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Fundamental Structureof 

Crystalline Solidsy

•  Non dense, random packing

Energy and PackingEnergy 

typical neighborbond length

•  Dense, ordered packing

rtypical neighborbond energy

Energy

typical neighborbond length

2

Dense, ordered packed structures tend to havelower energies.

rtypical neighborbond energy

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•  atoms pack in periodic, 3D arrays

Crystalline materials...

‐metalsmany ceramics

Materials and Packing

•  typical of:‐many ceramics

‐some polymers

•  atoms have no periodic packing

Noncrystalline materials...

crystalline SiO2

Si Oxygen

3

‐complex structures

‐rapid cooling

"Amorphous" = Noncrystalline

•  occurs for:

noncrystalline SiO2

Crystal Terminalogy

Crystal : Regular, periodic and repeated arrangement 

Lattice: Three dimensional arrayLattice: Three dimensional array of points coinciding with atom positions.

Unit cell: smallest repeatable entity that can be used to completely represent a crystalcompletely represent a crystal structure. It is the building block of crystal structure.

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7  crystal systems

A small group of lattice points which forms a repetitive pattern is called a unit cell.

14 Bravais /space lattices

a, b, and c are the lattice constants

Cl ifi ti f C t l S t

5

Classification of Crystal Systems:

The angles ( α,β,γ) between the lattice translation vectors and the distance(a,b,c) between two successive points along these vectors are used to classify the crystal structures into 7 systems

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

d

Simple Cubic Structure (SC)

•  Coordination # = 6(# nearest neighbors)

9(Courtesy P.M. Anderson)

Atomic Packing Factor (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

4

3π (0.5a) 31

atoms

unit cell

atom

volume

*assume hard spheres

a

R 0 5

10•  APF for a simple cubic structure = 0.52

APF = 

a 3

unit cell

volume

R=0.5a

contains 8 x 1/8 = 

1 atom/unit cell

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•  Atoms touch each other along cube diagonals.

‐‐Note:  All atoms are identical; the center atom is shadeddifferently only for ease of viewing.

Body Centered Cubic Structure (BCC)

•  Coordination # = 8

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

11(Courtesy P.M. Anderson)2 atoms/unit cell:  1 center + 8 corners x 1/8

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

a3

a

length = 4R =

Close‐packed directions:

3 aaR

a2

12

APF = 

4

3π ( 3 a/4 ) 32

atoms

unit cell atom

volume

a 3

unit cell

volume

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• Atoms touch each other along face diagonals.

‐‐Note:  All atoms are identical; the face‐centered atoms are shadeddifferently only for ease of viewing.

Face Centered Cubic Structure (FCC)

•  Coordination # = 12ex: Al, Cu, Au, Pb, Ni, Pt, Ag

13(Courtesy P.M. Anderson)4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

•  APF for a face‐centered cubic structure = 0.74

Atomic Packing Factor:  FCC

maximum achievable APF

Close‐packed directions: 

atoms

length = 4R = 2 a

Unit cell contains:

6 x 1/2 + 8 x 1/8  

=  4 atoms/unit cella

2 a

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APF = 

4

3π (  2 a/4 ) 34

atoms

unit cell atom

volume

a 3

unit cell

volume

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• 3D Projection

•  2D Projection

Hexagonal Close‐Packed Structure (HCP)

  3D Projection

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

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•  Coordination # = 12

•  APF = 0.74

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

• c/a = 1.633

a

Possible Bravais lattices:

•Cubic : Simple, BC, FC•Hexagonal : Simple•Tetragonal : Simple, BC•Orthorhombic : Simple Base C Body-C FCOrthorhombic : Simple, Base C, Body C, FC•Monoclinic : Simple, BC•Triclinic, Trigonal : Simple

Crystal System Axial relationships Interaxial Angles

Cubic a=b=c α=β=γ=90o

tetragonal a=b≠c α=β=γ=90o

Hexagonal a=b≠c α=β=90ο γ=120ο

Rhomohedral/trygonal

a=b=c α=β=γ≠90o

Orthorhombic a ≠b ≠c α=β=γ=90o

Monoclinic a ≠b ≠c α=γ=90o ≠ β

triclinic a ≠b ≠c α ≠ β ≠ γ ≠ 90o

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Allotropy

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ρ = n AV NA

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

V l / it ll

THEORETICAL DENSITY, ρ

Example: Copper

VcNAVolume/unit cell

(cm3/unit cell)Avogadro's number

(6.023 x 1023 atoms/mol)

Data from Table inside front page of Callister :

• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63 55 g/mol (1 amu = 1 g/mol)• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10 -7 cm)

Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3

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

Element Aluminum Argon Barium Beryllium

Symbol Al Ar Ba Be

At. Weight (amu) 26.98 39.95 137.33 9.012

Atomic radius (nm) 0.143 ------ 0.217 0.114

Density

(g/cm3) 2.71 ------ 3.5 1.85

Crystal Structure FCC ------ BCC HCP

Characteristics of Selected Elements at 20C

yBoron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt

B Br Cd Ca C Cs Cl Cr Co

9.01 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58 93

------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0 125

2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8 9

Rhomb ------ HCP FCC Hex BCC ------ BCC HCP

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Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen

Co Cu F Ga Ge Au He H

58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008

0.125 0.128 ------ 0.122 0.122 0.144 ------ ------

8.9 8.94 ------ 5.90 5.32 19.32 ------ ------

HCP FCC ------ Ortho. Dia. cubic FCC ------ ------

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ρmetals� ρceramics� ρpolymers Graphite/

Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

2 0

30B ased on data in Table B1, Callister

*GFRE CFRE & AFRE are Glass Platinum

DENSITIES OF MATERIAL CLASSES

(g/c

m3)

2 0 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

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

Tantalum Gold, W

Glass - soda Concrete

Si nitride Diamond Al oxide

Zirconia

Glass fibers

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ρ

1

2

0.3 0.4 0.5

Magnesium G raphite Silicon Concrete

H DPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE *

CFRE *

GFRE* Carbon fibers

A ramid fibers

a) crystal System? b) Crystal Structure?  c) Density ? if Aw=141 g/mol

3.2

TetragonalBCT

) y w g

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Crystallographic Points Coordinates

Fractional multiples of unit cell, edge lengths.Less than or equal 1With out separation with commas (,)

Point CoordinatesPoint coordinates for unit cell 

center are

a/2 b/2 c/2 ½½½

z

c111

a/2, b/2, c/2       ½½½

Point coordinates for unit cell corner are 111x

ya b000

24

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Crystallographic Directions1. Vector repositioned (if necessary) to pass

through origin.2 R d ff j ti i t f

z Algorithm

2. Read off projections in terms of unit cell dimensions a, b, and c

3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

ex: 1 0 ½ => 2 0 1 => [ 201 ]

x

y

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ex: 1 0 ½ => 2 0 1 => [ 201 ]

-1 1 1

families of directions <uvw>

where overbar represents a negative index[ 111 ]=>

How to draw the direction of [uvw]

1.Divide with the biggest integer among them.

2. Multiply with edge lengths.

3. Take corresponding length on ref. Axis/edges.

4. Farm a line by projection.

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3.9: Draw a [211] direction in Orthorhombic unit cell?

1 2 -1 1U V W

_

4. Move along the edges / axis

2 2/2 -1/2 1/2

3 a - b/2 c/2

3.9: Draw a [211] direction in Orthorhombic unit cell?

1 2 -1 1U V W

_

4. Move along the edges / axis

2 2/2 -1/2 1/2

3 a - b/2 c/2

a-b/2 c/2

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HCP Crystallographic Directions

1. Vector repositioned (if necessary) to pass through origin.

2 R d ff j ti i t f it

zAlgorithm

2. Read off projections in terms of unitcell dimensions a1, a2, a3, or c

3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvtw]a2

-a3a2

-a3

a1

a2

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[ 1120 ]ex: ½, ½, -1, 0 =>

dashed red lines indicate projections onto a1 and a2 axes a1

a3

a32

a2

2a1

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HCP Crystallographic Directions• Hexagonal Crystals

– 4 parameter Miller‐Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.

=v

u

)'u'v2(1

-

)'v'u2(3

1-=

]uvtw[]'w'v'u[ →

a2

z

31

=

=

=

'ww

t

v

)vu( +-

)uv2(3

-

-a3

a1

3.17

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Crystallographic Planes

• Crystallographic planes are specified by Miller indices (hkl)

• Indices are smallest Reciprocal of plane Intercepts with edges (Axis) of unit cell.

A t l ll l t h th• Any two planes parallel to each other are equivalent and have identical indices.

Procedure:1.Determine the intercepts made by the plane with the three axes. If the plane passes through the origin pick a parallel plane within the unit cell.2 C l l t th i l f th i t t2. Calculate the reciprocals of the intercepts.3. Divide by a common factor to convert to smallest possible integers.4. Enclose in parentheses no

commas i.e., (hkl)

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

1 1 03. Reduction 1 1 0

example a b c z

y

c

4. Miller Indices (110)

1. Intercepts 1 1 12. Reciprocals 1/1 1/1 1/1

1 1 1

Example 2 a b cx

ya b

4. Miller Indices (111)

1 1 13. Reduction 1 1 1

z

c

example

1. Intercepts 1/2 1 3/4a b c

x

ya b

•

••

4. Miller Indices (634)

1. Intercepts 1/2 1 3/42. Reciprocals 1/½ 1/1 1/¾

2 1 4/3

3. Reduction 6 3 4

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(001)(010),

Family of Planes {hkl}Same atomic arrangement

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

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(110) For FCC

Atomic Arrangements in planes

(110) For BCC

3.21: for cubic unit cella) (101), b) (211) c) (012) d) (313) e) (111) f) (212) g) (312) h) (301)

_ _ _ _ _ _ _

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4.3) c/a = 1.633in HCP

Atom at M is midway between the top and bottom so

Atom at M is midway between the top and bottom so

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3.27: fallowing are the planes

a) What is crystal system?b) What is crystal lattice?

Tetragonal

BCT

BCT

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4.20: fallowing are the planes

a) What is crystal system?

b) What is crystal lattice?

Orthorambic

FCO

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c) Determine atomic wt if ρ = 18.91 ?

ρ = n A# atoms/unit cell Atomic weight (g/mol)

ρ =VcNAVolume/unit cell

(cm3/unit cell)Avogadro's number

(6.023 x 1023 atoms/mol)

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Linear Density

[110]vectordirection oflength

vectordirection on centered atoms ofnumber Density Linear =

ex: linear density of FCCl in [110] direction

[110]

47

a

# atoms

length

12/a nma2

2LD −==

[111] 2/ 3 a

[100] 1/a [110] 1/ 2 a

BCC

[ ]

aR

FCC

[100] 1/a

a

2 a

[111] 1/ 3 a

[100] 1/a[110] 2/ a

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Planar Densities

planeofareaplane aon centered atoms ofnumber Density Planar =

planeofarea

(001) plane

# atoms

length

22 /a2 nma2

2PD −==

(111) 1/ 3a2

(100) 1/a2

(110) 2 /a2

BCC

( )

aR

FCC

(100) 2/a2

a

2 a

(111) 4/ 3 a2

(100) 2/a(110) 2/ a2

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

4

2D repeat unit(100) 1/a2

Radius of iron R = 0.1241 nm

R3

4a =

aR

51

= Planar Density =a2

1atoms

2D repeat unit=

nm2atoms12.1

m2atoms= 1.21 x 1019

12

R34area

2D repeat unit

Linear  Density of (111) Iron

[111] 2/ 3a [ ]

aR

Radius of iron R = 0.1241 nm

R3

4a =

52

2= =

nmatoms4.029

matoms4.029 x 109

R3linear Density =

atoms2D repeat unit

length2D repeat unit 3

4

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[110] 2/ a

Highest planar & Linear density for FCC / CCP (Cubic Closed Pack)

a

2 a

(111) 4/ 3 a2

Closed packed directions

Closed packed planes

• ABCABC... Stacking Sequence• 2D Projection

BBC

A

A

FCC STACKING SEQUENCE

A sites

B sites

C sitesB B

BB BC C

A

• FCC Unit Cell AB

11

BC

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AA

B

C

B

FCC Stacking sequence

C

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• ABAB... Stacking Sequence

• 3D Projection • 2D Projection

HEXAGONAL CLOSE‐PACKED STRUCTURE (HCP)

• 3D Projection • 2D Projection

A sites

B sites

A sitesBottom layer

Middle layer

Top layer

12

Bottom layer

B

A

A

B

A

A

A

• HCP Stacking sequence

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Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism)

liquid

iron system

Titanium  α, β‐Ti

carbon

diamond, graphite 

BCC

FCC

1538ºC

1394ºC

912ºC

δ-Fe

γ-Fe

60

BCC α-Fe

"Existence of an elemental solid into more than one physical forms is known as ALLOTROPY "

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Existence of substance into more than one crystalline forms is known as "POLYMORPHISM".

Example:1) Mercuric iodide (HgI2) forms two types of crystals.

a. Orthorhombicb. Trigonal

2) Calcium carbonate (CaCO3) exists in two types of t lli fcrystalline forms.a. Orthorhombic (Aragonite)b. Trigonal

Allotropy

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• Some engineering applications require single crystals:

--diamond singlecrystals for abrasives

--turbine blades

CRYSTALS AS BUILDING BLOCKS

• Crystal properties reveal featuresof atomic structure.

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

• Most engineering materials are polycrystals.Polycrystals Anisotropic

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

1 mm

Isotropic

64

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 typ. range from 1 nm to 2 cm

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

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Poly crystals

• Single Crystals

-Properties vary withdirection: anisotropic.Example: the modulus

Single vs PolycrystalsE (diagonal) = 273 GPa

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

• Polycrystals

-Properties may/may notvary with direction.

-If grains are randomly

200 μm

E (edge) = 125 GPa

66

If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)

-If grains are textured,anisotropic.