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.

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Concepts of Crystal Geometry

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Concepts of crystal geometry

Concepts of Crystal Geometry


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.


Concepts of crystal geometry

  • 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.


Concepts of crystal geometry

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


Concepts of crystal geometry

Figure 9-2(b)


Concepts of crystal geometry

Figure 9-2(c)


Concepts of crystal geometry

Figure 9-2(d)


Concepts of crystal geometry

Figure 9-2(e)


Concepts of crystal geometry

Figure 9-2(f)


Concepts of crystal geometry

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

cubic structure.


Concepts of crystal geometry

Figure 9-4 Hexagonal

close-packed structure

Figure 9-5 Stacking of

close-packed spheres.


Concepts of crystal geometry

  • 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.


Concepts of crystal geometry

  • 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.


Concepts of crystal geometry

  • 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.


Concepts of crystal geometry

  • 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


Concepts of crystal geometry

  • 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


Concepts of crystal geometry

  • 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.


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