MM409: Advanced engineering materials
W.D. Callister, Materials science and engineering an introduction,
5th Edition, Chapter 3
• 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
• The atomic order in crystalline solids indicates that
small groups of atoms form a repetitive pattern.
• Unit cells subdivide the structure into small repeated
• A unit cell is chosen to represent the symmetry of the
• Unit cell is chosen to represent the symmetry of the
• Thus, the unit cell is the basic structural unit or
building block of the crystal structure.
Metallic crystal structure
The unit cell geometry is completely
defined in terms of six
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
3 integers or indices are used to designate directions
The basis for determining index values is the unit cell.
Coordinate system consists of three (x, y and z) axes.
A crystallographic direction is defined as a line
between two points, or a vector.
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 , , and 
directions with in a unit cell.
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
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
Figure: Close-packed plane
staking sequence for FCC.
Fig: Two-dimensional schemes of the structure of (a)
crystalline silicon dioxide and (b) noncrystalline silicon