Mackay icosahedra

Fig. 12.2 (a) Polyhedral view of [MnCglAIgGe]. (b) Ball-and-stick view of the shell structure around the [MntglAIgGe] polyhedral cluster. The top and side views are along (001) and (no), respectively. In (c) a Mackay icosahedron in a-MnAI(Si) is oriented along the corresponding directions. 

Figure 9-30. Icosahedral packing of spheres showing the third shell [53], This is popularly called the Mackay icosahedron. ...

The conformational spaces of clusters of Lennard-Jones atoms were searched by application of the diffusion equation method for clusters of various sizes m from m = 5 to 55. For example, for m = 55, with 3m - 6 = 159 degrees of freedom, there are about 1045 local minima, and the global minimum (a MacKay icosahedron) was found in about 400 s on an IBM 3090 computer.205... 

The above discussion considers a model for the stmcture of the crystalline cubic R-AlsCuTis. Now consider possible perturbations in this model to give the quasicrystalline T2-Al5CuTi3. In this coimection, the 54-vertex Mackay icosahedron (Figure 2) appears as a structural unit in certain quasicrystals. The Mackay icosahedron has a shell stmcture consisting of the following layers ... 

Figure 2 A view of the surface of a Mackay icosahedron showing the vertices of the larger icosahedron (layer h) as black circles and the vertices of the icosidodecahedron (layer c) as white circles. The vertices of the central icosahedron (layer a) are not visible...

Instead, it only shows that the low-energy region of the PES, associated with structures based on the Mackay icosahedron, is funnel-like. The graph probably only represents the bottom of a larger funnel leading down from the huge number of disordered minima. 

Figure 1.10. Thermodynamic properties of LJ55 in the canonical ensemble, (a) The heat capacity, C,. (b) The Landau free energy, Al( c)- (c) The probability of the cluster being in the Mackay icosahedron, with one or two defects, and liquid-like regions of configuration space, (d) Gn,( c). the number of minima with energy less than c. All these properties were calculated using the anharmonic form of the superposition method from a sample of 1153 minima generated from a microcanonical MD simulation. In (b) the results are compared with simulation data obtained by Lynden-Bell and Wales (data points) [218] with the zero of free energy chosen for clarity.

An interesting result from the work of Doye (2006) is that for the Gupta potential the uncentered Mackay icosahedron with N = 54 is more stable than the complete N = 55 Mackay icosahedron. This is because of the strain applied by the external atoms on the center, making it unfavorable to be occupied. 

So far, three types of i-QCs appear in the literature Mackay [17], Bergman [18], and Tsai types [19], which have been differentiated on the basis of the polyhedral cluster sequences observed in the respective 1/1 AC structures. These are commonly represented as shown in Fig. 2. An i-QC is concluded to be Mackay-type if its 1/1 AC contains a 54-atom multiply endohedral cluster ordered, from the center out, as a small icosahedron (12 atoms), a larger icosahedron (12), and an icosidodecahe-dron (30). This motif occurs in ACs that consist of transition metals and main-group elements on the right side of the periodic table such as Al-(Pd,Mn)-Si [17,20]. In... 

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. ... 

Figure 1.5 provides confirmation that the complete Mackay icosahedra are particularly stable. Between these magic numbers, the stability of the clusters first decreases as atoms are added to the surface of the icosahedron, and then increase as the overlayer nears completion. When especially stable... 

In the Mackay cluster [17], a central position is surrounded by three concentric shells, an icosahedron, an icosidodecahedron and another, larger icosahedron. These define 12 + 30 -I-12 = 54 atom positions. The Mackay cluster is a basic structural element, for instance of the cubic a-Al-Mn-Si [18]. Clusters referred to as Mackay-type are closely related to Mackay clusters, but may show shght deformation or differing site occupation, in particular of the inner shell, which is frequently not fully occupied due to steric constraints. The s-phases, which will be discussed extensively below (Section 2.2), are based on Mackay-type clusters. 

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. 

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