Miller Indices

1
Miller Indices Steriographic Projection
The Miller indices can be determined from the
steriographic projection by measuring the angles
relative to known crystallographic directions and
applying the law-of-cosines.
(Figure 2-39 Cullity)
For r, s, and t to represent the angles between
the normal of a plane and the a1, a2, and a3 axes
respectively, then
Where a, b, and c are the unit cell dimensions,
and a/h, b/k, and c/l are the plane intercepts
with the axes. The inner planar spacing, d, is
equal to the distance between the origin and the
plane (along a direction normal to the plane).
2
Vector Operations
Dot product
Cross product
b
a
a
Volume
3
Reciprocal Lattice
Unit cell a1, a2, a3
Reciprocal lattice unit cell b1, b2, b3 defined
by
b3
B
P
C
a3
a2
A
a1
O
4
Reciprocal Lattice
Like the real-space lattice, the reciprocal space
lattice also has a translation vector, Hhkl
Where the length of Hhkl is equal to the
reciprocal of the spacing of the (hkl) planes
Consider planes of a zone (i.e.. 2D reciprocal
lattice).
Next overhead and (Figures A1-4, and A1-5 Cullity)
5
Zone Axis
Planes could be translated so as not to intersect
at a common point.
6
Reciprocal Lattice
(Zone Axis)
Zone axis ua1 va2 wa3