Abstract
General analytical expressions are obtained for calculating the dimension of multi-shell clusters with a common center of shells in those cases when there is a metal atom in the center of the shells and when it is not. The shells can be in the shape of any body of Plato. It has been established that the gamma-copper cluster has the form of a cross-polytope of high dimension. The forms of clusters with ligands of the core of which is a chain of metal atoms or a metal polyhedron are geometrically investigated. It is shown that if the skeleton is a chain of metal atoms, then the cluster is polytope composed of two polytopes of higher dimension adjacent to each other along a flat section containing a chain. If the skeleton is a metal polyhedron, then a cluster of higher dimension has several ligand shells.
TopThe icosahedron is only one of Plato's bodies found in the skeleton of metal clusters (Gubin, 2019). Suppose there is a cluster, each shell of which is a convex regular three - dimensional polyhedron (Plato's body) with some possible number of vertices n and the same for all shells of this cluster. The flat sides of the shells are regular m - corner. All shells of this cluster differ only in size and have a common center. Assume that the number of flat edges in the shell is j. One denoted Sj a shell satisfying the indicated conditions. Then the following statement is true.
Key Terms in this Chapter
Linear Metal Chains: Metal chains in which metal atoms are located in a straight line.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.
Homo-Element Metal Chains: Clusters in which the core contains atoms of the same element.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Hetero-Element Metal Chains: Clusters in which the core contains atoms of different elements.
Nonlinear (Curved) Metal Chains: Clusters in which metal-metal valent bonds form an angle different from 180 degrees between them.
Metal Chains: Cluster compounds that have a skeleton in the form of metal chains, i.e. polymetallic chains formed by metal-metal localized covalent bonds.
Polytope: Polyhedron in the space of higher dimension.