Electronic vs Geometric Magic Numbers

 Magic numbers based on electronic shells were first observed in mass spectra of alkali metal clusters[214]. These features are now well understood in the framework of the self-consistent spherical jellium model, in which the nearly-free valence electrons are assumed to move in a homogeneous spherical ionic background. Further refinements, such as allowing the cluster to deform into an ellipsoidal shape for incomplete electronic shells, improve the agreement with experiment[94].

In experiments on large sodium clusters by Martin et al. electronic shell structure was found to persist up to about 1000 atoms and above this size geometric magic numbers were observed[53]. These magic numbers are associated with the completion of shells of the Mackay icosahedron. Further temperature-dependent experiments have shown that for N>1000 the geometric magic numbers disappear as the temperature is increased[215]. This has been attributed to the loss of icosahedral structure on melting (or surface melting) of the cluster and so has been used to examine the size dependence of the melting temperature. At sufficiently high temperatures, electronic magic numbers have been observed up to at least 3000 sodium atoms[216]. Similarly, experiments on large aluminium clusters have shown that as the temperature is increased the observed magic numbers change from geometric (due to octahedra[217]) to electronic[218].

The natural interpretation of these experiments is that geometric magic numbers are associated with solid-like clusters that are based on a regular packing and that electronic magic numbers are associated with liquid-like clusters. However, before this conclusion can be confirmed, another question must be addressed: could a regularly-packed structure also have the observed electronic shell structure? This has been examined in calculations by Mansikka-aho which have shown that the electronic shell structure differs from that observed experimentally for all the common cluster shapes except the icosahedron[219,220]. The icosahedron is sufficiently spherical that the electronic shell structure is similar to the experimental up to 1000 atoms. However, Pavloff and Creagh[221] have shown that the electronic supershell structure observed for sodium clusters[216] cannot be explained if the clusters have icosahedral structure, thus confirming that the sodium clusters which exhibit electronic magic numbers are liquid-like.

Our results allow us to explain the transition. For small Morse clusters with a long-ranged potential the lowest energy minimum has an amorphous structure typical of the liquid-like state. At these sizes, the cluster has a very low melting point, as shown in §4.5, because the lowest energy minimum lies at the bottom of a band of minima which are almost continuous in energy. Hence, the cluster can adopt structures that give the most favourable electronic energy without incurring excessive strain energy and the cluster would be expected to exhibit electronic magic numbers at all temperatures. The effect of size on the correlation diagram is to displace the `liquid-like' band of minima upwards with respect to the line of the Mackay icosahedron until in the bulk limit regular structures are the lowest in energy for all $\rho_0$. Increasing the size has an effect similar to decreasing the range of the potential--they both destabilize more strained structures. A corollary of this is the well-known increase in cluster melting temperature with size[222] that was first predicted by Pawlow[223]. As a consequence of this effect, for clusters with long-ranged potentials, there must be a critical size at which the Mackay icosahedron becomes lower in energy than the `liquid-like' minima. Above this size the cluster would be expected to exhibit geometric shell structure at temperatures below the melting point.

One alternative explanation for this transition that has been proposed by Stampfli and Bennemann simply considers the size- and temperature-dependence of the variation in the electronic and geometric effects[224,225]; they show that the latter is likely to dominate at large sizes. However, in their model they do not include the possibility that small clusters may never adopt regular structures.

The jellium-type models can provide a good description of the alkali metals because these elements most closely approximate free electron systems, and understanding the electronic effects becomes more difficult as one goes further from this limit. Our results lead us to expect that for metals with shorter-ranged potentials geometric magic numbers could be seen at much smaller sizes than for sodium. This may provide an explanation for the behaviour of group II metals: barium clusters of less than 50 atoms show magic numbers consistent with an icosahedral growth sequence[99] and magnesium[54] and calcium[55] have magic numbers due to Mackay icosahedra from 147 atoms upwards. However, it is hard to judge the role many-body forces may play in the energetic competition between regular and disordered structures in these systems.