cos a
0
¼
0
1 þ 4 þ 0
ð
Þ
1=2 ð0 þ 0 þ 1Þ
1=2
¼ 0
) a
0
¼ 90
And the angle between the edges with the end coordinates 3, 1, 0 and 0, 0, 1 is
given by
cos b
0
¼
0
9 þ 1 þ 0
ð
Þ
1=2 ð0 þ 0 þ 1Þ
1=2
¼ 0
) b
0
¼ 90
(c) The new unit cell with a′ = 16.16Å, b′ = 13Å, c′ = 7Å and a′ = b′ = 90°,
c′ = 45° is a monoclinic lattice and its volume is given by
V m ¼ a
0 b
0 c
0 sin 45
¼ 16:16 Â 13 Â 7 Â sin 45
¼ 1039:84 ˚
A
3
(d) Ratio of the two volumes ¼
1039:84
210
¼ 4:95 ffi 5
This implies that the new unit cell is non-primitive and the number of lattice
points in it is 5.
Example 8 A rhombohedral unit cell has a r = 5Å and a = 75°. Calculate: (a) the
volume of the unit cell (b) the dimension of the triply primitive hexagonal unit cell
that may be chosen (c) the volume of the hexagonal unit cell (d) the number of
lattice points in the hexagonal unit cell.
Solution: Given: a r = 5Å and a = 75°, triply primitive hexagonal unit cell.
(a) The volume of a rhombohedral unit cell is given by
V R ¼ a
3
R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À 3 cos 2 a þ 2 cos 3 a
p
¼ 125
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À 3 cos 2 75 þ 2 cos 3 75
p
¼ 125
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À 0:2 þ 0:035
p
¼ 114:2 ˚
A
3
(b) Take a plane projection of the rhombohedral unit cell and choose a triply
primitive hexagonal unit cell (Fig. 1.17) whose unit cell parameters are a =
b 6 ¼ c, a = b = 90° and c = 120°. In order to calculate a H and c H , take projection of a R on c-axis. Therefore,
1.2 Choice of Axes and Unit Cells
15
0
¼
0
1 þ 4 þ 0
ð
Þ
1=2 ð0 þ 0 þ 1Þ
1=2
¼ 0
) a
0
¼ 90
And the angle between the edges with the end coordinates 3, 1, 0 and 0, 0, 1 is
given by
cos b
0
¼
0
9 þ 1 þ 0
ð
Þ
1=2 ð0 þ 0 þ 1Þ
1=2
¼ 0
) b
0
¼ 90
(c) The new unit cell with a′ = 16.16Å, b′ = 13Å, c′ = 7Å and a′ = b′ = 90°,
c′ = 45° is a monoclinic lattice and its volume is given by
V m ¼ a
0 b
0 c
0 sin 45
¼ 16:16 Â 13 Â 7 Â sin 45
¼ 1039:84 ˚
A
3
(d) Ratio of the two volumes ¼
1039:84
210
¼ 4:95 ffi 5
This implies that the new unit cell is non-primitive and the number of lattice
points in it is 5.
Example 8 A rhombohedral unit cell has a r = 5Å and a = 75°. Calculate: (a) the
volume of the unit cell (b) the dimension of the triply primitive hexagonal unit cell
that may be chosen (c) the volume of the hexagonal unit cell (d) the number of
lattice points in the hexagonal unit cell.
Solution: Given: a r = 5Å and a = 75°, triply primitive hexagonal unit cell.
(a) The volume of a rhombohedral unit cell is given by
V R ¼ a
3
R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À 3 cos 2 a þ 2 cos 3 a
p
¼ 125
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À 3 cos 2 75 þ 2 cos 3 75
p
¼ 125
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À 0:2 þ 0:035
p
¼ 114:2 ˚
A
3
(b) Take a plane projection of the rhombohedral unit cell and choose a triply
primitive hexagonal unit cell (Fig. 1.17) whose unit cell parameters are a =
b 6 ¼ c, a = b = 90° and c = 120°. In order to calculate a H and c H , take projection of a R on c-axis. Therefore,
1.2 Choice of Axes and Unit Cells
15