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.

o

cba

90

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

o

cba

90

Figure 9-2(b)

o

cba

90

Figure 9-2(c)

o

cba

90

Figure 9-2(d)

o

cba

90

Figure 9-2(e)

o

cba

90

o

o

cba

120

90

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).

1,1

1,1

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.

)100(),010(),001(

• 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

)112(2/1

1,

11

,11

• 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.