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SOLID MATERIALS 32.13
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FIGURE 32.8 Unit cell of body-centered cubic structure. (a) The unit cell has 1 8 atom at each of 8 corners and 1 atom at the geometric center of the cell, for a total of 2 atoms; (b) the relationship of the lattice parameter a and atomic radius r.
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the axis and the origin measured in terms of multiples or fractions of the unit cell lengths a, b, and c used in the determination. The final steps in specifying the Miller indices are to reduce the three reciprocals to the lowest integers having the same ratio and then to enclose them in parentheses. As is true with direction indices, the sequence of integers relates to the distances along the x, y, and z axes, respectively. The following examples should make this procedure clear. Figure 32.9a identifies the front face of the crystal with the Miller indices (100). This notation is arrived at as follows: The front face intercepts the x axis at one a distance, and it does not intercept the y and z axes (or it intercepts at zero b and zero c units). If the side lengths are dropped, the intercepts are 1, 0, and 0. The reciprocals of these are also 1, 0, and 0. Since these are already the smallest integers, the Miller indices are specified by enclosing them in parentheses: (100). The commas are not included because they are simply part of the sentence structure. Figure 32.9b shows the (110) plane that is parallel to the z axis and is a face diagonal on the top and bottom faces of the unit cell. This plane intercepts the x axis at one a distance, the y axis at one b distance, and the z axis at zero c distance.The intercepts are 1, 1, and 0, and so are the reciprocals. Since these are the smallest integers, the Miller indices are specified as (110). Figure 32.9d shows the crystallographic plane that intercepts the x axis at 1 2 a, the y axis at one b, and the z axis at one c. The reciprocals are therefore 2, 1, and 1, and so this plane is identified as the (211) plane. Parentheses are used, as in the preceding examples, to specify a single plane or a family of parallel planes. Thus (100) represents all the planes that are parallel to the yz axes and intercept the lattice structure at one a, two a, three a, etc. distances. Wavy brackets, or braces, are used to designate all planes in a crystal that are equivalent. For example, the six face planes of a unit cell such as that in Fig. 32.9a are (100), (010), (001), (100), (010), and (001). The notation {100} includes all these six planes.
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32.4 CRYSTAL IMPERFECTIONS
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The previous discussions on crystal structure assumed a perfect lattice; that is, an atom occupied each and every lattice point, and the distances between equivalent
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SOLID MATERIALS 32.14
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FIGURE 32.9 Miller indices for some crystallographic planes.
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lattice points were all exactly the same. In the early 1900s it was found that real crystals did not have perfect properties. Notable among these properties was a mechanical strength much lower than crystals should have. As early as 1928 Prandtl suggested that slip (plastic deformation) in a crystal and the strength of a crystal are related to the presence of linear imperfections within the crystal. This type of imperfection is now called a dislocation. At the present time, the terms imperfection and defect refer to a deviation from a perfectly ordered lattice structure. Lattice imperfections are classified into three types: point defects, where the imperfection is localized about a single lattice point and involves only a few atoms; line defects, where the imperfection lies along a line of finite length involving a row (line) or many atoms; and planar defects or boundaries, where the imperfections involve entire planes or atoms such as the interface between adjacent crystals.
32.4.1 Point Defects Point defects are caused by (1) the absence of an atom from a lattice point, (2) the presence of an extra atom (usually a small foreign one) in the void spaces of the lattice,