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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.

- 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.

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

Figure 9-2(b)

Figure 9-2(c)

Figure 9-2(d)

Figure 9-2(e)

Figure 9-2(f)

Figure 9-3 a) Body-centered cubic structure; b) face-centered

cubic structure.

Figure 9-4 Hexagonal

close-packed structure

Figure 9-5 Stacking of

close-packed spheres.

- 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.

- 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.

- 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.

- 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

- 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

- 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.