This indicates that the reciprocal lattice translation has the same direction as
that of the direct lattice but the unit is in inverse mode. The two lattices are
shown in Fig. 2.11a.
2. For a plane rectangular lattice with a 6 ¼ b and c ¼ 90
, the following relationships exist:
a
Ã
: a ¼ a : a
Ã
¼ 2p
b
Ã
: b ¼ b : b
Ã
¼ 2p
a
Ã
: b ¼ b
Ã
: a ¼ 0
so that a
Ã
¼
2p
a
and b
Ã
¼
2p
b
These relationships suggest that
a
à is ? to b or a
à is jj to a
and b
à is ? to a or b
à is jj to b
They are shown in Fig. 2.11b.
However, for a general plane lattice with a 6 ¼ b and c = c° the relationships
between the direct and reciprocal lattices are not straightforward as for
orthogonal axes. Here, the angle between a* and b* will be 180° – c as shown in
Fig. 2.11c and their magnitudes are:
a
Ã
¼
2p
a cosð90 À cÞ
¼
2p
a sin c
and b
Ã
¼
2p
b cosð90 À cÞ
¼
2p
b sin c
Table 2.1 Modifications required for non-primitive unit cells
Direct lattice
parameter
Direct lattice
volume
Lattice
type
Reciprocal lattice - Lattice
parameters
Reciprocal lattice
volume
a, b, c
V
P and R a*, b*, c*
V*
A
a*, 2b*, 2c*
4V*
B
2a*, b*, 2 c*
4V*
C
2a*, 2b*, c*
4V*
F
2a*, 2b*, 2c*
8V*
I
2a*, 2b*, 2c*
8V*
50
2 Unit Cell Construction