Chapter 03 C R Y S T A L S

• How do atoms assemble into solid structures?

• How does the density of a material depend onits structure?

• When do material properties vary with thesample (i.e., part) orientation?

Structures of Metals and Ceramics

• How do the structures of ceramic materials differ from those of metals?

Chapter 3

• Non dense, random packing

• Dense, regular packing

Now, bonding energy is not only between two atoms, its from many atoms.

Dense, regular-packed structures tend to have lower energy.

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

ENERGY AND PACKING

average

Building 3D ‘ordered’ array of atoms for Dummies

(i) Construct lattice

(ii) Filling the lattice

with atoms or

molecules or group

of atoms/molecules

You could choose many

number of different unit

cells for the same building

process.

7 Crystal Systems

&

14 Crystal Lattices

Any crystalline structure (3D ordered array of atoms/molecules)

must fall into one of the systems and one of the crystal lattices.

Unit cellsOften called ‘lattice constants’

• tend to be densely packed.

• have several reasons for dense packing:

-Typically, only one element is present, so all atomicradii are the same.-Metallic bonding is non-directional.-Nearest neighbor distances tend to be small inorder to lower bond energy.

• have the simplest crystal structures.

We will look at three such structures...

METALLIC CRYSTALS

B B

B

BB

B BC C

CA

A

• Rare due to poor packing (only Po has this structure)• Close-packed directions are cube edges.

• Coordination # (CN) = 6(# of nearest neighboring atoms)

SIMPLE CUBIC (SC) STRUCTURE

• Unit cell (Bravais lattice): Simple cubic

1/8

CN is the one way to tell

how much the structure is packed with atoms.

= 0.52

ATOMIC PACKING FACTOR (APF)

APF = Volume of atoms* in unit cell

Volume of unit cell

*assume hard spheres

Here’s the better way to tell about packing.

APF =

a3

4

3π (0.5a)31

atoms

unit cellatom

volume

unit cell

volume

a

R=0.5a

1 atom/unit cell

There are 8 of 1/8 atoms.

Close-packed direction:

a= 2R

• Coordination # = 12

• Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shadeddifferently only for ease of viewing.

FACE-CENTERED CUBIC (FCC) Structure

• Unit cell (Bravais lattice): FCC

• γγγγ-Fe, Al, Ni, Cu, Ag, Pt, and Au

Grey and red atoms are same.

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cella

= 0.74

Close-packed directions: length = 4R

= 2 a

ATOMIC PACKING FACTOR: FCC Structure

APF =

a3

4

3π ( 2a/4)34

atoms

unit cell atom

volume

unit cell

volume

• Coordination # = 8

• Close packed directions are cube diagonals.

BODY-CENTERED CUBIC (BCC) Structure

• Unit cell (Bravais lattice): BCC

• αααα-Fe, Cr, Mo, W, and V

aR

= 0.68

Close-packed directions: length = 4R

= 3 a

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell

ATOMIC PACKING FACTOR: BCC

APF =

a3

4

3π ( 3a/4)32

atoms

unit cell atom

volume

unit cell

volume

Summary (Metal Cubic System + HCP)

Name of StructureUnit Cell

(Bravais lattice) CN APF

SC

FCC

BCC

SC

FCC

BCC

6

12

8

0.52

0.74

0.68

HCP hexagonal 12 0.74

Next slide

• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection

• 2D ProjectionA sites

B sites

A sites

Bottom layer

Middle layer

Top layer

HEXAGONAL CLOSE-PACKED (HCP) STRUCTURE

• Unit cell (Bravais lattice): Hexagonal

• Be, Mg, α-Ti, Zn, and Zr Unit cell: 1/3 of it

Closed Packed Planes (metals)

FCC – ABCABC HCP – ABABAB

B B

B

BB

B BC C

CA

A

A sites

B sites

C sites

ABC

Example: Copper

ρ = nA

VcNA

# atoms/unit cell Atomic weight (g/mol)

Volume/unit cell

(cm3/unit cell)

Avogadro's number

(6.023 x 1023 atoms/mol)

Data from Table inside front cover of texbook

• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10 cm)-7

Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10

-23cm3

Result: theoretical ρCu = 8.89 g/cm3

THEORETICAL DENSITY, ρ

Before we study crystal structure of ceramics,

We need to learn crystallographic notations

Crystallographic Points, Directions, and Planes

Points

Point Coordinates?

a, b, c : lattice constant

q r s : multiple or fraction of

lattice constant

(Example - cubic system)No parenthesis !

No comma ! In fact, we’ll only deal with

cubic in this course.

Examples Fraction possible

Crystallographic Points, Directions, and Planes

Directions (Cubic) [uvw] & <uvw> Miller Indices

[111]

[111]

[111]

[111]

[111][111]

[111]

Family: <111>

[111]

Cubic system

How about

tetragonal system?

[112]

[111]

[111]

Crystallographic Points, Directions, and Planes

Planes (Cubic) (hkl) & {hkl} Miller Indices

Crystallographic Points, Directions, and Planes

Planes (Cubic)(hkl) & {hkl} Miller Indices

Linear and Planar Densities

FCC crystal structure (metal)

LD = # of atoms centered on direction vector/length of direction vector

PD = # of atoms centered on a plane/area of plane

Closed packed direction Closed packed plane

Closed Packed Planes (metals)

FCC – (111) : ABCABC

B B

B

BB

B BC C

CA

A

A sites

B sites

C sites

ABC

Closed Packed Planes (metals)

HCP – (0001): ABABAB

B B

B

BB

B BC C

CA

A

Unit cell:

hexagonal

Now we learn crystal structure of ceramics.

• Bonding:--Mostly ionic, some covalent.--% ionic character increases with difference in electronegativity.

He -

Ne -

Ar -

Kr -

Xe -

Rn -

Cl 3.0

Br 2.8

I 2.5

At 2.2

Li 1.0

Na 0.9

K 0.8

Rb 0.8

Cs 0.7

Fr 0.7

H 2.1

Be 1.5

Mg 1.2

Sr 1.0

Ba 0.9

Ra 0.9

Ti 1.5

Cr 1.6

Fe 1.8

Ni 1.8

Zn 1.8

As 2.0

C 2.5Si 1.8

F 4.0

Ca 1.0

Table of Electronegativities

CaF2: large

SiC: small

• Large vs small ionic bond character:

CERAMIC CRYSTALS

• Charge Neutrality:--Net charge in thestructure should

be zero.

--General form: AmXp

m, p determined by charge neutrality

• Rcation/Ranion (Ratio of ionic radii) ⇒ determines CN (next slide)

--maximize the # of nearest oppositely charged neighbors(while maintaining charge neutrality and stability)

- -

- -+

unstable

- -

- -+

stable

- -

- -+

stable

CaF2:Ca2+

cation

F-

F-

anions+

IONIC BONDING & STRUCTURE

# of atoms

Q: How many anions can you arrange around a cation?

rcationranion

rcationranion

Coord #

< .155 .155-.225 .225-.414 .414-.732 .732-1.0

ZnS (zincblende)

NaCl (sodium chloride)

CsCl (cesium chloride)

2 3 4 6 8

COORDINATION # AND IONIC RADII

• Coordination # increases with

• Structure of FCC metalsBravais lattice: FCCCoordination #: 12

• Structure of NaClBravais lattice: FCCCoordination #: 6

FCC Bravais lattice (Metal vs. Ionic Material)

Different crystal structures with the same Bravais lattice (unit cell)

APF (or Ionic pakcing factor (IPF)) metals vs ionic material

• Structure of NaClBravais lattice: FCCCoordination #: 6

• Structure of FCC metalsBravais lattice: FCCCoordination #: 12

a = 2r Na+ + 2rCl

- a = 2r√2

Note the difference

in closed-packed

direction.

Semiconducting Materials (Covalent bonding)

Allotropes & Polymorphs

Diamond

Graphite

Fullerene (C60)Carbon nanotube

Allotropes of carbonDifferent stable (or metastable)

crystal structures of the same

compounds

Different stable (and metastable)

crystal structures of single element

• atoms pack in periodic, 3D arrays• typical of:

Crystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packing• occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

Si Oxygen

crystalline SiO2

noncrystalline SiO2"Amorphous" = Noncrystalline

Crystalline vs. Amorphous

Single-crystalline vs. Polycrystalline

Grain

boundaries

• Most engineering materials are polycrystals.

• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If crystals are randomly oriented,overall component properties are not directional.

• Crystal sizes typ. range from 1 nm to 2 cm(i.e., from a few to millions of atomic layers).

1 mm

POLYCRYSTALS

• Single Crystals

-Properties vary withdirection: anisotropic.

-Example: the modulusof elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may notvary with direction.-If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)

-If grains are textured,anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 µm

SINGLE VS POLYCRYSTALS

XXXX----ray Diffraction to determine Crystal Structureray Diffraction to determine Crystal Structureray Diffraction to determine Crystal Structureray Diffraction to determine Crystal Structure

• Incoming X-rays diffract from crystal planes.

X-ray

SourceDetector

Extra distance travelled by wave 2spacing

between

planes

Beams 1 & 2 have to be in phase

to be diffracted.

(next slide)

variables

Bragg’s law

• Bragg’s law is a necessary but not sufficient condition for diffraction.

=λ 2 d sin θθθθn

Extra distance travelled by beam 2 have to be an integer

multiple of λ.

n: order of reflection

θθθθ-2θθθθ scan

X-ray

source Detector

Typically X-ray

source and detector

are both rotating.

If sample S is

polycrystalline,

X-ray data will

resemble the date below.

ρmetals� ρceramics� ρpolymers

ρ (g/cm3)

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

20

30Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers

in an epoxy matrix). 10

3

4

5

0.3

0.4 0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

Tantalum Gold, W Platinum

Graphite

Silicon

Glass-soda Concrete

Si nitride Diamond Al oxide

Zirconia

HDPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Why?Metals have...• close-packing(metallic bonding)

• large atomic mass

Ceramics have...• less dense packing(covalent bonding)

• often lighter elements

Polymers have...• poor packing(often amorphous)

• lighter elements (C,H,O)

Composites have...• intermediate values

DENSITIES OF MATERIAL CLASSES

• Atoms may assemble into crystalline oramorphous structures.

• We can predict the density of a material,provided we know the atomic weight, atomicradius, and crystal geometry (e.g., FCC,BCC, HCP).

• Material properties generally vary with singlecrystal orientation (i.e., they are anisotropic),but properties are generally non-directional(i.e., they are isotropic) in polycrystals withrandomly oriented grains.

SUMMARY