Ch03 Crystalline Solids

7/25/2019 Ch03 Crystalline Solids

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AL- FATIHAH


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2MME 2503Materials Science & Engineering

CHAPTER 3:

THE STRUCTURE OF CRYSTALLINE SOLIDS

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?


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MME 2503Materials Science & Engineering 3

MATERIALS AND PACKING

atoms pack in periodic, 3D arraysCrystallinematerials...

-metals

-many ceramics

-some polymers

atoms have no periodic packing

Noncrystallinematerials...

-complex structures-rapid cooling

crystalline SiO2

noncrystalline SiO2"Amorphous" = Noncrystalline

Si Oxygen

typical of:

occurs for:


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Chapter 3 -MME 2503Materials Science & Engineering

4

CRYSTAL SYSTEMS

7 crystal systems

14 crystal lattices

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

a, b, and care the lattice constants


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Chapter 3 -MME 2503Materials Science & Engineering 5

CRYST L SYSTEMS


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MME 2503Materials Science & Engineering 6

METALLIC CRYSTAL STRUCTURES

Tend to be densely packed. Reasons for dense packing:

- Typically, only one element is present, so all atomic

radii are the same.

- Metallic bonding is not directional.

- Nearest neighbor distances tend to be small in

order to lower bond energy.

- Electron cloud shields cores from each other.

Have the simplest crystal structures.


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7

SIMPLE CUBIC STRUCTURE (SC)

Rare due to low packing density(only Po has this structure)

Close-packed directionsare cube edges.

Coordination #= 6

(# nearest neighbors)

MME 2503Materials Science & Engineering


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8

ATOMIC PACKING FACTOR (APF)

APF for a simple cubic structure = 0.52

APF =a3

4

3p (0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

APF =Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

close-packed directions

a

R=0.5a

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



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9

BODY CENTERED CUBIC STRUCTURE (BCC)

Coordination # = 8

Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded

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

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



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10

ATOMIC PACKING FACTOR: BCC

a

APF =

4

3p( 3a/4 )32

atoms

unit cell atom

volume

a3

unit cell

volume

length = 4R=

Close-packed directions:

3 a

APF for a body-centered cubic structure = 0.68

aR

a2

a3



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11

FACE CENTERED CUBIC STRUCTURE (FCC)

Coordination # = 12

Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded

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

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

MME 2501 - Engineering Materials


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MME 2501 - Engineering Materials 12

ATOMIC PACKING FACTOR: FCC APF for a face-centered cubic structure = 0.74

maximum achievable APF

APF =

43p ( 2a/4 )34

atoms

unit cell atomvolume

a3

unit cell

volume

Close-packed directions:

length = 4R= 2 a

Unit cell contains:6 x1/2 + 8 x1/8

= 4 atoms/unit cella

2 a


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Chapter 3 -MME 2501 - Engineering Materials

13

A sites

B B

B

BB

B B

Csites

C C

CA

B

Bsites

ABCABC... Stacking Sequence

2D Projection

FCC Unit Cell

FCC Stacking Sequence

B B

B

BB

B B

Bsites

C C

CA

C C

CA

A

BC


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HEXAGONAL CLOSE-PACKED STRUCTURE

(HCP)

Coordination # = 12

ABAB... Stacking Sequence

APF = 0.74

3D Projection 2D Projection

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

c/a= 1.633

c

a

A sites

Bsites

A sites Bottom layer

Middle layer

Top layer


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THEORETICAL DENSITY, R

where n= number of atoms/unit cell

A=atomic weight

VC= Volume of unit cell = a3for cubic

NA= Avogadros number

= 6.023 x 1023atoms/mol

Density = =

VCNA

nA =

CellUnitofVolumeTotal

CellUnitinAtomsofMass


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Chapter 3 -

Ex: Cr (BCC)

A=52.00 g/mol

R= 0.125 nm

n= 2


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THEORETICAL DENSITY, R

theoretical

a= 4R/ 3 = 0.2887 nm

actual

aR

=a3

52.002

atoms

unit cell mol

g

unit cell

volume atomsmol

6.023x1023

= 7.18 g/cm3

= 7.19 g/cm3


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17

DENSITIES OF MATERIAL CLASSES

metals>

ceramics>

polymers

Why?

(

g/cm

)3

Graphite/Ceramics/

Semiconductor

Metals/Alloys

Composites/fibers

Polymers

1

2

2 0

30

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% 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

TantalumGold, WPlatinum

Graphite

Silicon

Glass-sodaConcrete

Si nitrideDiamond

Al oxide

Zirconia

HDPE, PSPP, LDPE

PC

PTFE

PET

PVCSilicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Metalshave... close-packing

(metallic bonding)

often large atomic massesCeramicshave...

less dense packing

often lighter elements

Polymershave...

low packing density(often amorphous)

lighter elements (C,H,O)

Compositeshave... intermediate values

In general


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CRYSTALS AS BUILDING BLOCKS

Someengineering applications require single crystals:

Properties of crystalline materials

often related to crystal structure.

--Ex: Quartz fractures more easily

along some crystal planes thanothers.

--diamond single

crystals for abrasives

--turbine blades


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POLYCRYSTALS

Mostengineering materials are polycrystals.

Nb-Hf-W plate with an electron beam weld.

Each "grain" is a single crystal. If grains are randomly oriented,

overall component properties are not directional.

Grain sizes typically range from 1 nm to 2 cm

(i.e., from a few to millions of atomic layers).

1 mm

Isotropic

Anisotropic


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SINGLE VS POLYCRYSTALS

Single Crystals

-Properties vary withdirection: 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.

200 mm

E (diagonal) = 273 GPa

E (edge) = 125 GPa


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Chapter 3 -

Two or more distinct crystal structures for thesame material (allotropy/polymorphism)

titanium

, -Ti

carbon

diamond, graphite


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POLYMORPHISM

BCC

FCC

BCC

1538C

1394C

912C

-Fe

-Fe

-Fe

liquid

iron system


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Chapter 3 -

Point coordinates for unit

cell center are

a/2, b/2, c/2

Point coordinates for unitcell corner are 111

Translation: integer multipleof lattice constants identical position inanother unit cell


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POINT COORDINATESz

x

y

a b

c

000

111

y

z

2c

b

b


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CRYSTALLOGRAPHIC DIRECTIONS

1. Vector repositioned (if necessary) to passthrough origin.

2. Read off projections in terms of

unit cell dimensions a, b, and c

3. Adjust to smallest integer values

4. Enclose in square brackets, no commas

[uvw]

ex:1, 0, => 2, 0, 1 => [201]

-1, 1, 1

z

x

Algorithm

where overbar represents a

negative index

[111]=>

y


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CRYSTALLOGRAPHIC PLANES


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Chapter 3 -

Miller Indices: Reciprocals of the (three)axial intercepts for a plane, cleared of

fractions & common multiples. All parallel

planes have same Miller indices.

Algorithm1. Read off intercepts of plane with axes in

terms of a, b, c

2. Take reciprocals of intercepts3. Reduce to smallest integer values

4. Enclose in parentheses, no

commas i.e., (hkl)


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CRYSTALLOGRAPHIC PLANESz

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c


1. Intercepts 1 1

2. Reciprocals 1/1 1/1 1/1 1 0

3. Reduction 1 1 0

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

2 0 03. Reduction 2 0 0

example a b c


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z

x

ya b

c


example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/ 1/1 1/

2 1 4/3

3. Reduction 6 3 4


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Chapter 3 -

Linear Density of Atoms LD =


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LINEAR DENSITY

ex: linear density of Al in [110]

direction

a= 0.405 nm

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD -==


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Chapter 3 -

We want to examine the atomic packingof 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.


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Chapter 3 -

Solution: At T < 912C iron has the BCCstructure.


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PLANAR DENSITY OF (100) IRON

(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

1

2

R3

34area

2D repeat unit


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Chapter 3 -

Solution (cont): (111) plane


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PLANAR DENSITY OF (111) IRON

1 atom in plane/ unit surface cell

333

2

2

R3

16R

3

42a3ah2area =

===

atoms in plane

atoms above plane

atoms below plane

ah23=

a2

1

= =nm2

atoms7.0

m2

atoms0.70 x 1019

3 2R3

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit


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Chapter 3 -

Diffraction gratings must have spacingscomparable to the wavelength of diffractedradiation.

Cant resolve spacings Spacing is the distance between parallel planes of

atoms. MME 2501 - Engineering Materials

35

X-RAY DIFFRACTION


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X-RAYS TO DETERMINE CRYSTAL

STRUCTURE

X-rayintensity(fromdetector)

q

qc

d=n

2sinqc

Measurement ofcritical angle, qc,allows computation of

planar spacing, d.

Incoming X-rays diffractfrom crystal planes.

Adapted from Fig. 3.19,

Callister 7e.

reflections mustbe in phase fora detectable signal

spacingbetweenplanes

d

q

qextra

distancetravelledby wave 2


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X-RAY DIFFRACTION PATTERN

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2q

Diffraction pattern for polycrystalline -iron (BCC)

Intensity(relative)

z

x

ya b

c

z

x

ya b

c


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SUMMARY

Atoms may assemble into crystallineoramorphousstructures.

We can predict the densityof a material, provided we

know the atomic weight, atomic radius, and crystal

geometry(e.g., FCC, BCC, HCP).

Common metallic crystal structures are FCC, BCC, and

HCP. Coordination numberand atomic packing factor

are the same for both FCC and HCP crystal structures.

Crystallographic points, directionsand planesare

specified in terms of indexing schemes.

Crystallographic directions and planes are related

to atomic linear densitiesand planar densities.


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MME 2501 Engineering Materials 39

SUMMARY

Some materials can have more than one crystal

structure. This is referred to as polymorphism(or

allotropy).

Materials can be single crystalsor polycrystalline.Material properties generally vary with single crystal

orientation (i.e., they are anisotropic), but are generally

non-directional (i.e., they are isotropic) in polycrystals

with randomly oriented grains.

X-ray diffractionis used for crystal structure andinterplanar spacingdeterminations.

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Ch03 Crystalline Solids