3.6.2 In 3-D
Packing
Efficiency
¼
Volume of the atoms ðin the unit cellÞ
Volume of the unit cell
ð3:22Þ
Being a ratio, the packing efficiency is a dimensionless quantity.
Solved Examples
Example 1 Determine the efficiencies of 2-D arrangement of identical spheres on
square and hexagonal unit cells.
Solution: Identical spheres in square and hexagonal unit cells (Fig. 3.12). From the
figure, we can determine the following:
Area of the circle = pR
2
Area of the square = 4R
2
Therefore,
Efficiency
ð
Þ S ¼
Area of the circle
Area of the square
¼
pR
2
4R
2
¼
p
4
¼ 78.5%
Similarly, for hexagonal unit cell,
Area of the circle = pR
2
Area of the triangle ABC ¼
1
2
 base  height
¼
1
2
 a  a sin 60
ðwhere a ¼ 2R and R ¼ a sin 60
Fig. 3.12 Equal spheres arranged in a square b hexagonal lattice in 2-D
3.6 Packing Efficiency
121
Packing
Efficiency
¼
Volume of the atoms ðin the unit cellÞ
Volume of the unit cell
ð3:22Þ
Being a ratio, the packing efficiency is a dimensionless quantity.
Solved Examples
Example 1 Determine the efficiencies of 2-D arrangement of identical spheres on
square and hexagonal unit cells.
Solution: Identical spheres in square and hexagonal unit cells (Fig. 3.12). From the
figure, we can determine the following:
Area of the circle = pR
2
Area of the square = 4R
2
Therefore,
Efficiency
ð
Þ S ¼
Area of the circle
Area of the square
¼
pR
2
4R
2
¼
p
4
¼ 78.5%
Similarly, for hexagonal unit cell,
Area of the circle = pR
2
Area of the triangle ABC ¼
1
2
 base  height
¼
1
2
 a  a sin 60
ðwhere a ¼ 2R and R ¼ a sin 60
Fig. 3.12 Equal spheres arranged in a square b hexagonal lattice in 2-D
3.6 Packing Efficiency
121