ch-3 compatibility mode

8/22/2019 Ch-3 Compatibility Mode

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The Structure of

CHAPTER 3

Chapter 3-

Crystalline Solids


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ISSUES TO ADDRESS...ISSUES TO ADDRESS...ISSUES TO ADDRESS...ISSUES TO ADDRESS...

How do atoms assemble into solid structures?

(focus on metals)

How does the density of a material depend on

Chapter 3-

When do material properties vary with the

sample (i.e., part) orientation?

1


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Level of Structure in Metals

Macrostructure 10 XMicrostructure 1500 X

Chapter 3-

Substructure 105 X (SEM)

Crystal Structure X ray

Electronic Structure Spectrometer

Nuclear Structure Nuclear

Spectrometer


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Material Structure Level &

Chapter 3-Crystallization


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How do atoms assemble into solid

structures? (focus on metals)

SOLID MATERIALS

Chapter 3-

Crystalline Materials Non-Crystalline Materials

Single Crystal Poly-Crystal


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atoms have no periodic packing

occurs for:

Noncrystalline materials...

-complex structures

-rapid cooling

MATERIALS AND PACKING

Chapter 3-

Si Oxygen

noncrystalline SiO2

"Amorphous" = Noncrystalline


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Single Crystal (Quartz)

Chapter 3-


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atoms pack in periodic, 3D arrays

typical of:

Crystalline materials...

-metals-many ceramics

-some polymers

MATERIALS AND PACKING

Chapter 3-crystalline SiO2

Si Oxygen


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tend to be densely packed.

have several reasons for dense packing:

-Typically,Typically,Typically,Typically, onlyonlyonlyonly oneoneoneone elementelementelementelement isisisis present,present,present,present, sosososo allallallall atomicatomicatomicatomic

METALLIC CRYSTALS

Chapter 3- 4

rararara areareareare eeee samesamesamesame....----MetallicMetallicMetallicMetallic bondingbondingbondingbonding isisisis notnotnotnot directionaldirectionaldirectionaldirectional....

----NearestNearestNearestNearest neighborneighborneighborneighbor distancesdistancesdistancesdistances tendtendtendtend totototo bebebebe smallsmallsmallsmall inininin

orderorderorderorder totototo lowerlowerlowerlower bondbondbondbond energyenergyenergyenergy....

have the simplest crystal structures.


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Metallic Solids (Imp. Props)

High Strength

Ductile

High Density

Chapter 3-

Good electrical and thermalconductivity

Good workability


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Crystal Structure Terminology

Crystal Lattice (Regular periodic

arrangement of atoms)

Chapter 3-

Unit cell

Lattice parameters

cba ,, & ,,


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7 Crystal Systems

CUBIC

TETRAGONAL ORTHORHOMBIC

Chapter 3-

MONOCLINIC

TRICLINIC

HEXAGONAL


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CUBIC

Crystal Structure

cba ==

Chapter 3-

90===


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TETRAGONAL

CRYSTAL STRUCTURE

cba =

Chapter 3-

90===


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ORTHORHOMBIC

Crystal Structure

cba

Chapter 3-

90===


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RHOMBOHEDRAL

Crystal Structure

cba ==

Chapter 3-

90==


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MONOCLINIC

Crystal Structure

cba

Chapter 3-

== 90


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TRICLINIC

Crystal Structure

cba

Chapter 3-

90


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HEXAGONAL

Crystal Structure

Chapter 3-

120,90 ===


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SIMPLE CUBIC STRUCTURE

Chapter 3-


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Rare due to poor packing (only Po has this structure)

SIMPLE CUBIC STRUCTURE (SC)

Chapter 3- 5Ex.: Po structure


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Effective No. of Atoms in a unit cell

a

Chapter 3-

close-packed directions

R=0.5a

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


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APF =Volume of atoms in unit cell*

ATOMIC PACKING FACTOR

Chapter 3- 6

o ume o un ce

*assume hard spheres


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Data for Simple Cubic Structure

Coordination No. (CN) 6

a-r Relation a = 2r

Chapter 3-

Effective No. of atoms per unit cell

(EN) 1/8 X 8 = 1

Atomic Packing Factor (APF) 0.5


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Body Centered Cubic (BCC)

Chapter 3-

BODY CENTERED CUBIC


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--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

BODY CENTERED CUBIC

STRUCTURE (BCC)

Chapter 3- 7Ex Structure of Cr, Iron


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Data for BCC Structure


a-r Relation ar 34 =

Chapter 3-


(EN) 1/8 X 8 + 1 = 2



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Face Centered Cubic FCC

Chapter 3-


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--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

FACE CENTERED CUBIC

STRUCTURE (FCC)

Chapter 3- 9Ex.: Structure of Al, Cu, Au


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Data for FCC Structure


a-r Relation ar 24 =

Chapter 3-


(EN) 1/8(8)+1/2(6) = 4



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Hexagonally Packed Structure

HCP

Chapter 3-


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CLOSELY PACKED STRUCTURE (HCP)

B

Chapter 3-A

Stacking

A-B-A-B-A-B


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ABAB... Stacking Sequence

3D Projection

HEXAGONAL CLOSE-PACKED

STRUCTURE (HCP)

Chapter 3- 12

B sites

A sites

Ex.: Structure of Mg & Al


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Data for HCP Structure


a-r Relation a = 2r

Chapter 3-


(EN) = 1/6(12) + (2) + 1(3) = 6

Atomic Packing Factor (APF) = 0.74

CLOSELY PACKED STRUCTURE (FCC)


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CLOSELY PACKED STRUCTURE (FCC)

B

Stacking

A-B-C-A-B-C-A-B-C

Chapter 3-AC


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AB

ABCABC... Stacking Sequence

2D Projection

FCC STACKING SEQUENCE

Chapter 3-


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Crystallographic Points

Study of

Chapter 3-

Crystallographic Directions

Crystallographic Planes


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Crystallographic Points

Chapter 3-


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z

b

a

Pointq r s

Chapter 3-

x

yc

qa

rb

sc

q r s

L t 1 P i t i it ll


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z

a = 0.48 nm

b = 0.46 nm

Locate 1 Point in a unit cell

Chapter 3-

x

y

c =

0.40 nm 0.12

0.46

0.20

1


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Crystallographic Points in a SC Unit Cell

z

b

a 58

Chapter 3-

x

yc

2 3

4

1


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Concept of

Chapter 3-


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Equivalent Crystallographic Pointsare the points to arrive at same

Chapter 3-

corresponding point by unittranslations.


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0 0 0 All Corner Points

Set of Equivalent Points in a SC Unit Cell

Chapter 3-


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Crystallographic Points in a BCC Unit Cell

z

b

a 58

Chapter 3-

x

yc

2 3

4

1

9


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Set of Equivalent Points in a BCC Unit Cell


All Body Centered

Chapter 3-


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Lattice Points in a FCC Unit Cellz

b

a 5

7

8

Chapter 3-

x

yc

2 3

41

9

10

11

Set of Equivalent Points in a FCC Unit Cell


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Set of Equivalent Points in a FCC Unit Cell


0 All z-face centered

Chapter 3-

0 All y-face centered

0 All x-face centered


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Crystallographic Direction

Chapter 3-

ec or

Procedure to Draw a Crystal Direction


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Procedure to Draw a Crystal Direction

in a Unit Cell Note intercepts on x-, y- & z-axis in terms of edge lengths.

Divide the intercepts by corresponding edge lengths

Reduce them to smallest integers

Enclose these inte ers in S r bracket e. . [u v w].

Chapter 3-

Note[u v w] for one dirn, for family of dirns.

Negative index is noted by a bar on the top.


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Crystallographic Direction in a cubic Unit Cell

z

a

a

Chapter 3-

x

yO

A

OA [1 0 0]

C t ll hi Di ti i bi U it C ll


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z

a

a


Chapter 3-

x

yO

A

OA [1 1 0]

C t ll hi Di ti i bi U it C ll


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z

a

a


Chapter 3-

x

yO

OA [1 1 1]

Di ti i bi U it C ll


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z

a

a

Direction in a cubic Unit Cell

Chapter 3-

x

ya/2

OA

OA [2 0 1]



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z

a

a


Chapter 3-

x

y

a/2

O

A

a/2

OA [1 2 1]


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Equivalent Directions in a cubic Unit CellOR Family of Directions

are the directions which are crystallographically

Chapter 3-

equ va en ; .e. av ng same a om c pac ng

(spacing)

Equivalent Directions in a cubic Unit Cell


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q

along edges of cubeza

a

Chapter 3-

x

yO

A

OA [1 0 0]



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a

Chapter 3-

x

yO

A

AO [1 0 0]



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a

Chapter 3-

x

yO

A

OA [0 1 0]



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a

Chapter 3-

x

yO

A

AO [0 1 0]



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aA

Chapter 3-

x

yO

OA [0 0 1]



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aA

Chapter 3-

x

yO

AO [0 0 1]


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Equivalent Directionsalong edges

Famil Re resentation

Chapter 3-



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z

a

a

along face-diagonal of cube

Chapter 3-

x

yO

A

OA [1 1 0]



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z

a

a


Chapter 3-

x

yO

A

AO [1 1 0]



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z

a

a


Chapter 3-

x

y

O

A

OA [1 1 0]



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z

a

a


Chapter 3-

x

y

O

A

AO [1 1 0]



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z

a

a


A

Chapter 3-

x

yO

OA [1 0 1]



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z

a

a


A

Chapter 3-

x

yO

AO [1 0 1]



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z

a

a


A

Chapter 3-

x

y

O

OA [1 0 1]



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z

a

a


A

Chapter 3-

x

y

O

AO [1 0 1]



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z

a

a

along face-diagonal of cubeA

Chapter 3-

x

y

O

OA [0 1 1]



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z

a

a

along face-diagonal of cubeA

Chapter 3-

x

y

O

AO [0 1 1]



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z

a

a


A

Chapter 3-

x

y

O

OA [0 1 1]



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z

a

a


A

Chapter 3-

x

y

O

AO [0 1 1]


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Equivalent Directionsalong Face Diagonals


Chapter 3-



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z

a

a

along Body-Diagonal of cube

A

Chapter 3-

x

yO

OA [1 1 1]



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z

a

a


A

Chapter 3-

x

yO

AO [1 1 1]




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z

a

a

along Body Diagonal of cube

A

Chapter 3-

x

y

O

OA [1 1 1]




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z

a

a

along ody iagonal of cube

A

Chapter 3-

x

y

O

AO [1 1 1]




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z

a

a

g y g

A

Chapter 3-

x

yO

OA [1 1 1]




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z

a

a

g y g

A

Chapter 3-

x

yO

AO [1 1 1]




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z

a

a

g y g

A

Chapter 3-

x

y

O

OA [1 1 1]




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z

a

a

g y g

A

Chapter 3-

x

y

O

AO [1 1 1]

Equivalent Directions


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q

along Body Diagonals


Chapter 3-

Important Observation in Cubic Unit Cell


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In case of cubic unit cell,directions having same indices

irrespective of ORDER & SIGNare equivalent.

Chapter 3-

E.g. [1 1 0], [0 1 1], [1 0 1] are

equivalent

Draw a [1 1 0] direction in a Cubic Unit Cell


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z

a

a

Chapter 3-

x

yO

A

-a

[1 1 0] O

A

[1 1 0]

Shift of Origin to accommodate ve direction

in unit cell itselfO

) [ ]


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z

a

a

5

3

) [u v w] [- + +]

[+ - +]

+ + -

Chapter 3-

x

y

a

12

4

[- - +]

[+ - -]

[- + -]

[- - -]


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Cr stallo ra hic Planes

Chapter 3-

Procedure to Draw a Crystallographic

Plane in a Unit Cell


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Note intercepts on x-, y- & z-axis in terms of edgelengths.

Divide the intercepts by corresponding edge lengths Write the reciprocals

Chapter 3-

Enclose these integers in parenthesis e.g. (h k l). Note

(h k l) for one plane {h k l} for family of planes

Negative indices are noted by a bar on the top

Crystal Planes in a cubic Unit Cell along

faces of unit cellz


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z

a

a

B

Chapter 3-

x

y

A

D

ABCD (1 0 0)

z


faces of unit cell


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z

a

a

C

D

Chapter 3-

x

yA

B

ABCD (0 1 0)

z


faces of unit cell


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z

a

a

A

D

Chapter 3-

x

y

ABCD (0 0 1)

z

Crystal Planes in a cubic Unit Cell

along face-Diagonal of cube


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z

a

a

B

C

Chapter 3-

x

y

A

D

ABCD (1 1 0)

z




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z

a

a

C

D

Chapter 3-

x

y

B

A

ABCD (1 1 0)

ABCD (1 1 0)OR

z




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z

a

a C

B

Chapter 3-

x

y

A

D

ABCD (0 1 1)

z




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z

a

a

C

B

Chapter 3-

x

y

A

D

ABCD (0 1 1)

ABCD (0 1 1)OR

z




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z

a

a

BC

Chapter 3-

x

yAD

ABCD (1 0 1)

ABCD (1 0 1)OR

z




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z

a

a

AB

Chapter 3-

x

y

O

ABCD (1 0 1)

C

D

z


along 3 face diagonals of a cube


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za

aC

Chapter 3-

x

y

A

B

ABC (1 1 1)

z

Determine the Miller Indices for the plane

ABCD


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za

a

Chapter 3-

x

y

A

a/2

BDa

ABCD (0 1 2)

z

Construct a (0 1 1) plane within a

Cubic Unit Cell


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a

a

Chapter 3-

x

yO

z

For (0 1 1) plane adjacent Unit Cell in ve y-

direction is required


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a

a

Chapter 3-

x

y

A

-yO

z

(0 1 1) plane w.r.t Cubic Unit Cell


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a

a

(0 1 1)

Chapter 3-

x

y-y

0

O

a

Transfer the coordinate System to

accommodate y axis


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a

a

Chapter 3-

x

y

z

O

-y

a

(0 1 1) plane within a Cubic Unit Cell

C


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a

a

C

B

Chapter 3-

x

y

z

O

-y

A

D

z


Cubic Unit Cell


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a

a

C

O

Chapter 3-

x

y

z


Cubic Unit Cell


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a

a

O

A

B

Chapter 3-

x

y

C

(1 1 1)

Equivalent Planes OR Family of Planes

are the planes which are Crystallographically


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p y g p y

equivalent; i.e. having the same atomic packing

(spacing)

Chapter 3-

{1 0 0}

{1 1 0} {1 1 1}

In case of cubic unit cell, planes

Important Observation in Cubic Unit Cell


110/143

, phaving same indices irrespective

of ORDER & SIGN are equivalent.

Chapter 3-

E.g. (0 0 1), (0 1 0), (1 0 0) areequivalent

Important Observation in Cubic Unit Cellz

a

a


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B aC

Chapter 3-

x

y

A

DABCD (1 0 0

OA [1 0 0

O

Miller indices are same for a plane and

direction, which are perpendicular to each

other


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Density of a crystal Structure

Chapter 3-

Theoretical Density of a crystal Structure

nA


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Ac

Th

NV

nA=

Chapter 3-

n Effective No. of atoms per unit cell

A Atomic weight, gram per mole

Vc Volume of unit cell

Na Avogadro's No. = 6.023 XXXX 1023

Data from Table inside front cover of Callister

Theoretical Density of Cu


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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 = 10-7 cm)

Chapter 3-

R = 0.128 x 10- cm

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

-23cm3

Compare to actual: Cu = 8.94 g/cm3

Result: theoretical Cu = 8.89 g/cm3

metals ceramics polymersGraphite/

Ceramics/Semicond

Metals/Alloys Composites/fibersPolymers

30Why?

DENSITIES OF MATERIAL CLASSES


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3)

20Based on data in Table B1, Callister

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

60% volume fraction of aligned fibersin an epoxy matrix).10

SteelsCu,Ni

Tin, Zinc

Silver, Mo

TantalumGold, WPlatinum

Zirconia

Why?Metals have... close-packing

(metallic bonding) large atomic mass

Chapter 3- 16

(g

/c

1

2

3

4

0.3

0.40.5

Magnesium

Aluminum

Titanium

Graphite

Silicon

Glass-sodaConcrete

Si nitrideDiamondAl oxide

HDPE, PSPP, LDPE

PC

PTFE

PETPVCSilicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

... less dense packing

(covalent bonding) often lighter elements

Polymers have... poor packing

(often amorphous) lighter elements (C,H,O)

Composites have... intermediate values

Atomic arrangements

(110) plane of FCC


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A B C

D E F

A

BC

Chapter 3-

DE

F

Atomic packing in(110) planeReduced sphereFCC unit cell with(110) plane

Atomic packing in (100) plane of FCC


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Chapter 3-

Atomic Packing in (111) plane of BCC

Atomic arrangements

(110) plane of BCC


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A/

B/

C/E/

A/ B/

C/

D/

Chapter 3-

D/

Reduced sphereBCC unit cell with(110) plane

Atomic packing in(110) plane of aBCC structure

E/

Atomic Packing Factor: BCC APF for a body-centered cubic structure = 0.68

a3


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a

a2

a3

Chapter 3-119

APF =

4

3 ( 3a/4 )3

2

atoms

unit cell atom

volume

a3

unit cell

volume

length = 4R =

Close-packed directions:

3aaR

From

Fig. 4.2(a)

Callisters MaterialsScience and Engineering,

Adapted Version.

APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

maximum achievable APF


120/143

Close-packed directions:

length = 4R = 2a

Unit cell contains:

2a

Chapter 3-120

APF =

4

3

( 2a/4 )34

atoms

unit cell

atom

volume

a3

unit cell

volume

6 x1/2 + 8 x1/8

= 4 atoms/unit cellaFrom

Fig. 4.1(a),

Callisters Materials Science and

Engineering,

Adapted Version.


121/143

Chapter 3-

Linear Density formula


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VectorDirectionoLen thvectordirectiononcenteredatomsofNumberLD

.........=

Chapter 3-

z

a

a

Linear density in a SC unit cell along

Edge


123/143

a

a

Chapter 3-

x

y

LD = 1/a = 1/2R

z

Linear density in a SC unit cell along

Face Diagonal


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a

a

Chapter 3-x

y

a

LD = 1/ 2a = 1/2 2R

za

Linear density in a SC unit cell along Body

Diagonal LD


125/143

a

Chapter 3-

x

y

a

LD

= 1/ 3a = 1/2 3R

z

a

Linear density in a BCC unit cell along Body

Diagonal LD


126/143

a

Chapter 3-

x

y

a

LD

= 2/ 4R = 1/ 2R

Linear Density

Linear Density of Atoms LD =Unit length of direction vector

Number of atoms


127/143

ex: linear density of Al in [110]direction

=

[110]

Chapter 3-127

a# atoms

length

13.5 nm

a2

2LD

========

z

a

Linear density in a FCC unit cell along

Face Diagonal LD


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a

a

Chapter 3-x

y

a

LD = 2/ 2a = 1/2R


129/143

PLANAR DENSITY

Chapter 3-

Planar Density Formula


130/143

PlaneaoAreaPlaneaoncenteredAtomsofNumberPD

.........=

Chapter 3-

Planar Density in a SC unit cell

PD {1 0 0}

za


131/143

a

Chapter 3-

x

z

PD{1 00}= 1 /a2 = 1/4R2


PD {1 1 0}

za


132/143

a

Chapter 3-

x

z

PD{1 1 0}= 1 /a 2 a =

1/4R2 2


PD {1 1 1}

za


133/143

a

Chapter 3-

x

z

PD{1 1 1}= 1/2 /( 3 a2/2) =

1/4R2 3

Planar Density in a BCC unit cell

PD {1 0 0}

za


134/143

a

Chapter 3-

x

z

PD{1 00}= 1 /a2 = 3/16R2


PD {1 1 0}

za


135/143

a

Chapter 3-

x

z

PD{1 10}=2/ 2a2 = 3 2/16R2


PD {1 1 1}

za


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a

Chapter 3-

x

z

PD{1 1 1}= 1/2 /( 3 a2/2) =

3 /16R2

Planar Density in a FCC unit cell

PD {1 0 0}

za


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a

Chapter 3-

x

z

PD{1 00}= 2 /a2 = 1/4R2


PD {1 1 0}

za


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a

Chapter 3-

x

z

PD{1 10}=2/ 2a2 = 2/8R2


PD {1 1 1}

za

a


139/143

a

Chapter 3-

x

z

PD{1 1 1}= 2 /( 3 a2/2) = 1/2 3 R2

Polymorphism & Allotropy

Metals/Nonmetals may have more than one crystalstructure. This phenomenon is called as polymorphism

d th diti i ll d ll t


140/143

and the condition is called as allotropy

Chapter 3-

e.g. 1. Carbon: Graphite at ambient condition

Diamond at extreme pressure

2. Iron: BCC Iron at room temperature

FCC Iron at 912 oC.

Demonstrates "polymorphism"The same atoms can

have more than one

crystal structure

DEMO: HEATING AND

COOLING OF AN IRON WIRE

T t C


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crystal structure.Temperature, C

1536BCC Stable

Liquid

Chapter 3- 22

BCC Stable

FCC Stable

914

1391

shorter

longer!shorter!

longer

Tc 768 magnet falls off

heat up

cool down

ANISOTROPY

Directionality of a property is called as an

ANISOTROPY


142/143

ANISOTROPY.

If properties are independent of direction, then

Chapter 3-

.

Degree of anisotropy is dependent on structuresymmetry.

Higher the symmetry, lower the degree of anisotropy

and lower the symmetry, higher the degree of anisotropy.

Atoms may assemble into crystalline or

amorphous structures.

SUMMARY


143/143

We can predict the density of a material,provided we know the atomic weight, atomic

Chapter 3-

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.

23

ch-3 compatibility mode

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