Mackay icosahedral structure

Energy calculations for small clusters of atoms indicate that a cluster of 55 atoms should be reasonably stable (Mackay 1962, Allpress and Sanders 1970, Hore and Pal 1972). In addition, calculations suggest that the 55-atom cluster will take up an icosahedral structure in preference to the cubic cuboctahedral structure (Hore and Pal 1972). 

Many of the magic number combinations observed in the CMS of inert gas atoms have been identified with stable structures having an icosahedral symmetry (Echt et al. 1981). The Mackay icosahedra series (Hoare 1979 Mackay 1962) exhibits completion of the first three solvation shells as = 13, 55, and 147, respectively, such that the completion of solvation shells at n = 13, 55, etc., can arise from structures with a cuboctahedron symmetry (Hoare 1979). However, theoretical studies indicate that the icosahedral structures are more stable than those with cuboctahedral symmetry (Hoare 1979). The theoretical studies of Farges et al. (1986) and Northby (1987) provide insight into the growth of icosahedral structures. 

Many other kinds of clusters differ sharply from this simple relationship. Clusters of rare-gas atoms such as argon tend to have structures based on icosahedral geometry. This geometry cannot be the basis for a lattice it simply does not have the translational symmetry necessary to build a lattice. The spacings between neighbors in shells distant from the central atom differ from those near the core. The cluster sizes for which complete, filled icosahedra can be made, namely 13, 55, 137,..., are called magic numbers and the structures are called Mackay icosahedra. Not all the most stable structures of such clusters have icosahedral structures, and the specific structure of the most stable form depends on the forces that bind the cluster together. ° 4i por example, a number of clusters bound by Lennard-Jones or Morse potentials, potentials V(R) that depend only on the distance R between pairs of particles. The first of these has the form... 

For N < 1,000, Lennard-Jones clusters follow an icosahedral pattern growth with magic numbers corresponding to Mackay icosahedra (Mackay 1962) for N = 13, 55,147, 309, etc. In between these magic numbers, most of the structures are Mackay-like with incomplete outer layers. Exceptions occur when there are alternative structures with complete shells. These are mostly Marks decahedra (Doye 2003) but there are instances of an fee truncated octahedron and a Leary tetrahedron (Noya and Doye 2006). The preference for icosahedral structures of Lennard-Jones clusters at small sizes is thought to be due to a trade-off between optimal bond distance and strain (Hartke 2002 Krainyukova 2006). 

More precisely, if ,c = 0, the polymer forms a non-icosahedral structure, e.g., it is decahedral or fcc-like ,c = 1 indicates icosahedral geometry with Mackay overlayer or a complete icosahedron which might possess a few monomers bound in anti-Mackay type. Finally, for n,c > 2, the monomers form an icosahedral core with a considerably extended anti-Mackay overlayer. The probabilities(7) for the different values of njc as a function of temperature provide the necessary information to reveal structural transitions. 

Structural phase diagram for a flexible, elastic polymer with 90 monomers, parametrized by temperature T and nonbonded Interaction range 8. The transition lines (solid lines) were obtained by inflection-point microcanonical analysis. The crossover between collapse transition and nucleation cross is enlarged in the inset. The dashed vertical line separates solid phases that cannot be discriminated thermodynamically. The bottom figure shows for T = 02 the (canonical) probability that a conformation in these solid phases contains nic icosahedral cores, thereby separating fee or decahedral crystalline structures with Oje = 0 from Mackay icosahedral shapes (Ok > 1). From [136]. 

Mackay called attention to yet another limitation of the 230 space-group system. It covers only those helices that are compatible with the three-dimensional lattices. All other helices that are finite in one or two dimensions are excluded. Some important virus structures with icosahedral symmetry are among them. Also, there are very small... 

Mackay has considered larger icosahedral assemblies obtained by adding further complete layers of atoms to theA = 55 icosahedron shown in Figure 1 such structures contain N atoms where = lOn — 15 + 11 — 3) = 147, 309, 561, etc., where n is an integer greater than or equal to 4. Baker and Hoare have devised structures containing numbers of atoms intermediate between the A, -values quoted above which are close competitors for the absolute minimum potential-energy description. ... 

Mackay [9-73, 9-74] called attention to yet another limitation of the 230-space-group system. It covers only those helices that are compatible with the three-dimensional lattices. All other helices that are finite in one or two dimensions are excluded. Some important virus structures with icosahedral symmetry are among them. Also, there are very small particles of gold that do not have the usual face-centered cubic lattice of gold. They are actually icosahedral shells. The most stable configurations contain 55 or 147 atoms of gold. However, icosahedral symmetry is not treated in the International Tables, and crystals are only defined for infinite repetition. 

The exceptional case of the 38-mer shows a significantly different behavior. A single icosahedral core forms and in the interval 0.08 < 7 <0.19 icosahedra with Mackay overlayer are dominant. Although the energetic fluctuations are weak, near T 0.08, a surprisingly strong structural crossover to non-icosahedral stmctures occurs the formation of a maximally compact fee truncated octahedron. 

The magic 55-mer exhibits a very pronounced transition from unstmetured globules to icosahedral conformations with complete Mackay overlayer at a comparatively high temperature T 0.33). Below this temperature, the ground-state structure has already formed and is sufficiently robust to resist the thermal fluctuations. 

The second interesting result is the different ground states, zks in the case of unbormded clusters with untruncated LJ potential, the ground-state conformation is decahedral for N = 102 [Fig. 6.5(c)]. We can also identify the solid-solid transition toward Mackay structures The decahedral-icosahedral crossover occurs at the temperature T 0.02 (see Fig. 6.12), with a relatively prominent signal in the specific heat. 

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