V atom ¼
4
3
pR
3
Therefore, efficiencies in two hexagonal cases are:
Efficiency
ð
Þ SH ¼
3 Â 4pR
3
À
Á =3
12
ffiffi ffi
3
p
R
3
¼
p
3
ffiffi ffi
3
p ¼ 60%
and Efficiency
ð
Þ HCP ¼
3 Â 4pR
3
À
Á =3
12
ffiffi ffi
2
p
R
3
¼
p
3
ffiffi ffi
2
p ¼ 74%
Example 4 The ionic radii of Na
+ and Cl
− are 0.98 Å and 1.81 Å, respectively.
Calculate the packing efficiency of NaCl structure.
Solution: Given: The ionic radius of Na
+ = 0.98 Å, Cl
− = 1.81 Å. Efficiency of
(NaCl) = ?
The NaCl crystal structure is shown in Fig. 3.15.The Na
+ and Cl
− ions are
supposed to touch each other, so that the lattice parameter is:
a ¼ 2 radius of Na
þ
þ radius of Cl
À
ð
Þ
¼ 2ð0:98 ˚
A þ 1:81 ˚
AÞ ¼ 5:58 ˚
A
Therefore, the volume of the unit cell is
V ¼ a
3
¼ 5.58
ð
Þ
3 ˚
A
3
Further, from the figure, we observe that there are four NaCl molecules in a unit
cell. Therefore, the packing efficiency of NaCl with 4 ions each is
Fig. 3.15 Sodium chloride
structure
3.6 Packing Efficiency
125