There exist some geometrical relationships among the zone axis and the plane in
the zone. Let us derive them with the help of reciprocal lattice concept using vector
algebra.
1. The Weiss Zone Law
This states that the indices of a zone axis [uvw] and a crystal plane (hkl) in the zone
must obey the algebraic relation:
hu þ kv þ 1w ¼ 0
ð4:9Þ
Proof If the plane (hkl) and the zone axis [uvw] are in the same zone, then the
reciprocal lattice vector d* hkl must be perpendicular to the vector R uvw . Therefore,
we can write.
d
à hkl:R uvw ¼ 0
or ha
Ã
þ kb
Ã
þ 1c
Ã
ð
Þ :ðua þ vb þ wcÞ ¼ 0
) hu þ kv þ lw ¼ 0
where a*.a = b*.b = c*.c = 1 and a*.b = a*.c = 0, etc.
2. Zone Axis Lying at the Intersection of Two Planes
This states that the indices of the zone axis [uvw] of two intersection planes
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:
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
ð4:10Þ
where the determinant
k 1 l 1
k 2 l 2
¼ k 1 l 2 - l 1 k 2
ð
Þ , etc.
Proof Since the zone axis R uvw is lying at the intersection of the planes h 1 k 1 l 1
ð
Þ
and h 2 k 2 l 2
ð
Þ, therefore, R uvw will be perpendicular to their normal, that is, d
Ã
h 1 k 1 l 1
and d
Ã
h 2 k 2 l 2
. From simple geometry, the zone axis is expressed as.
R uvw ¼
d
Ã
h 1 k 1 l 1
 d
Ã
h 2 k 2 l 2
V
Ã
ð4:11Þ
where V* is the volume of the reciprocal unit cell. Equation 4.11 can further be
written as
164
4 Unit Cell Representations of Miller Indices