MM409: Advanced engineering materials
Crystallography
W.D. Callister, Materials science and engineering an introduction,
5th Edition, Chapter 3
Crystal structure
• The solid materials may be classified according to the
regularity with which atoms or ions are arranged with respect
to one another
• A crystalline materials is one in which the atoms are situated
in a repeating or periodic array over large atomic distances
• In crystalline structures, atoms are thought of as being solid
spheres having well-defined diameters
• This is termed the atomic hard sphere model in which spheres
representing nearest-neighbor atoms touch one another
An example of the
hard sphere model
Unit cells
• The atomic order in crystalline solids indicates that
small groups of atoms form a repetitive pattern.
• Unit cells subdivide the structure into small repeated
entities.
• A unit cell is chosen to represent the symmetry of the
crystal structure.
• Unit cell is chosen to represent the symmetry of the
crystal structure
• Thus, the unit cell is the basic structural unit or
building block of the crystal structure.
Metallic crystal structure
BCC
FCC
Density computations
Crystal systems
The unit cell geometry is completely
defined in terms of six
parameters:
3 edge lengths, a, b and c
3 interaxial angles ,  and 
These are termed as ‘lattice parameters’
of the crystal structure.
Fig: A unit cell with x, y, and z coordinate
axes, showing axial lengths (a, b, and c)
and interaxial angles (, , and )
Crystallographic directions and planes
When dealing with crystalline materials, it is often
becomes necessary to specify some particular
crystallographic plane of atoms or a crystallographic
direction.
3 integers or indices are used to designate directions
and planes.
The basis for determining index values is the unit cell.
Coordinate system consists of three (x, y and z) axes.
Crystallographic directions
A crystallographic direction is defined as a line
between two points, or a vector.
Steps:
1. A vector of convenient length is positioned
such that it passes through the origin of
the coordinate system
2. The length of the vector projection on each
of the 3 axes is determined; a, b & c
3. Reduce them to the smallest integer
values; u, v & w
4. The 3 indices are enclosed in square
brackets, thus: [uvw].
The [100], [110], and [111]
directions with in a unit cell.
Crystallographic planes
Crystallographic planes are specified by three Miller
indices as (hkl).
Any two planes parallel to each other are equivalent
and have identical indices.
A unit cell with x, y, and z
coordinate axes, showing axial
lengths (a, b, and c) and
interaxial angles (, , and ).
Steps in determining (hkl)
1. Define origin of axis
2. At this point the crystallographic plane either intersects or
parallels each of the 3 axes; the length of the planar intercepts
for each axis is determined in terms of the lattice parameter a,
b and c
3. Reciprocal of these numbers are taken
4. These numbers are changed to set of smallest integers
5. Enclose integer indices within parentheses (hkl)
Fig: Representations of a
series each of (110) and
(111) crystallographic
planes.
Atomic arrangements
Atomic arrangement depends on crystal structure
Fig: (a) Reduced-sphere FCC unit cell with
(110) plane. (b) Atomic packing of an FCC
(110) plane. Corresponding atom positions
from (a) are indicated
Fig: (a) reduced-sphere BCC unit cell
with (110) plane. (b) Atomic packing of
a BCC (110) plane. Corresponding
atom positions from (a) are indicated
Closed-packed crystal structures
ABC, ABA, ACB, ACA
Figure: Close-packed plane
staking sequence for
hexagonal close-packed.
Figure: Close-packed plane
staking sequence for FCC.
Noncrystalline solids
Fig: Two-dimensional schemes of the structure of (a)
crystalline silicon dioxide and (b) noncrystalline silicon
dioxide.