Global Optimization
Polymer Crystallization




Cluster Structure

One of the unique properties of clusters is the possibility of structures containing five-fold axes - unlike crystalline structures there is no requirement for translational periodicity. Two examples are shown below. The Mackay icosahedron has six five-fold axes through the twelve vertices, and the Marks decahedron has a single five-fold axis. Both structures are common in a wide variety of atomic clusters.

Fig. 1. Three examples of the structures clusters can adopt: (a) a 38-atom truncated octahedron, (b) a 55-atom Mackay icosahedron, and (c) a 75-atom Marks' decahedron. These clusters have the optimal shape for the three main types of regular packing seen in clusters: face-centred cubic, icosahedral and decahedral, respectively.

When a cluster's interactions can be described by a pair potential the main contribution to the energy comes from nearest neighbours. Therefore, one might expect those structures with the greatest number of nearest-neighbour contacts to be lowest in energy. This would favour structures, such as the Mackay icosahedron, which are both spherical and have {111} surface facets. However, there is another factor. In icosahedra and decahedra the structures must be strained in order to eliminate the gaps that result if the structures are composed of regular tetrahedra (see below). There is an energetic penalty associated with this strain.

Fig. 2 Examples of the strain involved in packing tetrahedra. (a) Five regular tetrahedra sharing a common edge leave a gap of 7.36 degrees. (b) Twenty regular tetrahedra sharing a common vertex leave gaps amounting to a solid angle of 1.54 steradians.

As the strain energy increases with size, it is common to see a cluster's structure change from icosahedral to decahedral to face-centred-cubic as the size increases. A similar series of structural changes can be induced by narrowing the width of the potential well, because this also disfavours strained structures.

As I have worked on the structure of a variety of clusters, I have obtained a large database of structures. These are accessible from the Cambridge Cluster Database.

Associated Publications