Also, volume of the atom in three cases is:
V atom ¼
4
3
pR
3
Therefore, efficiencies in three cubic cases are:
Efficiency
ð
Þ SC ¼
1 Â 4pR
3
À
Á
=3
8R
3
¼
p
6
¼ 52%
Efficiency
ð
Þ BCC ¼
2 Â 4pR
3
À
Á =3
64R
3
=3
ffiffi ffi
3
p ¼
ffiffi ffi
3
p p
8
¼ 68%
and Efficiency
ð
Þ FCC ¼
4 Â 4pR
3
À
Á
=3
16
ffiffi ffi
2
p
R
3
¼
p
3
ffiffi ffi
2
p ¼ 74%
Example 3 Determine the number of atoms in the unit cells of simple hexagonal
(SH) and hexagonal close-packed (HCP) crystal structures. Calculate the packing
efficiency in each case.
Solution: In a hexagonal crystal system, the number of atoms per unit cell is given
by:
n ¼
N c
6
þ
N f
2
þ N b
Fig. 3.13 Sectional view of conventional unit cell showing relationships between the lattice
constant a and the atomic radius R in a sc b bcc, and c fcc unit cell
3.6 Packing Efficiency
123
V atom ¼
4
3
pR
3
Therefore, efficiencies in three cubic cases are:
Efficiency
ð
Þ SC ¼
1 Â 4pR
3
À
Á
=3
8R
3
¼
p
6
¼ 52%
Efficiency
ð
Þ BCC ¼
2 Â 4pR
3
À
Á =3
64R
3
=3
ffiffi ffi
3
p ¼
ffiffi ffi
3
p p
8
¼ 68%
and Efficiency
ð
Þ FCC ¼
4 Â 4pR
3
À
Á
=3
16
ffiffi ffi
2
p
R
3
¼
p
3
ffiffi ffi
2
p ¼ 74%
Example 3 Determine the number of atoms in the unit cells of simple hexagonal
(SH) and hexagonal close-packed (HCP) crystal structures. Calculate the packing
efficiency in each case.
Solution: In a hexagonal crystal system, the number of atoms per unit cell is given
by:
n ¼
N c
6
þ
N f
2
þ N b
Fig. 3.13 Sectional view of conventional unit cell showing relationships between the lattice
constant a and the atomic radius R in a sc b bcc, and c fcc unit cell
3.6 Packing Efficiency
123