Concepts of Crystal Geometry

crystal are arranged in a regular, repeated three-dimensional pattern. 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.

The unit cell is the smallest repetitive unit that there are 14 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

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.

x and z axes and intersects the y axis at one interatomic 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.

Any one of which can have the indices (100), (010), 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 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.