Discussion

In this paper we have attempted to find the global minima for Morse clusters as a function of $\rho_0$ and the number of atoms. The global potential energy minimum represents the equilibrium structure at zero Kelvin, but to predict the structure at non-zero temperatures we must consider the free energy, and the effect of other low energy minima. This can be done by summing the density of states over all the relevant minima.[104] An illustration of this approach for understanding the structure of Morse clusters has been given previously,[38,105] revealing that for $\rm M_{75}$ at $\rho_0$=6, the equilibrium structure changed from decahedral to icosahedral at very low temperature. This transition is simply a consequence of the larger entropy for the icosahedral region of configuration space--there are far more low energy icosahedral minima.

This effect is likely to be general. At the magic numbers for a morphology the entropy will be low because there is a single unique low entropy structure and a large gap to other isomers with the same morphology. Therefore the finite temperature equivalent of the structural phase diagram of Fig. 4 is likely to show weakened magic number effects and so have smoother boundaries between the different morphologies. Furthermore, the energy gap between the lowest energy ordered minimum and the liquid-like band of minima increases as the range becomes shorter, and hence the melting temperature increases with $\rho_0$.[42] Therefore the region of the phase diagram where disordered polytetrahedral structures have the lowest free energy is likely to spread up from the bottom as the temperature increases.

In this paper we have considered only isotropic pairwise additive interactions. As noble gas clusters and clusters of $\rm C_{60}$ molecules can be reasonably modelled by such potentials we would expect the structures we have found at the appropriate values of $\rho_0$ to be similar to the actual structures of these clusters. Our results lead us to predict that neutral clusters of $\rm C_{60}$ molecules exhibit decahedral and fcc structures at small sizes because of the short range of the intermolecular potential. This basic conclusion has been confirmed in studies using more realistic potentials.[72,106]

In contrast, making predictions for metal clusters is problematic because the range of the potential is only one factor determining the structure and many-body terms, in particular, may also be important.[107,108] These terms may affect the relative surface energies of $\{111\}$ and $\{100\}$ faces, and so alter the energetic competition between icosahedral, decahedral and fcc structures.[81] For example, in a study of lead clusters cuboctahedra are always found to be lower in energy than icosahedra because the surface energies of $\{111\}$ and $\{100\}$ faces are nearly equal.[83]

Nevertheless our results are of value to the field of metal cluster structure. Firstly, they enable particularly stable structural forms to be identified. For example, in our previous paper on Morse clusters we identified the 38-atom octahedron and the 75-atom Marks decahedron as particularly stable. Subsequently, they have both been observed experimentally;[11,33] it even being possible to isolate fractions of the latter when passivated by surfactants. This correspondence between the Morse structures and those of real systems encourages us to believe that some of the general principles that determine stability in our simple model system do carry over to real clusters.

Secondly, the Morse structural database should be useful in providing candidate structures for comparison with the indirect structural information yielded by experiments on size-selected clusters. Finally, the database can also provide plausible starting structures for theoretical studies with more realistic, but computationally expensive, descriptions of the interactions; this expense would prevent the type of extensive searches that have been performed in this paper. Indeed, we have used the database in this way in studies of metal clusters modelled by the Sutton-Chen family of potentials[20] and clusters of $\rm C_{60}$ molecules.[72,106] For these reasons, the coordinates for all the global minima given in this and previous papers[38,43] will be made available on the world-wide-web.[109]