Results

All the global minima that we have found are catalogued in Table I along with their energies, point groups, number of nearest neighbours, strain energies and the values of $\rho_0$ for which they are probably the lowest energy minimum.

  

Figure 4: Zero temperature `phase diagram' showing the variation of the lowest energy structure with N and $\rho_0$.The data points are the values of $\rho_0$ at which the global minimum changes. The lines joining the data points divide the phase diagram into regions where the global minima have similar structures. The solid lines denote the boundaries between the four main structural types--icosahedral, decahedral, close-packed and those associated with low $\rho_0$ (L)--and the dashed lines are internal boundaries within a structural type, e.g. between icosahedra with Mackay and anti-Mackay overlayers, or between decahedra with different length decahedral axes.

The results are summarized in Fig. 4 which provides a zero temperature `phase diagram', showing how the global minimum depends upon N and $\rho_0$. The structural behaviour of Morse clusters with fewer than eight atoms is rather uninteresting because the global minimum is independent of $\rho_0$.For all $N\ge 8$, however, the global minimum changes at least once as a function of $\rho_0$.For $N\ge 13$, icosahedral, decahedral and fcc structures all exist, and the form of the phase diagram is in good agreement with the predictions we made earlier based upon the decomposition of the potential energy. For most sizes the structure changes from icosahedral to decahedral to close-packed as the range of the potential is decreased. For N<23, however, the transition from decahedral to close-packed occurs at larger values of $\rho_0$ than we consider in this study. There are also a number of sizes (N=38-40, 52, 53, 59 and 61) for which there is a transition directly from an icosahedral to a close-packed structure; this occurs when n_nn for the lowest energy close-packed structure is greater than or equal to that for the lowest energy decahedron.

The boundaries between the different morphologies are sensitive functions of N. Such size dependence is observed for many properties of clusters, and gradually lessens as N increases (because the addition of a single atom becomes a smaller perturbation) until the bulk limit is reached. The decahedral to close-packed boundary is particularly sensitive, because the range of $\rho_0$ for which the decahedron is most stable changes dramatically even when the difference in n_nn between the decahedral and close-packed structures changes by one.

  

Figure 5: Plots of $\Delta_2 E$ as a function of N for (a) $\rho_0$=3, (b) $\rho_0$=6, (c) $\rho_0$=10 and (d) $\rho_0$=14.

Sizes for which a morphology is the lowest in energy for a particularly large range of $\rho_0$indicate that the structure is especially stable. The optimal geometries shown in Fig. 1 are good examples. Another indicator of special stability is provided by $\Delta_2 E(N)=E(N+1)+E(N-1)-2 E(N)$.Peaks in $\Delta_2 E$ have been found to correlate well with the magic numbers (sizes at which clusters are particularly abundant) observed in mass spectra.[71] Plots of $\Delta_2 E$ are shown in Fig. 5 for a number of values of $\rho_0$. Unsurprisingly, the plot for $\rho_0$=6 is very similar to that for LJ clusters with peaks due to especially stable icosahedral clusters. At higher values of $\rho_0$ peaks corresponding to close-packed and decahedral clusters begin to occur. The plot at $\rho_0$=14 is very similar to that recently obtained for $\rm C_{60}$ clusters using the Girifalco intermolecular potential.[72] If the energy is `normalized' by subtracting a suitable function of N, particularly stable sizes again stand out (Fig. 6).

  

Figure 6: Comparison of the energies of icosahedral (solid line with diamonds), decahedral (dashed line with crosses) and close-packed (dotted line with squares) $\rm M_{N}$ clusters at $\rho_0$=6. The energy zero is E_MI, the interpolated energy of Mackay icosahedra. E_MI=-3.0354+0.2624 N^1/3+8.8400 N^2/3-6.8381N and was obtained by fitting to the first four Mackay icosahedra (N=13, 55, 147 and 309).

In the following subsections we will look at the growth sequences for each morphology in more detail. We also examine the unusual structures that occur for the larger clusters at low $\rho_0$, which, as we will see, involve a mixture of order and disorder.