Structure, Bonding, and Properties
1.3. COORDINATION NUMBER AND COORDINATION POLYHEDRON
Interaction between adjacent atoms determines the fundamental properties of the crystalline materials. The coordination number (c.n.) and the coordination polyhedron can be used to characterize the immediate surroundings of an atom. The coordination number is defined as the number of coordinated atoms that are the closest neighbors. Thus, any atom (cation or anion) in the unit cell has a coordination number. However, there is not a sharp cut in counting the neighbors that directly interact with the center atom, because of the long-range interaction. In general, the interatomic distance is taken as a reference for defining the nearest neighbors. In metallic antimony, for example, each Sb atom has three neighbors at a distance of 0.291 nm and three at a distance of 0.336nm, which is only
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STRUCTURE, BONDING, AND PROPERTIES
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15% longer than the first shell of atoms. In this case the symbol 3
+
3 is used to represent the coordination number of the antimony, where the first 3 stands for the atoms in the first shell and the second 3 for those in the second shell. This notation system is frequently used in describing a complex compound. In white tin, an atom has four neighboring atoms at a distance of O.302nm, two at O.318nm, and four at O.377nm; thus, a mean or"effective" coordination number is introduced, which is the sum of all the surrounding atoms with proper weighting factors. The atoms are counted as fractional atoms with a weighting factor between 0 and 1. The weighting factor is close to zero if the atom is farther away from the center atom.
In general, interaction between atoms is short-ranged. The coordination domain, to be defined below, is useful in describing the interaction range of an atom. If an atom is considered as the center, its coordination domain is constructed by connecting the atom with all of its surrounding atoms; the set of planes perpendicular to the connecting lines and passing through their midpoints forms the domain of interaction, which is a convex
0
D 4
Linear arrangement
~
Angular arrangement Triangle Square Tetragonal~ CJJ
Trigonal bipyrarnid Tetragonalpyrarnid Octahedron Trigonal prism
Capped trigonal prism Cube Square antiprism Dodecahedron
Triply capped trigonal prism Anticuboctahedron Cuboctahedron Figure 1.4. The most important coordination polyhedra in describing crystal structures.
polyhedron. In this way, a polyhedron face can be assigned to every atom in the unit cell, the area of the polyhedron face directly facing the atom is a measure of the weighting factor in the previous calculation. For a crystal consisting of ions, only the anions immediately adjacent to a cation and the cations immediately adjacent to an anion are considered in the calculation.
For the convenience of describing the atom distribution around a center atom, a coordination polyhedron is introduced by connecting the centers of mutual adjacent coordinated atoms. For every coordination number, typical shapes of coordination polyhedra exist (Fig. 1.4). These polyhedra are the fundamental building blocks for constructing many existing structures. The symmetry and coordination number associated with a polyhedron are the fundamental characteristics that determine the geometry and chemical composition of a compound that can form. In some cases, small displacements of atoms may convert one polyhedron into another, resulting in a change in the interactive energy and the property of a crystal.
A larger structural unit cell can be constructed by using polyhedra. Since the polyhedra are faceted building blocks, two polyhedra can be joined at a common vertex, a common edge, or a common face (Fig. 1.5). The common atoms of two connected polyhedra are called bridging atoms. In face-sharing polyhedra, the central atoms are closest to one another, but they are farthest apart in vertex-sharing polyhedra. The transformation among these connecting configurations results in phase transition and changes the structure of the unit cell. This is the fundamental of many structural evolution. As shown in Chapters 2-5, many different compounds with unique structures are built using these blocks under different connecting configurations.
In constructing a polyhedron, the atom located at the center can be a cation or anion, forming at least two different types of polyhedra. The usual choice is to make the center atom a cation. This type of coordinated polyhedra is usually taken as the fundamental block in constructing the structure of a compound. The coordinated polyhedra with coordination numbers 2, 3, 4, and 6 seem more essential for structure formation. In the
Figure 1.5. Some examples for the connections of polyhedra: (a) two tetrahedra sharing a vertex; (b) two tetrahedra sharing an edge; (c) two octahedra sharing a vertex; and (d) two octahedra sharing a face.
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CHAPTER 1
1 1
1 1 1
1 _______________ I
c
I 1
1 1
L ____ ______ ----I
Figure 1.6. The crystal structure of fergusonite-(Y). The coordination polyhedra of Nb atoms are drawn as tetrahedra (left) and eight-coordinated polyhedra (right), respectively.
rare earth and transition metal ternary oxides, for example, the transition metal cations form the coordinated octahedra with a coordination number of 6. They can share vertexes or edges to build up a frame in which the rare earth cations located on the vertexes or the edges have a coordination number larger than 6. In YNb04 (Fig. 1.6) the Nbcan have regular tetrahedra or distorted eight-coordination polyhedra. It seems that coordination numbers less than 8 indicate tightly bonded groups and coordination numbers more than 8 imply a pure ionic interaction with its neighbors. Transition metal cations have a strong ligand field effect with the coordinated oxygen anions, but rare earth cations only have electron Coulomb interaction with their neighbors.