a:b  c ¼ 2 ^ i þ ^ j
À
Á : 2 ^ j  ^ k
À
Á
¼ 2 2 ^ i þ ^ j
À
Á : ^ i
ðwhere ^ i: ^ i ¼ 1; ^ i: ^ j ¼ 0Þ
¼ 2 2 þ 0
ð
Þ¼4
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
2 ^ j  ^ k
À
Á ¼ p ^ i
and
b
Ã
¼ 2p
c  a
a:b  c
¼
2p
4
^ k  2 ^ i þ ^ j
À
Á
Â
à ¼
p
2
2 ^ j À ^ i
Â
à ¼
p
2
À ^ i þ 2 ^ j
Â
Ã
Example 3 A two-dimensional lattice has its basis vectors as a = 2 ^ i and b = ^ i + 2 ^ j.
Determine its corresponding reciprocal lattice vectors.
Fig. 2.13 Graphical construction of a 2-D reciprocal lattice
54
2 Unit Cell Construction