This document provides an overview of crystal structures and bonding in materials. It discusses topics such as the differences between crystalline and amorphous solids, unit cells, lattice structures, metallic crystal structures like body centered cubic and face centered cubic, atomic packing factors, and anisotropic vs isotropic materials. The key concepts covered include how crystal structures are composed of a periodic arrangement of points in a lattice, with atoms attached at each lattice point, and how properties can differ based on crystal structure and orientation.

1. Bonding &
Structure
Course : Engineering Materials
Course Code: ID-301
Credit Hours: 3
Topic :
Department of Chemical Engineering
NFC Institute of Engineering & Technology, Multan

2. Agenda
• Why we study structures
• Crystal Structure
• Types of Solids
• Lattice
• Unit Cell
• Lattice Parameters & Constants
• Classification of Lattices
• Metallic Crystal Structures
• Atomic Packing Factor
• Coordination Number
• Stacking Sequences
• Comparisons
• Anisotropy
• Isotropy

3. Why we Study Crystal Structure ?
• Properties Changes
• Slip Systems
Like Diamond & Graphite
Deformation Occurs by shear stress
DiamondGraphite

4. Crystal Structure
• The structure of all crystals can be described in term
of a lattice, with the group of atoms attached to every
lattice point
• Basis (Motif)
• Lattice + Motif = Crystal

5. Types of Solids
• Crystalline
• Amorphous
Long Range Order, Repetitive Manner, Periodic
Arrangement, High Bond Energy, Closed Packed
Structure, Sharp Melting Point, Cleavage Property,
(Ex: NaCl, CsF, Diamond ..)
Short Range Order, Random Manner, Non Periodic
Arrangement, Lower Bond Energy, Less Dense Packed,
Less Melting Point, Cleavage Property, (Ex: Glass,
Rubber, Plastic ..)

6. • Poly-crystalline
• Single Crystal
• If a crystalline material consists of only one large crystal, we refer
to it as a single crystal. Single crystals are useful in many electronic
and optical applications
• Atoms are in a repeating or periodic array over the entire extent of
the material
• Object composed of randomly oriented crystals, formed by rapid
solidification
• Comprised of many small crystals or grains

7. Formation of Polycrystalline
Material

8. Difference Between Lattice &
Crystal
• A 3D arrangement of atoms
• It’s physical object (ex: weight, density..)
• A 3D periodic arrangement of points
• It’s geometric Concept (ex: triangle,
Square..)

9. Energy and Packing of
Crystalline &
Amorphous Structure
Dense, regular-packed structures tend to
have lower energy

10. Lattice + Motif (Basis) =
Crystal ?
• A three dimensional periodic
array of points conceding
with atoms position.
Lattice

11. Unit Cell
• A region of space which can
generate the entire lattice or
(crystal) by repetition
through lattice translation.
• The small repeat entities of
crystal structure called unit
cell.
• Its help us to describe the
crystal structure may be
primitive and non primitive
Characterized by
1) Types of atoms & their
radii
2) Cell dimension
3) No: of atoms/unit cell
4) Coordination number
5) APF (Atomic packing
factor)

12. Square Unit Cell
Hexagonal
Unit Cell

13. Crystal Lattice
Squares Rectangles Hexagons

14. Lattice Parameters
• Interaxial angles
1
2
What do
you think
which
steps are
correct ?
Lattice
Parameter
Lattice
Parameter Unit Cell
Unit Cell
Lattice
Lattice
Motif Crystal
Crystal

15. Classification of Lattices
• 7 Crystal Systems or unit cell & 14 Bravais Systems
• Crystal Systems are
1) Cubic
2) Tetragonal
3) Orthorhombic
4) Rhombohedral
5) Hexagonal
6) Monoclinic
7) Triclinic

17. Metallic Crystal Structure
• The atomic bonding in this group of materials is metallic
and thus non-directional in nature
• For metals, using the hard-sphere model for the crystal
structure, each sphere represents an ion core
• Three principle crystal structures for metals are:
1) Body Centered Crystal (BCC)
2) Face Centered Crystal (FCC)
3) Hexagonal Closed Packed (HCP)

18. Body Centered Cubic
No: of atoms/unit
cell : 2
Coordination no : 8 APF = 0.68
Ex: Cr, Molybedenum,
Tantalum. Fe (alpha)

19. Face Centered Crystal
No: of atoms/unit
cell : 4
Coordination no :
12
APF = 0.74
Ex: Al, Cu, Pb, Ni,
Ag, Pt

20. Hexagonal Closed Packed
No: of atoms/unit
cell : 6
Coordination no :
12
APF = 0.74
Ex: Beryllium, Cadmium,
Titanium, Magnesium

21. Atomic Packing Factor
• Packing Efficiency
• It tell us how tightly atoms are packed
• The packing factor or atomic packing fraction is the
fraction of space occupied by atoms, assuming that the
atoms are hard spheres. The general expression for the
packing factor is
APF =
𝑵𝒐 𝒐𝒇 𝒂𝒕𝒐𝒎𝒔 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒄𝒆𝒍𝒍 𝒙 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒆𝒂𝒄𝒉 𝒂𝒕𝒐𝒎
𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒖𝒏𝒊𝒕 𝒄𝒆𝒍𝒍

22. Steps To Calculate Atomic
Packing Factor
1
2
4
3
Calculate Number of Atoms
Calculate Volume of Atoms
Calculate Atomic Radius
Steps
Calculate Area of Cube

23. Atomic Packing Factor (BCC)
APF =
𝑁𝑜 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑡𝑜𝑚
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙
• Coordinate Number : 8
• Number of Atoms : 2
• Volume of Atom :
4
3
𝜋𝑟3
= 8.373𝑟3
• Volume of Unit Cell : 𝑎3
• Atomic Radius :
√3
4
a

24. Atomic Packing Factor (BCC)
• Radius Calculation :
𝐶2
= 𝐴2
+ 𝐵 2
𝐴𝐷2
= 𝐴𝐶2
+ 𝐶𝐷2
As; AD = 4r
𝐴𝐶2
= 𝐴𝐵2
+ 𝐵𝐶2
𝐴𝐶2
= 𝑎2
+ 𝑎2
𝐴𝐷2 = 𝐴𝐶2 + 𝐶𝐷2
4𝑟2
= 2𝑎2
+ 𝑎2
16𝑟2
= 3𝑎2
r =
√3
4
a a =
4𝑟
√4 v=12.32𝑟3

25. • Volume of atom = 8. 𝟑𝟕𝟑𝒓 𝟑
• Volume of unit cell = 1𝟐. 𝟑𝟐𝒓 𝟑
Atomic Packing Factor (BCC)
APF =
𝑁𝑜 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑡𝑜𝑚
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙
APF =
8.373𝑟3
12.32𝑟3
APF = 0.68 or 68%

26. Atomic Packing Factor (FCC)
APF =
𝑁𝑜 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑡𝑜𝑚
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙
• Coordinate Number : 12
• Number of Atoms : 4
• Volume of Atom :
4
3
𝜋𝑟3
8.373𝑟3
• Volume of Unit Cell : 𝑎3
• Atomic Radius :
𝑎
2√2

27. Atomic Packing Factor (FCC)
APF =
𝑁𝑜 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑡𝑜𝑚
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙
APF =
4 𝑥 (
𝑎
2 2
)3 4
3
𝑥 3.14
𝑎3
APF = 0.74 or 74%

28. Atomic Packing Factor (HCP)
• Coordinate Number : 12
• Number of Atoms : 6
• Volume of Atom :
4
3
𝜋𝑟3
8.373𝑟3
• Volume of Unit Cell : 𝑉 = 𝐵 𝑥 𝐻
• Atomic Radius : 𝑎 = 2𝑟
• c/a : 1.633

29. Atomic Packing Factor (HCP)
APF =
𝑁𝑜 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑡𝑜𝑚
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙
• Volume of unit cell :
V = B x H
Base of triangle =
1
2
x a x asin60
=
3𝑎2
4
x 6
=
3 3
2
𝑎2
x c

30. Atomic Packing Factor (HCP)
APF =
6 𝑥
4
3
𝑥 3.14 𝑥 𝑟3
3 3
2
𝑎2 𝑥 𝑐
𝑎 = 2𝑟 𝑎𝑛𝑑
𝑐
𝑎
= 1.633
=
6 𝑥
4
3
𝑥 3.14 𝑥 (
𝑎
2
)3
3 3
2
𝑎2 𝑥 𝑐
=
12.55𝑎
10.392𝑐
= 1.207 0.612
APF = 0.74 or 74%
APF =
𝑁𝑜 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑡𝑜𝑚
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙

31. FCC Stacking Sequence

32. Closed
Packed
Crystal

33. Comparison of FCC,
BCC, HCP
Crystal
Structure
Coordination
Number
Packing
Factor
Closed Packed
Direction
BCC
FCC
HCP
8
12
12
0.68
0.74
0.74
Body
Diagonal
Face
Diagonal
Hexagonal
Side

34. • An anisotropic material is a material which does not behave
the same way in all directions. Take wood for example. Wood
is very strong along the grain. Against the grain, however, it
will easily break.
• A different chemical bonding in all directions is also a
condition for anisotropy
• Poly Crystal (Property Vary with Direction)
Anisotropy

35. Figure 7-1
Anisotropic
materials:
(a) rolled material,
(b) wood,
(c) glass-fiber cloth
in an epoxy matrix,
and
(d) a crystal with
cubic unit cell.

36. • The opposite of an anisotropic material is an isotropic
material. Most metals (steel, aluminum) are isotropic
materials. They respond the same way in all directions.
• Substances in which measured (physical)properties are
independent of the direction of measurement are isotropic.
• Ex : Single Crystal ( Properties do not vary with
direction)
• Real Materials (Isotropic)
Isotropy