chap-2-crystal structure

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Fundamental Structure

of

Crystalline Solids

3

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have

lower energies.

Energy and Packing

Energy

r

typical neighbor

bond length

typical neighbor

bond energy

Energy

r

typical neighbor

bond length

typical neighbor

bond energy

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4

• atoms pack in periodic, 3D arrays

Crystalline materials...

-metals

-many ceramics

-some polymers

• atoms have no periodic packing

Noncrystalline materials...

-complex structures

-rapid cooling

crystalline SiO2

"Amorphous" = Noncrystalline

Materials and Packing

Si Oxygen

• typical of:

• occurs for:

noncrystalline SiO2

Crystal Terminalogy

Crystal : Regular, periodic and repeated arrangement

Lattice: Three dimensional array

of points coinciding with atom

positions.

Unit cell: smallest repeatable

entity that can be used to

completely represent a crystal

structure. It is the building

block of crystal structure.

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6

7 crystal systems

14 Bravais /space lattices – Crystal structure

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

a, b, and c are the lattice constants

Classification of Crystal Systems:

The angles ( α,β,γα,β,γα,β,γα,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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10

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

• Coordination # = 6

(# nearest neighbors)

(Courtesy P.M. Anderson)

Simple Cubic Structure (SC)

11

• APF for a simple cubic structure = 0.52

APF =

a 3

4

3

π (0.5a) 31

atoms

unit cell

atom

volume

unit cell

volume

Atomic Packing Factor (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

a

R=0.5a

contains 8 x 1/8 =

1 atom/unit cell

Effective no of atoms / UC

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12



• Atoms touch each other along 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 (α), Tantalum, Molybdenum

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

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Atomic Packing Factor: BCC

a

APF =

4

3

π ( 3 a/4 ) 32

atoms

unit cellatom

volume

a 3

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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• Atoms touch each other along face diagonals.

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

differently only for ease of viewing.

Face Centered Cubic Structure (FCC)

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

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

15

• APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

maximum achievable APF

APF =

4

3

π ( 2 a/4 ) 34

atoms

unit cell atom

volume

a 3

unit cell

volume

Close-packed directions:

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 = 0.74

• 3D Projection

• 2D Projection

Hexagonal Close-Packed Structure

(HCP)

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

• c/a = 1.633

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

Possible Bravais lattices:

•Cubic : Simple, BC, FC•Hexagonal : Simple•Tetragonal : Simple, BC•Orthorhombic : 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/try

gonal

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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Example: CopperExample: CopperExample: CopperExample: Copper

ρ = n A

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 page of Callister :Data from Table inside front page of Callister :Data from Table inside front page of Callister :Data from Table inside front page of Callister :

• crystal structure = FCC: • crystal structure = FCC: • crystal structure = FCC: • crystal structure = FCC: 4 atoms/unit cell4 atoms/unit cell4 atoms/unit cell4 atoms/unit cell• atomic weight = • atomic weight = • atomic weight = • atomic weight = 63.55 g/mol63.55 g/mol63.55 g/mol63.55 g/mol (1 amu = 1 g/mol)(1 amu = 1 g/mol)(1 amu = 1 g/mol)(1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10• atomic radius R = 0.128 nm (1 nm = 10• atomic radius R = 0.128 nm (1 nm = 10• atomic radius R = 0.128 nm (1 nm = 10 ----7777 cm)cm)cm)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

THEORETICAL DENSITY, ρρρρ

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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 ------ ------

Characteristics of Selected Elements at 20C

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a) crystal System?

b) Crystal Structure?

c) Density ? if Aw=141 g/mol

3.23.23.23.2

Tetragonal

BCT

Crystallographic Points Coordinates

� Fractional multiples of unit cell, edge lengths.

� Less than or equal 1

� With out separation with commas (,)

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Point CoordinatesPoint coordinates for unit cell

center are

a/2, b/2, c/2 ½½ ½

Point coordinates for unit cell corner are 111

z

x

y

a b

c

000

111

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Crystallographic Directions (Lines)

1. Vector repositioned (if necessary) to pass

through 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 ½ => 2 0 1 => [ 201 ]

-1 1 1

families of directions <uvw>

z

x

Algorithm

where overbar represents a negative index[ 111 ]=>

y

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3.14

A)

B)

C)

D)

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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.

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

4. Move along the edges / axis

1 2 -1 1

2 2/2 -1/2 1/2

3 a - b/2 c/2

U V W

_

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

4. Move along the edges / axis

1 2 -1 1

2 2/2 -1/2 1/2

3 a - b/2 c/2

U V W

_

a-b/2

c/2

HCP Crystallographic Directions

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Crystallographic Planes

• Crystallographic planes are specified by

Miller indices (hkl)

• Indices are Reciprocal of plane Interceptswith edges (Axis) of unit cell.

• Any two planes parallel to each other and placed equal distance 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. 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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4. Miller Indices (110)

1. Intercepts 1 1 ∞

2. Reciprocals 1/1 1/1 1/∞

1 1 03. Reduction 1 1 0

example a b c


1. Intercepts 1 1 1

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

1 1 13. Reduction 1 1 1

Example 2 a b c

z

x

y

a b

c

35

z

x

y

a b

c

•

••


example

1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾

2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes {hkl}

Same atomic arrangement

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

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3.22: Determine miller indices for the planes?

2. Intercepts (in a,b,c) 1 1 -1

Plane A

3. Reciprocals 1 1 -1

Miller Indices (111)

_

Plane B

1. Intercepts a b -c

h k l

3.19: a) Draw (021) plane for orthogonal unit cell?_

3 Intercept par b/2 -c

1. Indices 0 2 -1

2. Reciprocals 1/2 -1

h k l

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3.19: b) (200) monoclinic

3.21: for cubic unit cell

a) (101), b) (211) c) (012) d) (313) e) (111) f) (212) g) (312) h) (301)_ _ _ _ _ _ _

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(110) For FCC

Atomic Arrangements

(110) For BCC

41

ex: linear density of FCC in [110]

direction

Linear Density

a

[110]vectordirection oflength

vectordirection on centered atoms ofnumber Density Linear =

# atoms

length

12/a nm

a2

2LD

−==

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[111] 2/ 3 a

[100] 1/a

[110] 1/ 2 a

BCC

aR

a

2 a

FCC

[111] 1/ 3 a

[100] 1/a

[110] 2/ a

Planar Densities

plane of area

plane aon centered atoms ofnumber Density Planar =

# atoms

length

22 /a2 nm

a2

2PD

−==

(001) plane

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(111) 1/ 3a2

(100) 1/a2

(110) 2 /a2

BCC

aR

a

2 a

FCC

(111) 4/ 3 a2

(100) 2/a2

(110) 2/ a2

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

Radius of iron R = 0.1241 nm

R3

4a =

2D repeat unit

= Planar Density =a2

1

atoms

2D repeat unit

=

nm2

atoms12.1

cm2

atoms= 1.21 x 1013

1

2

R3

4area

2D repeat unit

(100) 1/a2

aR

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Linear Density of [111] Iron

[111] 2/ 3a

aR

Radius of iron R = 0.1241 nm

R3

4a =

2

= =

nm

atoms4.029

m

atoms4.029 x 109

R3

linear Density =

atoms

2D repeat unit

length

2D repeat unit 3

4

a

2 a

[110] 2/ a

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

(111) 4/ 3 a2

Closed packed directionsClosed packed directionsClosed packed directionsClosed packed directions

Closed packed planesClosed packed planesClosed packed planesClosed packed planes

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11

• ABCABC... Stacking Sequence• ABCABC... Stacking Sequence• ABCABC... Stacking Sequence• ABCABC... Stacking Sequence• 2D Projection• 2D Projection• 2D Projection• 2D Projection

A sites

B sites

C sites

B B

B

BB

B BC C

CA

A

• FCC Unit Cell AB

C

FCC STACKING SEQUENCE

FCC Stacking sequence

A

B

C

A

B

C

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12

• ABAB... Stacking Sequence• ABAB... Stacking Sequence• ABAB... Stacking Sequence• ABAB... Stacking Sequence

• 3D Projection• 3D Projection• 3D Projection• 3D Projection • 2D Projection• 2D Projection• 2D Projection• 2D Projection

A sites

B sites

A sitesBottom layer

Middle layer

Top layer

HEXAGONAL CLOSE-PACKED

STRUCTURE (HCP)

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A

B

A

B

A

A

• HCP Stacking sequence

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Polymorphism • Two or more distinct crystal structures for the

same material (allotropy/polymorphism)

Titanium α, β-Ti

carbon

diamond, graphite

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

δδδδ-Fe

γγγγ-Fe

αααα-Fe

liquid

iron system

"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 crystalline forms.a. Orthorhombic (Aragonite)b. Trigonal

Allotropy

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• • • • SomeSomeSomeSome engineering applications require single crystals:engineering applications require single crystals:engineering applications require single crystals:engineering applications require single crystals:

• Crystal properties reveal features• Crystal properties reveal features• Crystal properties reveal features• Crystal properties reveal featuresof atomic structure.of atomic structure.of atomic structure.of atomic structure.

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

--------diamond singlediamond singlediamond singlediamond singlecrystals for abrasivescrystals for abrasivescrystals for abrasivescrystals for abrasives

--------turbine bladesturbine bladesturbine bladesturbine blades

CRYSTALS AS BUILDING BLOCKS

59

• Most engineering 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 typ. range from 1 nm to 2 cm

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

1 mm

Polycrystals

Isotropic

Anisotropic

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61

• 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.

200 µm

Single vs PolycrystalsE (diagonal) = 273 GPa

E (edge) = 125 GPa

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22

• Demonstrates "polymorphism"The same atoms can have more than one crystal structure.

DEMO: HEATING AND

COOLING OF AN IRON WIRE

Temperature, C

BCC Stable

FCC Stable

914

1391

1536

shorter

longer!shorter!

longer

Tc 768 magnet falls off

BCC Stable

Liquid

heat up

cool down

Non crystalline solid

• atoms have no periodic packing

Noncrystalline materials...

-complex structures

-rapid cooling

"Amorphous" = Noncrystalline

= super cooled liquid

• occurs for:

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ρmetals ρceramics ρpolymers

16

ρ (

g/c

m3

)Graphite/

Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

2 0

30B ased 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

G raphite

Silicon

Glass - soda Concrete

Si nitride Diamond Al oxide

Zirconia

H DPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE *

CFRE *

GFRE*

Glass fibers

Carbon fibers

A ramid fibers

DENSITIES OF MATERIAL CLASSES

3.27: fallowing are the planes

a) What is crystal system?

b) What is crystal lattice?

Tetragonal

BCT

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BCT

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

VcNA

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

Volume/unit cell

(cm3/unit cell)

Avogadro's number

(6.023 x 1023 atoms/mol)

chap-2-crystal structure

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