Example 4 A tetragonal crystal has the c/a ratio as 1.5. Determine the interplanar
spacing in (100), (110) and (111) planes.
Solution: Given: A tetragonal crystal system with c/a = 1.5. d 100 ¼ ?0:1cmd 110 ¼
?0:1cmd 111 ¼ ?
For a tetragonal crystal system, we know that a = b 6 ¼ c, therefore, c = 1.5a or
c = 1.5 (if a = 1). Further, the interplanar spacing for tetragonal crystal system is
given by
d ðhklÞ =
h
2 + k
2
a 2 +
l
2
c 2
! À1=2
Therefore,
d ð100Þ ¼ 1
½ Š
À1=2 ¼ 1 ˚
A
d ð110Þ ¼ 2
½ Š
À1=2 ¼ 0:707 ˚
A
d ð111Þ ¼ 2 þ
1
1:5
! À1=2
¼ 2 þ 0:66
½
Š
À1=2 ¼ 0:61 ˚
A
Example 5 Lattice parameters of an orthorhombic crystal are 2Å, 3Å and 4Å,
respectively. Determine the interplanar spacing in (100), (110) and (111) planes.
Solution: Given: An orthorhombic crystal system with a = 2Å, b = 3Å, c = 4Å.
d 100 ¼ ?0:1cmd 110 ¼ ?0:1cmd 111 ¼ ?
For an orthorhombic crystal system, we know that a = b = c and the interplanar
spacing is given by
d ðhklÞ =
h
2
a 2 þ
k
2
b
2
þ
l
2
c 2
! À1=2
Therefore,
d ð100Þ ¼
1
4
! À1=2
¼ 2 ˚
A
d ð110Þ ¼
1
4
þ
1
9
! À1=2
¼
13
36
! À1=2
¼ 1:66 ˚
A
d ð111Þ ¼
1
4
þ
1
9
þ
1
16
! À1=2
¼
61
144
! À1=2
¼ 1:54 ˚
A
Example 6 Lattice parameters of a hexagonal crystal are a = b = 4Å, and c = 6Å,
respectively. Determine the interplanar spacing in 10 10
ð
Þ, 01 10
ð
Þ, 1100
ð
Þ and
11 20
ð
Þ planes.
Solution: Given: A hexagonal crystal system with a = b = 4Å, c = 6Å. Determine
the interplanar spacing in 10 10
ð
Þ, 01 10
ð
Þ, 1100
ð
Þ and 11 20
ð
Þ planes.
4.4 Interplanar Spacing
161