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Properties of engineering materials
Chapter 3 -
materials
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Metals (ferrous and non ferrous)
Ceramics
Polymers
Composites
Advanced (biomaterials, semi
conductors)
Chapter 3 - 2
Properties
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Optical
Electrical
Thermal
Mechanical
Magnetic
Deteriorative
Chapter 3 - 3
The Structure of Crystalline Solids
- Crystalline : atoms/ crystals
- Non crystalline or amorphous
- Solidification?
Chapter 3 - 4
Packing of crystals:
Non dense =random
Dense = regular
Chapter 3 - 5
Unit cell and lattice parameters
• Unit cell : small repeated
entities
• It is the basic structural unit or
building block of the crystal
structure
• Its geometry is defined in
terms of 6 lattice parameters:
3 edges length (a, b, c)
3 interaxial angles
Chapter 3 - 6
Unit cell parameters
Chapter 3 - 7
Chapter 3 - 8
Metallic crystals
• 3 crystal structures:
– Face-Centered Cubic FCC
– Body-Centered Cubic BCC
– Hexagonal close-packed HCP
• Characterization of Crystal structure:
– Number of atoms per unit cell
– Coordination number (Number of nearest
neighbor or touching atoms)
– Atomic Packing Factor (APF) : what fraction of
the cube is occupied by the atoms
Chapter 3 - 9
The Body-Centered Cubic Crystal
Structure
• Coordination no. = ?
• Number of atoms per unit
cell = ? ?
• what fraction of the cube is
occupied by the atoms or
Atomic packing factor
Chapter 3 - 10
The Body-Centered Cubic Crystal
Structure
Chapter 3 - 11
APF = 0.68 solve ?
Relation between r and a
Chapter 3 - 12
Solution
Chapter 3 - 13
The Face-Centered Cubic Crystal
Structure
• Coordination no. = ?
• Number of atoms per unit
cell = ? ?
• what fraction of the cube is
occupied by the atoms or
Atomic packing factor
Chapter 3 - 14
The Face-Centered Cubic Crystal
Structure
Chapter 3 - 15
APF = 0.74 solve ?
Relation between r and a
Chapter 3 - 16
Solution
Chapter 3 - 17
The Hexagonal Close-Packed Crystal
Structure
Chapter 3 - 18
HCP
• The coordination number and APF
for HCP crystal structure are same
as for FCC: 12 and 0.74,
respectively.
• HCP metal includes: cadmium,
magnesium, titanium, and zinc.
Chapter 3 - 19
Examples
Chapter 3 - 20
Polymorphism
Chapter 3 - 21
THEORETICAL DENSITY
Chapter 3 - 22
Crystallographic Points,
Directions, and Planes
Chapter 3 - 23
POINT COORDINATES
Point position
specified in terms
of its coordinates
as fractional
multiples of the
unit cell edge
lengths
( i.e., in terms of X
a b and c)
Z
1,1,1
0,0,0
Y
Chapter 3 - 24
Solve point coordinates (2,3,5,9)
Chapter 3 - 25
solution
Chapter 3 - 26
Crystallographic directions
1- A vector of convenient length is positioned such that
it passes through the origin of the coordinate system.
Any vector may be translated throughout the crystal
lattice without alteration, if parallelism is maintained.
2. The length of the vector projection on each of the
three axes is determined; these are measured in
terms of the unit cell dimensions a, b, and c.
3. These three numbers are multiplied or divided by a
common factor to reduce them to the smallest
integer.
4. The three indices, not separated by commas, are
enclosed as [uvw]. The u, v, and w integers
correspond to the reduced projections along x, y, and
z axes, respectively.
Chapter 3 - 27
Example
Chapter 3 - 28
Solution
Chapter 3 - 29
Z
[0 1 0]
[1 0 1]
Y
X
Chapter 3 - 30
Planes ( miller indices)
1- If the plane passes through the selected origin, either another parallel plane
must be constructed within the unit cell by an appropriate translation, or a
new origin must be established at the corner of another unit cell.
2. At this point the crystallographic plane either intersects or parallels each of
the three axes; the length of the planar intercept for each axis is determined
in terms of the lattice parameters a, b, and c.
3. The reciprocals of these numbers are taken. A plane that parallels an axis
may be considered to have an infinite intercept, and, therefore, a zero index.
4. If necessary, these three numbers are changed to the set of smallest integers
by multiplication or division by a common factor.
5. Finally, the integer indices, not separated by commas, are enclosed within
parentheses, thus: (hkl).
Chapter 3 - 31
example
Chapter 3 - 32
Solution
Chapter 3 - 33
Example : solve
Z
Y
X
Chapter 3 - 34
solve
z
c
z
c
a
x

y
a

b
x

y
b
Chapter 3 - 35
Crystallographic Planes
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
a
1
1/1
1
1
4.
Miller Indices
(110)
example
1. Intercepts
2. Reciprocals
3.
Reduction
a
1/2
1/½
2
2
4.
Miller Indices
(100)
b
1
1/1
1
1
c

1/
0
0
c
y
b
a
x
b

1/
0
0
c

1/
0
0
z
c
y
a
b
x
Chapter 3 - 36
Crystallographic Planes
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
4.
Miller Indices
a
1/2
1/½
2
6
(634)
b
1
1/1
1
3
c
c
3/4
1/¾
4/3

4 a
x


y
b
Chapter 3 - 37
FCC Stacking Sequence
repeated every third plane
Chapter 3 - 38
(HCP)
A sites
B sites
A sites
repeated every second plane
Chapter 3 - 39
Polycrystals
• Most engineering materials are polycrystals.
1 mm
Chapter 3 - 40
The various stages in solidification of polycrystalline
materials.
Polycrystalline crystalline solids
composed of many crystals
or grains.
Various stages in solidification:
a. Small crystallite nuclei
b. Growth of the crystallites;
of obstruction of some
grains that are adjacent to
one another is also shown.
c. Upon completion of
solidification, grains that are
adjacent to one another is
also shown.
d. Grain structure as it would
appear under the microscope.
Chapter 3 - 41
Anisotropy
• Physical properties (e.g., Elastic modulus, index
of refraction) of single crystals of some
substances depend on crystallographic direction
in which measurements are taken (i.e.,
anisotropy).
• It results from variation of atomic or ionic spacing
with crystallographic direction.
• Isotropic materials: Substances in which
measured properties are independent of the
direction of measurement.
Chapter 3 - 42
Chapter 3 - 43
X – ray diffraction
Chapter 3 - 44
X ray diffraction
Chapter 3 - 45
Chapter 3 - 46
Case study: explain why?
Group of four . Dead line: 30/12/ 2015
BCC crystal structure, h+ k+ l must be even
FCC, h, k, and l must all be either odd or even
Chapter 3 - 47
Chapter 3 - 48
Linear and Planar Densities
• Atomic linear density:
LD = number of atoms centered on direction vector /
length of direction vector
• Planar density: Fraction of total crystallographic
plane area that is occupied by atoms
(represented as circles).
• PD = number of atoms centered on a plane / area of
plane
• Linear and planar densities are important for
deformation
Chapter 3 - 49
Indices?
Chapter 3 - 50
notes
for cubic crystals
- Planes having the same indices, irrespective of the
order and sign are equivalent. For example, both (123)
and (312) belong to the family ( 123)
Chapter 3 - 51
Chapter 3 - 52