The potential

The Morse potential[49] may be written as

$$V_M = \epsilon\sum_{i<j} e^{\rho_0(1-r_{ij}/r_0)}(e^{\rho_0(1-r_{ij}/r_0)}-2),$$ (1)

where $\epsilon$ is the pair well depth and r₀ the equilibrium pair separation. We denote an N-atom cluster bound by the Morse potential as M_N. In reduced units ($\epsilon=1$ and r₀=1) there is a single adjustable parameter, $\rho_0$, which determines the range of the interparticle forces. Fig. 2 shows that decreasing $\rho_0$ increases the range of the attractive part of the potential and softens the repulsive wall, thus widening the potential well. Values of $\rho_0$ appropriate to a range of materials have been catalogued elsewhere.[50] The LJ potential has the same curvature at the bottom of the well as the Morse potential when $\rho_0=6$.Girifalco has obtained an intermolecular potential for $\rm C_{60}$ molecules[51] which is isotropic and short-ranged relative to the equilibrium pair separation, with an effective value of $\rho_0=13.62$.[52] The alkali metals have longer-ranged interactions, for example $\rho_0$=3.15 has been suggested for sodium.[53] Fitting to bulk data gives a value of $\rho_0$=3.96 for nickel.[14]

  

Figure 2: The Morse potential for different values of the range parameter $\rho_0$ as indicated.

At absolute zero the structure with the lowest free energy is simply the global potential energy minimum of the PES. At higher temperatures, entropic factors must also be considered. Although we only perform a comprehensive survey of the zero Kelvin geometries, the structural effects of temperature are considered in the subsequent discussion.

To understand the structural effects of the range parameter, $\rho_0$, it is instructive to look more closely at the form of the potential. The energy can be partitioned into three contributions:

$$V_M=-n_{nn}\epsilon+E_{\rm strain}+E_{nnn}.$$ (2)

The number of nearest-neighbour contacts, n_nn, the strain energy, $E_{\rm strain}$, and the contribution to the energy from non-nearest neighbours, E_nnn, are given by

where x_ij=r_ij/r₀-1, and x₀ is a nearest-neighbour criterion. x_ij is the strain in the contact between atoms i and j.

The dominant term in the energy comes from n_nn. E_nnn is a smaller term and its value varies in a similar manner to n_nn. It is only likely to be important in determining the lowest energy structures when other factors are equal. For example, bulk fcc and hexagonal close-packed (hcp) lattices both have twelve nearest neighbours per atom. Next-nearest neighbour interactions are the cause of the lower energy of the hcp crystal when a pair potential such as the LJ form is used.[54,55]

$E_{\rm strain}$, which measures the energetic penalty for the deviation of a nearest-neighbour distance from the equilibrium pair distance, is a key quantity in our analysis. It must not be confused with strain due to an applied external force. For a given geometry, $E_{\rm strain}$ grows rapidly with increasing $\rho_0$because the potential well narrows. To a first approximation the strain energy grows quadratically with $\rho_0$.[38] Hence, decreasing the range destabilizes strained structures.

From the above analysis we can see that minimization of the potential energy involves a balance between maximizing n_nn and minimizing $E_{\rm strain}$. The interior atoms of the three morphologies (Fig. 1) all have twelve nearest neighbours, and so differences in n_nn arise from surface effects. The optimal shape for each morphology results from the balance between maximizing the proportion of $\{111\}$ faces (an atom in a $\{111\}$ face is 9-coordinate, but in a $\{100\}$ face only 8-coordinate) and minimizing the fraction of atoms in the surface layer. As Mackay icosahedra (Fig. 1(b)) have only $\{111\}$ faces and are approximately spherical, the icosahedra have the largest n_nn. Complete Mackay icosahedra occur at $N=13, 55, 147, \ldots$A pentagonal bipyramid has only $\{111\}$ faces, but because it is not very spherical more stable decahedral forms are obtained by truncating the structure parallel to the five-fold axis to reveal five $\{100\}$ faces and then introducing re-entrant $\{111\}$ faces between adjacent $\{100\}$ faces. The resulting structure is called a Marks decahedron (Fig. 1(c)) and was first predicted by the use of a modified Wulff construction.[56] Decahedra generally have lower values of n_nn than icosahedra because of the $\{100\}$ faces. The tetrahedron and octahedron are fcc structures that have only $\{111\}$ faces, but they are not very spherical. The optimal fcc structure is the truncated octahedron with regular hexagonal faces (Fig. 1(a)). At larger sizes than those considered in this study, the optimal structure involves further facetting, so that it more closely approximates the Wulff polyhedron.[57] Of the three morphologies fcc structures have the smallest values of n_nn.

  

Figure 3: Examples of the strain involved in packing tetrahedra. (a) Five regular tetrahedra around a common edge produce a gap of $7.36^\circ$. (b) Twenty regular tetrahedra about a common vertex produce gaps equivalent to 1.54 steradians.

A Mackay icosahedron can be decomposed into twenty fcc tetrahedra, but as we see from Fig. 3(b), when twenty regular tetrahedra are packed around a common vertex large gaps remain. To bridge these gaps the tetrahedra have to be distorted, thus introducing strain. The distance between adjacent vertices of the icosahedron is 5% larger than the distance between a vertex and the centre. Similarly, a pentagonal bipyramid can be decomposed into five fcc tetrahedra sharing a common edge. Again, a gap remains if regular tetrahedra are used (Fig. 3(a)) and consequently decahedral structures are strained, although not as much as icosahedra. In contrast, close-packed structures can be unstrained.

Having deduced the relative values of n_nn and $E_{\rm strain}$ for icosahedral, decahedral and fcc structures, we can predict the effect of the range on the competition between them. For a moderately long-ranged potential, the strain associated with the icosahedron can be accommodated without too large an energetic penalty and so such structures are the most stable. As the range of the potential is decreased, the strain energy associated with icosahedra increases rapidly, and there comes a point where decahedra become more stable. Similarly, for a sufficiently short-ranged potential fcc structures become more stable than decahedral structures.

The above decomposition of the potential energy also helps us to understand the effect of size on the energetic competition between the three morphologies. The differences in n_nn, which arise from the different surface structures, are approximately proportional to the surface area ($\propto N^{2/3}$). The strain energies, however, are proportional to the volume ($\propto N$). Therefore, the differences in $E_{\rm strain}$ increase more rapidly with size than the differences in n_nn, thus explaining the change in the most stable morphology from icosahedral to decahedral to fcc with increasing size. The effect of increasing the size is similar to the effect of decreasing the range of the potential: both destabilize strained structures.