Lesson 8 Lattice Planes and Directions Suggested Reading...Planes in Unit Cells Some important...
Transcript of Lesson 8 Lattice Planes and Directions Suggested Reading...Planes in Unit Cells Some important...
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Analytical Methods for Materials
Lesson 8Lattice Planes and Directions
Suggested Reading• Chapters 2 and 6 in Waseda
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Directions and Miller Indices • Draw vector and define the tail
as the origin.
• Determine the length of the vector projection in unit cell dimensions
– a, b, and c.
• Remove fractions by multiplying by the smallest possible factor.
• Enclose in square brackets
• Negative indices are written with a bar over the number..
[111]
[110]
x
y
z
[201]
1 12 3
13
12
10 0 0
1[ 26 3 6 ]
P
a b c
OriginPoint P
ab
c
O
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Families of Directions(i.e., directions of a form)
• In cubic systems, directions that have the same indices areequivalent regardless of their order or sign.
[10The
0],famil
[100]y of <100> d
[010], [010]
ir
[0
ections
01], [
is:
001]
[001]
[010]
[100] [001]
[100]
x
z
y[010]
We enclose indices in carats rather than brackets to indicate
a family of directions
All of these vectors have the same
“size” and# lattice
points/length
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[100]
[100] [010] [001]
<100>
CUBIC <aaa>
[100] [010] [001]
TETRAGONAL <aac>[100] [010][100] [010]
ORTHORHOMBIC <abc>[100]
In non-cubic systems,
directions with the same
indices may notbe equivalent.
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Directions in Crystals
Directions and their multiples are identical
x
z y
[010]
[020]
[030]
[110]
[220]
Ex.:[220] 2 [110]
Vectors and multiples of vectors
have the same # lattice points/length
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Miller Indices for Planes Specific crystallographic plane: (hkl)
Family of crystallographic planes: {hkl}
– Ex.: (hkl), (lkh), (hlk) … etc.
– In cubic systems, planes having the same indices are equivalent regardless of order or sign.
• In hexagonal crystals, we use a four index system (h k i l).– We can convert from three to four indices
• h+k = -i
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ALL MEMBERS HAVE SAME ARRANGEMENT OF LATTICE POINTS
{h k l}
We use Miller indices to denote planes
FAMILY OF PLANES
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1. Identify the coordinate intercepts of the plane (i.e., the coordinates at whichthe plane intersects the x, y, and z axes).
If plane is parallel to an axis, the intercept is taken as infinity ().
If the plane passes through the origin, consider an equivalent plane in an adjacentunit 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.
PROCEDURES FOR INDICES OF PLANES(Miller indices)
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MILLER INDICES FOR A SINGLE PLANE
x
y
z
x y zIntercept 1 1
Reciprocal 1/1 1/1 1/Clear 1 1 0
INDICES 1 1 0
The {110} family of planes(110), (011), (101), (110), (011), (101)(110), (110), (101), (101), (011), (011)
(110)
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MILLER INDICES FOR A SINGLE PLANE – cont’d
x
y
zx y z
Intercept 1 1 1Reciprocal 1/1 1/1 1/1
Clear 1 1 1INDICES 1 1 1
x y zIntercept -1 -1 -1
Reciprocal -1/1 -1/1 -1/1Clear -1 -1 -1
INDICES 1 1 11111 1 1 (( ) )
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2 2 0
2 2 0
2/1 2/1 1/
1/2 1/2
MILLER INDICES FOR A SINGLE PLANE – cont’d
x y zIntercept
Reciprocal
Clear
INDICES
(220)
x
y
z
Planes and their multiples are not identical( (110)220)
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Planes in Unit CellsSome important aspects of Miller indices for planes:
1. Planes and their negatives are identical.This was NOT the case for directions.
2. Planes and their multiples are NOT identical.This is opposite to the case for directions.
3. In cubic systems, a direction that has the same indices as a plane is to that plane. This is not always true for non-cubic systems.
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Planes of a Zone
• A zone is a direction [uvw]
• Planes belonging to a particular zone share a direction.
This direction is known as a zoneaxis.
y
z
(220)
(010)
ZONE AXIS [uvw] = [001]
(110)
001 1 10
001 220
110 001 1 0 1 0 0 1 0
220 001 2 0 2 0 0 1 0
0
lies on
lies on
hkl uvw
Weiss Zone Law
• If a direction [uvw] lies in a plane {hkl}:
[uvw]·{hkl} = uh + vk + wl = 0
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ZONE AXIS
(110)
(220)
(010)
[ ] [001]uvw
This rule holds for all
crystal systems
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How to Determine the Zone Axis
• Take the cross product of the intersecting planes.
y
z ZONE AXIS [uvw] = [001]
(110)
(h1k1l1) × (h2k2l2) = [uvw]
1 1 0 1 12 2 0 2 2
1 0 0 2 1 2 1 0 0 2 1 2[0 ] [ 040 0 1]
u u
u u
v w
w w
v
v v
(220)
(010)
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Indexing in Hexagonal Systems
• The regular 3 index system is not suitable.
• Planes with the same indices do not necessarily look like.
• 4 index system introduced.– Miller-Bravais indices
a1
a2
a3
c
120°
120° 120°
a1
a2
a3
c (0001)
(1210)(1100)
(10 11)[100]
[010]
[001][110]
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Indexing in Hexagonal Systems
• Planes:– (hkl) becomes (hkil)– i = -(h+k)
• Directions:– [UVW] becomes [uvtw]– U = u-t ; u = (2U – V)/3– V = v-t ; v = (2V – U)/3– W=w ; t=-(U+V)
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a1
a2
a3
a1
a2
a3
PLANES
(100)
(010) (110)
(010)(110)
(100)
(1010)
(1100)(0110)
(1100) (0110)
(1010)
DIRECTIONS
a1
a2
a3
a1
a2
a3
[120] [110]
[100]
[210]
[110]
[1010]
[2110][1120]
[1100][0110]
Miller Indices Miller-Bravais Indices(hkil)(hkl)
(uvtw)(UVW)
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a1
c
a2
a3
010 12 10
100 21 10
110 1120
Some typical directions in an HCP unit cell using three- and four-axis systems.
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Inter-planar Spacings
• The inter-planar spacing in a particular direction is the distance between equivalent planes of atoms.
Each material has a set of characteristic inter-planar spacings.They are directly related to crystal size (i.e. lattice parameters) and atom location.
a
x
y(100)
(110)
(210)
z
Assuming no intercept on z-axis
d210
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Interplanar Spacing – cont’d
2 2 2
2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2 2
2 2 2 3
2 2 2
2 2 2 2
2
1
1 43
1
sin 2 cos cos11 3cos 2cos
1
1 1si
h k ld a
h hk k ld a c
h k ld a c
h hk k hk kl hld a
h k ld a b c
d
CUBIC:
HEXAGONAL:
TETRAGONAL:
RHOMBOHEDRAL:
ORTHORHOMBIC:
MONOCLINIC:
2 2 2 2
2 2 2 2
2 2 211 22 3 12 23 132 2
sin 2 cosn
1 1 2 2 2
h k l hla b c ac
S h S k S l S hk S kl S hld V
TRICLINIC*:
2 2 2 2 2 2 2 2 211 22 33
2 2 212 23 13
2 2 2
sin ; sin ; sin
cos cos cos ; cos cos cos ; cos cos cos
1 cos cos cos 2cos cos cos
S b c S a c S a b
S abc S a bc S ab c
V abc