VðCl
À
Þ ¼
4p r Cl
ð Þ
3
3
¼
4p
3
a
2
3 ¼
pa
3
6
Similarly,
VðCs
þ
Þ ¼
4p r Cs
ð Þ
3
3
¼
4p r Cl
ð Þ
3
3
ffiffi ffi
3
p À 1
3 ¼
pa
3
6
ffiffi ffi
3
p À 1
3
Therefore, the packing efficiency of CsCl with one ion each is
Efficiency
ð
Þ CsCl ¼
1
ð Þpa
3
6
þ
1
ð Þpa
3
6
ffiffi ffi
3
p À 1
À
Á 3
a 3
¼ 0:52 þ 0:21 ¼ 0:76 ¼ 76%
Multiple Choice Questions (MCQ)
1. In a unit cell of plane hexagonal lattice with a = b = 10 Å and c = 120°, an
atom has its fractional coordinates x = 0.5 and y = 0.5. Draw four unit cells on
its edges anticlockwise starting from its right, their fractional coordinates in the
same order are:
(a) (1.5, 0.5), (0.5, 1.5), (−0.5, 0.5), (0.5, −0.5)
(b) (0.5, 0.5), (0.5, 0.5), (0.5, 1.5), (0.5, 1.5)
(c) (1.5, 1.5), (1.5, 1.5), (0.5, 1.5), (−0.5, −1.5)
(d) (0.5, 1.5), (0.5, 1.5), (0.5, 1.5), (0.5, 1.5)
2. In a Triclinic unit cell with a = 5 Å, b = 6 Å, c = 8 Å and a, b, c not equal to
90°, an atom has its fractional coordinates x = 0.6 and y = 0.5 and z = 0.25.
Draw six unit cells on its faces, their fractional coordinates counterclockwise
are:
(a) (1.6, 0.5, 0.25), (1.6, 0.5, 1.25), (1.6, 0.5, 0.25), (0.6, 1.5, 0.25), (1.6, 0.5,
1.25), (0.6, −0.5, 0.25)
(b) (1.6, 0.5, 0.25), (0.6, 1.5, 0.25), (0.6, 0.5, 1.25), (−0.4, 0.5, 0.25), (0.6,
−0.5, 0.25), (0.6, 0.5, −0.75)
(c) (0.6, 0.5, 0.25), (1.6, 0.5, 0.25), (1.6, 1.5, 0.25), (0.6,- 0.5, 0.25), (0.6, 0.5,
1.25), (−1.6, 0.5, 0.25)
(d) (1.6, 0.5, 0.75), (1.6, 0.5, 0.25), (0.6, 0.5, 0.25), (1.6, 1.5, 0.25), (−0.6, 0.5,
0.25), (1.6, 0.5, 0.25)
3. The set of fractional coordinates of equivalent positions related to a 2-fold
rotational symmetry parallel to z-axis in 2-D are:
(a) (x, y), (x, y)
3.6 Packing Efficiency
127
À
Þ ¼
4p r Cl
ð Þ
3
3
¼
4p
3
a
2
3 ¼
pa
3
6
Similarly,
VðCs
þ
Þ ¼
4p r Cs
ð Þ
3
3
¼
4p r Cl
ð Þ
3
3
ffiffi ffi
3
p À 1
3 ¼
pa
3
6
ffiffi ffi
3
p À 1
3
Therefore, the packing efficiency of CsCl with one ion each is
Efficiency
ð
Þ CsCl ¼
1
ð Þpa
3
6
þ
1
ð Þpa
3
6
ffiffi ffi
3
p À 1
À
Á 3
a 3
¼ 0:52 þ 0:21 ¼ 0:76 ¼ 76%
Multiple Choice Questions (MCQ)
1. In a unit cell of plane hexagonal lattice with a = b = 10 Å and c = 120°, an
atom has its fractional coordinates x = 0.5 and y = 0.5. Draw four unit cells on
its edges anticlockwise starting from its right, their fractional coordinates in the
same order are:
(a) (1.5, 0.5), (0.5, 1.5), (−0.5, 0.5), (0.5, −0.5)
(b) (0.5, 0.5), (0.5, 0.5), (0.5, 1.5), (0.5, 1.5)
(c) (1.5, 1.5), (1.5, 1.5), (0.5, 1.5), (−0.5, −1.5)
(d) (0.5, 1.5), (0.5, 1.5), (0.5, 1.5), (0.5, 1.5)
2. In a Triclinic unit cell with a = 5 Å, b = 6 Å, c = 8 Å and a, b, c not equal to
90°, an atom has its fractional coordinates x = 0.6 and y = 0.5 and z = 0.25.
Draw six unit cells on its faces, their fractional coordinates counterclockwise
are:
(a) (1.6, 0.5, 0.25), (1.6, 0.5, 1.25), (1.6, 0.5, 0.25), (0.6, 1.5, 0.25), (1.6, 0.5,
1.25), (0.6, −0.5, 0.25)
(b) (1.6, 0.5, 0.25), (0.6, 1.5, 0.25), (0.6, 0.5, 1.25), (−0.4, 0.5, 0.25), (0.6,
−0.5, 0.25), (0.6, 0.5, −0.75)
(c) (0.6, 0.5, 0.25), (1.6, 0.5, 0.25), (1.6, 1.5, 0.25), (0.6,- 0.5, 0.25), (0.6, 0.5,
1.25), (−1.6, 0.5, 0.25)
(d) (1.6, 0.5, 0.75), (1.6, 0.5, 0.25), (0.6, 0.5, 0.25), (1.6, 1.5, 0.25), (−0.6, 0.5,
0.25), (1.6, 0.5, 0.25)
3. The set of fractional coordinates of equivalent positions related to a 2-fold
rotational symmetry parallel to z-axis in 2-D are:
(a) (x, y), (x, y)
3.6 Packing Efficiency
127