NATIONAL INSTITUTE OF

TECHNOLOGY, JAMSHEDPUR

ASSIGNMENT

MATERIAL SCIENCE

ATOMIC PACKING FACTOR

SUBMITTED TO: SUBMITTED BY:

RANJIT SIR ROLL NO.-
7,37,67,97

TEAM MEMBERS
NAME REGISTRATION NO.
SARVESH YADAV (2020UGCS007)
ANKUR KUMAR (2020UGCS037)
ANAND KUMAR SAW (2020UGCS067)
SAHIL KUMAR (2020UGCS097)
TOPIC:

ATOMIC PACKING FACTOR

CONTENT
1) ATOMIC PACKING FACTOR

2) UNIT CELL

3) SIMPLE CUBIC

4) BODY CENTERED CUBIC

5) FACE CENTERED CUBIC

6) HEXAGONAL CLOSED PACKED

7) DIAMOND CUBIC(D.C.) STRUCTURE

8) CONCLUSION
 ❑ Atomic Pacing Factor of Simple Cubic
Atomic packing factor:
= (No. of atoms/unit cell) × (volume of each atom)
The packing factor or atomic packing fraction is
the fraction of space occupied by atoms, ( volume of a unit cell)
assuming that the atoms are hard spheres.
= 1×(4/3)×π×r3

Expression to calculate packing factor is (2r)3

= (No. of atoms/unit cell) × (volume of each atom) = 0.52

( volume of a unit cell ) Atomic Packing factor of Simple Cubic = 52%

UNIT CELL Body Centered cubic:

❑ The unit cell is subdivision of a lattice that still
retain the overall characteristics of the entire
lattice.

❑ A lattice is a collection of point called lattice

point which are arranged in a periodic pattern.

Number of atoms per unit cell ❑ Body centered cubic structure has a unit cell
having an atom at its centre and eight atoms at
❑ A specific number of lattice points define each of
corners.
the unit cell.
❑ Each corner atom is bonded to 8 other atoms and
❑ For example the corner of the cell are easily 1 atom is present in the centre.
identified as are the body centered and face
centered position. ❑ As (8x1/8)+1

❑ So no. of atom in BCC unit cell=2

❑ 4r=√3a
Simple cubic:
❑ Examples of metals with the bcc structure
are alpha-iron, tungsten, chromium, and beta-
titanium.

❑ The BCC structure has a coordination number of

eight.

❑ Materials having BCC structure are soft

and malleable.
❑ Unit cell of Simple cubic structure has
eight corner atoms which are bonded to eight ❑ Atomic Packing factor
other atoms.
= (No. of atoms/cell)×(volume of each atom)
❑ All sides of cube are equal in length a=b=c
( volume of a unit cell )
❑ Here, a=2r
= 2×(4/3)×π×r3
3 3
❑ Volume of a unit cell=a =(2r)
(4r/√3)3
❑ No. of atoms per unit cell = (1/8)×8 =1
 = 0.68 hexagon around a central atom. In between these
planes is a half hexagon of 3 atoms.
Atomic Packing factor of BCC =68 %
❑ There are two lattice parameters in HCP, a and H,
Face Centered Cubic: representing the basal and height parameters
respectively.

❑ a=2r ; H=4*(√(2/3))r

❑ A FCC has a unit cell having an atom at the

centre of each face of unit cell and 8 corner
atoms.

❑ Each of 8 corner atom contribute 1/8 and each

❑ face atom contribute 1/2 in each unit cell.

❑ As (8 x1/8)+((1/2 )x 6)=4

❑ No. of atoms in FCC unit cell=4

❑ H/a ratio of 1.633
❑ 4r=√2×a
❑ No. of atoms per unit cell= (1/6)×12+(½)×2+3=6
❑ Examples: Aluminium and Copper has FCC
structure

❑ The FCC structure has a coordination number of

twelve(12).

❑ FCC structure are usually harder.

❑ Atomic Packing factor

= (No. of atoms/cell)×(volume of each atom)

(volume of a unit cell)

= 4×(4/3)×π×r3

(2√2r)3

= 0.74 ❑ ABAB... Stacking Sequence of HCP

3D Projection 2D Projection
Atomic Packing factor of FCC=74%

HEXAGONAL CLOSED
PACKED:
❑ Cell of an HCP lattice is visualized as a top and
bottom plane of 7 atoms, forming a regular
❑ Co-ordination No. in this structure =12
 ❑ Examples: Cadmium , Zinc , Titanium are some
of the elements with Hexagonal closed packing
structure. COORDINATION NUMBER

❑ Atomic Packing Factor = ❑ In a diamond crystal, the carbon atom are linked
by the directional covalent bond
(No. of atoms/unit cell)×(volume of each atom)
❑ Carbon atoms from covalent bonds with four
( volume of a unit cell ) other carbon atoms that occupy four corners of
the cube in a tetrahedral structure.
= (6)×(4/3)×π×r3
❖ In the diamond lattice each atom has four nearest
6×√3/4×a2H
neighbour with which it forms covalent bonds
= (6)×(4/3)×π×r 3 ,thus the coordination number of diamond
crystal is 4.
6×√3/4×(2r)24r√(2/3)

= π/√18 = 0.74

Atomic Packing Factor of HCP=74%

Diamond cubic (D.C.) structure:

❑ The diamond lattice can be considered to be
formed by interpenetrating two FCC lattices
along the body diagonal by ¼ cube edge.

❑ One sub lattice has its origin at the point (0,0,0 )

and the other at the point quarter of the way
ATOMIC RADIUS
along the body diagonal (at the point a/4,a/4,a/4).
We know that the length of body diagonal in a
❑ The diamond cubic structure is loosely packed cube is √3a. The atoms present at quarter way along the
,since each atom has four nearest neighbor . body diagonal and that at the corners are considered for
calculating the relation between radius of atom and edge
NUMBER OF ATOMS PER UNIT CELL length of cube. Distance between these atoms is √3/4*a.

❑ In the eight corner each atom is shared by eight Therefore, we can notice that
unit cells
2r = √3/4*a
1/8 *8=1
⇨ r = √3/8*a
❑ The each 6 faces ever single atom which is
shared by 2 unit cells

½*6=3

❑ And in every unit cell it has 4 atoms inside 4.

❑ 1+3+4=8

PACKING FACTOR

❑ Packing factor

= number of the atoms in the unit cell×volume of

one atom/volume of the unit cell
 =8*(4/3*π𝒓𝟑)/𝒂𝟑

=(π√3)/16

=0.34

❑ Packing factor =34%

❑ Thus it is loosely packed structure

❑ Example: Carbon ,silicon ,germanium ,and grey

tin crystallize in the diamond structure.

CONCLUSION:
Overall, we can notice that the order of atomic packing
fraction is as follows:

FCC=Hexagonal>BCC>Simple Cubic>Diamond cubic

Therefore, it can be said that the density of FCC or

Hexagonal is maximum while density of diamond is
minimum and we can also say that atoms of FCC crystal is
more closely packed as compared to other crystal structure.

BIBLIOGRAPHY
❖ Materials Science by R.S Khurmi, S. Chand
Publication.

❖ Materials Science and Engineering by V

Raghavan, Eastern Economy Edition PHI
publication.

❖ Materials Science and Engineering by William

Callister, Wiley Publication.