Mackay cluster

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. 

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. 

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... 

Fig. 2 Schematics of the multiply endohedral clusters in Mackay-, Bergman-, and Tsai-type 1/1 ACs...

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). 

Finally, we note a number of higher nuclearity iron-containing clusters. Hieber (74) reported a complex formulated as [Sn2Fe5(CO)2o], 62, although no structural details were presented. Hieber also described a complex formulated as [PbFe3(CO)12], but later work by Whitmire (71) indicates that this compound is probably the tetrairon species, 56. Mackay and Nicholson (75) have described the synthesis and structures of three polynuclear species [Fe2(CO)7 //-E(Fe2(CO)8) 2] (63, E = Ge 64, E = Sn) and [Fe3(CO)10- //-Ge(Fe2(CO)8) 2], 65, from reactions involving germanium or tin... 

There are a sufficiently large number of complexes, mainly of germanium, containing more than one type of transition metal, to warrant a separate discussion some examples have already been mentioned (viz. 38 and 57). Mackay and Nicholson (89,91) have described the reaction between [Fe2-(CO)8(//-GeH2)2] and [Co2(CO)8], which affords the mixed cobalt-iron clusters [Fe2(CO)8 /i-Ge(Co2(CO)7) 2], 82, and [Fe2(CO)7 /i-Ge(Co2-(CO)7) 2], 83, the latter having been characterized by X-ray diffraction. This is isoelectronic with the iron-germanium cluster, 63, and both adopt very similar structures. 

Wang F, Weidt S, Xu J, Mackay CL, Langridge-Smith PRR, Sadler PJ (2008) Identification of clusters from reactions of ruthenium arene anticancer complex with glutathione using nanoscale liquid chromatography fourier transform ion cyclotron mass spectrometry combined with 180-labeling. J Am Soc Mass Spectrom 19 544-549... 

K. H. Kuo, Mackay, Anti-Mackay, Double-Mackay, Pseudo-Mackay, and Related Icosahedral Shell Clusters. Struct. Chem. 2002, 13,221-230. 

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... 

In 1962 Mackay introduced some amazing structures built with hard spheres as possible models for atomic clusters. Their external shape was that of a regular IC, and their size could be extended from the atomic scale to infinity. Despite the perfect fivefold symmetry of these structures, their internal local order was nearly that of a crystal. 

Solid-liquid coexistence in Ar, in the sense of Berry et al. - would contrast with MD results of several other clusters of size 20 (Beck et al. ), where no such phenomenon is observed. However, IV = 55 is the next Mackay icosahedronafter Ar, making the 55-particle cluster an attractive candidate for the two-phase phenomenon. 

There is at least one major area of activity pertaining directly to the environment for which the reader will seek in vain. The complexity of environmental problems and the availability of personal computers have led to extensive studies on models of varying sophistication. A discussion and evaluation of these lie well beyond the competence of an old-fashioned experimentalist this gap is left for others to fill but attention is drawn to a review that covers recent developments in the application of models to the risk assessment of xenobiotics (Barnthouse 1992), a book (Mackay 1991) that is devoted to the problem of partition in terms of fugacity — a useful term taken over from classical thermodynamics — and a chapter in the book by Schwarzenbach et al. (1993). Some superficial comments are, however, presented in Section 3.5.5 in an attempt to provide an overview of the dissemination of xenobiotics in natural ecosystems. It should also be noted that pharmacokinetic models have a valuable place in assessing the dynamics of uptake and elimination of xenobiotics in biota, and a single example (Clark et al. 1987) is noted parenthetically in another context in Section 3.1.1. In similar vein, statistical procedures for assessing community effects are only superficially noted in Section 7.4. Examples of the application of cluster analysis to analyze bacterial populations of interest in the bioremediation of contaminated sites are given in Section 8.2.6.2. 

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... 

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.

FIG. 8. Examples of the smallest Mackay icosahedra, from sizes of 13, 55, 147, 309 and 561. The larger clusters show the formation of crystal surfaces [93]. 

Among other results, we conclude that the Bergman [15] and Mackay [16] poly-hedra, which perform a decoration on the icosahedral tiling 7 pr) jjj pjg model Af(T ( ) can not be energetically stable constituents, i.e. the clusters, because these are cut by both the fivefold and the twofold stable surfaces. 

Boudard model. Not only have we shown that the atomic decorations in a shape of Bergman and Mackay polyhedra can not be stable clusters but also we have drawn the appearance of the cut polyhedra on the surfaces that matches the image of the real material on the fivefold surface. 

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... 

Even in dilute solutions they associate (49, 50). Published sizes of the micelles vary from 2 to 4 run. Sophisticated analytical techniques such as small-angle X-ray diffraction (SAXS), small-angle neutron scattering (SANS), and NMR were used to study the asphaltene particle or micelle sizes (51). MacKay (15) reported that a MWtof 10,000 g/mol would correspond to a 2 to 4-nm cluster. This is very much smaller than a 1-pm water droplet, and considered to be 1/100 to 1/1000 the droplet diameter. This topic is worthy of a review on its own. However, the colloidal properties of asphaltenes, micelles, and... 

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