V
Ã
¼ a
à b
à c
à 1 À cos
2
a
Ã
À cos
2
b
Ã
À cos
2
c
Ã
þ 2 cos a
à cos b
à cos c
Ã
À
Á 1=2
¼ 0:23 Â 0:115 Â 0:67 1 À 0 À 0 À
1
4
þ 0
1=2
¼ 1:77 Â 10
À3 3
4
1=2
¼ 1:53 Â 10
À3 ˚
A
3
Example 6 The primitive translation vectors of a simple hexagonal (Fig.2.16)
space lattice may be taken as
A ¼
a
2
^ i þ
ffiffi ffi
3
p
a
2
^ j
; B ¼ À
a
2
^ i þ
ffiffi ffi
3
p
a
2
^ j
and C ¼ c ^ k
where ^ i, ^ j and ^ k are unit vectors.
Determine the primitive translation vectors of the reciprocal lattice and the unit
cell volume.
Solution: Given: Direct lattice parameters A, B and C; determine the reciprocal
lattice parameters A*, B*, C* and volume V*. Volume of the direct unit cell is
given by
Fig. 2.15 Showing direct cells and corresponding reciprocal cells
2.2 Construction of Reciprocal Lattice
57