Crystal Structure Continued!
NOTE!! Again, much discussion & many
figures in what follows was constructed
from lectures posted on the web by Prof.
Beşire GÖNÜL (Turkey). She has done an
excellent job covering many details of
crystallography & she illustrates with many
very nice pictures of lattice structures. Her
lectures are posted Here:
http://www1.gantep.edu.tr/~bgonul/dersnotlari/.
Her homepage is Here:
http://www1.gantep.edu.tr/~bgonul/
The Wigner-Seitz Method to Construct a
Primitive Cell
• A simple, geometric method to construct a Primitive Cell
is called the Wigner-Seitz Method. The procedure is:
1. Choose a starting lattice point.
2. Draw lines to connect that point
to its nearest neighbors.
3. At the mid-point & normal to
these lines, draw new lines.
4. The volume enclosed is called
a Wigner-Seitz cell.
Illustration for the 2D
parallelogram lattice.
3 Dimensional Wigner-Seitz Cells
Face Centered
Cubic (FCC)
Wigner-Seitz Cell
Body Centered
Cubic (BCC)
Wigner-Seitz Cell
3
Lattice Sites in a Cubic Unit Cell
• The standard notation is shown in the figure. It
is understood that all distances are in units of the
cubic lattice constant a, which is the length of a
cube edge for the material of interest.
Directions in a Crystal: Standard Notation
• See Figure. Choose an origin, O. This
choice is arbitrary, because every lattice
point has identical symmetry. Then,
consider the lattice vector joining O to any
point in space, say point T in the figure. As
we’ve seen, this vector can be written
T = n1a1 + n2a2 + n3a3
[111] direction
• In order to distinguish a Lattice Direction from a
Lattice Point, (n1n2n3), the 3 integers are enclosed in
square brackets [ ...] instead of parentheses (...),
which are reserved to indicate a Lattice Point. In
direction [n1n2n3], n1n2n3 are the smallest integers
possible for the relative ratios.
Examples
210
X = 1, Y = ½, Z = 0
[1 ½ 0]
[2 1 0]
X=½ ,Y=½,Z=1
[½ ½ 1]
[1 1 2]
Negative Directions
• When we write the direction [n1n2n3]
depending on the origin, negative
directions are written as
R = n1a1 + n2a2 + n3a3
With a bar above the negative integers.
To specify the direction, the smallest
possible integers must be used.
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Examples of Crystal Directions
X = 1, Y = 0, Z = 0
[100]
X = -1, Y = -1, Z = 0
[110]
Examples
A vector can be moved to the origin.
X = -1, Y = 1, Z = -1/6
[-1 1 -1/6]
[6 6 1]
Crystal Planes
• Within a crystal lattice it is possible to identify
sets of equally spaced parallel planes.
These are called lattice planes.
• In the figure, the density of lattice points on each
plane of a set is the same & all lattice points are
contained on each set of planes.
The set of
planes for a
2D lattice.
b
b
a
a
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.
• To find the Miller indices of a plane, take the
following steps:
1. Determine the intercepts of the plane along
each of the three crystallographic directions.
2. Take the reciprocals of the intercepts.
3. If fractions result, multiply each by the
denominator of the smallest fraction.
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Example 1: (100) Plane
Axis
X
Y
Z
Intercept
points
1
∞
∞
Reciprocals
Smallest
Ratio
(1,0,0)
1/1 1/ ∞ 1/ ∞
1
Miller İndices
0
0
(100)
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Example 2: (110) Plane
(0,1,0)
(1,0,0)
Axis
X
Y
Z
Intercept
points
1
1
∞
Reciprocals
1/1
Smallest
Ratio
1
1/ 1 1/ ∞
Miller İndices
1
0
(110)
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Example 3: (111) Plane
(0,0,1)
(0,1,0)
(1,0,0)
Axis
X
Y
Z
Intercept
points
1
1
1
Reciprocals
1/1
1/ 1
1/ 1
Smallest
Ratio
1
1
1
Miller İndices
(111)
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Example 4: (210) Plane
Axis
X
Y
Z
Intercept
points
1/2
1
∞
Reciprocals
(0,1,0)
(1/2, 0, 0)
Smallest
Ratio
1/(½) 1/ 1 1/ ∞
2
Miller İndices
1
0
(210)
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Example 5: (102) Plane
Axis
a
b
c
Intercept
points
1
∞
½
Reciprocals
1/1
Smallest
Ratio
1
1/ ∞ 1/(½)
0
2
Miller İndices (102)
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Example 6: (102) Plane
Axis
a
b
c
Intercept
points
-1
∞
½
Reciprocals
1/-1
Smallest
Ratio
-1
1/ ∞ 1/(½)
0
2
Miller İndices (102)
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Examples of Miller Indices
[2,3,3]
2
• See the figure.
• Consider the plane
shaded in yellow:
c
b
2
a
3
Plane intercepts axes at 3a , 2b , 2c
1 1 1
Reciprocal numbers are: 3 , 2 , 2
Miller Indices of the plane: (2,3,3)
Indices of the direction: [2,3,3]
Examples of Miller Indices
(111)
(110)
(100)
(200)
(100)
Examples of Miller Indices
Indices of a Family of Planes
• Sometimes. when the unit cell has rotational
symmetry, several nonparallel planes may be
equivalent by virtue of this symmetry, in which
case it is convenient to lump all these planes in
the same Miller Indices, but with curly brackets.
{100}  (100), (010), (001), (0 1 0), (00 1 ), ( 1 00)
{111}  (111), (11 1 ), (1 1 1), ( 1 11), ( 1 1 1 ), ( 1 1 1), ( 1 1 1 ), (1 1 1 )
• So, indices {h,k,l} represent all of the planes
equivalent to the plane (hkl) through rotational
symmetry.
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