Crystallographic Points, Directions, and Planes.
Points, Directions, and Planes in Terms of Unit Cell Vectors
ISSUES TO ADDRESS...
All periodic unit cells may be described via
these vectors and angles, if and only if
v
c
• How to define points, directions, planes, as well as
linear, planar, and volume densities
• a, b, and c define axes of a 3D coordinate system.
• coordinate system is Right-Handed!
v
b
But, we can define points, directions and
planes with a “triplet” of numbers in units
of a, b, and c unit cell vectors.
– Define basic terms and give examples of each:
v
a
• Points (atomic positions)
• Vectors (defines a particular direction - plane normal)
• Miller Indices (defines a particular plane)
• relation to diffraction
• 3-index for cubic and 4-index notation for HCP
MSE 280: Introduction to Engineering Materials
For HCP we need a “quad” of numbers, as
we shall see.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
POINT Coordinates
Crystallographic Directions
To define a point within a unit cell….
Express the coordinates uvw as fractions of unit cell vectors a, b, and c
(so that the axes x, y, and z do not have to be orthogonal).
Procedure:
1. Any line (or vector direction) is specified by 2 points.
c
•
b
v
c
v
b
origin
v
a
2.
a
pt. coord.
pt.
©D.D. Johnson 2004, 2006-08
Determine length of vector projection in each of 3 axes in
units (or fractions) of a, b, and c.
•
x (a)
y (b)
z (c)
0
0
0
1
0
0
1
1
1
1/2
0
1/2
The first point is, typically, at the origin (000).
X (a), Y(b), Z(c)
1
1
0
3.
Multiply or divide by a common factor to reduce the
lengths to the smallest integer values, u v w.
4.
Enclose in square brackets: [u v w]:
[110] direction.
5. Designate negative numbers by a bar [ 1 1 0]
•
Pronounced “bar 1”, “bar 1”, “zero” direction.
6.
“Family” of [110] directions is designated as <110>.
DIRECTIONS will help define PLANES (Miller Indices or plane normal).
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
1
Self-Assessment Example 1: What is crystallographic direction?
Magnitude along
X
Along x: 1 a
c
(a) What is the lattice point given by point P?
−112
Along y: 1 b
b
Self-Assessment Example 2:
Y
a
Along z: 1 c
DIRECTION =
(b) What is crystallographic direction
for the origin to P?
Z
[ 1 12]
[1 1 1]
Example 3: What lattice direction does the lattice point 264 correspond?
The lattice direction [132] from the origin.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
Symmetry Equivalent Directions
©D.D. Johnson 2004, 2006-08
Families and Symmetry: Cubic Symmetry
z
z
Note: for some crystal structures, different
directions can be equivalent.
x
x
z
Rotate 90 o about y-axis
Families of crystallographic directions
e.g. <1 0 0>
(001)
Symmetry operation can
generate all the directions
within in a family.
Angled brackets denote a family of crystallographic directions.
©D.D. Johnson 2004, 2006-08
y
y
(100)
[1 0 0], [ 1 0 0], [0 1 0], [0 1 0], [0 0 1], [0 0 1 ]
MSE 280: Introduction to Engineering Materials
(010)
Rotate 90 o about z-axis
e.g. For cubic crystals, the directions are all
equivalent by symmetry:
y
x
MSE 280: Introduction to Engineering Materials
Similarly for other
equivalent directions
©D.D. Johnson 2004, 2006-08
2
Designating Lattice Planes
How Do We Designate Lattice Planes?
Example 1
Why are planes in a lattice important?
Planes intersects axes at:
• a axis at r= 2
• b axis at s= 4/3
• c axis at t= 1/2
(A) Determining crystal structure
* Diffraction methods measure the distance between parallel lattice planes of atoms.
• This information is used to determine the lattice parameters in a crystal.
* Diffraction methods also measure the angles between lattice planes.
(B) Plastic deformation
* Plastic deformation in metals occurs by the slip of atoms past each other in the crystal.
* This slip tends to occur preferentially along specific crystal-dependent planes.
(C) Transport Properties
* In certain materials, atomic structure in some planes causes the transport of electrons
and/or heat to be particularly rapid in that plane, and relatively slow not in the plane.
• Example: Graphite: heat conduction is more in sp 2-bonded plane.
• Example: YBa 2Cu3O7 superconductors: Cu-O planes conduct pairs of electrons
(Cooper pairs) responsible for superconductivity, but perpendicular insulating.
+ Some lattice planes contain only Cu and O
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
How do we symbolically designate planes in a lattice?
Possibility #1: Enclose the values of r, s, and t in parentheses (r s t)
Advantages:
• r, s, and t uniquely specify the plane in the lattice, relative to the origin.
• Parentheses designate planes, as opposed to directions given by [...]
Disadvantage:
• What happens if the plane is parallel to --- i.e. does not intersect--- one of the axes?
• Then we would say that the plane intersects that axis at ∞ !
• This designation is unwieldy and inconvenient.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
How Do We Designate Lattice Planes?
Planes intersects axes at:
• a axis at r= 2
• b axis at s= 4/3
• c axis at t= 1/2
Self-Assessment Example
What is the designation of this plane in Miller Index notation?
How do we symbolically designate planes in a lattice?
Possibility #2: THE ACCEPTED ONE
1.
2.
3.
4.
What is the designation of the top face of the unit cell
in Miller Index notation?
Take the reciprocal of r, s, and t.
•
Here: 1/r = 1/2 , 1/s = 3/4 , and 1/r = 2
Find the least common multiple that converts all reciprocals to integers.
•
With LCM = 4, h = 4/r = 2 , k= 4/s = 3 , and l= 4/r = 8
Enclose the new triple (h,k,l) in parentheses: (238)
This notation is called the Miller Index.
* Note: If a plane does not intercept an axes (I.e., it is at ∞), then you get 0.
* Note: All parallel planes at similar staggered distances have the same Miller index.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
3
Families of Lattice Planes
Crystallographic Planes in FCC: (100)
Given any plane in a lattice, there is a infinite set of parallel lattice planes
(or family of planes) that are equally spaced from each other.
• One of the planes in any family always passes through the origin.
z
The Miller indices (hkl) usually refer to the plane that
is nearest to the origin without passing through it.
• You must always shift the origin or move the plane
parallel, otherwise a Miller index integer is 1/0!
y
• Sometimes (hkl) will be used to refer to any other plane
in the family, or to the family taken together.
• Importantly, the Miller indices (hkl) is the same vector
as the plane normal!
MSE 280: Introduction to Engineering Materials
Distance between (100) planes
x
Look down this direction
(perpendicular to the plane)
… between (200) planes
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
Crystallographic Planes in FCC: (110)
d100 = a
d200 =
a
2
©D.D. Johnson 2004, 2006-08
Crystallographic Planes in FCC: (111)
z
Look down this direction
(perpendicular to the plane)
y
Distance between (110) planes
d110 =
MSE 280: Introduction to Engineering Materials
a 2
2
©D.D. Johnson 2004, 2006-08
x
Distance between (111) planes
MSE 280: Introduction to Engineering Materials
d111 =
a 3
3
©D.D. Johnson 2004, 2006-08
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Comparing Different Crystallographic Planes
Note: similar to crystallographic directions, planes that are parallel to
each other, are equivalent
Distance between (110) planes
a
a
a 2
d110 =
=
=
2
2
12 + 12 + 0 2
-1
1
For (220) Miller Indexed planes you are getting planes at 1/2, 1/2, ∞.
The (110) planes are not necessarily (220) planes!
Miller Indices provide you easy
measure of distance between planes.
For cubic crystals:
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
Directions in HCP Crystals
1.
2.
3.
4.
To emphasize that they are equal, a and b is changed to a1 and a2.
The unit cell is outlined in blue.
A fourth axis is introduced (a 3) to show symmetry.
•
Symmetry about c axis makes a 3 equivalent to a1 and a2.
•
Vector addition gives a 3 = –( a1 + a2).
This 4-coordinate system is used: [a 1 a2 –( a1 + a2) c]
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
Directions in HCP Crystals: 4-index notation
What is 4-index notation for vector D?
Example
• Projecting the vector onto the basal plane, it lies
between a 1 and a2 (vector B is projection).
• Vector B = (a 1 + a 2), so the direction is [110] in
coordinates of [a1 a2 c], where c-intercept is 0.
• In 4-index notation, because a3 = –( a1 + a2), the
vector B is 1 [112 0] since it is 3x farther out.
3
a2
–2a 3
• In 4-index notation c = [0001], which must be
added to get D (reduced to integers) D = [1123]
Check w/ Eq. 3.7
or just use Eq. 3.7
B without 1/3
Easiest to remember: Find the coordinate axes that straddle the vector
of interest, and follow along those axes (but divide the a 1, a2, a3 part of vector
by 3 because you are now three times farther out!) .
Self-Assessment Test: What is vector C?
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
5
Directions in HCP Crystals: 4-index notation
Example
Miller Indices for HCP Planes
Check w/ Eq. 3.7: a dot-product projection in hex coords.
4-index notation is more important for planes in HCP, in order
to distinguish similar planes rotated by 120 o.
What is 4-index notation for vector D?
As soon as you see [1100], you will know
that it is HCP, and not [110] cubic!
• Projection of the vector D in units of [a1 a 2 c] gives
u’=1, v’=1, and w’=1. Already reduced integers.
t
• Using Eq. 3.7:
Find Miller Indices for HCP:
1
1
u = [2u '−v '] v = [2v '−u '] w = w '
3
3
1.
1
1
1
1
u = [2(1) − 1] =
v = [2(1) − 1] =
w = w '= 1
3
3
3
3
• In 4-index notation:
112
[
1]
333
• Reduce to smallest integers:
[112 3]
r
s
2.
3.
4.
5.
Find the intercepts, r and s, of the plane with any two
of the basal plane axes (a1, a2, or a3), as well as the
intercept, t, with the c axes.
Get reciprocals 1/r, 1/s, and 1/t.
Convert reciprocals to smallest integers in same ratios.
Get h, k, i , l via relation i = - (h+k), where h is
associated with a1, k with a2, i with a3, and l with c.
Enclose 4-indices in parenthesis: (h k i l) .
After some consideration, seems just using Eq. 3.7 most trustworthy.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
Miller Indices for HCP Planes
Yes, Yes….we can get it without a 3!
What is the Miller Index of the pink plane?
1.
The plane’s intercept a1, a3 and c
at r=1, s=1 and t= ∞, respectively.
1.
The reciprocals are 1/r = 1, 1/s = 1, and 1/t = 0.
2.
They are already smallest integers.
3.
We can write (h k i l) = (1 ? 1 0).
4.
Using i = - (h+k) relation, k=–2.
5.
Miller Index is
MSE 280: Introduction to Engineering Materials
(1210)
©D.D. Johnson 2004, 2006-08
1.
The plane’s intercept a1, a2 and c
at r=1, s=–1/2 and t= ∞, respectively.
1.
The reciprocals are 1/r = 1, 1/s = –2, and 1/t = 0.
2.
They are already smallest integers.
3.
We can write (h k i l) =
4.
Using i = - (h+k) relation, i=1.
5.
Miller Index is
(12 ?0)
(12 10)
But note that the 4-index notation is unique….Consider all 4 intercepts:
• plane intercept a1, a2, a3 and c at 1, –1/2, 1, and ∞, respectively.
• Reciprocals are 1, –2, 1, and 0.
• So, there is only 1 possible Miller Index is (12 10)
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
6
Basal Plane in HCP
Another Plane in HCP
z
Name this plane…
• Parallel to a 1, a2 and a3
• So, h = k = i = 0
• Intersects at z = 1
plane = (0001)
a2
a2
a3
+1 in a1
a3
-1 in a2
a1
a1
(1 1 0 0) plane
h = 1, k = -1, i = -(1+-1) = 0, l = 0
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
SUMMARY
z
(1 1 1) plane of FCC
• Crystal Structure can be defined by space lattice and basis atoms
(lattice decorations or motifs).
• Only 14 Bravais Lattices are possible. We focus only on FCC, HCP,
and BCC, i.e., the majority in the periodic table.
y
x
z
(0 0 0 1) plane of HCP
SAME THING!*
• We now can identify and determined: atomic positions, atomic planes
(Miller Indices), packing along directions (LD) and in planes (PD).
• We now know how to determine structure mathematically.
So how to we do it experimentally? DIFFRACTION.
a2
a3
a1
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-08
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