اعداد جادویی هندسی خوشه های سدیم: تفسیر رفتار ذوب Geometric magic numbers of sodium clusters: Interpretation of the melting behaviour

Sodium clusters have become an extremely important model system for understanding the thermodynamic properties of clusters, because of the availability of high-quality experimental data produced by the Haberland group for sizes up to 360 atoms [1–۵]. They have been the first cluster system for which the rounding of phase transitions due to a cluster’s finite size [1], negative heat capacities in the microcanonical ensemble [3] and the liquid-gas transition [4] have been revealed experimentally. However, one of the biggest remaining puzzles arising from this data is the origin of the non-monotonic variation of the melting temperature with cluster size. Despite intensive theoretical effort [6–۱۲], the geometric or electronic effects underlying this variation have not been fully identified. Significant progress was made in Haberland et al.’s most recent paper, in which they observed that the energy and entropy changes on melting, rather than the melting temperature itself, provide more structural insight [13]. In particular, these two quantities exhibit pronounced maxima at certain ‘magic numbers’, some of which have a clear interpretation in terms of geometric structures, such as the complete Mackay icosahedra, but most remain unassigned. Therefore, a systematic theoretical investigation of the geometric structures of sodium clusters in this size range could be of particular importance in identifying the structures underlying these magic numbers. Previous work on the structure of sodium clusters has for the most part concentrated on clusters with less than 60 atoms [14–۱۹]. By contrast, in this Letter we have attempted to locate the lowest-energy structures of sodium clusters for all sizes up to N = 380 using the basinhopping global optimization method [20]. Such large sizes necessitate the use of a model potential, and we have considered two different forms for the interatomic interactions, namely the Gupta [21,22] and Murrell-Mottram (MM) [23–۲۵] potentials. The MM potential has more parameters, has been fitted to a wider range of properties, and exhibits good transferability [25]. Therefore, it is expected to be the more reliable of the two potentials, but it is also significantly more expensive to compute. The advantage of considering two potentials is that we can have greater confidence in those structural features that are common to both potentials. In Figure 1, we have plotted the energies of the putative global minima for the two potentials, and Figure 2 shows the structures of some of the magic number clusters. The energies and coordinates for all the structures are available at the Cambridge Cluster Database [26]. For N ≤ ۵۷ the Gupta global minima have been previously reported by Lai et al. [17]. The Haberland group found that for N < 100 many sodium clusters do not show a clear melting transition, but pass from solid to liquid without a pronounced latent heat [5]. Na55 stands in contrast to this trend having a particularly high melting temperature, but Na70 and Na92 also represent exceptions [13]. Both potentials exhibit a pronounced magic number at N = 55, which, as expected, corresponds to a complete Mackay icosahedron. Typically, there are two types of overlayer for growth 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 the Mackay icosahedron, and leads to the next Mackay icosahedron. By contrast, the second, the anti-Mackay overlayer, adds atoms in sites that are hexagonal close-packed with respect to the underlying fcc tetrahedra. Typically, growth starts off in the anti-Mackay overlayer because of a greater number of nearest-neighbour interactions, but then switches to the Mackay overlayer because it involves less strain [27,28]. Interestingly, structures that do not adopt either of these overlayers are prevalent for both potentials. The magic number at Na71, a possible explanation for the experimental feature at N = 70, provides a good example. Both potentials have the same C5 global minimum, where the five faces around the vertex of the 55-atom Mackay icosahedron are covered by a Mackay-like cap, but where both the overlayer and core have been twisted with respect to the ideal Mackay sites. This twist increases the coordination number of some of the surface atoms at the expense of increased strain and creates a structure where, unlike both the anti-Mackay and Mackay overlayers, the surface consists entirely of {111}-like faces. A similar structure is a magic number at N = 92 and involves the covering of ten faces with a Mackay overlayer, which then undergoes a twist distortion, giving rise to a structure with T point group symmetry, instead of C3v for the ideal Mackay geometry. These structures look like a hybrid of the 55- atom and 147-atom Mackay icosahedra, because they have triangular {111} faces of sizes corresponding to both the smaller and larger Mackay icosahedra.