The potential

 The Morse potential[77] may be written as  

$$V_M = \epsilon\sum_{i<j} e^{\rho_0(1-r_{ij}/r_0)}(e^{\rho_0(1-r_{ij}/r_0)}-2),$$ (2.1)

where $\epsilon$ is the pair well depth and r₀ the equilibrium pair separation. In reduced units ($\epsilon=1$ and r₀=1) the potential has a single adjustable parameter, $\rho_0$, which determines the range of the interparticle forces. Figure 2.2 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 wide range of materials have been catalogued elsewhere[78]. Here, we give some representative examples. The LJ potential, which provides a reasonable description of the rare gases, 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 C₆₀ molecules[79] which is isotropic and short-ranged relative to the equilibrium pair separation, with an effective value[80] of $\rho_0=13.62$.This potential was designed to describe the room temperature fcc solid phase in which the C₆₀ molecules are able to rotate freely. The alkali metals have long-ranged interactions, for example $\rho_0$=3.15 has been suggested for sodium[81]. Fitting to bulk data gives a value of $\rho_0$=3.96 of nickel[82].

  

Figure 2.2: The Morse potential for different values of the range parameter $\rho_0$ as marked.

The structure of a cluster is given by the structure(s) associated with the global free energy minimum. At zero temperature, the structure with the lowest free energy is simply the global minimum of the PES. At higher temperatures, entropic factors must also be considered. Although we only perform a comprehensive survey of the zero Kelvin structure, some examples of the structural effects of temperature are discussed in §2.4.

To understand what factors cause a structure to have low potential energy and how these change with the value 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 as follows,  

$$V_M=-n_{nn}\epsilon+E_{\rm strain}+E_{nnn}.$$ (2.2)

The number of nearest-neighbour contacts, n_nn, the strain energy, $E_{\mathrm{strain}}$, and the contribution to the energy from non-nearest neighbours, E_nnn, are given by
$$\begin{eqnarray} n_{nn}&=&\sum_{i<j, x_{ij}<x_0}1, \nonumber \\ E_{\rm strain}&=... ...\sum_{i<j, x_{ij}\gt x_0}e^{-\rho_0 x_{ij}}(e^{-\rho_0 x_{ij}}-2),\end{eqnarray}$$

(2.3)

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[83,84]. Non-nearest neighbour interactions also cause the contraction of nearest neighbour distances from the equilibrium pair value and this contraction increases with the range of the potential[41].

$E_{\mathrm{strain}}$, which measures the energetic penalty for the deviation of a nearest-neighbour distance from the equilibrium pair distance, is a key quantity in this analysis. It should not be confused with strain due to an applied external force. For a given geometry, $E_{\mathrm{strain}}$ grows rapidly with increasing $\rho_0$because the potential well narrows. This can be seen by taking a Taylor expansion of $E_{\rm strain}$ about x_ij=0:

$$E_{\rm strain}\approx \sum_{i<j, x_{ij}<x_0} \rho_0^2 x_{ij}^2 -\rho_0^3 x_{ij}^3 + 7 \rho_0^4 x_{ij}^4/12 + \ldots$$ (2.4)

This expansion is valid when $\rho_0 x_{ij}$ is small. To a first approximation the strain energy grows quadratically with $\rho_0$. Thus decreasing the range destabilizes strained structures.

From this 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 (Figure 2.1) 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 on a $\{111\}$ face is 9-coordinate, but on a $\{100\}$ face only 8-coordinate) and minimizing the fraction of atoms in the surface layer. As Mackay icosahedra (Figure 2.1b) have only $\{111\}$ faces and are approximately spherical, the icosahedral structures have the largest n_nn. Complete Mackay icosahedra occur at $N=13, 55, 147, \ldots$A pentagonal bipyramid has all $\{111\}$ faces, but as 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 (Figure 2.1c) and was first predicted by the use of a modified Wulff construction[85]. The decahedral structures have lower values of n_nn than icosahedral structures because of the presence 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 (Figure 2.1a). At larger sizes than those considered in this study, the optimal structure involves further facetting, so that it more closely approximates the Wulff polyhedron[86]. Of the three morphologies fcc structures have the lowest values of n_nn.

  

Figure 2.3: Examples of the strain involved in packing tetrahedra. (a) Five regular tetrahedra around a common edge. The angle of the gap is $7.36^\circ$. (b) Twenty regular tetrahedra about a common vertex. The gaps amount to 1.54 steradians.

However, icosahedral structures involve the most strain of the three structural types. The Mackay icosahedra can be considered to be made of twenty fcc tetrahedra which share a common vertex. As we see from Figure 2.3b, 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 into the structure. The distance between adjacent vertices of the icosahedron is 5% longer than the distance between a vertex and the centre. Similarly, a pentagonal bipyramid can be considered to be made up of five fcc tetrahedra sharing a common edge. Again, a gap remains if regular tetrahedra are used (Figure 2.3a) and consequently decahedral structures are necessarily strained, although to a lesser extent than icosahedral structures. By 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 the three morphologies. For a moderately long-ranged potential, the strain associated with the icosahedral structures can be accommodated without too large an energetic penalty and so they are the most stable. As the range of the potential is decreased, the strain energy associated with the icosahedral structures increases rapidly, and there comes a point where decahedral structures become more stable. Similarly, for a sufficiently short-ranged potential fcc structures become more stable than decahedral structures.

The 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 between the morphologies, which arise from their different surface structures, are approximately proportional to the surface area ($\propto N^{2/3}$). The strain energies, though, 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 changes 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. The most favourable morphology varies with both the size of the cluster and the range of the potential.