The Crystal Structure of Solids: TYPES OF SOLIDS
Amorphous, polycrystalline, and single crystal are the three general types of solids. Each type is characterized by the size of an ordered region within the material. An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity. Amorphous materials have order only within a few atomic or molecular dimensions, while polycrystalline materials have a high degree
of order over many atomic or molecular dimensions. These ordered regions. or single-crystal regions, vary in size and orientation with respect to one another. The single-crystal regions are called grains and are separated from one another by grain boundaries. Single-crystal materials, ideally, have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material. The advantage of a single-crystal material is that. in general, its electrical properties are superior to those of a nonsingle-crystal material, since grain boundaries tend to degrade the electrical characteristics. Two-dimensional representations of amorphous, polycrystalline, and single-crystal materials are shown in Figure 1.1.
SPACE LATTICES
Our primary concern will be the single crystal with its regular geometric periodicity in the atomic arrangement. A representative unit, or group of atoms, is repeated at regular intervals in each of the three dimensions to form the single crystal. The penodic arrangement of atoms in the crystal is called the lattice.
Primitive and Unit Cell
We can represent a particular atomic array by a dot that is called a lattice point. Figure 1.2 shows an infinite two-dimensional array of lattice points. The simplest means of repeating an atomic array is by translation. Each lattice point in Figure 1.2 can be translated a distance a, in one direction and a distance bl in a second noncolinear direction to generate the two-dimensiunal lattice. A third noncolinear translation will produce the three-dimensional lattice. The translation directions need not be perpendicular. Since the three-dimensional lattice is a periodic repetition of a group of atoms, we do not need to consider the entire lattice, but only a fundamental unit that is being repeated. A unit cell is a small volume of the crystal that can be used to reproduce the entire crystal. Aunit cell is not a unique entity. Figure 1.3 shows several possible unit cells in a two-dimensional lattice.
The unit ccll A can be translated in directions a2 and b2, the unit ccll B can be translated in directions a3 and b3. and the entire two-dimensional lattice can be constructed by the translations of either of these unit cells. The unit cells C and D in Figure 1.3 can also be used to construct the entire lattice by using the appropriate translations. This discussion of two-dimensional unit cells can easily be extended to three dimensions to describe a real single-crystal material.
Aprirnitive cell is the smallest unit cell that can be repeated to form the lattice. In many cases, it is more convenient to use a unit cell that is not a primitive cell. Unit cells may be chosen that have orthogonal sides, for example, whereas the sides of a primitive cell may be nonorthogonal. A generalized three-dimensional unit cell is shown in Figure 1.4. The relationship between this cell and the lattice is characterized by three vectors a, 6, and ?, which need not be perpendicular and which may or may not be equal in length. Every equivalent lattice point in the three-dimensional crystal can he found using the vector
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of order over many atomic or molecular dimensions. These ordered regions. or single-crystal regions, vary in size and orientation with respect to one another. The single-crystal regions are called grains and are separated from one another by grain boundaries. Single-crystal materials, ideally, have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material. The advantage of a single-crystal material is that. in general, its electrical properties are superior to those of a nonsingle-crystal material, since grain boundaries tend to degrade the electrical characteristics. Two-dimensional representations of amorphous, polycrystalline, and single-crystal materials are shown in Figure 1.1.
SPACE LATTICES
Our primary concern will be the single crystal with its regular geometric periodicity in the atomic arrangement. A representative unit, or group of atoms, is repeated at regular intervals in each of the three dimensions to form the single crystal. The penodic arrangement of atoms in the crystal is called the lattice.
Primitive and Unit Cell
We can represent a particular atomic array by a dot that is called a lattice point. Figure 1.2 shows an infinite two-dimensional array of lattice points. The simplest means of repeating an atomic array is by translation. Each lattice point in Figure 1.2 can be translated a distance a, in one direction and a distance bl in a second noncolinear direction to generate the two-dimensiunal lattice. A third noncolinear translation will produce the three-dimensional lattice. The translation directions need not be perpendicular. Since the three-dimensional lattice is a periodic repetition of a group of atoms, we do not need to consider the entire lattice, but only a fundamental unit that is being repeated. A unit cell is a small volume of the crystal that can be used to reproduce the entire crystal. Aunit cell is not a unique entity. Figure 1.3 shows several possible unit cells in a two-dimensional lattice.
The unit ccll A can be translated in directions a2 and b2, the unit ccll B can be translated in directions a3 and b3. and the entire two-dimensional lattice can be constructed by the translations of either of these unit cells. The unit cells C and D in Figure 1.3 can also be used to construct the entire lattice by using the appropriate translations. This discussion of two-dimensional unit cells can easily be extended to three dimensions to describe a real single-crystal material.
Aprirnitive cell is the smallest unit cell that can be repeated to form the lattice. In many cases, it is more convenient to use a unit cell that is not a primitive cell. Unit cells may be chosen that have orthogonal sides, for example, whereas the sides of a primitive cell may be nonorthogonal. A generalized three-dimensional unit cell is shown in Figure 1.4. The relationship between this cell and the lattice is characterized by three vectors a, 6, and ?, which need not be perpendicular and which may or may not be equal in length. Every equivalent lattice point in the three-dimensional crystal can he found using the vector
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