The document serves as a basic introduction to crystallography, covering essential concepts including crystal systems, lattice structures, unit cells, coordination numbers, and atomic packing factors. It details the characteristics of crystalline materials, such as their long-range order and repetitive patterns, and explores various cubic structures like simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc). Additionally, it discusses the atomic packing factor and stacking sequences relevant to these structures.

1. Lab# 02
Basic Introduction to
Crystallography
By Yousaf Haroon
Lab Engineer FMCE

2. Contents of lab 2
• Crystallography and crystal systems.
• Lattice and motif
• Unit cell
• Coordination number
• How to find CN of SC,FCC and BCC.
• Number of atoms in a unit cell ( SC,FCC,BCC)
• Atomic Packing Factor (SC, FCC,BCC)

3. The branch of science concerned with the structure and
properties of crystals.

4. Crystal structure
The atomic arrangement inside a crystalline
material is called crystal structure.

5. Why do we study
crystal structure?
1. Properties
2. Applications
Similar materials having different
properties and applications?

6. Fundamental concepts
Crystalline/Non crystalline materials
• A crystalline material is one in which the
atoms have a
1. Repeating or periodic array over large
atomic distances
2. Long-range order exists.
3. Repetitive three-dimensional pattern,
in which each atom is bonded to its
nearest-neighbor atoms.
4. All metals, many ceramic materials, and
certain polymers
Amorphous material?

7. Hard sphere model
• When describing crystalline
structures, atoms (or ions) 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.

8. Lattice and motif
Three dimensional (infinite)
array of regularly spaced points.
Crystal structure= lattice + motif
Lattice points are the
connection points in a
unit cell
Arrangement of
the building
blocks(atoms) of
the unit cell

9. CRYSTAL SYSTEMS
• Lattice parameters
•Axis and angles a, b, c &
•α,ꞵ,γ
•Seven possible
combinations of a, b, c
& α,ꞵ,γ resulting in
seven crystal systems

11. Unit cell
•Small repeating
entities
•Unit cells for most
crystal structures are
parallelepipeds or
prisms having three
sets of parallel faces.

12. Coordinate number
and point
coordinates
• Number of nearest-
neighbors or touching
atoms
• The position of any point
located within a unit cell
may be specified in terms
of its coordinates as
fractional multiples of the
unit cell edge lengths (i.e.,
in terms of a, b, and c)

13. Number of atoms in a unit cell

14. The Crystal Structure of Metals
Metals usually have crystal
structures:
1. Simple cubic
2. Body centered cubic
3. Face centered cubic

15. 15
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
Simple Cubic Structure (SC)
No. of atoms/unit cell : 8 corners x 1/8 = 1

16. 16
• Coordination # = 8
• Atoms touch each other along cube diagonals.
Body Centered Cubic Structure (BCC)
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8

17. 17
• Coordination # = 12
• Atoms touch each other along face diagonals.
Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

18. Hexagonal close-packed
Coordination # = 12
ex: Cd, Mg, Ti, Zn
6 atoms/unit cell: 3 center + ½ x 2
top and bottom + 1/6 x 12 corners

19. Atomic packing factor
•How closely atoms are packed in a unit cell.
•How much portion an atom occupy.
• The APF is the sum of the sphere volumes of all atoms
within a unit cell!

20. 20
• APF for a simple cubic structure = 0.52
APF =
a3
4
3
p (0.5a) 3
1
atoms
unit cell
atom
volume
unit cell
volume
Atomic Packing Factor (APF)
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
close-packed directions
a
R=0.5a
contains 8 x 1/8 =
1 atom/unit cell

21. 21
Atomic Packing Factor: BCC
a
APF =
4
3
p ( 3a/4)3
2
atoms
unit cell atom
volume
a3
unit cell
volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
a
R
a
2
a
3

22. 22
• APF for a face-centered cubic structure = 0.74
Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3
p ( 2a/4)3
4
atoms
unit cell atom
volume
a3
unit cell
volume
Close-packed directions:
length = 4R = 2 a
Unit cell contains:
6 x1/2 + 8 x1/8
= 4 atoms/unit cell
a
2 a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 3e.

23. Stacking sequences

24. 24
A sites
B B
B
B
B
B B
C sites
C C
C
A
B
B sites
• ABCABC...
Stacking Sequence
• FCC Unit Cell
FCC Stacking Sequence
B B
B
B
B
B B
B sites
C C
C
A
C C
C
A
A
B
C

25. 25
• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
Adapted from Fig. 3.3(a),
Callister & Rethwisch 3e.
Hexagonal Close-Packed Structure (HCP)
6 atoms/unit cell
• c/a = 1.633
c
a
A sites
B sites
A sites Bottom layer
Middle layer
Top layer