Efficiency
ð
Þ NaCl ¼
4Â 4pR
3
Na
À
Á
3
þ
4Â 4pR
3
Cl
À
Á
3
a 3
¼
16p
3
0:98
ð
Þ
3 þ 1:81
ð
Þ
3
5:58
ð
Þ
3
"
#
¼ 0:663 ¼ 66:3%
Example 5 Calculate the packing efficiency of CsCl structure where the Cs
+ ion is
at the body-centered position and Cl
− ions are at the corner positions of the unit cell
such that the central Cs
+ ion touches the neighboring 8 Cl
− ions.
Solution: Given: CsCl structure where the Cs
+ ion is at the body-centered position
and touches the 8 Cl
− ions. Efficiency of CsCl = ?
The CsCl crystal structure is shown in Fig. 3.16. Since the Cs
+ ions touches the
8 Cl
− ions, the lattice parameter a is given by
a ¼ 2r Cl ; or r Cl ¼ a=2
Similarly,
ffiffi ffi
3
p
a ¼ 2r Cl þ 2r Cs
or
ffiffi ffi
3
p  2r Cl ¼ 2r Cl þ 2r Cs
or
ffiffi ffi
3
p
r Cl ¼ r Cl þ r Cs
or r Cs ¼ r Cl ð
ffiffi ffi
3
p À 1Þ
Now, the volume of the Cs
+ ion is
Fig. 3.16 a Caesium chloride structure with axial directions b Cs and Cl atoms in (1 10) plane
126
3 Unit Cell Calculations