Materials c

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MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-10

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 viathese 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 uni ts

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.

ab

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?

ab

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


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Symmetry Equivalent Directions

Note: for some crystal structures, differentdirections 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?

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.



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. 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 multipl e 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?





z

x

y

Look down this direction

(perpendicular to the plane)

Crystallographic Planes in FCC: (100)

d 100 aDistance between (100) planes

Distance to (200) plane d 200 a

2




d 110 a 2

2

Distance between (110) planes




z

x

y

Look down this direction(perpendicular to the plane)

d 111 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.

d 110 a

1212

02

a

2

a 2

2

Distance between (110) planes

For any vector, v

cos(vx)+cos(vy)+cos(vz)=1




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. 3.7

or 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

3 1]

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

u 13

[2u 'v ' ] v 13

[2v 'u ' ] w w '

u

1

3[2(1)1]

1

3v

1

3[2(1)1]

1

3w 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 isassociated 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






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 FCC

z

x

y

z

a1

a2

a3

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

SUMMARY



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 .

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Materials c