The document discusses Miller indices, which are used to uniquely identify crystallographic planes in a crystal structure. It provides the following key points: - Miller indices were introduced in 1839 and allow quantification of intercepts of crystal planes with the main crystallographic axes. - To determine Miller indices, the intercepts of a plane with the a, b, and c axes are found and reciprocals are taken before reducing to lowest terms. - General principles include that a zero index means parallel to an axis, smaller indices are more parallel, and larger indices are more perpendicular. - Miller indices can be used to represent both individual planes and families of equivalent planes related by symmetry. The direction

1. MILLER INDICES
• 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.

2. • Rules for Miller Indices
i. Determine the intercepts (a,b,c) of the face along the crystallographic
axes, in terms of unit cell dimensions.
ii. Take the reciprocals
iii. Clear fractions
iv. Reduce to lowest terms
v. 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}.

3. • 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 a plane, we obtain a plane located at the
same distance on the other side of the origin.

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

5. • 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

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

7. • 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:
Intercept on X
axis
Intercept on Y
axis
Intercept on Z
axis
 -b c/2
 -1 1/2

8. • 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.

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