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

a

b

c

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• a, b, and c define axes of a 3D coordinate system.• coordinate system is Right-Handed!

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.

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).

a

b

c

origin

pt. coord.

x (a) y (b) z (c)

0 0 0

1 0 0

1 1 1

1/2 0 1/2

pt.

Crystallographic Directions

Procedure:1. Any line (or vector direction) is specified by 2 points.

• The first point is, typically, at the origin (000).

2. 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.

a b

c

DIRECTIONS will help define PLANES (Miller Indices or plane normal).

[1 1 0]5. Designate negative numbers by a bar • Pronounced “bar 1”, “bar 1”, “zero” direction.

6. “Family” of [110] directions is designated as <110>.

Self-Assessment Example 1: What is crystallographic direction?

a b

c Along x: 1 a

Along y: 1 b

Along z: 1 c

[1 1 1]DIRECTION =

Magnitude alongX

Y

Z

Self-Assessment Example 2:

(a) What is the lattice point given by point P?

(b) What is crystallographic direction for the origin to P?

The lattice direction [132] from the origin.

Example 3: What lattice direction does the lattice point 264 correspond?

[1 12]

112

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], [ 0 0], [0 1 0], [0 0], [0 0 1], [0 0 ] 111

Families of crystallographic directions e.g. <1 0 0>

Angled brackets denote a family of crystallographic directions.

Families and Symmetry: Cubic Symmetry

x

y

z

(100)

Rotate 90o about z-axis

x

y

z

(010)

x

y

z

(001)Rotate 90o about y-axis

Similarly for other equivalent directions

Symmetry operation can generate all the directions within in a family.

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

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?

1. Take the reciprocal of r, s, and t. • Here: 1/r = 1/2 , 1/s = 3/4 , and 1/r = 2

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

3. Enclose the new triple (h,k,l) in parentheses: (238)4. 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.

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?

Families of Lattice Planes

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!

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

x

y

Look down this direction(perpendicular to the plane)

Crystallographic Planes in FCC: (100)

d100 aDistance between (100) planes

Distance to (200) plane d200 a

2

Crystallographic Planes in FCC: (110)

d110 a 2

2

Distance between (110) planes

Crystallographic Planes in FCC: (111)

z

x

y

Look down this direction(perpendicular to the plane)

d111 a 3

3Distance between (111) planes

Note: similar to crystallographic directions, planes that are parallel to each other, are equivalent

Comparing Different Crystallographic Planes

-1

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: Miller Indices provide you easy measure of distance between planes.

d110 a

12 12 02a

2a 2

2

Distance between (110) planes

Directions in HCP Crystals

1. To emphasize that they are equal, a and b is changed to a1 and a2.2. The unit cell is outlined in blue.3. 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).

4. This 4-coordinate system is used: [a1 a2 –( a1 + a2) c]

Directions in HCP Crystals: 4-index notation

Example What is 4-index notation for vector D?

• 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 since it is 3x farther out.

• In 4-index notation c = [0001], which must be added to get D (reduced to integers) D =

[1123]

Self-Assessment Test: What is vector C?

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!).

1

3[112 0]

Check w/ Eq. 3.7or just use Eq. 3.7

a2

–2a3

B without 1/3

Directions in HCP Crystals: 4-index notation

Example

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:

[112 3]

[1

3

1

3

2

31]

Check w/ Eq. 3.7: a dot-product projection in hex coords.

u 13

[2u'v'] v 13

[2v'u'] w w '

u 13

[2(1) 1]13

v 13

[2(1) 1]13

w w '1

• In 4-index notation:

• Reduce to smallest integers:

After some consideration, seems just using Eq. 3.7 most trustworthy.

Miller Indices for HCP Planes

As soon as you see [1100], you will know that it is HCP, and not [110] cubic!

4-index notation is more important for planes in HCP, in order to distinguish similar planes rotated by 120o.

1. 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.

2. Get reciprocals 1/r, 1/s, and 1/t.3. Convert reciprocals to smallest integers in same ratios.4. 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.5. Enclose 4-indices in parenthesis: (h k i l) .

Find Miller Indices for HCP:

r s

t

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 (1210)

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 10)

(12 ?0)

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)

a1

a2

a3

• Parallel to a1, a2 and a3 • So, h = k = i = 0

• Intersects at z = 1

Name this plane…

Basal Plane in HCP

plane (0001)

z

a1

a2

a3

(1 1 0 0) plane

+1 in a1

-1 in a2

h = 1, l = 0i = -(1+-1) = 0,k = -1,

Another Plane in HCP

(1 1 1) plane of FCCz

x

y

z

a1

a2

a3

(0 0 0 1) plane of HCPSAME THING!*

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.