Crystalline Solids Unit Cells and Basic Structures

11 Crystalline Solids Unit Cells and Basic Structures 

Cubic Cell Name Simple Cubic Atemsper Unit Cell 1 Structure Coordination Number 6 Edge Length In terms of r 2r Packing Elficientqr (fraction of volume occupied) 52% 

A FIGURE 11.44 Ihe Cubic Crystalline Lattices The different colors used for the atoms in this figure are for clarity only. All atoms within each structure are identical. 

A In the body-centered cubic lattice, the atoms touch only along the cube diagonal. The edge length is 4r/V.  

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A unit cell is the basic repeating structural unit of a crystalline solid. Figure 11.14 shows a unit cell and its extension in three dimensions. Each sphere represents an atom, ion, or molecule and is called a lattice point. In many crystals, the lattice point does not actually contain such a particle. Rather, there may be several atoms, ions, or molecules idaitically arranged about each lattice point. For simplicity, however, we can assume that each lattice point is occupied by an atom. This is certainly the case with most metals. Every aystalline solid can be described in terms of one of the seven types of unit cells shown in Hgure 11.15. The geometry of the cubic unit cell is particularly simple because aU sides and aU angles are equal Any of the unit cells, when repeated in space in all three dimensions, forms the lattice structure characteristic of a crystalline solid. 

A unit cell is the basic repeating structural unit of a crystalline solid. Figure 12.13 shows a unit cell and its extension in three dimensions. Each sphere represents an atom, ion, or molecule and is called a lattice point. For the purpose of clarity, we will limit our discussion in this section to metal crystals in which each lattice point is occupied by an atom. 

Atoms and ions can be considered to be spheres which pack together in special and reproducible patterns to form solid state crystalline materials. The patterns of the spheres repeat in all three directions. The simplest, basic repeating unit in a crystalline solid is called the unit cell. It is of importance to realize that all crystalline solids, no matter how complex, are described by unit cells. The unit cell must be consistent with the chemical formula of the solid, must indicate the coordination number and geometry of each type of atom or ion, and must generate the crystal structure by simple translation or displacement of the unit cell in three dimensions. 

In this chapter, we start by introducing the most basic structural imit of a solid material — the unit cell. From there, we explain how crystalline solids form different shapes and are organized into different systems. At the end of the chapter, we explain some of the important and useful characteristics of solid materials, including their use as superconductors, and semiconductors. 

For crystalline solid compounds, species densities and concentrations can also be calculated from information on the crystal structure and unit cell size as well as the lattice occupancy of the species in question. It is sometimes more convenient to calculate species concentrations this way, particularly when structural information is more readily available than mass density information. The basic idea is to calculate density/concentration from the weight and volume of the crystallographic unit cell. Example 2.7 illustrates this approach. 

A central tenet of materials science is that the behavior of materials (represented by their properties) is determined by their structure on the atomic and microscopic scales (Shackelford, 1996). Perhaps the most fundamental aspect of the structure-property relationship is to appreciate the basic skeletal arrangement of atoms in crystalline solids. Table 2.21 illustrates the fundamental possibilities, known as the 14 Bravais lattices. All crystalline structures of real materials can be produced by decorating the unit cell patterns of Table 2.21 with one or more atoms and repetitively stacking the unit cell structure through three-dimensional space. 

The basic foundation of modeling any crystalline solid lies in the definition of its unit cell. Generally, structures derived from X-ray and neutron diffraction patterns provide good starting points for the structural optimization and subsequent determination of any property of interest. To simulate bulk systems, the model cell is tessellated in space to form an infinite lattice as shown in Figure 15.1. This replication of the ion positions or the so-called periodic boundary condition, produces mirror images of the ions at positions defined by the equation... 

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