Example 7
a
2 b0
Â
Ã
Â
Ã
are the coordinates of the lattice site nearest to the origin. Find
the indices of its direction.
Solution: Following the above said procedure, we have
(i)
Coordinates
a
2
b
0
(ii)
Division by unit translation
a
2a ¼
1
2
b
b ¼ 1
0
c ¼ 0
(iii)
After clearing fraction
1
2
0
⟹ The required Miller indices of the given direction are [120].
Example 8 The coordinates of a lattice site nearest to the origin are: x = −2a,
y = 1b and z ¼
3c
2 : Determine the indices of direction.
Solution: Following the above said procedure, we have
(i)
Coordinates
–2a
1b
3c
2
(ii)
Division by unit translation
À2a
a ¼ À2
1b
b ¼ 1
3c
2c ¼
3
2
(iii)
After clearing fraction
–4
2
3
⟹ The required Miller indices of direction are 423
½
.
Example 9 In a cubic system, different family of directions are symbolically
represented as <100>, <110> and <111>. Write all their members.
Solution: The family <100> comprises of 6 crystallographically equivalent directions. They are:
100
½
, 100
½
, 010
½
, 0 10
½
, 001
½
and 00 1
½ .
Similarly, the family <110> comprises of 12 crystallographically equivalent
directions. They are:
110
½ ; 110
½
; 1 10
½ ; 1 10
½
; 101
½
; 101
½
; 10 1
½
; 10 1
½
; 011
½
; 0 11
½ ; 01 1
½
and 0 1 1
½
:
Further, the family <111> comprises of 8 crystallographically equivalent
directions. They are:
111
½ ; 111
½
; 1 11
½
; 11 1
½
; 1 11
½
; 11 1
½
; 1 1 1
½
and 1 1 1
½
:
Example 10 Classify the given members of the family of directions <100>,
<110> and <111> according to a tetragonal crystal system.
4.1 Miller Indices of Atomic Sites, Planes and Directions
143
a
2 b0
Â
Ã
Â
Ã
are the coordinates of the lattice site nearest to the origin. Find
the indices of its direction.
Solution: Following the above said procedure, we have
(i)
Coordinates
a
2
b
0
(ii)
Division by unit translation
a
2a ¼
1
2
b
b ¼ 1
0
c ¼ 0
(iii)
After clearing fraction
1
2
0
⟹ The required Miller indices of the given direction are [120].
Example 8 The coordinates of a lattice site nearest to the origin are: x = −2a,
y = 1b and z ¼
3c
2 : Determine the indices of direction.
Solution: Following the above said procedure, we have
(i)
Coordinates
–2a
1b
3c
2
(ii)
Division by unit translation
À2a
a ¼ À2
1b
b ¼ 1
3c
2c ¼
3
2
(iii)
After clearing fraction
–4
2
3
⟹ The required Miller indices of direction are 423
½
.
Example 9 In a cubic system, different family of directions are symbolically
represented as <100>, <110> and <111>. Write all their members.
Solution: The family <100> comprises of 6 crystallographically equivalent directions. They are:
100
½
, 100
½
, 010
½
, 0 10
½
, 001
½
and 00 1
½ .
Similarly, the family <110> comprises of 12 crystallographically equivalent
directions. They are:
110
½ ; 110
½
; 1 10
½ ; 1 10
½
; 101
½
; 101
½
; 10 1
½
; 10 1
½
; 011
½
; 0 11
½ ; 01 1
½
and 0 1 1
½
:
Further, the family <111> comprises of 8 crystallographically equivalent
directions. They are:
111
½ ; 111
½
; 1 11
½
; 11 1
½
; 1 11
½
; 11 1
½
; 1 1 1
½
and 1 1 1
½
:
Example 10 Classify the given members of the family of directions <100>,
<110> and <111> according to a tetragonal crystal system.
4.1 Miller Indices of Atomic Sites, Planes and Directions
143