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EEE 352: Lecture 04
Crystal Directions; Wave Propagation
(110)
(001)
Electron beam generated pattern in TEM
Crystal Directions
We have referred to various directions in the crystal as (100), (110),
and (111).
What do these mean?
How are these directions determined?
Consider a cube
and a plane
1/2
1
2/3
The plane has intercepts:
x
= 0.5
a
,
y
= 0.667
a
, z =
a
.
Crystal Directions
We want the NORMAL to the surface.
So we take these intercepts
(in units of
a
), and invert them:
Then we take the lowest common set of integers:
These are the MILLER INDICES of the plane.
The NORMAL to the plane is the (4,3,2) direction, which is
normally written just (432).
(A negative number is indicated by a
bar over the top of the number.)
⎟
⎠
⎞
⎜
⎝
⎛
⇒
⎟
⎠
⎞
⎜
⎝
⎛
1
,
2
3
,
2
1
,
3
2
,
2
1
()
2
,
3
,
4
1
,
2
3
,
2
1
,
3
2
,
2
1
⇒
⎟
⎠
⎞
⎜
⎝
⎛
⇒
⎟
⎠
⎞
⎜
⎝
⎛
Crystal Directions
1
1
1
1
(111)
1
(110)
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Crystal Directions
1/2
1
1
(100)
1/2
(220)
Crystal Directions
(110)
(001)
Electron beam generated pattern in TEM
So far, we have discussed the concept of crystal directions:
1
1
1
1
(111)
1
(110)
We want to say a few more things about this.
x
y
z
Consider this plane.
The intercepts are 0,0,
∞
,
which leads to 1,1,0 for
Miller indices.
What are the normals to
the plane?
3
x
y
z
1
1
The intercepts are 1, 1,
∞
,
which leads to
The normal direction is
(
)
0
1
1
[
]
0
1
1
ˆ
ˆ
→
−
y
x
The intercepts are 1, 1,
∞
,
which leads to
The normal direction is
(
)
10
1
[
]
10
1
ˆ
ˆ
→
+
−
y
x
x
y
z
1
1
We can easily shift the planes by one lattice vector in
x
or
y
x
y
z
1
1
[ ]
0
1
1
[]
10
1
These are two different
normals to the same plane.
Directions have square brackets […]
Planes have parentheses (…)
x
y
z
1
1
Now consider the following plane:
[ ]
0
1
1
[ ]
110
This direction lies
in
the original plane.
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x
y
z
1
1
Now consider the following plane:
[]
001
This direction lies
in
the original plane.
x
y
z
1
1
[ ]
10
1
points into the page
[ ]
110
[ ]
001
[ ]
111
Crystal Directions
We want the NORMAL to the surface.
So we take these intercepts
(in units of
a
), and invert them:
Then we take the lowest common set of integers:
These are the MILLER INDICES of the plane.
The NORMAL to the plane is the [4,3,2] direction, which is
normally written just [432].
(A negative number is indicated by a
bar over the top of the number.)
⎟
⎠
⎞
⎜
⎝
⎛
⇒
⎟
⎠
⎞
⎜
⎝
⎛
1
,
2
3
,
2
1
,
3
2
,
2
1
2
,
3
,
4
1
,
2
3
,
2
1
,
3
2
,
2
1
⇒
⎟
⎠
⎞
⎜
⎝
⎛
⇒
⎟
⎠
⎞
⎜
⎝
⎛
This inversion
creates units of 1/cm
These “numbers”
define a new VECTOR
in this “reciprocal
space”
Crystal Directions
1/2
1
1
[100]
1/2
[220]
These vectors
are normal to the
planes, but are
defined in this
new space.
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This note was uploaded on 09/19/2009 for the course EEE 352 taught by Professor Ferry during the Spring '08 term at ASU.
 Spring '08
 Ferry

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