Example 5 A monoclinic unit cell of 1st setting has the following parameters: a =
5Å, b = 10Å, c = 15Å, a = b = 90° and c = 120°. Determine its corresponding
reciprocal lattice parameters and the two cell volumes.
Solution: General equation of unit cell volume is given by
V ¼ abc 1 À cos
2
a À cos
2
b À cos
2
c þ 2 cosa cosb cosc
À
Á 1=2
¼ 5 Â 10 Â 15 1 À 0 À 0 À
1
4
þ 0
1=2
¼ 750
3
4
1=2
¼ 649:52 ˚
A
3
a
Ã
¼
bcsina
V
¼
10 Â 15sin90
649:52
¼ 0:23 ˚
A
À1
b
Ã
¼
acsinb
V
¼
5 Â 15sin90
649:52
¼ 0:115 ˚
A
À1
c
Ã
¼
absinc
V
¼
5 Â 10sin120
649:52
¼ 0:067 ˚
A
À1
and
sina
Ã
¼
V
abc sinb sinc
¼
649:52
750 sin90
 sin120
¼ 1; a
Ã
¼ 90
sinb
Ã
¼
V
abc sina sinc
¼
649:52
750 sin90
 sin120
¼ 1; b
Ã
¼ 90
sinc
Ã
¼
V
abc sina sinb
¼
649:52
750 sin90
 sin90
¼ 0:866; c
Ã
¼ 60
Finally,
Fig. 2.14 Showing lattice
vectors in the direct lattice
56
2 Unit Cell Construction
5Å, b = 10Å, c = 15Å, a = b = 90° and c = 120°. Determine its corresponding
reciprocal lattice parameters and the two cell volumes.
Solution: General equation of unit cell volume is given by
V ¼ abc 1 À cos
2
a À cos
2
b À cos
2
c þ 2 cosa cosb cosc
À
Á 1=2
¼ 5 Â 10 Â 15 1 À 0 À 0 À
1
4
þ 0
1=2
¼ 750
3
4
1=2
¼ 649:52 ˚
A
3
a
Ã
¼
bcsina
V
¼
10 Â 15sin90
649:52
¼ 0:23 ˚
A
À1
b
Ã
¼
acsinb
V
¼
5 Â 15sin90
649:52
¼ 0:115 ˚
A
À1
c
Ã
¼
absinc
V
¼
5 Â 10sin120
649:52
¼ 0:067 ˚
A
À1
and
sina
Ã
¼
V
abc sinb sinc
¼
649:52
750 sin90
 sin120
¼ 1; a
Ã
¼ 90
sinb
Ã
¼
V
abc sina sinc
¼
649:52
750 sin90
 sin120
¼ 1; b
Ã
¼ 90
sinc
Ã
¼
V
abc sina sinb
¼
649:52
750 sin90
 sin90
¼ 0:866; c
Ã
¼ 60
Finally,
Fig. 2.14 Showing lattice
vectors in the direct lattice
56
2 Unit Cell Construction