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LECTURE #05
Chapter 3:
Lattice Positions, Directions and Planes Learning Objective
• To describe the geometry in and around a
unit cell in terms of directions and planes.
Relevant Reading for this Lecture...
• Pages 64-83.
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Why are Crystal Planes &
Directions Important?
• Materials structures and
properties are related to them!
• Crystals deform on specific
planes and directions – they don’t
just ‘break’
break randomly
VIDEO
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Why are Crystal Planes & Directions Important?
• Materials structures and properties are related to
them! direction
di ti
After
plane
stress
• Crystals deform on specific planes and directions –
They don’t just ‘break’ randomly
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How do we express Planes and Directions in Crystals ?
• We use Miller indices: – hkl or uvw are the generic letters we use.
– hkl and uvw are called indices. They will be numbers that are related
to coordinate systems.
– No commas between the numbers.
z
• h represents the plane perpendicular to the x-axis;
k represents the plane perpendicular to the y-axis;
l represents the plane perpendicular to the z-axis.
• u represents the vector parallel to the x-axis;
y
v represents the vector parallel to the y-axis;
x
w represents the vector parallel to the z-axis.
– Negative values are expressed with a bar over the number
• Ex.: -3 is expressed as 3 (“bar 3”)
• Crystallographic directions:
– [1 11] ; one step in +x dir.; one step in –y dir.; one step in +z dir.
– [010] ; zero step in x dir.; one step in +y dir.; zero step in z dir.
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Point Coordinates
z
1,1,1
c
000
a
x
y
b
Express as fractions of unit vectors.
vectors
Point coordinates for unit cell corner
are 1,1,1 (or a, b, c)
What about this
position?
1, 1 , 0 (or a, b , 0)
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2
• For the above 1,1,1 coordinate – what is its direction?
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Let’s Relate Miller Indices to Vectors
“My crimes have both directions and
magnitude.” Vector from Despicable Me
Where is the origin
position in a crystal?
What do you look for?
Representation of a lattice and unit cell
Vector – points in a specific direction (hence you need an origin)
Vector – has a unit of length or magnitude
We will use vectors to define directions and lengths in crystal systems
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Directions in Crystals
Directions and their multiples are identical
[110]
Ex.:
[220]  2  [110]
[220]
[030]
z
y
[330]
[020]
x
[010]
(“Translational Symmetry”)
Translation: integer multiple of lattice constants 
identical position in another unit cell: (111), (222),
(333), etc.
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How to apply Miller Indices for How to apply Miller Indices for Directions
Directions
1 0

• Draw vector and define the tail as the origin.
z
1
0 0
1
1 0 
2  [ 2 0
• D
Determine the length of the t
i th l
th f th
vector projection in unit cell dimensions
2
1
2
1]
Example in class
[111]
P
– a, b, and c.
[021]
y
• Remove fractions by multiplying by the smallest possible factor.
[110]
• Enclose in square brackets
a b c
x
• In cubic crystals, directions and their negatives are equivalent but NOT the same.
0

1
0 0
1
2
Point P (head)
0 Origin
(tail)
0 1 2
2  [0 2 1]
1
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In class example #1:
What are the indices of the line/vector connecting points O and P? What are the
indices of the line connecting points Q and R?
z
Q
P
R
y
O
x
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In class example #1: SOLUTION
What are the indices of the line/vector connecting points O and P? What are the
indices of the line connecting points Q and R?
z
a b c
0 1 1 2 Point P
1
0 0 Origin O
2

 12
1 12
[1 2 1]
2 
Q
P
R
O
y
a
0

x
b
0
c
0 Point R
3
4 Origin Q
1
2
 1 2 1  3 4
4  [2 4
3]
1
10
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In class example #2:
In a cubic unit cell, draw correctly a vector with indices [146].
z
O
y
x
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In class example #2: SOLUTION
In a cubic unit cell, draw correctly a vector with indices [146].
This step is the
opposite of
clearing
fractions!
z
Select your origin. Put it
wherever you want to.
indices [1
6
6
1
6
O
Div. by 6
1
y
4
6

2
3
1
6
4 6]
4
6
6
6
These fractions denote
how far to step in the
x, y, or z directions
(away from the origin).
x
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NOTE: It would be “wise” to select the origin so that you can complete the
desired steps within the cell that you are using!
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Families of Directions
• In cubic systems, directions that have the same indices are
equivalent regardless of their order or sign.
[001]
z
[100]
[0 10]
y
x
[010]
[100]
[00 1]
The family of < 100 > directions is:
[100], [ 100]
We enclose indices in
carats rather than brackets
to indicate a family of
directions
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[010], [0 10]
[001], [00 1]
Families of Directions
• In non‐cubic systems, directions that have the same indices
are not necessarily equivalent.
z
z
CUBIC
a=b=c
a
[010]
a
a
c
[010]  [001]
[010]
y
b
x
x
a
ORTHORHOMBIC
abc
[010]  [001]
y
z
TETRAGONAL
a  b  c
c
[010]  [001]
[010]
a
a
y
x
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Crystallographic Planes
A specific direction is normal (90o) to its
specific, equivalent plane. For example [100]
is normal to (100) but [100] is not normal to
(010)
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Adapted from Fig. 3.9, Callister 7e.
Miller Indices for Planes
 Specific crystallographic plane: (hkl)
 Family of crystallographic planes: {hkl}
– (hkl), (lkh), (hlk) … etc.
– In cubic systems, planes having the same indices are equivalent regardless of order or sign AND directions are normal to the planes
of order or sign. AND directions are normal to the planes
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PROCEDURES FOR INDICES OF PLANES
(Miller indices)
1. Identify the coordinate intercepts of the plane (i.e., the coordinates at
which the plane intersects the x, y, and z axes).
 If plane is parallel to an axis (DOES NOT INTERSECT IT), the
intercept is taken as infinity ().
 If the plane passes through the origin, consider an equivalent plane in
an adjacent unit cell or select a different origin for the same plane.
2. Take reciprocals of the intercepts.
3. Clear fractions to the lowest integers.
4. Cite specific planes in parentheses, (h k l), placing bars over negative
indices.
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MILLER INDICES FOR A SINGLE PLANE
z
x
y
z
Intercept

1

Reciprocal
1/
1/1
1/
Clear
0
1
0
INDICES
0
1
0
y
((010))
x
The cube faces are from the {100} family of planes
(100), (010), (001), ( 100), (0 10), (00 1),
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MILLER INDICES FOR A SINGLE PLANE – cont’d
z
x
y
z
Intercept
1
1

Reciprocal
1/1
1/1
1/
Clear
1
1
0
INDICES
1
1
0
y
((110))
x
The {110} family of planes
(110), (011), (101), ( 1 10), (0 1 1), ( 10 1)
( 110), (1 10), ( 101), (10 1), (01 1), (0 11)
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MILLER INDICES FOR A SINGLE PLANE – cont’d
z
x
y
z
Intercept
1/2
1/2

Reciprocal
2/1
2/1
1/
Clear
2
2
0
INDICES
2
2
0
y
((220))
x
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Crystallographic Planes
example
1. Intercepts
2. Reciprocals
3. Reduction
b
1
1/1
/
1
3
a
1/2
1/(½)
/(½)
2
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4. Miller Indices
c
3/4
1/(¾)
/(¾)
4/3
4
(634)
z
c
a



y
b
x
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General Rules for Crystal Directions, Planes, and Miller Indices
• x, y, and z are the axes (on an
arbitrary origin).
– In some crystal systems the axes
are not mutually perpendicular.
• a, b, c and α, β, γ are lattice
parameters.
– length of unit cell along side of
unit cell.
• h, k, l are the Miller indices for planes
and directions.
– Ex., (hkl) and [hkl]
Unit cell


c

a
b
Geometry of a general unit cell
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Comment: HCP Crystallographic Directions
z
• In general we can define
Miller indices just like we
d ffor the
do
th other
th crystals.
t l
c
• However, sometimes in
engineering practice, a
4-indice system is used.
x
(a1)
• It is called Miller-Bravais
indices. There are
equations to convert.
a
120°
a
a
y
(a2)
a=b≠c
α = β = 90°; γ = 120°
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Comment: HCP Crystallographic Directions
• Miller‐Bravais indices (i.e., uvtw) are related to the direction indices (i.e., UVW) as follows.
[UVW ]  [uvtw]
1
 2U  V 
3
1
v   2V  U 
3
t   U  V 
u
w W
Fig. 3.8(a), Callister 7e.
I WILL NOT TEST YOU ON THIS!!!
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I only show it because some of you will end up working with
hexagonal metals like Ti or Mg after you graduate.
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Comment: HCP Crystallographic Directions
DIRECTIONS
(UVW)
(uvtw)
a3
a3
[ 110]
[ 120]
[ 1100]
[0 110]
a2
[110 ]
a1
[100 ]
[ 210 ]
a2
[1120]
a1
[2 1 10]
[10 10]
I WILL NOT TEST YOU ON THIS!!!
I only show it because some of you will end up working with
hexagonal metals once you graduate.
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Summary
•
•
Miller Directions
– Indices for a direction are enclosed in square brackets.
– Negative values are expressed with a bar over the number.
– An example of a Miller Direction: [1 11]
one step in +x dir.; one step in –y dir.; one step in +z dir.
Miller Planes
– Intercepts for a specific crystallographic plane are enclosed in
parenthesis.
– When identifying the coordinate intercepts of the plane (i.e., the
coordinates at which the plane intersects the x, y, and z axes):
 If plane is parallel to an axis (DOES NOT INTERSECT IT), the
intercept is taken as infinity ().
 If the plane passes through the origin, consider an equivalent plane
in an adjacent unit cell or select a different origin for the same
plane.
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