1. Concepts of Crystal Geometry

2. 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.

3. 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.

4. 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

5. Figure 9-2(b)

6. Figure 9-2(c)

7. Figure 9-2(d)

8. Figure 9-2(e)

9. Figure 9-2(f)

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

11. Figure 9-4 Hexagonal
close-packed structure
Figure 9-5 Stacking of
close-packed spheres.

12. 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.

13. 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.

14. 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.

15. 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

16. 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

17. 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.