Lecture Notes on Structure of Matter by Mohammad Jellur Rahman, Department of Physics, BUET, Dhaka-1000
27
Lecture 5:
Crystal planes and Miller Indices
Index system for crystal directions and planes
Crystal directions:
Any lattice vector can be written as that given by Eq.(1.2). The direction is then
specified by the three integers [n
1
n
2
n
3
]. If the numbers n
1
n
2
n
3
have a common factor, this factor is
removed. For example, [111] is used rather than [222], or [100], rather than [400]. When we speak
about directions, we mean a whole set of parallel lines, which are equivalent due to transnational
symmetry. Opposite orientation is denoted by the negative sign over a number. For example:
Crystal planes
: The orientation of a plane in a lattice is specified by
Miller indices
. They are
defined as follows. We find intercept of the plane with the axes along the primitive translation
vectors
a
1
,
a
2
and
a
3
. Let’s these intercepts be
x
,
y
, and
z
, so that
x
is fractional multiple of a
1
,
y
is a
fractional multiple of a
2
and
z
is a fractional multiple of a
3
. Therefore we can measure
x
,
y
, and
z
in
units a
1
, a
2
and a
3
respectively. We have then a triplet of integers (
x y z
). Then we invert it (1/
x
1/
y
1/
z
) and reduce this set to a similar one having the smallest integers by multiplying by a common
factor. This set is called Miller indices of the plane (
hkl
). For example, if the plane intercepts
x
,
y
,
and
z
in points 1, 3, and 1, the index of this plane will be (313).
The orientation of a crystal plane is determined by three points in the plane, provided they are not
collinear. If each point lay on a different crystal axis, the plane could be specified by giving the co-
ordinates of the points in terms of the lattice constants
a, b, c
. A notation conventionally used to
describe lattice points (sites), directions and planes is known as Miller Indices.
A crystal lattice may be considered as an assembly of equidistant parallel planes passing through the
lattice points and are called lattice planes. In order to specify the orientation one employs the so
called Miller indices.
For simplicity, let us start with a two dimensional lattice and then generalized to three dimensional
case.