L03A: Chapter 3 Structures of Metals & Ceramics

• The properties of a material depends on the arrangement of atoms within the solid.

• In a single crystal the atoms are in an ordered array called the structure. Single crystals are necessary for many applications and can be very large. For example, silicon crystals can be up to 2 feet in diameter: http://www.flickr.com/photos/davemessina/6231300549/

• A polycrystalline material consists of many crystals. Materials used for construction or fabrication are usually polycrystalline. For example: http://www.cartech.com/news.aspx?id=578

• In this chapter we examine typical crystal structures for metals, inorganic compounds, and carbon.

• You will see how to specify crystal planes and directions.• You will learn how to calculate some properties of crystals from their

structure, including the dependence on direction in their lattice.• Review calculation of areas for squares and rectangles, and calculation of

distances and areas for right triangles, e.g. at the Math Skills Review under Read, Study & Practice at WileyPLUS.com.

Last revised January 12, 2014 by W.R. Wilcox, Clarkson University

Amorphous and crystalline materials

• A material is crystalline if the atoms display long-range order, i.e. the same repeating arrangement over-and-over.

• The atoms in some materials do not have long-range order. These are called amorphous or glassy. Most polymers are amorphous, but so are some ceramics, metals, and forms of carbon.

crystalline SiO2amorphous SiO2

• Equilibrium structures are those with the minimum Gibbs energy G, although atomic movement in solids is so slow that equilibrium is often not reached at room temperature. (In thermodynamics you’ll see that G = H – TS)

Hard-sphere model of crystals• We may show the atoms as points or small spheres connected by lines, or

we may show them as hard spheres of defined diameter in contact with one another.

• For a metal with a face-centered cubic lattice:

Unit cell. When repeated, generates the entire crystal.

Metallic Crystal Structures• Bonding is not directional• Minimum energy when nearest-neighbor distances are small.• The electron cloud shields the positive cores from one another.• Metals have the simplest crystal structures. • We will examine the three most common.• Two of their unit cells are based on a cube:

Body-centered cubic (BCC) Face-centered cubic (FCC)

Virtual Materials Science and Engineering (VMSE): http://higheredbcs.wiley.com/legacy/college/callister/1118061608/vmse/xtalc.htm

APF calculation for a simple cubic structure

Atomic Packing Factor (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

* assuming hard spheres

a

R=0.5a

atoms/unit cell = 8 x 1/8 = 1

APF = a3

4

3(0.5a) 31

atoms

unit cellatom

volume

unit cell

volume= 0.52

The coordination number is the number of nearest neighbors. What is it here?

Coordination number = 8

Body Centered Cubic Structure (BCC)Examples: Cr, W, Fe(), Ta, Mo

1 center + 8 corners x 1/8 = 2

Atoms touch only along cube diagonals.

All atoms are identical and are colored differently only

for ease of viewing.

Click on image for animation (Courtesy P.M. Anderson)

Coordination number?

Number of atoms per unit cell?

How many touch the one in the center?

Atomic Packing Factor for BCC

aR length = 4R =

Close-packed directions:3 a

a

a 2

a 3

APF =

4

3 ( 3a/4)32

atoms

unit cell atom

volume

a3unit cell

volume= 0.68

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Theoretical Density

where n = number of atoms/unit cell A = atomic weight (g/mol) VC = Volume of unit cell = a3 for cubic NA = Avogadro constant = 6.022 x 1023 atoms/mol

VC NA

n A =

Density = =Cell Unit of VolumeTotal

Cell Unit in Atomsof Mass

• Cr is body-centered cubic

A = 52.00 g/mol

R = 0.125 nm

n = 2 atoms/unit cell

a = 4R/ 3 = 0.2887 nm

aR

= a3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.022 x 1023

Example: Theoretical Density of Chromium

theoretical

experimental

= 7.18 g/cm3

= 7.19 g/cm3

Face Centered Cubic Structure (FCC)Examples: Al, Cu, Au, Pb, Ni, Pt, Ag

6 face x 1/2 + 8 corners x 1/8 = 4

Atoms only touch along face diagonals.

How many atoms in the unit cell touch the atom in the center of

the front face?

How many additional atoms touch it in the unit cell in front of this one?

Coordination number = 8 + 4 = 12

How many atoms in one unit cell?

Atomic Packing Factor for FCC

This is the maximum achievable APF and is one of two close-packed structures.

Close-packed directions: length = 4R = 2 a

a

2 a

APF =

4

3( 2a/4)34

atoms

unit cell atom

volume

a3unit cell

volume= 0.74

Crystal Systems

a, b, and c are the lattice constants

Only for the cubic system are the angles all 90o and the lattice constants all the same.

Crystal structure• Seven different possible geometries for the unit cell.• There are 14 Bravais lattices, with each point representing the same atom or

collection of atoms.

• Pure metals are usually FCC, BCC or HCP.

• Except for hexagonal, number of atoms per unit cell:1/8 at corners1/2 at face centersAll of body centered

Point Coordinates in a Lattice

Point coordinates for the unit cell center are

a/2, b/2, c/2 ½ ½ ½

Point coordinates for unit cell corner are a, b, c 111

Translation by an integer multiple of lattice constants reaches an identical position in another unit cell

z

x

ya b

c

000

111

y

z 2c

b

b

Miller Indices for Crystallographic Directions

examples: 1, 0, ½ => 2, 0, 1 => [201]

1. If necessary, translate the vector so it starts at the origin.2. Read off the end of the vector in increments of unit cell dimensions a, b, and c.3. Adjust these to the smallest integer values.4. Enclose in square brackets without commas.

That is, [uvw]

Algorithm

-1, 1, 1 where the overbar represents a negative index

[ 111 ]=>

z

x

y

families of directions <uvw> , for example:

100100010001001010100 ][],[],[],[],[],[

VMSE with examples, problems, exercises

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example: linear density of Al in [110] direction FCC with a = 0.405 nm

Linear Density of Atoms (LD)LD = 

a

[110]

Length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD

Miller Indices for Crystallographic Planes• Reciprocals of the three axial intercepts for a plane, cleared of fractions &

common multiples. • All parallel planes have the same Miller indices.• Algorithm (procedure):

1. If the plane passes through the origin, translate so it does not.2. Read off the intercepts of the plane with the axes in increments

of the lattice constants (a, b, c). For example, 1, 2, 2

3. Take reciprocals of those intercepts. If it is parallel to an axis so that it doesn’t intersect it, the reciprocal is 0.

For example, 1, ½, ½ 4. Convert the numbers to the smallest possible integer values.

For example, 2, 1, 15. Enclose those numbers in parentheses, with no commas.

For example (211). 6. As with directions, a bar over a number indicates it is negative.

• VMSE with illustrations, problems, exercises• Families of equivalent planes. For a cubic structure, for example:

.etc),(),(),(),(),(),( 121112112112121211211

Three Low-index Planes

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Crystallographic Plane Examplesz

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c

4. Miller Indices (100)

1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c

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Planar Density or Packing

• Atoms per unit area• Very important for mechanical strength and for

chemical properties.• Essential step is to sketch the plane of interest, and

then use geometry to relate lattice constant to atomic radius.

• For example, iron foil can be used as a catalyst. The atomic packing of the exposed plane is important.

a) Draw (100) and (111) crystallographic planes

b) Calculate the planar density for each of these planes.

Planar Density of (100) -Iron (Ferrite)For T < 912C the equilibrium structure of iron is BCC.

Radius R = 0.1241 nm

= Planar Density = a2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

12

R3

34area

2D repeat unit

(100) R3

34a

2D repeat unit

(from slide 8)

Planar Density of (111) Ferrite

ah2

3

a2

2D re

peat

uni

t

1

= = nm2

atoms7.0m2

atoms0.70 x 1019

3 2R3

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit

333 2

2

R3

16R

34

2a3ah2area

Close-packed planes and structures• {111} in FCC have metal atoms as close together as possible.

Called close-packed. • So FCC structure of metals is also sometimes called “cubic close packed.”• In VMSE: http://higheredbcs.wiley.com/legacy/college/callister/1118061608/vmse/

xtalclose.htm

Watch close-packed {111} planes added to build FCC: http://departments.kings.edu/chemlab/animation/clospack.html ABCABCABC , where A, B and C are three possible positions of atoms.

• In L-03B we look at another close-packed structure for metals built of the same planes, but in a different order, ABABAB.

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