c H
3
¼ a R
ð Þ P ¼ a R cos 45
¼ 5 cos 45
¼ 3.535
or c H ¼ 3 Â 3.353 = 10:61 ˚
A
Again taking projection of a R on ab-plane, we have
a R
ð Þ P ¼ 5 cos 45
¼ 3.535
From Fig. 1.17, we can obtain
a H ¼ 3.535)
2
þ ð3.535)
2
þ 2 3.535
ð
Þ cos 60
À
Á
Â
à 1=2 ¼ 3.535 Â
ffiffi ffi
3
p ¼ 6.12 ˚
A
(c) Volume of the hexagonal unit cell,
V H ¼ 0:866 a
2
c
¼ 0:866 Â 6:12
ð
Þ
2 Â10:61
¼ 344:14 ˚
A
3
(d) Ratio of the two volumes = V H
V R
¼
344:14
114:2
ffi 3
) There are three lattice points in the hexagonal unit cell.
1.3 Crystal Systems in 2-D and 3-D Lattices
Arbitrary values of lattice parameters a, b and c (in a plane lattice) and a, b, c and a,
b, c (in a space lattice) produce unlimited number of 2-D and 3-D lattices,
respectively. However, when suitable restrictions on the lattice parameters are
imposed, the numbers of possible primitive lattices drastically reduce to four in 2-D
as shown in Fig. 1.18. They also represent the number of crystal systems in their
respective category.
Fig. 1.17 Showing hexagonal unit cells and projected rhombohedral positions
16
1 Unit Cell Composition