, Israel

PHYSICAL REVIEW B 67, 195408 ~2003!

Carbon nanotube closed-ring structures

Oded Hod and Eran RabaniSchool of Chemistry, The Sackler Faculty of Exact Science, Tel Aviv University, Tel Aviv 69978, Israel

Roi BaerInstitute of Chemistry and Lise Meitner Center for Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904

~Received 2 July 2002; revised manuscript received 28 January 2003; published 14 May 2003!

We study the structure and stability of closed-ring carbon nanotubes using a theoretical model based on theBrenner-Tersoff potential. Many metastable structures can be produced. We focus on two methods of gener-ating such structures. In the first, a ring is formed by geometric folding and is then relaxed into minimumenergy using a minimizing algorithm. Short tubes do not stay closed. Yet tubes longer than 18 nm arekinetically stable. The other method starts from a straight carbon nanotube and folds it adiabatically into aclosed-ring structure. The two methods give strikingly different structures. The structures of the second methodare more stable and exhibit two buckles, independent of the nanotube length. This result is in strict contradic-tion to an elastic shell model. We analyze the results for the failure of the elastic model.

DOI: 10.1103/PhysRevB.67.195408 PACS number~s!: 61.46.1w, 73.22.2f, 62.25.1g, 64.70.Nd

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I. INTRODUCTION

Since their discovery,1 carbon nanotubes have been prposed as ideal candidates for the fabrication of nanoetronic devices due to their unique physical properties. Cbon nanotubes can be either metallic or semiconducting,have relatively good thermal conductivity, and unusual mchanical strength.2 These unique properties make them ptential building blocks of future small electronic devices.3–7

Numerous studies in the past few years have focuon the mechanical properties of single-wall nanotub~SWNTs!,8–14 and the role of structural deformation on thelectronic properties of SWNTs.14–22 The experimental andtheoretical studies indicate that electronic properties, sucthe density of states near the Fermi level and/or the condtance of the tubes, are somewhat sensitive to mechandeformations. These mechanical deformations include being and twisting the SWNTs, bond rotation defects, and otvarious types of defects.

Most studies of deformed SWNTs focus on the rolerelatively small mechanical deformations, which are oserved under standard conditions. One exception is the sof Bernholc and collaborators11,12 who considered relativelylarge deformations such as twisting and bending a SWuntil a buckle was formed. They developed a continuelastic theory to treat these different distortions. With proerly chosen parameters the continuum elastic shell moprovided a remarkable accurate description of these defortions beyond Hooke’s law.12

More recently, Liuet al.14 have studied the effects of delocalized and localized deformations on the structural aelectronic properties of carbon nanotorus using the tigbinding molecular-dynamics method. Nanotoroidal strutures based on carbon nanotubes have been studied ipast.8–10,23 However, in order to stabilize these structurheptagon-pentagon defects were introduced into the hexnal atomaric structure of the nanotube, resulting in signcant changes of the electronic properties. Liuet al.14 wereable to systematically study the transition from delocaliz

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to-localized deformations. One of their major conclusiowas that delocalized deformations result in a small reducof the electrical conductance, while there is a dramaticduction in the conductance for localized deformations. Tstudy of Liuet al.14 is interesting from an additional perspetive, namely, that these closed-ring structures may be usebuilding blocks of small electromagnetic devices, and thelectronic properties are of primary interest for such applitions.

There are still several open issues related to the structand electronic properties of closed-ring carbon nanotubFor example, what is theglobal minimum-energy configura-tion of the closed-ring structure? Can a continuum elashell model predict the proper structure of these materiaWhat is the kinetic stability of these materials? In this wowe address these issues related to the structural properticlosed-ring carbon nanotubes. In Sec. II we extend the elashell model of Yakobsonet al.12 to the case of multibuckleformation required when closed-ring structures are formThe elastic theory provides a scaling law for the numberbuckles formed for a given nanotube length~L! and diameter(d). We find that unlike the case studied by Yakobsonet al.where the continuum shell model provided quantitativesults for the formation of a single buckle, it fails in the mutibuckle case. To illustrate this, we have used the TersBrenner potential to describe the interaction betweencarbon atoms,24 and simulated the formation of closed-rincarbon nanotube structures. In Sec. III we discuss two psible ways to generate the closed-ring structures. Theways lead to very different local minimum-energy configrations. Interestingly, for a large range of nanotube lengwe find that the lowest-energy configuration observedcharacterized by onlytwo, well localized, buckles, independent of the length of the corresponding nanotube.

In Sec. IV we study the relative stability of the closed-rinstructures. It is important to assess thekinetic stability ofthese structures since, as is well known, these structuresless stable thermodynamically compared to the correspoing nanotubes. We find that even at very high temperatu

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ODED HOD, ERAN RABANI, AND ROI BAER PHYSICAL REVIEW B67, 195408 ~2003!

the structures are very stable on the time scales of molecdynamics ~MD! and Monte Carlo~MC! simulation tech-niques. Therefore, we applied the nudged elastic band~NEB!method25–27 to obtain the minimum-energy paths betwedifferent closed-ring configurations, and the barrier for topening of the rings into the corresponding nanotubes. Cclusion and future directions are given in Sec. V.

II. ELASTIC SHELL MODEL

In this section we generalize the continuum elastic moof Yakobsonet al.12 to treat the case of a closed-ring nantube. This model was first used by Yakobsonet al. to studythe formation of instabilities in carbon nanotubes that wsubjected to large deformations. In bending, it was foundthe model provides quantitative results for the critical anin which the tube buckles. Furthermore, the critical curvatestimated from the shell model was in excellent agreemwith extensive simulation of SWNTs.11,12

When a SWNT is bent to form a nanoring, one expethat the resulting high-energy structure will relax intoclosed polygon that is characterized by many buckles.14 Thenumber of apexes will depend on the length, diameter,the mechanical properties of the tube. Yakobsonet al. esti-mated the critical strain for sideways buckling to be closethat of a simple rod:

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independent of the lengthl. Assuming that the tube bucklewhen the local strain on a bent tube is

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whereKc is the local critical curvature~and is close to that oan axial compression!, Yakobsonet al. found that the criticalcurvature for buckling isKc5p2dl22 for long tubes, andKc5(0.155 nm)d22 for short tubes. The results~for shorttubes! were found to be in excellent agreement with simutions on SWNTs of various diameters, helicities, alength.11

One can also define a critical angle for buckling, whichthe angle between the open ends of the nanotube at the bling point. This critical angle for buckling is given byuc5Kcl . The result for long tubes isuc5p2dl21, and forshort tubesuc5(0.155 nm)ld22.

For a closed-ring structure, the local curvature can beproximated byK52pL21, whereL is the total length of thenanotube. Assuming that the local strain on the closeddoes not change significantly when a buckle is formed,

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Equation~5! is the main result of this section. It predicthat the number of apexes increases with the square roothe total length of the closed-ring nanotube, and decreawith the square root of the diameter of the tube. In the limof L→` the number of apexes will also approach infinitresulting in a perfect ring structure.

III. SIMULATIONS OF CLOSED-RING CARBONNANOTUBE STRUCTURES

In this section we compare structures obtained from amistic simulations with those predicted from the continuuelastic shell model presented in the previous section. Wethe Brenner potential-energy surface to describe the intetions between the carbon atoms.24 This three-body potentiahas been used in the past to study structural propertiecarbon nanotubes, with excellent agreement betweensimulations and experiments for structural deformationscarbon nanotubes.11,12 As noted in Ref. 11, theexcellentagreement between the observed and the computed defostructures confirms the reliability of simulations based onBrenner potential.

The empirical form of the Brenner potential has beenjusted to fit thermodynamic properties of graphite and dmond, and therefore can describe the formation and/or breage of carbon-carbon bonds. In the original formulationthe potential, its second derivatives are discontinuous. Thfore, in order to carry out the conjugate gradient~CG! mini-mization, we have slightly modified the potential such thits derivatives are continuous.

Two different pathways were taken to construct a closring carbon nanotube. The first is based on a geometric cstruction of a high-energy nanoring configuration, followby an energy minimization procedure. The other is baseda reversible adiabatic folding of a carbon nanotube unticloses onto itself. The two different pathways of constructlead to different minimum-energy configurations, and threlative structural and kinetic stabilities are discussed innext section.

A. Optimizing a ring structure

In order to construct a closed-ring nanotube-based stture we geometrically fold a nanotube into a nanoring. Tstructure obviously is not a minimum-energy configuratioThere is an excess tension in the inner radius of the ring

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CARBON NANOTUBE CLOSED-RING . . . PHYSICAL REVIEW B67, 195408 ~2003!

to contraction of the carbon-carbon bonds, and there iscompensation from the outer radius where extension ofcarbon-carbon bonds occurs. In order to find a lominimum-energy structure of the ring configuration we ulize the CG method applied to the Brenner potential-enesurface.

The results of the CG minimization for a typical nanorinare shown in Fig. 1. The minimization path initially tendsrelieve the tension by quenching the inner and outer wallthe ring toward each other, resulting in an increase ofangle distortion energy at the sides of the ring. Apparenthis structure is not a local minimum on the potential-enesurface since further application of the CG minimization pcedure breaks the cylindrical symmetry of the ring, and pduces a polygonal shape, with a large number of apexethis structure most of the tension is concentrated inapexes, while the sides of the polygon resemble that onormal carbon nanotube. Further minimization results ireduction of the number of apexes. For most structures sied in this work the CG minimization procedure results inpolygonal structure with a critical angleuc'p/3 for eachbuckle ~see Fig. 2!.

We find that below a certain radius of about 2.8 nm (310370 nanoring! the rings formed using this proceduare unstable. As a consequence of the high initial energthe carbon nanoring, the CG minimization procedure resin ‘‘melting’’ the ring. For higher values of the initial nanor

FIG. 1. Conjugate gradient minimization path for a 103103270 nanoring. The notationn3m3N stands for an3m nanotubewith N unit-cell length.

FIG. 2. Polygon structures formed using the geometric consttion for different 10310 nanotube length. Note that the minimiztion to a polygonal structure is reminiscence of macroscopic behior, similar to what one finds when folding a straw.

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ing radius we find that there is a minimum-energy configration that corresponds to a polygonal structure.

In Fig. 2 we plot several polygonal structures that weformed following the above-outlined minimization procdure. The results obtained for four different tube lengthsshown in the figure: 10310390, 103103180, 103103270, and 103103360. The diameter of the nanotubed513.75 Å, and the length of a unit cell is 2.49 Å. Thlargest closed-ring structure studied is made of a tube tha'0.1-mm long ~this is the largest nanotube which formedpolygonal structure using this procedure!. As can be seen inFig. 2 this procedure produces closed-ring structure wfive–six buckles. The only systematic behavior observedthat the number of buckles, and thus the number of apeare independent of the length of the original nanotube. Silar results were obtained for a series of 535 tubes ~notshown!. However, unlike the case shown in Fig. 2 for a 1310 tube, the smaller diameter 535 tube (d56.875 Å)produced a perfect ring structure for a tube length corsponding to 180 unit cells or larger~see Fig. 5 below!. Ingeneral we find a strong correlation between the critilength required to form a perfect ring structure and theameter of the nanotube: nanotubes with smaller diamewill form perfect ring structure for smaller tube lengths. Wreturn to discuss the case of the 535 tubes below.

The fact that the number of apexes is independent oflength of the original nanotube~for tube lengths that formpolygonal structures! is surprising, since this means that thelastic shell model, which provided quantitative resultsthe critical strain and critical angle for the formation ofsingle buckle, fails when multiple buckles are formed. Telastic shell model predicts that the number of apexes grwith the square root of the length of the original nanotubfor long tubes. Furthermore, this model predicts that the ccal angle for buckling depends on the length of the apexAs can be seen in Fig. 2, this is not the case, and the critangle for buckling isuc'p/3, independent of the tubelength. Only above a certain length do we find that the tuform a perfect ring structure, and at this critical length tbuckling disappears. We note in passing that the smatubes studied (10310390 and 103103180) form six buck-les, where the length of each apex is smaller than the throld required for the continuum theory to hold~10 nm!. Be-low this threshold Yakobsonet al.12 found deviationsbetween the elastic shell model and the simulation reseven when a single buckle is formed. The deviations wattributed mainly to the fact that the average curvature isthan the local curvature.

B. Reversible adiabatic folding

In contrast to the method outlined in the above subsectwhere we started the optimization from a closed ring,now discuss results that are obtained from adiabaticbending an initially straight tube into a closed-ring structuThis procedure, as will be seen, gives still new structures,seen in the previous one.

Buckling formation due to critical strain folding was discussed by Brabec and collaborators11,12 and by Rochefort

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ODED HOD, ERAN RABANI, AND ROI BAER PHYSICAL REVIEW B67, 195408 ~2003!

et al.18 These groups reported a parabolic behavior ofenergy as a function of bending angle before the bucklemation and a linear behavior thereafter. We have performsimilar calculations where the initial carbon nanotube wfolded adiabatically into a closed-ring structure. The tuwas bent by forcing a torque on the edge atoms stepwise,after each angle step a full relaxation of the molecular strture under the constraint of fixed edge atom positions wachieved. The results for the total energy versus the benangle are shown in Fig. 3.

For small angles, the entire nanotube folds without a csiderable change in the circular cross section. This resultagreement with the calculations of Yakobsonet al.12 andRochefortet al.18 For larger bending angles the tube’s crosection reduces, and the symmetrically half-circular shtransforms into an ellipse. When the bending angle reachcritical valuetwo kinks are formed and the nanotube buckas indicated in Figs. 4~a! and 4~b!. The first buckling point@i.e., the formation of Fig. 4~b!# corresponds to mark numbeI on the energy diagram shown in Fig. 3. The reason thatkinks are formed while only one kink was formed in thsimulation results presented by Yakobsonet al. is that thetubes studied here are much longer than those studieYakobson et al. We note in passing that our proceduyielded only one buckle, in agreement with the resultsported by Yakobsonet al.when applied to shorter nanotube

Further bending of the nanotube beyond this critical anresults in a series of distortions in the energy curve as incated by the marksII 2IV in Fig. 3. These distortions do nocorrespond to a formation of a new buckle, but rather tsignificant change in the structure of thetwo existing buck-les, until a double kink is formed for each buckle. Thestructural changes involve atoms near the buckle are shin Figs. 4~c! and 4~d!.

The above-outlined procedure, in which a straight natube is folded into a closed-ring structure following the adbatic pathway, was repeated for a series of nanotube lenand for different nanotube diameters. The adiabatic fold

FIG. 3. Total energy versus bending angle for an adiabatic foing of a 535345 nanotube.

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results always in formation of only two buckles below acritical tube length, independent of the nanotube diameHowever, the critical angle at which the kinks are formdoes depend on the length and diameter of the nanotube.closed-ring structures obtained from the adiabatic foldingthree nanotubes are shown in Fig. 5. Common to all confirations shown which form buckles~and to others not shown!is that large portions of the folded nanotube resembleslightly bent nanotube, and most of the tension contributto higher energies is located in two, well-defined, bucregions. We refer to this structure as a ‘‘lipslike’’ structure fobvious reasons.

The longest nanotube shown in Fig. 5, namely, the 5353180 nanotube, does not form a lipslike structure, but ratthe adiabatic folding results in a perfect ring. This was athe case when a geometric construction followed by theminimization procedure was applied to this nanotube. Tresult is significant for two reasons:~i! The fact that two

-

FIG. 4. Buckling of a 535345 nanotube under adiabatic folding. Folding angles are~a! 191°, ~b! 191.5°, ~c! 348.5°, and~d!349°.

FIG. 5. Results of the total adiabatic folding of 535345, 535390, and 5353180 nanotubes. The 5353180 forms a per-fect ring structure. This result was also obtained using the geomconstruction.

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CARBON NANOTUBE CLOSED-RING . . . PHYSICAL REVIEW B67, 195408 ~2003!

different procedures that yield different configurationsshorter nanotubes result in the same final configurationsdicates that this indeed is the most stable structure for5353180. ~ii ! There is a critical length at which the nantube does not buckle, and the stable configuration cosponds to a perfect ring structure. This critical length wvary depending on the diameter of the nanotube. As theameter of the nanotube increases we find that the critlength for the formation of a perfect ring structure alsocreases.

As will be discussed below, the structure containing otwo buckles is more stable than the polygonal structustudied in the previous subsection. The reason is relatethe difference between the energy required to form an ational buckle compared to the energy required to furtbend an existing buckle, and to form a double-kink buckThe fact that the two-buckle structure is more stable thanone with many buckles indicates that the energy requirefurther bend an existing buckle is smaller than the enerequired to form an additional buckle. This observation isagreement with the results shown in Fig. 3, where the eneto form the first two buckles is slightly larger than that rquired to further bend the buckles to form a double-kistructure.

Despite the fact that these lipslike structures, as willdemonstrated below, are more stable than the polygostructures, we do not expect that the continuum elastic smodel will provide a qualitative description of the propertiof these configurations. The main reason is related to thethat the energy required to form a double-kink structuresignificantly higher than the energy regime at which an etic behavior is expected.

IV. STABILITY

In this section we discuss the kinetic stability of the strutures studied in Sec. III. Several techniques were usedexamine the kinetic stability. First, MC simulations were pformed at low and high temperatures. Since the closed-structures were found to be stable during the entire simtion run, we used the NEB method to calculate the enebarriers for transition between the different structures. Indition, we have also used the NEB method to studybreakage of a closed-ring structure into the correspondnanotube. All simulations were carried out with the modifiTersoff-Brenner potential. We note in passing that a versof this potential has also been applied to study unimolecdissociation of alkane chains, with remarkable successpredicting the dissociation rates, and thus the activabarriers.28

The large energy associated with the buckles formwhen a nanotube is folded into a closed-ring structure mlead one to assume that these structures are unstable kcally, and will break spontaneously even at low temperatuObviously, these structures are less stable thermodyncally compared to the corresponding nanotube, howetheir kinetic stability is an open problem.

Given these facts we have carried out MC simulationsstudy the stability of these structures on microscopic ti

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scales. The simulations were carried out at high temperatin order to accelerate any conformational change, if thoccur. Simulations of these materials at high temperaturepossible since the carbon-carbon bond is extremely stroand dissociation occurs only at very high temperatures.

First, we studied the stability of the polygonal structurewhich were found to be stable during the entire simulattime for temperatures as high as 2000 K. At 4000 K bonbegin to break, and at temperatures higher than 5000 Kclose-ring structure collapses. Similar results were obtaifrom MD runs under similar conditions. The MC runs belo2000 K indicate that the initial polygonal structure is stabduring the entire simulation, and we do not observe conmational changes to other polygonal structures. The simtions do show that local deformations occur at temperatubelow 2000 K. However, these typically relax back into tinitial polygonal structure. These defects were not obserat room temperature~300 K! on the time scale of the simulations. The structural changes observed above 2000 Kbefore the closed-ring collapses are mainly related to trations of the polygon into a ring. Namely, the buckling of thnanoring disappears.

A similar analysis was carried for the lipslike structurgenerated by adiabatically folding the nanotube intoclosed-ring structure, as described in Sec. III. At tempetures below 2000 K the lipslike structure is stable duringentire run, similar to the case of polygonal structures. Diffences were observed at relatively high temperatures, ab2000 K, where heptagon-pentagon defects were formethe buckles. Above 3000 K the lipslike structure is unstaeven on the short time scale of the simulations. This insbility results in bond breakage, and the lipslike structurecomes somewhat disordered.

The MC simulations indicate that the lipslike and polygnal structures are stable even at fairly high temperaturesing the entire simulation runs. Surprisingly, the high enerassociated with the buckling of the nanotube does not rein breakage when the temperature is relatively high~2000K!. Furthermore, on the time scale of the computer simutions no structural transformation between different polygnal structures was observed, indicating that the time scfor such processes are fairly long.

Since the MC and MD simulations are restricted to retively short time scales, we have adopted the NEmethod25–27 to study the minimum-energy path~MEP! be-tween different polygonal structures in order to addressissue of their kinetic stability. Specifically, we have applithe NEB method to study the MEP between the lipslike athe polygonal structures obtained using the two differconstruction paths described in Sec. III. In addition we hastudied the minimum energy required to open these closring structures into the corresponding nanotubes.

In Fig. 6 we show the minimum-energy path obtainusing the NEB method where the initial and final configutions were the lipslike and polygonal structures, respectivEleven images were used to construct the path. As clecan be seen in the figure, the lipslike structure is far mstable than the corresponding polygonal structure. Thisalso observed for other closed-ring tubes with different tudiameters and lengths, however, the difference between

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ODED HOD, ERAN RABANI, AND ROI BAER PHYSICAL REVIEW B67, 195408 ~2003!

energies of the two configurations decreases with increatube length. The minimum-energy path between the tclosed-ring structures indicates that upon each formationan additional buckle, there is an increase in the total eneThe increase in the total energy is somewhat compensatethe release of energy from the double kinks in the lipslstructure. The structure of the closed-ring tube at the tration state is characterized by four buckles. The activatenergy for this conformational change is approximatelyeV. The formation of the fifth buckle which gives the struture of the relaxed polygon results in a reduction of theergy of the closed ring. An additional energy gain is achievwhen the squashed five-apex polygon structure is furtherlaxed to the final polygonal structure. This final relaxationobviously not the bottleneck for this conformational chan

The major conclusion drawn from this simulation is ththe barrier height for the lips-to-polygon conformationtransformation is very high, much higher than the thermenergy at room temperature. Even at very high temperatuthe transformation between the two structures is unlikelyhappen. What will happen is that the more stable lipslstructure will dissociate before a conformational transformtion occurs.

Much higher barriers were observed when the polygostructure undergoes a unimolecular dissociation into theresponding nanotube. The most significant contributionthe barrier height corresponds to the breakage of carb

FIG. 6. Minimum-energy path for the internal conversion btween a polygonal structure obtained from the geometrical cstruction and the lipslike structure obtained from folding a carbnanotube onto itself. The results are for a 535345 structure.

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carbon bonds at the cross section of the nanopolygon. Oafter these bonds break is there a significant energy gwhen the kinks open up, and the polygon relaxes intofinal tube structure. Though energetically unfavorable wrespect to the nanotube structure, once created, the cloring structures are expected to have long-term stability.

V. CONCLUSIONS

In this work we have discussed the structural propertiesa class of closed-ring carbon nanotubes. First, we extenthe elastic shell model of Yakobsonet al.12 to treat the caseof multi-, uncorrelated, buckle formations. The model prvides a relation between the length and diameter of the tto the resulting closed-ring structure. Specifically, it predithat the number of apexes increases with the square roothe length of the tube~L! and decreases with the square roof the diameter of the tube (d).

To assess the accuracy of the elastic shell model we hperformed simulations using the modified Brenner potentTwo different pathways were taken to construct the closring structures: the geometric construction and the adiabfolding. The structures obtained using the adiabatic foldwere found to be more stable than those obtained usinggeometric construction. These lipslike structures have otwo buckles each characterized by a double kink. Given tthe energy of the double kink is much higher than the eneregime at which an elastic behavior is expected, it issurprising that the elastic model fails to predict tminimum-energy structure of the closed-ring configuratio

The large energy associated with the buckles formwhen a nanotube is folded into a lipslike structure may leone to assume that these structures are unstable kineticHowever, MC and MD simulations indicate that on the mcroscopic time scales of the simulations, the closed-rstructures are very stable, and dissociate only at very htemperatures. Nudged elastic band calculations indicatethe barrier for conformational changes is relatively high, athese closed-ring structures are expected to dissociate bthey undergo structural transformations.

ACKNOWLEDGMENTS

This research was supported by Israel Science Foundafounded by the Israel Academy of Sciences and Humani~Grant Nos. 34/00 and 9048/00!. E.R. would like to acknowl-edge the Israeli Council of Higher Education for the Alofellowship.

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