Miller indices are a symbolic representation of an atomic plane's orientation in a crystal lattice. They are defined as the reciprocals of the fractional intercepts the plane makes with the crystallographic axes. To determine the Miller indices of a plane, the intercepts of the plane along the crystallographic axes are found and the reciprocals of the intercepts are taken. The indices are then enclosed in parentheses with any fractions cleared and reduced to lowest terms. This provides a set of numbers that uniquely identifies the plane's orientation.

1. MILLER INDICES
Miller Indices are a symbolic vector representation for the
orientation of an atomic plane in a crystal lattice & are defined as:
The reciprocals of the fractional intercepts which the plane makes
with the crystallographic axes.

2. Introduction:
• Miller indices were introduced in 1839 by the British mineralogist William
Hallowes Miller. The method was also historically known as the Millerian
system, and the indices as Millerian.
• The orientation of a surface or a crystal plane may be defined by considering
how the plane (or indeed any parallel plane) intersects the main
crystallographic axes of the solid.
• The application of a set of rules leads to the assignment of the Miller Indices,
(hkl) ; a set of numbers which quantify the intercepts and thus may be used to
uniquely identify the plane or surface.
• To determine the crystallography planes we take a unit cell with three axes
coordinate system.

3. •Rules for Miller Indices:
i. Determine the intercepts (a,b,c) of the plane along the crystallographic axes,
in terms of unit cell dimensions.
ii. Take the reciprocals of the intercepts.
iii. Clear fractions and reduce to lowest terms by multiplying each intercepts by the
denominator of the smallest fraction.
iv. If a plane has negative intercept, the negative number is denoted by a bar (¯)
above the number.
Never alter negative numbers. For example, do not divide -1, -1, -1 by -1 to get 1,1,1.
vi. If plane is parallel to an axis, its intercept is zero and meets at infinity.
vii. The three indices are enclosed in parenthesis, (hkl). A family of planes is
represented by {hkl}.

4. • General Principles:
i. If a Miller index is zero, the plane is parallel to that axis.
ii. The smaller a Miller index, the more nearly parallel the plane is to the axis.
iii. The larger a Miller index, the more nearly perpendicular a plane is to that axis.
iv. Multiplying or dividing a Miller index by a constant has no effect on the
orientation of the plane
v. When the integers used in the Miller indices contain more than one digit,
the indices must be separated by commas. E.g.: (3,10,13)
vi. By changing the signs of all the indices of a plane, we obtain a plane located at
the same distance on the other side of the origin.

5. Find the Miller Indices for the vector shown in the
unit cell shown in fig. where, a=b=c.

6. • Step 1: The given vector is passing through the origin of the coordinate system.
• Step 2: Take the intercepts of the vector on the X, Y & Z axes.
• Step 3: Since a=b=c, the intercepts will be: ½, 1 & 0. Multiplying throughout by 2
and enclosing within square brackets we get, [120] to be the direction indices of the
given vector.
Intercept on X
Axis
Intercept on Y
Axis
Intercept on Z
Axis
a/2 b 0

7. Find the Miller Indices of plane shown in fig.
where a=b=c.
Fig. a Fig. b

8. • Step 1: The given plane passes through the origin. Hence, the origin is shifted to the
adjacent unit cell as shown in fig.(b).
• Step 2: Find the intercepts of the plane with the X, Y & Z axes:
• Step 3: Take the reciprocals of the intercepts, we get 0,-1 & 2.
• Step 4: Enclose the indices in round brackets (parenthesis) we get (0-12) to be the
Miller Indices of given plane.
Intercept on X
axis
Intercept on Y
axis
Intercept on Z
axis
 -b c/2
 -1 1/2

9. Some Examples to Solve:

10. • Family of Equivalent Planes:
Due to the symmetry of crystal structures the spacing and arrangement
of atoms may be the same in several planes. These are known as
equivalent planes, and a group of equivalent planes are known as a
family of planes. Families of planes are written in curly brackets.
{001} = (001), (010), (100), (00-1), (0-10), (-100)

11. Relationship between Crystallographic
Plane and Directions:
• Conventionally, a plane in analytical
geometry is expressed by a vector normal to
the plane under consideration. It may be
observed from fig, that the miller indices for
a plane and a vector normal to it are same.
• If (uvw) is the miller indices of a plane,
then the direction indices of a vector normal
to it is [uvw].

12. REFERENCE:
• Material Science And Engineering: An Introduction,
William Callister.