Concepts of Crystal Geometry

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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).

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.

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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

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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.