Solution: Given: A set of z4one axes: 111
½
Š, 1 11
½
Š, 11 1
½ Š, 1 21
½
Š, 121
½
Š and [112], a
plane (123).
We know that a zone axis is parallel to a plane if they satisfy the condition:
hu þ kv þ 1c ¼ 0
Now, checking with all the given planes one by one, we have
1  ðÀ1Þ þ 2  1 þ 3  1 ¼ À1 þ 2 þ 3 ¼ 4
1  1 þ 2  ðÀ1Þ þ 3  1 ¼ 1 À 2 þ 3 ¼ 2
1  1 þ 2  1 þ 3  ðÀ1Þ ¼ 1 þ 2 À 3 ¼ 0
1  1 þ 2  ðÀ2Þ þ 3  1 ¼ 1 À 4 þ 3 ¼ 0
1  ðÀ1Þ þ 2  2 þ 3  1 ¼ À1 þ 4 þ 3 ¼ 6
1 Â 1 þ 2 Â 1 þ 3 Â 2 ¼ 1 þ 2 þ 6 ¼ 9
⟹ The zone axes 11 1
½
Š and 1 21
½
Š lie parallel to the plane (123).
Example 3 Find the zone axes lying at the intersection of the two sets of planes:
(i) (342) and 10 3
ð
Þ, (ii) 21 3
ð
Þ and (110).
Solution: Given: Two sets of planes: (i) (342) and 10 3
ð
Þ, (ii) 21 3
ð
Þ and (110),
[uvw] = ?
We can determine the indices of the zone axis [uvw] by using the following relations:
u ¼
k 1 l 1
k 2 l 2
, v ¼
l 1 h 1
l 2 h 2
, w ¼
h 1 k 1
h 2 k 2
For the first set of planes: (342) and 10 3
ð
Þ, we have
u ¼
4 2
0 À3
= À 12; v ¼
2 3
À3 1
= 11, w ¼
3 4
1 0
= -4
⟹ [uvw]
1211 4
½
Š.
For the second set of planes: 21 3
ð
Þ and (110), we have
u ¼
1 À3
1 0
= 3 , v ¼
À3 2
0 1
= -3, w ¼
2 1
1 1
= 1
⟹ [uvw] 3 31
½
Š.
Example 4 Find the (hkl) plane containing two sets of directions: (i) [131] and
0 11
½
Š, (ii) [102] and 111
½
Š.
Solution: Given: Two sets of directions: (i) [131] and 0 11
½
Š, (ii) [102] and 111
½
Š,
(hkl) = ?
4.5 Zone and Zone Axis
167