Miller indices
The position and orientation of a crystal plane is determined by three numbers (integers) is
called Miller indices. It is denoted by (h k l).
Determination of Miller Indices:
➢ Taking any atom in the crystal as the origin and straight coordinate axes from this atom.
➢ Taking intercepts on the axis a, b, c in terms of the lattice constants.
➢ Taking the reciprocal of these intercepts.
➢ Taking the lowest common multiple of the denominator.
➢ Multiplying each reciprocal by the lowest common multiple of the denominator, thus obtained
into smallest set of integers. These integers are denoted by h, k, l.
➢ The result is conventionally enclosed in first parenthesis (h k l) which is Miller indices of the
plane. The meaning of these indices is that a set of parallel planes (h k l) cuts the a-axis into h
parts, the b-axis into k parts and the c-axis into l parts.
While finding Miller Indices of a plane, following points should be kept in mind:
➢ When the plane is parallel to one of the coordinate axes, it is said to meet that axis at infinity.
Since , the miller index for that axis is zero.
➢ If a plane cuts on axis on the negative side of the origin, the corresponding index is negative
and is indicated by a bar sign, above the index .
Example: The figure shows a plane whose intercepts are
. The Miller indices of the family to which this plane
belongs are obtaining by taking the reciprocals of these
numbers: and reducing these factions to the smallest set
of integers. This can be done by multiplying each of the
fractions by the lowest common multiple of the denominator 6.
In this case we get6, 3, 2.
Thus, the Miller indices of this plane are (6 3 2)
Miller indices for planes in each of the following sets of intercepts:
Solution: (i) It indicates a plane whose intercepts are . For c-axis the intercept is at
. The Miller indices of the family to which this plane belongs are obtaining by taking the
reciprocals of these numbers: . From this we can write: . For reducing these factions to
the smallest set of integers, multiply each of the fractions by the lowest common multiple of the
denominator2. In this case we get the Miller indices .
(ii)It indicates a plane whose intercepts are . The Miller indices of the family to which this
plane belongs are obtaining by taking the reciprocals of these numbers: , , and reducing these
factions to the smallest set of integers. This can be done by multiplying each of the fractions by
the lowest common multiple of the denominator 1. In this case we get the
Miller indices .
(iii)It indicates a plane whose intercepts are . The Miller indices of the family to which
this plane belongs are obtaining by taking the reciprocals of these numbers: and reducing these
factions to the smallest set of integers. This can be done by multiplying each of the fractions by
the lowest common multiple of the denominator 3. In this case we get the
Miller indices .
Drawing planes from Miller indices:
Direction of plane: Any vector drawn from the origin to a lattice point is defined as direction. It
is denoted by [h k l]. For example, in a cubic unit cell, if the origin is at the one corner and axes
are parallel to the edges, the body diagonal would be represented as [111].
[hkl]: The direction specified by this symbol is obtained as follows: Move from the origin over a
distance ha along the a-axis, kb along the b-axis and lc along the c-axis. The vector connecting the
origin with the point so obtained is then the direction specified by the symbol [hkl]. Thus, in a
cubic crystal, the direction of the X-axis is indicated by [100], the Y axis by [010] and the Z-axis
by [001]. Similarly, the direction of the negative X-axis is indicated by , the negative Y-axis
by and the negative Z-axis by .
