ua þ vb þ wc ¼
h 1 a
Ã
þ k 1 b
à þ l 1 c
Ã
ð
Þ Âh 2 a
Ã
þ k 2 b
à þ l 2 c
Ã
ð
Þ
V
Ã
¼
1
V
Ã
h 1 k 2 a
à  b
Ã
ð
Þþh 1 l 2 a
à  c
Ã
ð
Þþk 1 h 2 b
à  a
Ã
ð
Þþ
k 1 l 2 b
à  c
Ã
ð
Þþl 1 h 2 c
à  a
Ã
ð
Þþl 1 k 2 c
à  b
Ã
ð
Þ
"
#
¼
1
V
à a
à  b
Ã
ð
Þh 1 k 2 À k 1 h 2
ð
Þ þb
à  c
Ã
ð
Þk 1 l 2 À l 1 k 2
ð
Þ þc
à  a
Ã
ð
Þl 1 h 2 À h 1 l 2
ð
Þ
½
Š
Using the identities such as (b* Â a*) = - (a* Â b*), etc. and a ¼
b
à Âc
Ã
V
à ; etc. the
above equation reduces to
ua + vb + wc = c h 1 k 2 À k 1 h 2
ð
Þ þ a k 1 l 2 À l 1 k 2
ð
Þþb l 1 h 2 À h 1 l 2
ð
Þ
Comparing the coefficients of a, b and c, we obtain
u = k 1 l 2 À l 1 k 2
ð
Þ , v = l 1 h 2 À h 1 l 2
ð
Þ , w = h 1 k 2 À k 1 h 2
ð
Þ
3. A Plane (hkl) Containing Two Directions [u 1 v 1 w 1 ] and [u 2 v 2 w 2 ]
This states that if a plane (hkl) contains two directions [u 1 v 1 w 1 ] and [u 2 v 2 w 2 ], then
the indices of the plane can be obtained according to the following relations:
h =
v 1 w 1
v 2 w 2
, k =
w 1 u 1
w 2 u 2
, l =
u 1 v 1
u 2 v 2
Proof Since the plane (hkl) defined by its normal d* hkl is perpendicular to both
directions R u 1 v 1 w 1 and R u 2 v 2 w 2 , therefore, d* hkl can be expressed as
d
Ã
hkl ¼
R u 1 v 1 w 1 Â R u 2 v 2 w 2
V
ð4:12Þ
Here, V is the volume of direct unit cell. Equation 4.12 can further be written as
ua
Ã
þ vb
Ã
þ wc
Ã
¼
u 1 a þ v 1 b þ w 1 c
ð
Þ Âu 2 a þ v 2 b þ w 2 c
ð
Þ
V
¼
1
V
½u 1 v 2 a  b
ð
Þþu 1 w 2 a  c
ð
Þþv 1 u 2 b  a
ð
Þ
þ v 1 w 2 b  c
ð
Þþw 1 u 2 c  a
ð
Þþw 1 v 2 c  b
ð
ފ
¼
1
V
½ a  b
ð
Þ u 1 v 2 À v 1 u 2
ð
Þ þb  c
ð
Þ v 1 w 2 À w 1 v 2
ð
Þ
þ c  a
ð
Þ w 1 u 2 À u 1 w 2
ð
Þ Š
Using the identities such as a
Ã
¼
bÂc
V , etc. above equation reduces to
4.5 Zone and Zone Axis
165