Putative global minima of sodium clusters with up to 380 atoms have been
located for two model interatomic potentials. Structures based upon the Mackay
icosahedra predominate for both potentials, and the magic numbers for the
Murrell-Mottram model show excellent agreement with the sizes at which maxima
in the latent heat and entropy change at melting have been found in experiment.Comment: 4 pages, 2 figure

A. Aguado

Aguado

Calvo

Chacko

D. J. Wales

Doye

E. G. Noya

Fang

Gupta

Haberland

Heer

J. P.K. Doye

Kümmel

Manninen

Murrell

Northby

Pavloff

Reyes-Nava

Romero

Röthlisberger

Schmidt

Solov'yov

Wales

English

arXiv

Crossref

Geometric magic numbers of sodium clusters: Interpretation of the melting behaviour

Putative global minima 
of sodium clusters with up to 380 atoms
have been located for two model interatomic potentials in
order to identify the structures responsible for the size-dependence
of the thermodynamic properties in experiments. 
Structures based upon the Mackay icosahedra predominate for both potentials,
and the magic numbers for the Murrell-Mottram model show excellent
agreement with the sizes at which maxima in the latent heat and entropy 
change at melting have been found in experiment. In particular, 
the magic numbers at sizes intermediate between the complete Mackay icosahedra
are due to unusual twisted icosahedral structures

EDP Sciences OAI-PMH repository (1.2.0)

Geometric magic numbers of sodium clusters:  
Interpretation of the melting behaviour

Putative global minima of sodium clusters with up to 380 atoms have been located for two model interatomic potentials. Structures based upon the Mackay icosahedra predominate for both potentials, and the magic numbers for the Murrell-Mottram model show excellent agreement with the sizes at which maxima in the latent heat and entropy change at melting have been found in experiment

Noya, EG

Doye, Jonathan

Wales, DJ

ORA - Oxford University Research Archive

Geometric magic numbers of sodium clusters: Interpretation of the
  melting behaviour

Oxford University Research Archive

arXiv:cond-mat/0506329v1  [cond-mat.other]  14 Jun 2005Geometric magic numbers of sodium clusters: Interpretation of the melting behaviourEva G. Noya, Jonathan P. K. Doye, and David J. WalesUniversity Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom(Dated: February 2, 2008)Putative global minima of sodium clusters with up to 380 atoms have been located for twomodel interatomic potentials. Structures based upon the Mackay icosahedra predominate for bothpotentials, and the magic numbers for the Murrell-Mottram model show excellent agreement withthe sizes at which maxima in the latent heat and entropy change at melting have been found inexperiment.PACS numbers: 61.46.+w,36.40.MrThe melting of sodium clusters has been the subjectof numerous recent studies. Much of this interest hasarisen due to the availability of high-quality experimen-tal data, which has allowed detailed comparisons betweentheory and experiment. In particular, the Haberlandgroup [1, 2, 3] have measured the caloric curves of mass-selected positively-charged sodium clusters with up to360 atoms, from which the melting temperature (Tmelt)and latent heat can be extracted. The values of Tmelt forthese clusters are on average one third lower than thatfor bulk, and show variations of up to ± 50 K dependingon the cluster size. There have been a number of theo-retical studies that, using different levels of theory, haveinvestigated the origins of the size-dependence of Tmelt[4, 5, 6, 7, 8]. However, the peaks in the melting temper-ature do not seem to correlate either with the electronicor geometric shell closings of sodium clusters, and noneof those theoretical studies have been able to provide asatisfactory explanation for the non-monotonic variationof Tmelt. Significant progress was made in Haberlandet al.’s most recent paper, in which they observed thatthe energy and entropy changes on melting provide morestructural insight into the system than Tmelt itself [9]. Inparticular, these two quantities exhibit pronounced max-ima at certain ‘magic numbers’, some of which have aclear interpretation in terms of geometric structures, suchas the Mackay icosahedra, whilst others remain unas-signed. Therefore, a systematic investigation of the ge-ometric structure of sodium clusters in this size rangewould be of great help in the identification of the struc-tures underlying these magic numbers.Previous work on the structure of sodium clusters hasfor the most part concentrated on clusters with less than60 atoms [10, 11, 12, 13, 14, 15, 16]. By contrast, inthis Letter we have attempted to locate the lowest-energystructures of sodium clusters for all sizes up to N=380using the basin-hopping global optimization method [17].Such large sizes necessitate the use of a model potential,and we have considered two different forms for the in-teratomic interactions, namely the Gupta [18, 19] andMurrell-Mottram (MM) [20, 21, 22] potentials. The MMpotential has more parameters, has been fitted to a widerrange of properties, and exhibits good transferability [22].Therefore, it is expected to be the more reliable of thetwo potentials, but it is also significantly more expensiveto compute. The advantage of considering two potentialsis that we can have greater confidence in those structuralfeatures that are common to both potentials.In Figure 1, we have plotted the energies of the puta-tive global minima for the two potentials, and Figure 2shows the structures of some of the magic number clus-ters. The energies and coordinates for all the structuresare available at the Cambridge Cluster Database [23].For N ≤ 57 the Gupta global minima have been previ-ously reported by Lai et al. [14].The Haberland group found that for N < 100 manysodium clusters do not show a clear melting transition,but pass from solid to liquid without a pronounced la-tent heat [3]. Na55 stands in contrast to this trend hav-ing a particularly high melting temperature, but Na70and Na92 also represent exceptions [9]. Both potentialsexhibit a pronounced magic number at N=55, which,as expected, corresponds to a complete Mackay icosa-hedron. Typically, there are two types of overlayer forgrowth on the surface of a Mackay icosahedron. The first,the Mackay overlayer, continues the face-centred-cubic(fcc) packing of the twenty fcc tetrahedra making up theMackay icosahedron, and leads to the next Mackay icosa-hedron. By contrast, the second, the anti-Mackay over-layer, adds atoms in sites that are hexagonal close-packedwith respect to the underlying fcc tetrahedra. Typically,growth starts off in the anti-Mackay overlayer because ofa greater number of nearest-neighbour interactions, butthen switches to the Mackay overlayer because it involvesless strain [24, 25].Interestingly, structures that do not adopt either ofthese overlayers are prevalent for both potentials. Themagic number at Na71, a possible explanation for theexperimental feature at N=70, provides a good example.Both potentials have the same C5 global minimum, wherethe five faces around the vertex of the 55-atom Mackayicosahedron are covered by a Mackay-like cap, but whereboth the overlayer and core have been twisted with re-spect to the ideal Mackay sites. This twist increases thecoordination number of some of the surface atoms at theexpense of increased strain and creates a structure where,20.000.050.100.15(E−Emagic) / N1/3 (eV)0 100 200 300N0.000.050.100.15(E−Emagic) / N1/3 (eV)557192115135166186 2015571216 24126729792348116147178 216 232 2582003093571011313FIG. 1: Energy of the global minima found for the Gupta (upper panel) and MM (lower panel) potentials as a function ofsize. Energies are given relative to Emagic, which is a function fitted to the energies of the first four stronger magic numbers.EGuptamagic = 0.0403 − 0.2546N1/3 + 1.2134N2/3 − 1.1568N ; EMMmagic = −0.4788 + 0.5261N1/3 + 0.9852N2/3 − 1.1110N .unlike both the anti-Mackay and Mackay overlayers, thesurface consists entirely of {111}-like faces. A similarstructure is a magic number at N=92 and involves thecovering of ten faces with a Mackay overlayer, which thenundergoes a twist distortion, giving rise to a structurewith T point group symmetry, instead of C3v for the idealMackay geometry. These structures look like a hybrid ofthe 55-atom and 147-atom Mackay icosahedra, becausethey have triangular {111} faces of sizes correspondingto both the smaller and larger Mackay icosahedra.The Gupta potential does not exclusively exhibit struc-tures based on these twisted icosahedra in this size range.For example, at N=81 a structure where eight faces arecovered by an anti-Mackay overlayer is the global min-imum. This difference between the two potentials be-comes more prominent at larger sizes. For example, thereis a feature at N ≈ 116 in the experimental results thathas been interpreted in terms of a Mackay structure with15 of the faces of the underlying icosahedron covered [9].Both potentials have clear features near to this size. Forthe MM potential, there is a magic number at N=116 andat this size there are two minima with almost the sameenergy. The second-lowest minimum corresponds to atwisted form of the structure suggested by Haberland etal., and the lowest-energy isomer is based on a 116-atomIno decahedron but with the central ring of atoms twistedto remove any {100} faces. By contrast, the Gupta po-tential has a magic number at N=115 that correspondsto an Ih structure with a complete anti-Mackay overlayer.This is an unusual feature, since the anti-Mackay over-layer is usually observed during the initial growth on anicosahedron [24], but not when that overlayer is nearlycomplete. Moreover, this structure is very high in energyfor the MM potential.Experimentally, Na147 is a prominent magic number,and, again as expected, the Mackay icosahedron is theglobal minimum at this size for both potentials. However,for the Gupta potential a more stable structure can beobtained by removing the twelve vertex atoms, giving riseto a magic number at N=135 (Fig. 1). This feature is inclear contradiction with experiment.For growth on the 147-atom Mackay icosahedron, thedifferences between the results for the two potentials be-come even greater. For the MM potential structuresbased upon the twisted icosahedra continue to predom-inate. However, the Gupta potential initially exhibitsstructures with a Mackay overlayer, and then switchesover to an anti-Mackay overlayer near to the completionof this overlayer at N=267.The MM potential exhibits prominent magic numbersat N=178, 216, 232 and 258, with weaker features at184, 190, 200, 206, 222 and 238. These structures cor-3FIG. 2: A selection of sodium clusters that show enhanced stabilities for the (a) Gupta and (b) Murrell-Mottram potentials.Each structure is labelled by the number of atoms and its point group symmetry.respond to covering successive faces of the 147-Mackayicosahedron with Mackay-like overlayers, but where thecore and surface again undergo a twist distortion. The178-, 216-, 232- and 258-atom structures are equivalentto the 71-, 92-, 101- and 116-atom structures describedabove and correspond to covering all the faces surround-ing 1, 3, 4 and 6 vertices of the underlying icosahedron.These features are in good agreement with the experi-mental results, which have clear features at N=178 and216, and a smaller sub-peak at N=184. No experimen-tal features have yet been identified at N=232 and 258.However, in this size range the data is sparse, and theerror bars are of similar magnitude to the size variationof the properties. Therefore, it would be very interestingif further experiments were conducted at these sizes toexamine the predictions of the MM model.Interestingly, Haberland et al. suggested undistortedMackay structures to explain the magic numbers atN=178 and 216 [9]. However, it is more usually foundthat more stable structures are possible, when the five-coordinate atoms at the corners of the added triangularfaces are not occupied. For example, this leads to magicnumbers at N=173 and 213 for Lennard-Jones clusters[26]. The twist distortion of the icosahedra provides apossible explanation for this difference in magic numbers.As a consequence of the distortion, the coordination num-ber for the corner atoms increases from five to six, makingit more favourable for these sites to be occupied.The magic numbers for the Gupta potential are com-pletely different in this size range, because of the pref-erence for both undistorted icosahedral structures andempty vertex sites. The magic numbers at N=166, 186,201, 216 and 241 are all due to structures with a Mackayoverlayer. If it were favourable for the six-coordinate ver-tices to be occupied, these magic sizes would instead beat N=173, 196, 213, 230 and 258. Only if five-coordinatesites were also occupied would the magic numbers be 178,200, 216, 232 and 258. Analogous to the particular sta-bility of Na115 in the growth of the third shell, there is an-other magic number at N=267 whose structure involvesa complete anti-Mackay layer without vertices. Closeby(N = 268), Haberland et al. found a well-structured pho-toelectron spectrum, but they attributed this feature tothe existence of an electronic shell closing rather than to4high point group symmetry [9]. Furthermore, this com-plete anti-Mackay icosahedra again lies very high in en-ergy for the MM potential.As for the third shell, the complete Mackay icosahe-dron is not a magic number for the Gupta potential,but instead an icosahedron with twelve missing verticesis more stable, displacing the magic number to N=297.The MM potential still predicts the magic number to beat N=309, but the difference in stability between the297- and 309-atom structures is much smaller. Indeed,at N ≈ 360 structures with missing vertices actually be-come more stable. The similar behaviour of the two po-tentials suggests that the loss of vertex atoms is a robuststructural feature for sodium; the two potentials only dif-fer in the size at which this effect first appears. Theseresults suggest that a plausible explanation of the absenceof an experimental magic number at N=309, but the ap-pearance of a feature at N=298, is the greater stability ofa Mackay icosahedron that has lost its vertices. However,Haberland et al. found that the measured photoelectronspectrum for Na298 is not compatible with such a struc-ture [9]. Furthermore, on measuring the photoelectronspectrum of Na309 as a function of temperature, Haber-land et al. found that a structural transition occurred atabout 40 K below melting [9]. Parallel tempering simu-lations using the Gupta potential, however, did not showevidence of any transitions prior to melting for Na309.Therefore, our results are unable to offer a structural ex-planation compatible with all the experimental findingsassociated with the completion of the fourth icosahedralshell, and their origin remains somewhat mysterious.Finally, for growth of the fifth icosahedral shell, thesame patterns continue, i.e. Mackay overlayers for theGupta potential and twisted icosahedra for the MM po-tential. In this size range, experiments predict a peakat N=360, for which the MM magic number at N=357offers a possible explanation.In summary, our results support the conclusions de-rived from Haberland et al.’s recent analysis of the melt-ing behaviour of sodium clusters—namely that the clus-ters in this size range are predominantly icosahedral. Ofthe two potentials we have considered, the structures ob-tained for the MM potential appear more reliable andthe magic numbers are in excellent agreement with ex-periment. In particular, we suggest that the experimen-tal features at sizes intermediate between the completeMackay icosahedra are due to icosahedral structures witha Mackay overlayer, but where both the core and over-layer undergo a twist distortion to give structures thathave only {111}-like faces. It is noteworthy that suchfeatures cannot be captured by a pairwise-additive inter-atomic potential.The authors are grateful to the Ramón Areces Foun-dation (E.G.N.) and the Royal Society (J.P.K.D) for fi-nancial support. Fruitful discussions with H. Haberlandare acknowledged.[1] M. Schmidt, R. Kusche, B. v. Issendorff, and H. Haber-land, Nature 393, 238 (1998).[2] M. Schmidt, R. Kusche, T. Hippler, J. Donges, W. Kro-nmuler, B. v. Issendorff, and H. Haberland, Phys. Rev.Lett. 86, 1191 (2001).[3] M. Schmidt, J. Donges, T. Hippler, and H. Haberland,Phys. Rev. Lett. 90, 103401 (2003).[4] A. Aguado, L. M. Molina, J. M. Lopez, J. A. Alonso, andM. J. Stott, Eur. Phys. J. D 15, 221 (2001).[5] F. Calvo and F. Spiegelmann, J. Chem. Phys. 112, 2888(2000); 120, 9684 (2004).[6] K. Manninen, A. Rytkonen, and M. Manninen, Eur.Phys. J. D 29, 39 (2004).[7] J. A. Reyes-Nava, I. L. Garzon, and K. Michaelian, Phys.Rev. B 67, 165401 (2003).[8] S. Chacko, D. G. Kanhere, and S. A. Blundell, Phys. Rev.B 71, 155407 (2005).[9] H. Haberland, T. Hippler, J. Donges, O. Kotsko,M. Schmidt, and B. von Issendorff, Phys. Rev. Lett. 94,035701 (2005).[10] V. Bonacic-Koutecky, P. Fantucii, and J. Koutecky,Chem. Rev. 91, 1035 (1991).[11] U. Röthlisberger and W. Andreoni, J. Chem. Phys. 94,8129 (1991).[12] F. Calvo, S. Tran, S. A. Blundell, C. Guet, andF. Spiegelmann, Phys. Rev. B 62, 10394 (2000).[13] S. Kümmel, M. Brack, and P.-G. Reinhard, Phys. Rev.B 62, 7602 (2000).[14] S. K. Lai, P. J. Hsu, K. L. Wu, W. K. Liu, and M. Iwa-matsu, J. Chem. Phys. 117, 10715 (2002).[15] I. A. Solov’yov, A. V. Solov’yov, and W. Greiner, Phys.Rev. A 65, 053203 (2002).[16] K. Manninen, H. Häkkinen, and M. Manninen, Phys.Rev. A 70, 023203 (2004).[17] D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101,5111 (1997).[18] R. P. Gupta, Phys. Rev. B 23, 6265 (1981).[19] Y. Li, E. Blaisten-Barojas, and D. A. Papaconstantopou-los, Phys. Rev. B 57, 15519 (1998).[20] J. N. Murrell and R. E. Mottram, Mol. Phys. 69, 571(1990).[21] J.-Y. Fang, R. L. Johnston, and J. N. Murrell, Mol. Phys.78, 1405 (1993).[22] H. Cox, R. L. Johnston, and J. M. Murrell, J. Solid Stat.Chem. 145, 517 (1999).[23] The Cambridge Cluster Database,http://www-wales.ch.cam.ac.uk/CCD.html.[24] J. A. Northby, J. Chem. Phys. 87, 6166 (1987).[25] J. P. K. Doye and D. J. Wales, J. Chem. Soc., FaradayTrans. 93, 4233 (1997).[26] D. Romero, C. Barrón, and D. Gómez, Comp. Phys.Comm. 123, 87 (1999).

Geometric magic numbers of sodium clusters: Interpretation of the melting behaviour

Abstract

Similar works

Full text

Available Versions

Crossref

EDP Sciences OAI-PMH repository (1.2.0)

ORA - Oxford University Research Archive

Oxford University Research Archive