Abstract
The author has previously proved that diffraction patterns of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this chapter, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters are obtained from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three-dimensional, have a higher dimension. The Euler-Poincaré equation was used, and the internal geometry of clusters was investigated.
TopClasters Of Mackay
Mackay's cluster consists of two icosahedrons of different sizes with a common center (Mackay, 1962). A larger icosahedron is obtained by attaching a number of tetrahedrons and octahedrons to the surface of the smaller icosahedron. However, to determine the dimension of Mackay's cluster, only the result of this connection is important - the formation of a larger icosahedron. Moreover, each vertex of the larger icosahedron is located on a certain line passing through the common center of the icosahedrons. On the same line is located one of the vertices of the smaller icosahedron (Lord, Mackay, & Ranganathan, 2006). In all vertices of each icosahedron, one atom is located. In addition, there is one more atom in the middle of each edge. Since the icosahedron has 12 vertices and 30 edges, it turns out that Mackay's cluster has 54 atoms (Figure 1).
In this figure, the atoms in the midpoints of the edges of the larger icosahedron are not shown, since in determining the dimension the atoms located on the linear portions of the edges do not matter. Connect the edges of the vertices of both icosahedrons lying on the same line passing through the common center of the icosahedrons (dotted lines in Figure 1). This shows that the space between the icosahedrons is completely filled with triangular prisms, the bases of which are the triangular faces of the smaller and larger icosahedrons (Figure 2).
These prisms are adjacent to each other along flat quadrilateral side faces. The number of these prisms is equal to the number of triangular faces of the icosahedron - 20. To determine the dimension of the construction of two icosahedra with a common center, between which there are 20 triangular prisms can be determined by the Euler - Poincaré formula (Poincaré, 1895)
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(1) Key Terms in this Chapter
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Fractal: The set is self-similar, that is, uniformity at different scales.
Polytope: Polyhedron in the space of higher dimension.
Quasicrystal: A solid body, characterized by symmetry without translation in three-dimensional Euclidean space.
Golden Hyper-Rhombohedron: Polytope in four-dimensional space with facets as rhombohedron and metric characteristics associated the golden section.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.
Diffraction: A wide range of phenomena occurring in the propagation of waves in heterogeneous environments in the space.