Solution: Given: a = 2 ^ i and b = ^ i + 2 ^ j. Let us assume the third translation vector c of
the given lattice lies along the z-axis and it has a unit magnitude, that is, c = ^ k.
Therefore, the volume of the unit cell is
a:b  c ¼ 2 ^ i:ð ^ i þ 2 ^ jÞ Â ^ k ¼ 2 ^ i:ðÀ ^ j þ 2 ^ iÞ ¼ 2ð0 þ 2Þ ¼ 4
Further, we know that the reciprocal lattice vectors are given by
a
Ã
¼ 2p
b  c
a:b  c
;
b
Ã
¼ 2p
c  a
a:b  c
Since the reciprocal lattice vectors a* and b* lie in the same plane, therefore
a
Ã
¼ 2p
b  c
a:b  c
¼
2p
4
ð ^ i þ 2 ^ jÞ Â ^ k ¼
2p
4
ð2 ^ i À ^ jÞ ¼
p
2
2 ^ i À ^ j
Â
Ã
and
b
Ã
¼ 2p
c  a
a:b  c
¼
2p
4
^ k  2 ^ i
Â
à ¼ p ^ j
Example 4 A two-dimensional direct lattice is formed from a repetition of points
ABCD (Fig. 2.14) in which AB = CD = 3Å, AD = BC = 5Å and the angle BAD
(say c) = 60°. Draw a small area (about four unit cells) of the direct lattice.
Calculate the basis vectors and draw the corresponding reciprocal cells.
Solution: From Fig. 2.14, we suppose that a and b, respectively, are the lattice
vectors representing AB and AD in the direct lattice. Similarly, a* and b*are the
corresponding reciprocal lattice vectors. Then,
a
Ã
: a ¼ 2p and a
Ã
: b ¼ 0
This indicates that a* is perpendicular to b and is given by
a
Ã
¼
2p
asinc
¼
2p
3sin60
¼
2p  2
3 Â
ffiffi ffi
3
p ¼
4p
3
ffiffi ffi
3
p ˚
A
À1
Similarly, b* is perpendicular to a and is given by
b
Ã
¼
2p
bsinc
¼
2p
5sin60
¼
2p  2
5 Â
ffiffi ffi
3
p ¼
4p
5
ffiffi ffi
3
p ˚
A
À1
The angle between a* and b* will be 120°. Four cells of each, the direct lattice
and the reciprocal lattice are shown in Fig. 2.15.
2.2 Construction of Reciprocal Lattice
55