A New Technique to Determine the Coordination Number of BCC, FCC and HCP Structures of Metallic Crystal Lattices using Graphs

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1 A New Technique to Determine the Coordination Number of BCC, FCC and HCP Structures of Metallic Crystal Lattices using Graphs Sucharita Chakrabarti Department of Applied Science and Humanities (Mathematics) Guru Nanak Institute of Technology 157/F Nilgunj Road, Panihati, Kolkata, West Bengal, India. ABSTRACT: The study of crystal lattices, which is a part of crystallography, is important because the properties of materials are directly related to the crystal structure. The coordination number is one of the important parameters by which a crystal can be described. Graphical methods, on another hand, provide an excellent tool for describing the system or object under consideration and the subsequent analysis. In this paper as a first attempt the concept of graph theory is used to describe the metallic crystals lattice to find the coordination number, which is one of the important characteristics of a crystal structure. KEYWORDS: Crystal structure, Unit cell, Coordination number, BCC, FCC, HCP, Regular graphs 1. INTRODUCTION: Crystallography, which is the study of the properties of crystal, is an important subject for the chemists, physicists, geologists and mineralogists. The study of crystal lattices, which is a part of crystallography, is important because the properties of materials are directly related to the crystal structure. A crystal can be described by certain parameters such as coordination number, atomic packing factor, etc. Graph models can be used extensively in determining the parameters of crystals. However, still there is a gap between the results obtained in mathematical crystallography and their practical applications. To cover this technical gap the concept of graph theory is used to describe the metallic crystals lattice and find the coordination number, which is one of the important characteristics of a crystal structure. Three basic types of metallic crystals are considered, which are body-centered crystal (BCC), face-centered cubic (FCC) crystal and Hexagonal Closed Pack (HCP). For every type, an atom is considered to be a node and the bonds between the atoms are considered to be edges of the graph. The resultant graphs are then used to find coordination number of the crystal. This method is, therefore, considerably easier to analyze the crystal lattice than the conventional mathematical techniques. Unit cells of BCC, FCC and HCP crystals were successfully represented by simple, connected graphs. The crystal parameter namely coordination number has been successfully obtained from the graphs. When describing crystalline structure, atoms (or ions) are thought of as being solid sphere having welldefined diameters. This is termed the atomic hard sphere model in which spheres representing nearestneighbour atoms touch one another. Sometimes the term lattice is used in the context of crystal structures; in the sense lattice means a three-dimensional array of points coinciding with atom positions ( or sphere centers ). Centre of an atom is considered to be a node and the bonds (i.e., touching atoms) between the atoms are considered to be edges of the graph. In crystal structures we consider unit cubic cells. The number of nearest touching neighbour atoms of each atom in a cubic cell is same. So the graphs of BCC, FCC and HCP are regular graphs. The work carried out in this paper established for the first time that unit cells of crystals can be successfully represented by simple connected graphs. This may lead to a new area of study. Also, this will make the job of analyzing crystals easier. 18 Sucharita Chakrabarti

2 2. PRELIMINARIES: 2. i.crystal structures: Solid materials may be classified according to the regularity with which atoms or ions are arranged with respect to one another. A Crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances; that is, long range order exists, such that upon solidification, the atoms will position themselves in a repetitive three-dimensional pattern, in which each atom is bonded to its nearest-neighbour atoms. Some of the properties of crystalline solids depend on the crystal structure of the material, the manner in which atoms, ions, or molecules are spatially arranged. When describing crystalline structure, atoms (or ions ) are thought of as being solid sphere having well -defined diameters. This is termed the atomic hard sphere model in which spheres representing nearest-neighbour atoms touch one another (as shown in Fig 2.1). Sometimes the term lattice is used in the context of crystal structures; in the sense lattice means a three-dimensional array of points coinciding with atom positions (or sphere centers). Fig ii. Unit Cell: The atomic order in crystalline solids indicates that small groups of atoms form a repetitive pattern. Thus, in describing crystal structures, it is often convenient to subdivide the structure into small repeat entities called unit cells. Unit cells for most crystal structures are parallelepipeds or prisms having three sets of parallel faces; one is drawn within the aggregate of spheres, which in this case is a cube (as shown in Fig 2.2). Fig iii. Coordination Number: For metals, each atom has the same number of nearest neighbour or touching atoms, which is the coordination number of a crystal structure. 19 Sucharita Chakrabarti

3 2. iv. Body Centered Cubic (BCC)Crystal Structure: A metallic crystal structures has a cubic unit cell with atoms located at eight corners and a single atom at the cube centre (as shown in Fig 2.3). This is called a Body-Centered Cubic (BCC) crystal structures. Fig. 2.3 Face Centered Cubic (FCC) Crystal Structure: The crystal structure found for many metals has a unit cell of cubic geometry, with atoms located at each of the corners and the centers of all the cube faces (as shown in Fig 2.4). It is aptly called the Face-Centered Cubic (FCC) crystal structures. Fig. 2.4 Hexagonal Closed Pack (HCP): The top and bottom faces of the unit cell consist of six atoms that form regular hexagons and surround a single atom in the center. Another plane that provides three additional atoms to the unit cell is situated between the top and bottom planes. The atom in this mid plane have as nearest neighbours atom in both of the adjacent two planes. The equivalent of six atoms is contained in each unit cell; one-sixth of each of the 12 top and bottom face corner atoms, one-half of each of the 2 center face atoms, and all the 3 mid plane interior atoms (as shown in Fig 2.5). Fig. 2.5 Graphs: A graph G=(V(E), E(G)) consists of two finite sets: V(G), the vertex set of the graph, often denoted by just V, which is a nonempty set of elements called vertices, and E(G), the edge set of the graph, often denoted by just E, which is a possibly empty set of elements called edges, such that each edge e in E is assigned an unordered pair of vertices (u, v), called the end vertices of e. 20 Sucharita Chakrabarti

4 Degree of a Vertex: Let v be a vertex of the graph G. The degree d(v) of v is the number of edges of incident with v. Regular Graph: If for some positive integer k, d(v)= k for every vertex v of the graph G, then G is called k- regular. A regular graph is one that is k-regular for some k. 3. TECHNIQUE TO DETERMINE THE COORDINATION NUMBER USING GRAPH: Centre of an atom is considered to be a node and the bonds (i.e., touching atoms) between the atoms are considered to be edges of the graph. In crystal structures we consider unit cubic cells. The number of nearest touching neighbour atoms of each atom in a cubic cell is same. So the graphs of BCC, FCC and HCP are regular graphs. 3. i. GRAPHICAL REPRESENTATION OF BCC CRYSTAL STRUCTURES: In the graph below (Fig. 3.1), the red solid spheres are vertices and green lines are edges of the graph. In the unit cube ABCDEFGH the vertex is of degree 8, which is the coordination number of BCC crystal structure. As the graph of BCC crystal structure is regular, it is sufficient to find the degree of any one vertex. Fig ii. GRAPHICAL REPRESENTATION OF FCC CRYSTAL STRUCTURES: In the graph below (Fig. 3.2), the red solid spheres are vertices and green lines are edges of the graph. In the unit cube ABCDEFGH the vertex is of degree 12, which is the coordination number of FCC crystal structure. As the graph of FCC crystal structure is regular, it is sufficient to find the degree of any one vertex. 21 Sucharita Chakrabarti Fig.3.2

5 3. iii. GRAPHICAL REPRESENTATION OF HCP CRYSTAL STRUCTURES: In the graph below (Fig. 3.3), the green solid spheres are vertices and red lines are edges of the graph. The vertex a is of degree 12, which is the coordination number of HCP crystal structure. As the graph of HCP crystal structure is regular, it is sufficient to find the degree of any one vertex. Fig CONCLUSION AND OBSERVATIONS:The novel approach of representing the unit cell of a crystal by simple connected graphs has been successfully implemented. The coordination number of the atom could be obtained from the graph representing the unit cell of the crystal. The application of graphs in study and analysis of different crystals have been successfully achieved. In study of crystals, usually a small entity known as unit cell is used. A unit cell repeats itself within the crystal structure. Unit cells of crystals can be successfully represented by simple, connected graphs. The nodes of the graphs are used to represent the centre of an atom of the unit cell of the crystal and the edges of the graph are used to describe the bonds between the atoms within the unit cell. Coordination number of a crystal is defined as number of nearest neighbour or touching atoms, for each atom, can be obtained from the graph, which does not require manual analysis of the unit cell of the crystal. This has the advantage that the analysis of different crystals becomes easier and the chances of human error are less. 22 Sucharita Chakrabarti

6 Coordination number of BCC, FCC and HCP crystal structures can be easily obtained from the corresponding graphs by counting the degree of vertex. Computer programs can be used for drawing graphs that represent the unit cell of a crystal. 5. ACKNOWLEDGEMENT: This work has been supported by University Grants Commission under its Minor Research Project scheme (No. F. PSW-172/13-14(ERO) dated 2ND September, 2014). The author is thankful to Mr. Adhish Kumar Chakrabarty for encouragement and support. 6. REFERENCES: [1]. Pillai, S. O Solid State Physics,, New Age International Publishers [2]. Callister, Jr., William D. and Rethwisch, David G Callister s Materials Science and Engineering, Wiley [3]. Gross, J. L., Yellen, J. and Zhang, Ping 2006 Handbook of Graph Theory, CRC Press [4]. Sands, D. E Introduction to Crystallography, Dover Publication INC [5]. Deo, N Graph Theory, Prentice Hall India 23 Sucharita Chakrabarti