concepts of crystal geometry
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Concepts of Crystal Geometry. X-ray diffraction analysis shows that the atoms in a metal crystal are arranged in a regular, repeated three-dimensional pattern. The most elementary crystal structure is the simple cubic lattice (Fig. 9-1). Figure 9-1 Simple cubic structure. - PowerPoint PPT PresentationTRANSCRIPT
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Concepts of Crystal Geometry
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• X-ray diffraction analysis shows that the atoms in a metal crystal are arranged in a regular, repeated three-dimensional pattern.
• The most elementary crystal structure is the simple cubic lattice (Fig. 9-1).
Figure 9-1 Simple cubic structure.
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• We now introduce atoms and molecules, or “repeatable structural units”. • The unit cell is the smallest repetitive unit that there are 14 space lattices. • These lattices are based on the seven crystal structures.
• The points shown in Figure 9-1 correspond to atoms or groups of atoms. • The 14 Bravis lattices can represent the unit cells for all crystals.
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Figure 9-2 (a) The 14 Bravais space lattices (P = primitive or simple; I = body-centered cubic; F = face-center cubic; C = base-centered cubic
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Figure 9-2(b)
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Figure 9-2(c)
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Figure 9-2(d)
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Figure 9-2(e)
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Figure 9-2(f)
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Figure 9-3 a) Body-centered cubic structure; b) face-centered cubic structure.
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Figure 9-4 Hexagonal close-packed structure
Figure 9-5 Stacking of close-packed spheres.
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• Three mutually perpendicular axes are arbitrarily placed through one of the corners of the cell.
• Crystallographic planes and directions will be specified with
respect to these axes in terms of Miller indices.
• A crystallographic plane is specified in terms of the length of its intercepts on the three axes, measured from the origin of the coordinate axes.
• To simplify the crystallographic formulas, the reciprocals of these intercepts are used.
• They are reduced to a lowest common denominator to give the Miller indices (hkl) of the plane.
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• For example, the plane ABCD in Fig. 9-1 is parallel to the x and z axes and intersects the y axis at one interatomic distance ao. Therefore, the indices of the plane are , or (hkl)=(010).
1,1
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Figure 9-1 Simple cubic structure.
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• There are six crystallographically equivalents planes of the type (100).• Any one of which can have the indices (100), (010), (001), depending upon the choice of axes. • The notation {100} is used when they are to be considered as a group,or family of planes.
)100(),010(),001(
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• Figure 9.6(a) shows another plane and its intercepts.
Figure 9-6(a) Indexing of planes by Miller indices rules in the cubic unit cell
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• As usual, we take the inverse of the intercepts and multiply them by their common denominator so that we end up with integers. In Figure 9.6(a), we have
)112(2/1
1,
11
,11
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• Figure 9.6(b) shows an indeterminate situation. Thus, we have to translate the plane to the next cell, or else translate the origin.
Figure 9-6(b) Another example of indexing of planes by Miller rules in the cubic unit cell.