Crystallographic Points, Directions, and Planes.
ISSUES TO ADDRESS...
• How to define points, directions, planes, as well as
linear, planar, and volume densities
– Define basic terms and give examples of each:
• 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
©D.D. Johnson 2004, 2006-10
Points, Directions, and Planes in Terms of Unit Cell Vectors
All periodic unit cells may be described via
these vectors and angles, if and only if

c
• a, b, and c define axes of a 3D coordinate system.
• coordinate system is Right-Handed!

b

a
But, we can define points, directions and
planes with a “triplet” of numbers in units
of a, b, and c unit cell vectors.
For HCP we need a “quad” of numbers, as
we shall see.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
POINT Coordinates
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).
pt. coord.
pt.

c

b
origin

a
MSE 280: Introduction to Engineering Materials
x (a)
y (b)
z (c)
0
0
0
1
0
0
1
1
1
1/2
0
1/2
©D.D. Johnson 2004, 2006-10
Crystallographic Directions
Procedure:
1. Any line (or vector direction) is specified by 2 points.
c
•
b
2.
a
The first point is, typically, at the origin (000).
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)
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, 2006-10
Self-Assessment Example 1: What is crystallographic direction?
Magnitude along
X
Along x: 1 a
c
Along y: 1 b
b
Y
a
Along z: 1 c
DIRECTION =
MSE 280: Introduction to Engineering Materials
Z
[1 1 1]
©D.D. Johnson 2004, 2006-10
Self-Assessment Example 2:
(a) What is the lattice point given by point P?
112
(b) What is crystallographic direction
for the origin to P?

[1 12]

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, 2006-10
Symmetry Equivalent Directions
Note: for some crystal structures, different
directions can be equivalent.
e.g. For cubic crystals, the directions are all
equivalent by symmetry:
[1 0 0], [ 1 0 0], [0 1 0], [0 1 0], [0 0 1], [0 0 1 ]
Families of crystallographic directions
e.g. <1 0 0>
Angled brackets denote a family of crystallographic directions.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
Families and Symmetry: Cubic Symmetry
z
z
(010)
Rotate
90o
about z-axis
y
y
(100)
x
x
z
Rotate 90o about y-axis
(001)
Symmetry operation can
generate all the directions
within in a family.
y
x
MSE 280: Introduction to Engineering Materials
Similarly for other
equivalent directions
©D.D. Johnson 2004, 2006-10
Designating Lattice Planes
Why are planes in a lattice important?
(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 sp2-bonded plane.
• Example: YBa2Cu3O7 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, 2006-10
How Do We Designate Lattice Planes?
Example 1
Planes intersects axes at:
• a axis at r= 2
• b axis at s= 4/3
• c axis at t= 1/2
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, 2006-10
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
How do we symbolically designate planes in a lattice?
Possibility #2: THE ACCEPTED ONE
1.
2.
3.
4.
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, 2006-10
Self-Assessment Example
What is the designation of this plane in Miller Index notation?
What is the designation of the top face of the unit cell
in Miller Index notation?
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
Families of Lattice Planes
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.
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, i.e.,∞!
•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
©D.D. Johnson 2004, 2006-10
Crystallographic Planes in FCC: (100)
z
y
Distance between (100) planes
x
Look down this direction
(perpendicular to the plane)
MSE 280: Introduction to Engineering Materials
Distance to (200) plane
©D.D. Johnson 2004, 2006-10
d100  a
d200 
a
2
Crystallographic Planes in FCC: (110)
Distance between (110) planes
d110 
MSE 280: Introduction to Engineering Materials
a 2
2
©D.D. Johnson 2004, 2006-10
Crystallographic Planes in FCC: (111)
z
Look down this direction
(perpendicular to the plane)
y
x
Distance between (111) planes
MSE 280: Introduction to Engineering Materials
d111 
a 3
3
©D.D. Johnson 2004, 2006-10
Note: similar to crystallographic directions, planes that are parallel to
each other, are equivalent
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
Comparing Different Crystallographic Planes
-1
Distance between (110) planes
a
a
a 2
d110 


2
2
12  12  0 2
1
For any vector, v
cos(vx)+cos(vy)+cos(vz)=1
For (220) Miller Indexed planes you are getting planes at 1/2, 1/2, ∞.
The (110) planes are not necessarily (220) planes!
For cubic crystals:
MSE 280: Introduction to Engineering Materials
Miller Indices provide you easy
measure of distance between planes.
©D.D. Johnson 2004, 2006-10
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 (a3) to show symmetry.
•
Symmetry about c axis makes a3 equivalent to a1 and a2.
•
Vector addition gives a3 = –( a1 + a2).
This 4-coordinate system is used: [a1 a2 –( a1 + a2) c]
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
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 a1 and a2 (vector B is projection).
• Vector B = (a1 + a2), 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
–
2a3
• In 4-index notation c = [0001], which must be
added to get D (reduced to integers) D = [1123]

Check w/ Eq. 3.7
B without 1/3
or just use Eq. 3.7
Easiest to remember: Find the coordinate axes that straddle the vector
of interest, and follow along those axes (but divide the a1, 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, 2006-10
Directions in HCP Crystals: 4-index notation
Example
Check w/ Eq. 3.7: a dot-product projection in hex coords.
What is 4-index notation for vector D?
• Projection of the vector D in units of [a1 a2 c] gives
u’=1, v’=1, and w’=1. Already reduced integers.
• Using Eq. 3.7:
1
3
1
3
u  [2u 'v '] v  [2v 'u '] w  w '
1
3
u  [2(1)  1] 

1
1
1
v  [2(1)  1] 
w  w ' 1
3
3
3
• In 4-index notation:

[
112
1]
333
• Reduce to smallest integers:
[1123]
 Eq. 3.7 most trustworthy.
After some consideration, seems just using
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
Miller Indices for HCP Planes
4-index notation is more important for planes in HCP, in order
to distinguish similar planes rotated by 120o.
As soon as you see [1100], you will know
that it is HCP, and not [110] cubic!
t
Find Miller Indices for HCP:
1.
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) .
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
Miller Indices for HCP Planes
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, 2006-10
Yes, Yes….we can get it without a3!
1.
The plane’s intercept a1, a2 and c
at r=1, s=–1/2 and t= ∞, respectively.
2.
The reciprocals are 1/r = 1, 1/s = –2, and 1/t = 0.
3.
They are already smallest integers.
4.
We can write (h k i l) =
5.
Using i = - (h+k) relation, i=1.
6.
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, 2006-10
Basal Plane in HCP
Name this plane…
• Parallel to a1, a2 and a3
• So, h = k = i = 0
• Intersects at z = 1
plane  (0001)
a2

a3
a1
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
Another Plane in HCP
z
a2
+1 in a1
a3
-1 in a2
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, 2006-10
z
(1 1 1) plane of FCC
y
x
z
SAME THING!*
(0 0 0 1) plane of HCP
a2
a3
a1
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10
SUMMARY
• 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.
• 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.
MSE 280: Introduction to Engineering Materials
©D.D. Johnson 2004, 2006-10