the icosahedral structures represent neitherglobal minima of potential energy nor those of the free eergy at this given temperature.41

Therefore, the third and most plausible explanation fortwo structural phases reflected in the (C60)n spectra is a tran-sition from metastablenonequilibrium icosahedral to thclose-packed decahedral and tetrahedral equilibrium sttures presumably lowest in free energy at this temperat

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Thus, it seems that with the structural transition, thermonamical equilibrium has been established on the experimtal time scale.

How can one imagine the trapping of the cluster structin a metastable state? Doye and co-workers have introdua terminology for characterizing potential energy surfa~PES’s! of clusters that can give important clues towardsunderstanding of the kinetics and thermodynamics in thsmall systems.58,61,62A PES can, for example, be characteized as having a single funnel, a double funnel, or evemultiple funnel landscape. This can be visualized incalled disconnectivity graphs in which a set of low-lyinlocal minima and transition states are mapped. If a cluhas a single funnel landscape, as is the case for mosclusters, the PES is dominated by one large basin withglobal energy minimum at its bottom. Hence, the optimstructure for LJ clusters is usually easy to find, because this a superior downhill force towards the global minimumExceptions from this relatively simple case are the LJ clters LJ38, LJ75-LJ77, and LJ98, whose PES’s have a doubfunnel landscape.33,58,61 This causes a completely differenthermodynamic behavior. The funnel leading to the glominimum ~namely, the 38-cuboctahedron, the 75-Mardecahedron, and the 98-Leary tetrahedron! is deep but nar-row and therefore difficult to find. The funnel leading to thicosahedral structures is broad, i.e., easy to find. Furtmore, the structural similarity between icosahedral andordered structures is much greater than between disordand, for example, closed-packed structures.63 This gives atopological argument for the higher accessibility of the icohedral structure region from the disordered state. For threasons trapping in the icosahedral funnel is much mlikely than the direct relaxation to the global minimum struture.

When the potential becomes harder and shorter in raas is the case for (C60)n clusters, the PES becomes more fland rugged.64 Therefore, in general, the number of funneincreases. Nevertheless the growth simulations of Baland co-workers show that trapping in icosahedral structuis still preferred, although they are thermodynamically muless favorable than for LJ clusters.28,41For such sticky poten-tials structural rearrangements are more difficult to achiev65

because thelocal potential barriers become sharper ahigher.64 Furthermore, there are large interfunnel barrieevident on the PES of (C60)n clusters,

28 additionally indicat-ing the difficulty of major structural transformations. Thus,the structure is once trapped in a metastable structural regit is expected to take a relatively long time and/or requihigh temperatures to escape.

In order to quantify the dependence of structural trantions and decay processes in (C60)n clusters on heating timeand temperature in greater detail, we focused on the behaof selected cluster sizes. For this purpose mass spectrarecorded in sequences of small temperature steps. Afterwthe temperature-dependent ratios of selected peak vawere calculated. An example using this method is shownFig. 6 ~a!. Here the decay behavior of (C60)55 on the basis ofthe peak ratios (C60)56/(C60)55, (C60)55/(C60)54, and(C60)54/(C60)53 was investigated. At temperatures lower th

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Th5400 K all ratios are, as expected, equal to 1. Whentemperature is increased above 400 K, the peak r(C60)56/(C60)55 decreases monotonically due to the great dference in stability between these two clusters. In contrthe value of the ratio (C60)55/(C60)54 first increases becaus(C60)55 emerges as a magic cluster. However, at temperataboveTh5500 K the probability that one vertex moleculeevaporated becomes significant and thus the ratio decreThe evaporation of one vertex molecule is also reflectedthe ratio (C60)54/(C60)53 that rises in the temperature regiobetweenTh5500 K and ca. 560 K. The subsequent decreindicates the loss of a second vertex molecule. In this manit is possible to monitor the temperature dependence ofsuccessive loss of one and two C60 molecules from theclosed-shell 55-icosahedron prior to the complete decay.

In a similar way the temperature dependence of the sttural transition can be investigated. For this purpose it wobe favorable to find neighboring peak-pairs each represena prominent icosahedral and close-packed~decahedral or tet-rahedral! structure. For this reason we have selected the p

FIG. 6. ~a! Dependence of the experimentally determined peratios (C60)56/(C60)55 (*), (C60)55/(C60)54 (L), and(C60)54/(C60)53 (1) on the heating cell temperatureTh (ts50.5 ms).~b! Dependence of the experimentally determined vues of (C60)39/(C60)38 (n) and (C60)49/(C60)48 (L) on the tem-peratureTh (ts50.5 ms). The fitted curves in~a! and ~b! are onlyto guide the eye.

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ratios (C60)39/(C60)38 and (C60)49/(C60)48 as suitable candi-dates for investigation. For (C60)39 the optimal icosahedrastructure is an especially stable half-shell closing@Fig. 3~b!#.For (C60)38 the global energy minimum is represented by t38-cuboctahedron. Similarly, for (C60)49 the lowest-energyicosahedral structure is a subshell closing and in the cas(C60)48 the global minimum is the particularly stable dechedral structure shown in Fig. 5.

In Fig. 6 ~b! the temperature dependence of the peak r(C60)49/(C60)48 is presented. BelowTh5400 K the ratiotakes again the value 1. With increasing temperature the rincreases due to the emerging icosahedral subshell closin(C60)49 against the relatively unstable structure of (C60)48,which in this temperature region is also expected to be icohedral. The maximum value of the ratio is reached at ab510 K, followed by a rapid decrease. This decrease sigthe occurrence of the transition towards the high-temperadecahedral structure. Remarkably, within a temperature sof about 50 K the absolute value of (C60)49/(C60)48 switchesfrom ca. 1.7 to 0.3, which indicates a complete reversathe stability ratio. An analogous temperature evolution is aobserved for the peak ratio (C60)39/(C60)38. In this case therise ~values greater than 1! is not so steep as fo(C60)49/(C60)48 and the transition to the close packedshifted by about 30 K to lower temperature@Fig. 6~b!#. Thisfits to the general trend that higher temperatures are neto cause a structural transition in larger clusters. A simsize dependence was already observed in the melting beior of sodium clusters.43 However, in each individual casthe topology of the PES plays an important role and thefore the transition temperatures can locally fluctuate. Forample, the barrier height between the different structuralgions and the length of the transition path vary from clusto cluster.58,63 Because the dimension of the configuratispace increases as 3n26, it is expected that, in general, thlengths of the transition paths also get longer and therehigher temperatures and/or longer heating times are requto reach the thermodynamical equilibrium state. In this ctext we have investigated the correlation with temperatand heating time experimentally using the smallest andlargest clusters for which a structural transition could beserved, namely, the clusters (C60)38 and (C60)98. In Fig. 7the peak ratios (C60)37/(C60)38 and (C60)99/(C60)98 are plot-ted versus the heating cell temperature for twodifferentheat-ing timests50.5 ms and 1.0 ms. These two ratios are coposed of an especially stable high-temperature structure~38-cuboctahedron, 98-Leary tetrahedron! and a cluster that isneither especially stable nor particularly unstable inicosahedral and the high-temperature structural phHence, with increasing temperature a monotonic decreasexpected and observed for both peak ratios. The declinthe curves can be interpreted as a disappearance of the ihedral structures and emergence of the corresponding htemperature structures.

Although a theoretical description of such an interfunntransition does not seem to be possible in a simple wMiller et al.have recently demonstrated that the icosahedfcc interfunnel transition in LJ38 obeys an Arrhenius law.

66

Therefore, we have fitted an Arrhenius decay curve, Eq.~6!,

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to the experimental data points in order to get a quantitaestimate of the temperature- and time-dependent behaHowever, it has to be emphasized that in this wayArrhenius-like interfunnel behavior cannot be proved f(C60)n clusters. For the Arrhenius constantCn we used thevalue obtained for the dimer (C60)2 multiplied with the size-dependent factorn2/3 that considers the effective surface frearrangement steps. In this manner the values for the forArrhenius activation energiesEn @D2 in Eq. ~6!# are found tobe E3850.86 eV andE9850.94 eV. For comparison, thevalue obtained for the Arrhenius activation energy of ticosahedral to fcc transition in LJ38 is E38 (LJ)53.19«.

66

With the C60 pair well depth of«50.275 eV@Eq. ~4!# thiscorresponds toE38(LJ)50.877 eV.

Furthermore, the temperature shift expected from Eq.~6!due to the longer heating time ofts51.0 ms~dotted lines!fits well with the corresponding measured values of the pratios (C60)37/(C60)38 (n) and (C60)99/(C60)98 (*). The ex-pected @~Eq. 6!# and observed temperature shifts areDT'16.5 K for the peak ratio (C60)37/(C60)38 and DT'18 K in the case of (C60)99/(C60)98. This illustrates howthe temperature required for attaining the high-temperastructures is lowered with increasing heating time.

IV. THEORETICAL RESULTS

It has to be pointed out that the clusters ultimately beloing to a magic number peak in the mass spectrum originfrom larger clusters which have lost a large number of mecules by evaporation. Therefore the measured temperadependencies could be distorted by ensemble effects.ideal computational support for the performed experimewould therefore be a molecular dynamics~MD! simulationof the shrinking of a large ensemble~e.g., n51 –200! oforiginally disordered clusters by multiple evaporation at costant temperature. After a given heating time~e.g., ts50.5 ms) the size distribution of the ensemble would

FIG. 7. Experimentally determined values of the peak rat(C60)37/(C60)38 and (C60)99/(C60)98 recorded at two different heating times: ~1! ts50.5 ms (L, 37/38; 1, 99/98), ~2! ts51.0 ms (n, 37/38; *, 99/98). The fitted curves correspondEq. ~6! for ts50.5 ms~solid lines! and ts51.0 ms~dotted lines!.

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inspected for magic numbers. However, such a computais hardly feasible for the long experimental time scalTherefore we restricted our MD calculations to single clussizes and shorter heating timestMD . The purpose of thesesimulations is to give some hints how the temperature depdence of the icosahedral and nonicosahedral structuresbe explained.

In order to conserve the cluster size we placed the clusin a box with a size of 80380380 Å3, whose reflectingwalls prevented the molecules from evaporating. Furthmore, this box roughly simulates the situation within a hevaporating ensemble in which the clusters continuouslycay by evaporation and are generated by the decay of laclusters. This situation is approximated by the evaporaand back bouncing of the molecules from the contaiwalls. In the MD simulations the intermolecular C60 interac-tion was modeled by the two-body PPR potential, as welby a Morse potential67 VM :

VM5«$er(12r /r 0)@er(12r /r 0)22#%. ~8!

Here « is the pair well depth,r 0 the equilibrium pair separation, andr the so-called range parameter. Such spherpotentials should be an appropriate description for the boing conditions in (C60)n clusters, since the C60 moleculesrotate freely even at relatively low temperatures.68

In relation to the experiments for single cluster sizes psented in Fig. 6~b!, we selected cluster sizes with particularstable decahedral and close-packed structures, namen548 and 38, for the simulations. Because of the significenergy differences to the icosahedral configurations, thecurrence of a structural transition can be identified very eily. All MD simulations were started from completely random cluster configurations. The temperatureT was heldconstant by means of an Andersen thermostat.69 We used avelocity Verlet algorithm with a time step ofdt520 fs. Wechecked that the results remain unchanged when ushorter time stepsdt, obtained in selected MD runs withdt510 fs and 5 fs.

Information about the current cluster structure and thesociated potential energy values were obtained duringsimulations by systematic quenching every 1–10 ns. In F8 the time evolution of the potential energy for a 4Pacheco-Prates-Ramalho-cluster (PPR48) heated to 750 K isshown. The total heating time wastMD510 ms. Startingfrom disordered configurations the cluster structure deoped after ca. 0.5ms into the icosahedral region~funnel!.This can be seen from the sharp lower edge in potenenergy atEico5194.43«PPR. This energy value correspondto the lowest-energy icosahedral structure. The other envalues within the icosahedral region represent suboptiicosahedral structural isomers. In this simulation the clusstructure stayed in the icosahedral funnel for more th5 ms, before the free energy barrier to the structural regof the decahedral global minimum (Eglob5197.904«PPR)was finally overcome. This MD run demonstrates tha(C60)n cluster can indeed be trapped for a relatively lotime span in the metastable icosahedral region even at atemperature. In order to determine how sensitive this resu

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to the shape of the potential we tested various intermolecpotentials with different range and repulsion in our Msimulations. This can easily be done using the Morse potial ~8! with «50.28 eV by adjusting the range parameterr.With decreasingr, the attractive range of the potential increases and the repulsive wall softens. In this way varipair potentials can be fitted by the Morse potential, leadingthe same optimal structures and predicted magic numbe30

For example, the LJ potential corresponds to a range pareter ofrLJ56.0, the PPR potential torPPR511.28, and theGF potential torGF513.62. It should be mentioned that thPPR potential already contains a Morse potential.14

We did not perform growth simulations, as in the workBaletto et al.,41 but started the simulations from disordereconfigurations. Thereby it was noticed that for range paraeters larger thanrPPR the probability that the structure initially develops into close-packed and decahedral structregions increases. Otherwise, when the range parameterdecreased, the probability for icosahedral structures wascreased. For a range parameter betweenr510.5 and 10.8 inabout 70–80 % of all performed MD runs, the cluster struture initially developed into the icosahedral region. This fiwell with the findings from the growth simulations41 and theexperimental low-temperature observations. As the shapVM(r510.8) is still very close to the PPR potential, winvestigated this type of Morse clusters more extensivelyFig. 9, representative70 MD runs for the clusterM48 (r510.8) atfour different simulation temperatures for the reltively long heating time oftMD5100 ms are shown. In eachcase the structure initially developed into the icosahedstructural region. In the lowest-temperature simulation pformed at T5575 K @Fig. 9~a!# the deepest icosahedraminimum (Eico5188.144«) was reached after a simulatio

FIG. 8. Time evolution of the potential energy of PPR48 duringa canonical MD simulation atT5750 K. The plotted energy valuewere obtained after quenches every nanosecond. The energy scin units of the PPR pair potential depth«PPR50.2667 eV. Thesimulation was started from a random disordered structure. Thafter the cluster structure developed first into the icosahedral fuwith the deepest minimumEico5197.904«PPR. Here the clusterstayed for'6 ms before the global decahedral minimum (Eglob5194.43«PPR) was found.

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time of about 5 ms. For the rest of the simulation~ca.95 ms) the cluster stayed in the icosahedral funnel, as agseen from the sharp edge in potential energy. Moreover,is reflected in Fig. 9~e! in which the probability distributionof the potential energy values~structures! is shown. Thedominating peak in Fig. 9~e! belongs to the lowest-energicosahedral isomer ofM48. In contrast to that, the icosahedral minimum, second lowest in potential energy~Ico2!, hasonly a minor probability.

Surprisingly, in theT5675 K simulation of Fig. 9~b! atransition from the icosahedral to the decahedral funnel ctaining the global energy minimum (Eglob5190.73«) occursafter a simulation time of 34.7ms. Afterwards the structurestays in the decahedral funnel until the end of the appheating time. Thus, in the corresponding probability distribtion of Fig. 9~f! now the energy value of the decahedglobal minimum has the largest weight. When the tempeture in the simulations is further increased toT5725 K@Figs. 9~c,g!# and 775 K@Figs. 9~d,h!# the cluster configura-tion switches more and more between the icosahedral anddecahedral regime. This means that thermodynamical elibrium between both structural regions is reached on

FIG. 9. ~a!–~d! Time evolution of the potential energy~quenchstep: 10 ns! for the Morse clusterM48 (r510.8) during canonicalMD simulations at the temperatures~a! 575 K, ~b! 675 K, ~c! 725K, and ~d! 775 K. In all MD runs the total heating time wastMD5100 ms. ~e!–~i! Probability distributions of the potential energiefor the corresponding MD runs~a!–~d! in the left column~see text!.

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simulated time scale. Interestingly, in the 775 K simulatithe icosahedral structures have again the highest occupprobability. This seems plausible, because at very high tperatures, icosahedral and finally disordered~liquidlike!structures must become lowest in free energy due tolarger entropy contributions.34–37,39

With respect to the experiment, the results of the Msimulations from Fig. 9 can be interpreted as follows. Ttrapping in the metastable icosahedral structural [email protected]~a,e!# corresponds to the situation in the low-temperatu490 K (C60)n spectra. The transition to the high-temperatustructures of Figs. 4~c,d! is reflected in the simulations atT5675 K and 725 K. In this case at the end of the heattime tMD5100 ms, a decahedral configuration would be dtected with the highest probability.

As seen, the temperatures required in the MD simulatifor the observation of the different structural phasesabout 75–100 K higher than the corresponding temperatin the experiment. This can be explained by the five timshorter time scale (tMD,1/5ts) of the simulation but also bythe lack of calculations to describe the processes in a lacluster ensemble.

The transitionback to icosahedral structures at very higtemperatures, as predicted by the MD run atT5775 K@Figs. 9~d,h!#, could not be observed in the experiment.

We have also performed extensive MD simulations for38-molecule clusterM38 (r510.6) and obtained a verysimilar temperature behavior for the structure as in Fig~not shown!. The initial transition from the icosahedral funnel to the global fcc minimum was observed at a simulattemperature ofT5650 K. At this T the 38-cuboctahedronalso has the highest probability in the corresponding enedistribution. At temperatures aboveT5700 K icosahedralstructures were again preferred as in the case of Fig. 9~h!.

It has become clear that the shape of the intermolecpotential plays a crucial role in determining both the optimstructures and the kinetical/thermodynamical behavior(C60)n clusters. In general, the energy differences betwthe global minima and the energy values of the lowest icohedral structures decrease with the increasing range ofpotential.30 Furthermore, the transition size from icosahedto nonicosahedral structures in small clusters shifts to larsizes. In this context it is also worth noting that the MorclusterM55 is the only large cluster (n.20) with an icosa-hedral global minimum in the investigated parameter ranof r510.5–10.8. Thereforeno structural transition is ex-pected forM55 with increasing temperature, in agreemewith the experimental observations shown in Figs. 4 and F6~a!. Principally the same is valid for the magic numbersn513 and 19, the icosahedron and the double icosahedAlso for these clusters no structural transitions are obserin the experiments with increasing temperature. The douicosahedron becomes the global minimum ofM19 for r,10.7. However, at the experimental temperatures the icohedral structure should become lowest in free energy efor considerably higher range parameters.28

Summarizing, it is still uncertain which potential is thmost appropriate for (C60)n clusters. Because of the approxmations made in the MD simulations it is only possible

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make some very crude conclusions concerning the interlecular potential. Therefore the simulations have to be sonly as one part in a big mosaic consisting of all twork13–30,41,65that has been done in this field. Reviewing tpresent picture it can be stated that the PPR potential givbetter description of bulk C60 than the Girifalco potential, buit could be still too short ranged. For a Morse potential wr in the range 10.5–10.8 the equilibrium between the icohedral and the high-temperature structures seems to bebalanced, which is important for the observation of bostructural phases.

Furthermore, it is interesting that the experimental specan be well described by an artificial restriction to the diffeent structural phases, as can be seen from Fig. 10. WMorse clusters withr510.8 were restricted to icosahedrstructures@Fig. 10~a!# nearly all magic numbers of the 490spectra were reproduced. The high-temperature magic nbers were found in the simulated spectrum of the 98-Letetrahedron sequence@Fig. 10~b!# and in that of the globaenergy minima@Fig. 10~c!#.

FIG. 10. Simulated magic number spectra for Morse clus(r510.8), by plots of the second energy differenceD2EM5EM(n11)1EM(n21)22EM(n). In ~a! the cluster structure was restricted to icosahedra, in~b! to a sequence based on the 98-Leatetrahedron, and in~c! the global minimum energy values werused.

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V. CONCLUSIONS

In the present work we could give important experimenand theoretical clues towards the understanding of the lostanding discussion on the structure of (C60)n clusters. Be-ginning with the first results for laser heated (C60)n clustersfrom the year 1993~Ref. 9! it seems now clear that threason for the observed icosahedral structures was kintrapping in metastable states due to the very short heatime ('15 ns). Although in this initial work it was concluded that ‘‘It would not have been surprising to have fouthat C60 molecules pack together in clusters as if they wehard spheres.’’9 it was not expected that such long temperitimes would be required to achieve that goal.

The exact temperature dependence of the structural tsition that was measured for selected (C60)n clusters hasshown reasonable correlations with the temperature shand activation energies expected from a simple Arrhenmodel. This experimental information, together with tqualitative results from the molecular dynamics simulatiostrongly indicates that the observed change in magic nbers is indeed due to a structural transition from metastaicosahedral structures to configurations that are probalowest in free energy at the experimental temperature.

However, there are still many open questions and prlems to solve. The most important point is the shape orealistic intermolecular potential. So far no potential hyielded unusually stable global minima forall observedhigh-temperature magic numbers. But this requiremenalso not necessary, since these structures only have to bglobal minima of free energy at the corresponding high teperatures. Such free energy calculations would howeververy challenging task. At least, the vibrational entropy of t98-Leary tetrahedron is larger than for the alternative dehedral and close-packed structures and lower than in theof an icosahedron.28 Since the potential energy of an icoshedron is however much greater than for the 98-tetrahedit seems plausible that there is a temperature window whthe tetrahedral structure is stabilized, according to the resfrom Fig. 9.

Finally, it has to be pointed out that the performed simlations are not able to reproduce the whole complexity ofexperiment but can only give some important hints baseda simplified system. Therefore, even though it was possto simulate the structural transitions of single clustersclosed in a container, it would still be interesting to compathe results with a simulation of a whole cluster ensembleorder to investigate the effects of combined evaporatshrinking and structural transition.

ACKNOWLEDGMENTS

The authors thank Dr. Jonathan, P. K. Doye, Dr. EKoch, and Professor David Tomanek for useful discussio

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*Author to whom correspondence should be adressed. Presendress: Max-Planck-Institut fu¨r Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany. Email [email protected]; Permanent address: Central Labtory of Photoprocesses, Bulgarian Academy of Sciences, 1Sofia, Bulgaria.

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structural transition in c clusters - max planck societystructural transition in —c60–n clusters...

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