Chapter 3-

ISSUES TO ADDRESS...• How do atoms assemble into solid structures? (for now, focus on metals)

• How does the density of a material depend on its structure?

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

1

Why do we care about crystal structures, directions, planes ?

Physical properties of materials depend on the geometry of crystals

Chapter 3-2

Electronic Structure Bonding State of aggregation

The Big Picture

Bohr atom

Bohr-Sommerfeld

Quantum numbers

Aufbau principle

Multielectron atoms

Periodic table patterns

Octet stability

Primary:•Ionic•Covalent•Metallic

Gas

Liquid

Solid

Classification of Solids:1. Bonding type2. Atomic arrangement

Chapter 3-3

Atomic Arrangement

SOLID: Smth. which is dimensionally stable, i.e., has a volume of its own

classifications of solids by atomic arrangement

ordered disorderedatomic arrangement regular random*order long-range short-rangename crystalline amorphous

“crystal” “glass”

Chapter 3-

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have lower energies.

Energy and PackingEnergy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

CO

OL

ING

Chapter 3-

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

3

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" = NoncrystallineAdapted from Fig. 3.18(b), Callister 6e.

Adapted from Fig. 3.18(a), Callister 6e.

MATERIALS AND PACKING

LONG RANGE ORDER

SHORT RANGE ORDER

Chapter 3-6

6

 Metallic Crystal Structures • How can we stack metal atoms to minimize empty

space?

2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

Chapter 3-7

Robert Hooke – 1660 - Cannonballs

“Crystal must owe its regular shape to the packing of spherical particles”

Chapter 3-8

observed that quartz crystals had the same angles between corresponding faces regardless of their size.

Niels Steensen ~ 1670

Chapter 3-9

If I see something has a

macroscopic shape very regular

and cubic, can I infer from that if I

divide, divide, divide, divide,

divide, if I get down to atomic

dimensions, will there be some

cubic repeat unit?

SIMPLE QUESTION:

Chapter 3-10

Christian Huygens - 1690

Studying calcite crystals made drawings of atomic packing and bulk shape.

Chapter 3-11

BERYL

Be3Al2(SiO3)6

Chapter 3-12

Early Crystallography

René-Just Haüy (1781): cleavage of calcite

• Common shape to all shards: rhombohedral

• How to model this mathematically?

• What is the maximum number of distinguishable shapes that will fill three space?

• Mathematically proved that there are only 7 distinct space-filling volume elements

Chapter 3-13

The Seven Crystal Systems

Specification of unit cell parameters

BASIC UNIT

Chapter 3-14

Does it work with Pentagon?

Chapter 3-15

August Bravais• How many different ways can I put atoms into these seven crystal

systems, and get distinguishable point environments?

And, he proved mathematically that there are 14 distinct ways to arrange points in space.

When I start putting atoms in the cube, I have three distinguishable arrangements.

SC BCC FCC

Chapter 3-16

Chapter 3-17

Last Day: Last Day: Atomic Arrangement

SOLID: Smth. which is dimensionally stable, i.e., has a volume of its own

classifications of solids by atomic arrangement

ordered disorderedatomic arrangement regular random*order long-range short-rangename crystalline amorphous

“crystal” “glass”

Chapter 3-

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

3

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" = NoncrystallineAdapted from Fig. 3.18(b), Callister 6e.

Adapted from Fig. 3.18(a), Callister 6e.

MATERIALS AND PACKING

LONG RANGE ORDER

SHORT RANGE ORDER

Chapter 3-

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have lower energies.

Energy and PackingEnergy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

CO

OL

ING

Chapter 3-20

Three Types of Solids according to atomic

arrangement

Chapter 3-

Unit Cell Concept• The unit cell is the smallest structural unit or building

block that uniquely can describe the crystal structure. Repetition of the unit cell generates the entire crystal. By simple translation, it defines a lattice .

• Lattice: The periodic arrangement of atoms in a Xtal.

Lattice Parameter : Repeat distance in the unit cell, one for in each dimension

b

a

Chapter 3-

Crystal Systems

• Units cells and lattices in 3-D:– When translated in each lattice parameter direction, MUST

fill 3-D space such that no gaps, empty spaces left.

Lattice Parameter : Repeat distance in the unit cell, one for in each dimension

a

b c

Chapter 3-23

The Importance of the Unit Cell

• One can analyze the Xtal as a whole by investigating a representative volume.

• Ex: from unit cell we can – Find the distances between nearest atoms for calculations of

the forces holding the lattice together

– Look at the fraction of the unit cell volume filled by atoms and relate the density of solid to the atomic arrangement

– The properties of the periodic Xtal lattice determine the allowed energies of electrons that participate in the conduction process.

Chapter 3-

 Metallic Crystal Structures

• How can we stack metal atoms to minimize empty space?

2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

Chapter 3-

Crystal Systems

7 crystal systems

14 crystal lattices

Fig. 3.4, Callister 7e.

Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.

a, b, and c are the lattice constants

Chapter 3-26

Chapter 3-5

• Rare due to poor packing • Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)

(Courtesy P.M. Anderson)

SIMPLE CUBIC STRUCTURE (SC)

Closed packed direction is where the atoms touch each other

Chapter 3-6

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

close-packed directions

a

R=0.5a

contains 8 x 1/8 = 1 atom/unit cell

Adapted from Fig. 3.19, Callister 6e.

ATOMIC PACKING FACTOR

Chapter 3-

• Coordination # = 8

7(Courtesy P.M. Anderson)

• Close packed directions are cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

BODY CENTERED CUBIC STRUCTURE (BCC)

ex: Cr, W, Fe (), Tantalum, Molybdenum

2 atoms/unit cell: 1 center + 8 corners x 1/8

Chapter 3-8

• APF for a body-centered cubic structure = 0.68

Close-packed directions: length = 4R

= 3 a

ATOMIC PACKING FACTOR: BCC

a

a 2

a 3

aR

APF = a3

4

3( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

Chapter 3-9(Courtesy P.M. Anderson)

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

FACE CENTERED CUBIC STRUCTURE (FCC)

• Coordination # = 12

Adapted from Fig. 3.1, Callister 7e.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

Chapter 3-

ATOMIC PACKING FACTOR: FCC

?HW

Chapter 3-

HW1. Finish reading Chapter 3.2.On a paper solve example problems:3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.93. Fill in the blanks in Table below

33

Unit Cell Volume a3

Lattice Points per cell 1

Nearest Neighbor Distance a

Number of Nearest Neighbors 6

Atomic Packing Factor 0.52

BCC FCCSC

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Fill h

ere!

Chapter 3-14

Example: Copper

n AVcNA

# 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 Callister (see next slide):• 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

Compare to actual: Cu = 8.94 g/cm3Result: theoretical Cu = 8.89 g/cm3

THEORETICAL DENSITY,

Chapter 3 -

• Ex: Cr (BCC)

A = 52.00 g/mol

R = 0.125 nm

n = 2

theoretical

a = 4R/ 3 = 0.2887 nm

actual

aR

= a3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.023 x 1023

Theoretical Density,

= 7.18 g/cm3

= 7.19 g/cm3

Chapter 3-15

Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen

Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H

At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008

Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------

Density (g/cm3) 2.71 ------ 3.5 1.85 2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8.9 8.94 ------ 5.90 5.32 19.32 ------ ------

Crystal Structure FCC ------ BCC HCP Rhomb ------ HCP FCC Hex BCC ------ BCC HCP FCC ------ Ortho. Dia. cubic FCC ------ ------

Adapted fromTable, "Charac-teristics ofSelectedElements",inside frontcover,Callister 6e.

Characteristics of Selected Elements at 20C

Chapter 3-

metals ceramics polymers

16

(g

/cm

3)

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

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 Data from Table B1, Callister 6e.

DENSITIES OF MATERIAL CLASSES

Chapter 3-

POLYMORPHISM & ALLOTROPY

• Some materials may exist in more than one crystal structure, this is called polymorphism.

• If the material is an elemental solid, it is called allotropy. An example of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon.

Chapter 3-39

Polymorphism • Two or more distinct crystal structures for the same

material (allotropy/polymorphism)   titanium

  , -Ti

carbon

diamond, graphite

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system

Chapter 3-

Crystallographic Points, Directions, and Planes

• It is necessary to specify a particular point/location/atom/direction/plane in a unit cell

• We need some labeling convention. Simplest way is to use a 3-D system, where every location can be expressed using

three numbers or indices. – a, b, c and α, β, γ

x

y

z

βα

γ

Chapter 3-

Crystallographic Points, Directions, and Planes

• Crystallographic direction is a vector [uvw]– Always passes thru origin 000– Measured in terms of unit cell dimensions a, b, and c– Smallest integer values

• Planes with Miller Indices (hkl)– If plane passes thru origin, translate– Length of each planar intercept in terms of the lattice

parameters a, b, and c.– Reciprocals are taken– If needed multiply by a common factor for integer

representation

Chapter 3-

Section 3.8 Point CoordinatesPoint coordinates for unit cell

center are

a/2, b/2, c/2 ½ ½ ½

Point coordinates for unit cell corner are 111

Translation: integer multiple of lattice constants identical position in another unit cell

z

x

ya b

c

000

111

y

z

2c

b

b

Chapter 3-

Crystallographic Directions

1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a, b, and c3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

families of directions <uvw>

z

x

Algorithm

where overbar represents a negative index

[ 111 ]=>

y

Chapter 3-

ex: linear density of Al in [110] direction 

a = 0.405 nm

Linear Density

• Linear Density of Atoms LD = 

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD

Chapter 3-

Crystallographic Planes

Adapted from Fig. 3.9, Callister 7e.

Chapter 3-

Crystallographic Planes• Miller Indices: Reciprocals of the (three) axial

intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.

• Algorithm 1. If plane passes thru origin, translate2. Read off intercepts of plane with axes in terms of a, b, c3. Take reciprocals of intercepts4. Reduce to smallest integer values5. Enclose in parentheses, no commas i.e., (hkl)

Chapter 3-

Crystallographic Planesz

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

Chapter 3-

Crystallographic Planesz

x

ya b

c

4. Miller Indices (634)

example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes {hkl}

(100), (010),(001),Ex: {100} = (100),

Chapter 3-

Crystallographic Planes

• We want to examine the atomic packing of crystallographic planes

• Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.

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

for Fe.

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

Chapter 3-

Planar Density of (100) IronSolution:  At T < 912C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a

Adapted from Fig. 3.2(c), Callister 7e.

2D repeat unit

= Planar Density = a2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

12

R3

34area

2D repeat unit

Chapter 3-

Planar Density of (111) Iron

?

HW

Chapter 3-

Single Crystals and Polycrystalline Materials

• In a single crystal material the periodic and repeated arrangement of atoms is PERFECT This extends throughout the entirety of the specimen without interruption.

• Polycrystalline material, on the other hand, is comprised of many small crystals or grains. The grains have different crystallographic

orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

Chapter 3-

Example of Polycrystalline Growth

Chapter 3-17

• Some engineering applications require single crystals:

• Crystal properties reveal features of atomic structure.

(Courtesy P.M. Anderson)

--Ex: Certain crystal planes in quartz fracture more easily than others.

--diamond single crystals for abrasives

--turbine bladesFig. 8.30(c), Callister 6e.(Fig. 8.30(c) courtesyof Pratt and Whitney).(Courtesy Martin

Deakins,GE Superabrasives, Worthington, OH. Used with permission.)

CRYSTALS AS BUILDING BLOCKS

Chapter 3-18

• 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).

Adapted from Fig. K, color inset pages of Callister 6e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

POLYCRYSTALS

Chapter 3-19

• Single Crystals-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 m

Data from Table 3.3, Callister 6e.(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Adapted from Fig. 4.12(b), Callister 6e.(Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

SINGLE VS POLYCRYSTALS

Chapter 3-

Anisotropy and Texture• Different directions in a crystal have a different APF.

• For example, the deformation amount depends on the direction in which a stress is applied, other properties are thermal conductivity, optical properties, magnetic properties, hardness, etc.

• In some polycrystalline materials, grain orientations are random, hence bulk material properties are isotropic, i.e. equivalent in each direction

• Some polycrystalline materials have grains with preferred orientations (texture), so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties.

Chapter 3-

d=n/2sinc

x-ray intensity (from detector)

c20

• Incoming X-rays diffract from crystal planes.

• Measurement of: Critical angles, c, for X-rays provide atomic spacing, d.

Adapted from Fig. 3.2W, Callister 6e.

X-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase to detect signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“2”

Chapter 3-

• Atoms may assemble into crystalline or amorphous structures.

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

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

23

SUMMARY (I)

Chapter 3-

Summary (II)• Allotropy• Amorphous• Anisotropy• Atomic packing factor (APF)• Body-centered cubic (BCC)• Coordination number• Crystal structure• Crystalline• Face-centered cubic (FCC)• Grain• Grain boundary• Hexagonal close-packed (HCP)• Isotropic• Lattice parameter• Non-crystalline• Polycrystalline• Polymorphism• Single crystal• Unit cell