Crystalline Structure-Perfection: Seven Systems and Fourteen Lattices

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CHAPTER Three Crystalline Structure-Perfection

Three point one Seven Systems and Fourteen Lattices

Three point two Metal Structures

With the categories of engineering materials firmly established, we can now begin characterizing these materials. We will begin with atomic-scale structure, which for most engineering materials is crystalline; that is, the atoms of the material are arranged in a regular and repeating manner.

Common to all crystalline materials are the fundamentals of crystal geometry. We must identify the seven crystal systems and the fourteen crystal lattices. Each of the thousands of crystal structures found in natural and synthetic materials can be placed within these few systems and lattices.

The crystalline structures of most metals belong to one of three relatively simple types. Ceramic compounds, which have a wide variety of chemical compositions, exhibit a similarly wide variety of crystalline structures. Some are relatively simple, but many, such as the silicates, are quite complex. Glass is noncrystalline, and its structure and the nature of noncrystalline materials are discussed in Chapter Four. Polymers share two features with ceramics and glasses. First, their crystalline structures are relatively complex. Second, because of this complexity, the material is not easily crystallized, and common polymers may have as much as fifty percent to one hundred percent of their volume noncrystalline. Elemental semiconductors, such as silicon, exhibit a characteristic structure (diamond cubic), whereas semiconducting compounds have structures similar to some of the simpler ceramic compounds.

Within a given structure, we must know how to describe atom positions, crystal directions, and crystal planes. With these quantitative ground rules in hand, we conclude this chapter with a brief introduction to x-ray diffraction, the standard experimental tool for determining crystal structure.

The central feature of crystalline structure is that it is regular and repeating. In order to quantify this repetition, we must determine which structural unit is being repeated. Actually, any crystalline structure could be described as a pattern formed by repeating various structural units. As a practical matter, there will generally be a simplest choice to serve as a representative structural unit. Such a choice is referred to as a unit cell. The geometry of a general unit cell is shown in Figure Three point two. The length of unit-cell edges and the angles between crystallographic axes are referred to as lattice constants, or lattice parameters. The key feature of the unit cell is that it contains a full description of the structure as a whole because the complete structure can be generated by the repeated stacking of adjacent unit cells face to face throughout three-dimensional space.

The description of crystal structures by means of unit cells has an important advantage. All possible structures reduce to a small number of basic unit-cell geometries, which is demonstrated in two ways. First, there are only seven unique unit-cell shapes that can be stacked together to fill three-dimensional space. These are the seven crystal systems defined and illustrated in Table Three point one. Second, we must consider how atoms (viewed as hard spheres) can be stacked together within a given unit cell. To do this in a general way, we begin by considering lattice points, theoretical points arranged periodically in three-dimensional space, rather than actual atoms or spheres. Again, there are a limited number of possibilities, referred to as the fourteen Bravais lattices, defined in Table Three point two. Periodic stacking of unit cells from Table Three point two generates point lattices, arrays of points with identical surroundings in three-dimensional space. These lattices are skeletons

SECTION Three point one Seven Systems and Fourteen Lattices

The fourteen Crystal Bravais Lattices

Three point two Metal Structures

Three point three Ceramic Structures

POWERS OF ONE

Fullerenes

EXAMPLE three point three

SOLUTION

Using data from Appendices one and two, calculate the density of magnesium oxide.

Polymeric Structures

Orientation of unit cell

SOLUTION

Three point five Semiconductor Structures

THE MATERIAL WORLD

Crystal

EXAMPLE three point six

SOLUTION

EXAMPLE three point seven

Three point six Lattice Positions, Directions, and Planes

Miller indices (hkl):

EXAMPLE three point eight

SOLUTION

EXAMPLE three point nine

SOLUTION Sketching this case gives

SOLUTION

EXAMPLE three point eleven

SOLUTION

EXAMPLE three point thirteen

SOLUTION

EXAMPLE three point fourteen

SOLUTION

EXAMPLE three point fifteen

SOLUTION

Calculate the linear density of atoms along the one one one direction in silicon.

Calculate the planar density of atoms in the one one one plane of silicon.

Three point seven X-Ray Diffraction

Calculate the linear density of ions along the one one one direction for C A O. (See Example three point sixteen.)

POWERS OF ONE

EXAMPLE THREE point twenty

SOLUTION

PRACTICE PROBLEM THREE point twenty-two

Summary

Key Terms

CHAPTER three References

Overview

The chapter presents an overview of crystalline structures in engineering materials, emphasizing the classification into seven crystal systems and fourteen lattices. Key features such as unit cells, lattice parameters, and the geometry of metals and ceramics are discussed, along with methods for their analysis.

Key Points

1The chapter categorizes crystalline materials into seven systems and fourteen lattices
2Metals typically exhibit three simple crystal structures
3Unit cells serve as building blocks for describing crystal structures
4X-ray diffraction is the primary tool for determining crystal geometry
5The complexity of crystal structures varies significantly among materials like metals, ceramics, and polymers.

Details

Category: Technology and Engineering

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