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Solid State Physics for Materials
Miller Indices
Ababay K. Worku (PhD)
June, 2022
Bahir Dar, Ethiopia
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Miller Indices
 Definition
 Notation Summary
 Miller indices are used to specify directions
and planes.
 (h,k,l) represents a point – note the exclusive use of
commas
 These directions and planes could be in
lattices or in crystals.
 Negative numbers/directions are denoted with a bar on
top of the number
 The number of indices will match with the
dimension of the lattice or the crystal.
• E.g. in 1D there will be 1 index and 2D there
will be two indices etc.
 [hkl] represents a direction
 <hkl> represents a family of directions
 (hkl) represents a plane
 {hkl} represents a family of planes
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Miller Indices (Cont…)
 Miller indices are used to specify directions and planes.
 These directions and planes could be in lattices or in crystals.
 (It should be mentioned at the outset that special care should be given to see if
the indices are in a lattice or a crystal).
 The number of indices will match with the dimension of the lattice or the
crystal: in 1D there will be 1 index and 2D there will be two indices etc.
 Some aspects of Miller indices, especially those for planes, are not intuitively
understood and hence some time has to be spent to familiarize oneself with the
notation.
Miller Indices
Lattices
Crystals
Miller Indices
Directions
Planes
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Miller Indices (Cont…)
 Miller indices for DIRECTIONS
 A vector r passing from the origin to a lattice point can be
written as: r = r1 a + r2 b + r3 c
Where, a, b, c → basic vectors
r  r1 a  r2 b  r3 c
 Basis vectors are unit lattice translation vectors which define
the coordinate axis (as in the figure below).
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Miller Indices (Cont…)
 Miller Indices for directions in 2D
Miller indices → [53]
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Miller Indices (Cont…)
 How to find the Miller Indices for an arbitrary direction?  Procedure
 Consider the example below
 Subtract the coordinates of the end point from the starting point of the
vector denoting the direction  If the starting point is A(1,3) and the final
point is B(5,1)  the difference would be (4, 4)
 Enclose in square brackets, remove comma and
write negative numbers with a bar  [4 4]
 Factor out the common factor  4[11]
 If we are worried about the direction and magnitude then we
write  4[11]




If we consider only the direction then we write  [11]
Needless to say the first vector is 4 times in length
The magnitude of the vector [11]  [11]
is (1) 2  (1) 2  2
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Miller Indices (Cont…)
 A crystal lattice may be considered as an assembly of a
number of equidistant parallel planes passing through the
lattice points and are called lattice planes.

For a given lattice, these sets of planes can be selected in
a number of ways.
 The inter-planar spacing for a set of parallel planes is
fixed but for different sets of planes the inter-planar
spacing varies as also the density of lattice points.
 The equation of plane in three dimensions having the
intercepts a, b and c (Fig. ) along the axes x, y, z
respectively will be
Figure: Intercepts along x, y and z axes
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Miller Indices (Cont…)
•
Let
then the above
•
Then the equation can be written as
•
Where
are intercepts
along x, y and z axes respectively.
equation becomes
•
This equation describes the first lattice plane,
nearest to the origin, in a set of parallel,
identical and equally spaced planes
•
The set of three integers h, k and l are expressed as
(h,k,l) called Miller indices
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Miller Indices (Cont…)
 To obtain Miller indices of a plane the following
procedure will be adopted:
1. Determine the intercepts of the plane along x, y, z axes in
terms of lattice parameters.
2. Divide these intercepts by the proper unit translations.
3. Find their reciprocals.
4. If Fraction results, multiply each of them by the smallest
common divisor
5.
Put the final integers in parenthesis (hkl) to get the
required Miller Indices.
•
Example: In a crystal, a plane cuts intercepts of 2a, 3b
and 6c along three crystallographic axes. Determine the
miller indices
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Miller Indices (Cont…)
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Miller Indices (Cont…)
•
What’s the Miller Index of this plane?
•
This plane cuts all three crystallographic axes.
•
Intercepts = (1,1,1) →(111)
 Importance of Miller Indices
 In Materials Science it is important to have a
notation system for atomic planes since these planes
influence
o Optical properties
o Reactivity
o Surface tension
o Dislocations