ha
à + kb
à + lc
à = c
à u 1 v 2 À v 1 u 2
ð
Þ + a
à v 1 w 2 À w 1 v 2
ð
Þ + b
à w 1 u 2 À u 1 w 2
ð
Þ
Comparing the coefficients of a*, b* and c*, we obtain
h = v 1 w 2 À w 1 v 2
ð
Þ , k = w 1 u 2 À u 1 w 2
ð
Þ , l = u 1 v 2 À v 1 u 2
ð
Þ
4. The Addition Rule
The Miller indices of a plane (hkl) sandwiched between two planes of indices
h 1 k 1 l 1
ð
Þ and h 2 k 2 l 2
ð
Þ in a zone can be determined according to the following
relations:
h ¼ mh 1 Æ nh 2
k ¼ mk 1 Æ nk 2
l ¼ ml 1 Æ nl 2
where m and n can be any positive integer such as 1, 2, etc.
Solved Examples
Example 1 Identify the planes from (112), (321), (123), (212) and 23 1
ð
Þ
belonging to the zone 1 11
½
Š.
Solution: Given: A set of planes: (112), (321), (123), (212) and 23 1
ð
Þ, the zone axis
is 1 1 1
½
Š.
We know that a plane and a direction can belong to the same zone if they satisfy
the condition:
hu + kv + lc = 0
Now, checking with all the given planes one by one, we have
1  1 þ 1  ðÀ1Þ þ 2  ðÀ1Þ ¼ 1 À 1 À 2 ¼ À2
3  1 þ 2  ðÀ1Þ þ 1  ðÀ1Þ ¼ 3 À 2 À 1 ¼ 0
1  1 þ 2  ðÀ1Þ þ 3  ðÀ1Þ ¼ 1 À 2 À 3 ¼ À4
2  1 þ 1  ðÀ1Þ þ 2  ðÀ1Þ ¼ 2 À 1 À 2 ¼ À1
2  1 þ 3  ðÀ1Þ þ ðÀ1Þ Â ðÀ1Þ ¼ 2 À 3 þ 1 ¼ 0
⟹ The planes (321) and 23 1
ð
Þ belong to the zone 1 1 1
½
Š.
Example 2 Identify the zone axes from 111
½
Š, 1 11
½
Š, 11 1
½
Š, 1 21
½ Š, 121
½
Š and [112]
which are parallel to the plane (123).
166
4 Unit Cell Representations of Miller Indices