Higher Dimensions of Clusters of Intermetallic Compounds: Dimensions of Metallic Nanoclusters

Introduction

Currently, a large amount of scientific literature is devoted to the study of nano objects (Gubin, 2019; Haberland, 1994; Gusev, & Rempel, 2000; Suzdalev, 2005). A systematic study of the geometry of the structures of chemical compounds (Zhizhin, 2017, 2018) showed that almost all elements of the periodic system form molecules of higher dimension. It is natural to assume that clusters, as larger than formation molecules, including a large number of atoms, can have a higher dimension. However, until recently, clusters as three-dimensional objects (Lord, Mackay, & Ranganathan, 2006; Pauling, 1960). There are abstract methods for describing clusters, strictly speaking, not related to specific chemical compounds (Diudea & Nagy, 2007; Ashrafi, Cataldo, Iraumanesh, & Ori, 2013). In these works, proceeding from the well - known three - dimensional polyhedrons of Plato and Archimedes, they are transformed by various operations: construction of a polyhedron by the midpoints of edges, truncation, construction of a dual polyhedron, adding vertices of a polyhedron, etc. At the same time, such transformations are in no way connected with real chemical compounds. In addition, consideration of the transformed bodies is carried out in a certain abstract space, as if “forgetting” about the real dimension of chemical compounds (MacMullen, & Schulte, 2002). This cluster research direction is most clearly formulated in a generalizing monograph of Diudea (2018). In the preface to it, it is directly emphasized that the cluster models built in it are not associated with specific crystallographic objects: real crystals, networks, and quasicrystals. This paper discusses clusters of real chemical compounds. Moreover, in this work, consideration of clusters is limited to a special type of chemical compounds - intermetallic compounds, since the study of intermetallic compounds has had a significant impact on the development of scientific views in recent decades. In particular, the discovery of so-called quasicrystals is associated with intermetallic compounds, i.e. crystals supposedly devoid of translational symmetry (Shechtman, Blech, Gratias, & Cahn, 1984). Although it was later shown that quasicrystals have translational symmetry, but in the space of higher dimension (Shevchenko, Zhizhin, & Mackay, 2013 a, b; Zhizhin, 2014c; Zhizhin, & Diudea, 2017).