crystal and amorphous structure in materials

8/6/2019 Crystal and Amorphous Structure in Materials

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Lecture 3 -1

At the end of the lecture, you will know: The difference between crystalline and non-crystallinematerials.

How do atoms assemble into solid structures?

Draw and understand the relationship betweenface-centered, body-centered and hexagonal close packing.

Distinguish between single crystals and polycrystallinematerials. Explain polymorphism or allotropy in materials. Write atom position, direction indices and Miller indices for

cubic crystal system.

Chapter 3: Crystal and Amorphous Structuresin Materials

How do the crystal structures of ceramic materials differ from those for metals?

Halina Misran 2009


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Non dense, random packing

Dense, ordered packing

Dense, ordered packed structures tend to havelower energies.

Energy and PackingEnergy

r

typical neighborbond length

typical neighborbond energy

Energy

r

typical neighborbond length

typical neighborbond energy

Halina Misran 2009 Lecture 3 -2


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atoms pack in periodic, 3D arraysCrystalline materials...

-metals-many ceramics-some polymers

atoms have no periodic packingNoncrystalline materials...

-complex structures-rapid cooling

crystalline SiO 2

noncrystalline SiO 2

"Amorphous " = Noncrystalline

Materials and Packing

Si Oxygen

typical of:

occurs for:

long-range order (LRO)

short-range order (SRO)Halina Misran 2009 Lecture 3 -3


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Atoms, arranged in repetitive 3-dimensional pattern -give rise to crystal structure .

Properties of solids materials of engineering importancedepends upon crystal structure and bonding force . An imaginary network of lines, with atoms at intersection

of lines, representing the arrangement of atoms is calledspace lattice .

Unit cell is that block of atoms which repeats itself toform space lattice.

Unit Cell

SpaceLattice



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Crystal Systems and Bravais Lattice

a, b, c, , and are the lattice constants

7 Crystal Systems7 different types of crystal systems arenecessary to create all point lattices.1) cubic2) tetragonal3) orthorhombic4) rhombohedral5) hexagonal6) monoclinic7) triclinic

14 Unit Cellsaccording to Bravais (1811-1863)14 standard unit cells can describe allpossible lattice networks.



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1. Cubic Crystal Systema = b = c

= = = 90 0

Simple (SC) Body Centered(BCC)

Face Centered(FCC)

Simple Body CenteredHalina Misran 2009

a =b c

= = = 900

2. Tetragonal Crystal System

Lecture 3 -6


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3. Orthorhombic Crystal System a b c

= = = 900

Simple Base CenteredFace CenteredBody Centered

Simple

a = b = c = = 90 0

Halina Misran 2009

4. Rhombohedral Crystal System


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5. Hexagonal Crystal System a = b c

= = 90 0 , = 120Simple

SimpleBaseCentered

a b c = = 90 0

6. Monoclinic Crystal System


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7. Triclinic Crystal System

a

b

c 90 0

Simple


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Rare due to low packing density(only Po has this structure) Close-packed directions are cube edges.

Coordination # = 6(# nearest neighbors)

(Courtesy P.M. Anderson)

Simple Cubic Structure (SC)

1 atom/unit cell : 8 corners x 1/8


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APF for a simple cubic structure = 0.52

APF =a 3

4

3p (0.5 a ) 31

atomsunit cell

atomvolume

unit cellvolume

Atomic Packing Factor (APF)

APF =Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

close-packed directions

a

R =0.5 a

contains 8 x 1/8 =1

atom/unit cell


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Coordination # = 8

Adapted from Fig. 3.2,Callister & Rethwisch 3e.


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 ( ), Tantalum, Molybdenum

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


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aa

c

60120

Atomic Packing Factor: HCP

TBA in the lecture.


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Atomic Packing Factor: BCC

a

APF =

4

3p ( 3 a /4) 32

atomsunit cell atom

volume

a 3

unit cell

volume

length = 4 R =Close-packed directions:

3 a

APF for a body-centered cubic structure = 0.68

a R Adapted from

Fig. 3.2(a), Callister &Rethwisch 3e.

a 2

a 3


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Coordination # = 12

Adapted from Fig. 3.1, Callister & Rethwisch 3e.


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)

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

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


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APF for a face-centered cubic structure = 0.74Atomic Packing Factor: FCC

maximum achievable APF

APF =

43

p ( 2 a /4) 34atoms

unit cell atomvolume

a 3unit cellvolume

Close-packed directions:length = 4 R = 2 a

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

= 4 atoms/unit cella

2 a

Adapted fromFig. 3.1(a),Callister &Rethwisch 3e.


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Coordination # = 12

ABAB... Stacking Sequence

APF = 0.74

3D Projection 2D Projection

Adapted from Fig. 3.3(a),Callister & Rethwisch 3e.

Hexagonal Close-Packed Structure(HCP)

ex: Cd, Mg, Ti, Zn Ideal ratio of c / a = 1.633

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

(2 x 6 x 1/6) + (2 x ) + 3= 6 atoms/unit cell


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Principal Metallic Crystal Structures 90% of the metals have either Body Centered Cubic (BCC), Face

Centered Cubic (FCC) or Hexagonal Close Packed (HCP) crystalstructure.

HCP is denser version of simple hexagonal crystal structure. Most metals crystallize into HCP, atoms are closer and bond

tighter. Thus, lower energy and more stable.

BCC Structure FCC Structure HCP Structure3-7


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Atom Positions in Cubic Unit Cells Cartesian coordinate system is use to locate atoms.

In a cubic unit celly axis is the direction to the right .x axis is the direction coming out of the paper.z axis is the direction towards top.Negative directions are to the opposite of positivedirections.

Atom positions are located using unit distances along theaxes.

3-15


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Directions in Cubic Unit Cells Direction indices are position coordinates of unit cell where the

direction vector emerges from cell surface, converted to integers.Example (1,0,0)

In cubic crystals, Direction Indices are vector components ofdirections resolved along each axes, reduced to the smallestintegers. Example [100]

3-16

[110]

[111]

= (1,1/2,0)Must be integer= 2(1,1/2, 0)

= [210]

Neworigin(-1,-1,0)

[ 0]


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(0,0,0) (0,1,0)

x

y

z

(0,-1,0)

(1,0,0)

(0,0,1)

(-1,0,0)

(0,0,-1)

Directions are crystallographically equivalent when:Atom spacing along each direction is the same.[100], [010], [001], [00], [00], [00] = family of indices


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ExampleDraw the following direction vectors in cubic unit cells:

[102][313][212][301]

TBA in lecture.

22


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Direction Indices - Example Determine direction indices of the given vector.

Origin coordinates are (3/4, 0, 1/4).

Emergence coordinates are (1/4, 1/2, 1/2).

Subtracting emergence coordinates from origin := ( , , ) ( , 0, )

= ( - , , )Multiply by 4 to convert all fractionsto integers 4 x (-, , ) = (-2, 2, 1)

Therefore, the direction indices are[ 2 2 1 ]

3-18


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

cleared of fractions & common multiples. All parallel planes havesame 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 values4. Enclose in parentheses, no

commas for example (111)

z

x

y

MillerIndices =(111)


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z

x

y a b

c

4. Miller Indices (110)

example a b c z

x

y a b

c


1. Intercepts 1 1

2. Reciprocals 1/1 1/1 1/

3. Reduction 1 1 0

1. Intercepts 1

2. Reciprocals 1/1 1/ 1/

3. Reduction 1 0 0

example a b c

Crystallographic Planes Miller Indices


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z

x

y a 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

(001)(010),

Family of Planes { hkl }

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

Crystallographic Planes Miller Indices


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Miller Indices - Examples Plot the plane (101)

Taking reciprocals of the indices weget (1 1).

The intercepts of the plane are x=1,y= (parallel to y) and z=1.

TBA in lecture.

********************************************** Plot the plane (2 2 1)

Taking reciprocals of the indices

we get (1/2 1/2 1).The intercepts of the plane arex=1/2, y= 1/2 and z=1.

TBA in lecture.

3-22


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Example

28

Plot the plane (110)The reciprocals are (1,-1, )The intercepts are x=1, y=- 1 and z= (parallel to z axis)

y

x

z


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Miller Indices Important Relationship Direction indices of a direction perpendicular to a crystal

plane are same as miller indices of the plane.

Example:

Interplanar spacing between parallel closest planes withsame miller indices is given by

(110) plane

[110] indices

x

y

z

3-24

dhkl

= a

h2 + k2 + l2dhkl = interplanar spacing between parallel closest

planea = lattice constant

h, k, l = Miller indices of cubic plane being considered


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Crystallographic Planes



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HCP Crystallographic Directions

1. 4 parameter Miller-Bravais indices (hkil)

are used and 4 axes (a 1, a 2, a 3 and c) arerelated as follows :2. Read off projections in terms of unit

cell dimensions a 1, a 2, a 3, or c 3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvtw ]

[1120]ex: , , -1, 0 =>dashed red lines indicateprojections onto a 1 and a 2 axes a 1

a 2

a 3

-a 3

2

a 2

2a 1

Fig. 3.24(a), Callister & Rethwisch 3e.

-a 3

a 1

a 2

z


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Hexagonal Unit Cell - Examples

Basal Planes:-Intercepts a1 =

a2 = a3 = c = 1

(hkil) = (0001) Prism Planes :-

For plane ABCD,Intercepts a1 = 1

a2 = a3 = -1c =

(hkil) = (1010)

3-26


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Directions in HCP Unit Cells Indicated by 4 indices [uvtw]. u,v,t and w are lattice vectors in a

1, a

2, a

3and c

directions respectively. Example: For a 1, a 2, a 3 directions, the direction

indices are [ 2 1 1 0], [1 2 1 0] and [ 1 1 2 0]respectively.

3-27


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Crystallographic Planes (HCP)

In hexagonal unit cells the same idea is used

example a 1 a 2 a 3 c

4. Miller-Bravais Indices (1011)

1. Intercepts 1 -1 1

2. Reciprocals 1 1/ 1 0

-1-1

11

3. Reduction 1 0 -1 1

a 2

a 3

a 1

z

Adapted from Fig. 3.24(b),Callister & Rethwisch 3e.


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Comparison of FCC and HCP crystals Both FCC and HCP are close packed and have APF 0.74.

FCC crystal is close packed in (111) plane while HCP isclose packed in (0001) plane.


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Volume Density, r

where n = number of atoms/unit cellA = atomic weightV C = Volume of unit cell = a 3 for cubicN A = Avogadros number

= 6.022 x 10 23 atoms/mol

Density = r =

V C N A

n Ar =

CellUnitofVolumeTotalCellUnitinAtomsofMass


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Ex: Cr (BCC)A = 52.00 g/molR = 0.125 nmn = 2 atoms/unit cell

r theoretical

a = 4 R / 3 = 0.2887 nm

r actual

a R

r =a 3

52.002

atoms

unit cell molg

unit cell

volume atoms

mol

6.022x10 23

Volume Density, r for BCC

= 7.18 g/cm 3

= 7.19 g/cm 3


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Planar Density p

Example: In Iron (BCC, a = 0.287), the (110) plane intersects centerof 5 atoms (Four and 1 full atom).Equivalent number of atoms = (4 x ) + 1 = 2 atomsArea of 110 plane =

r p =

Atoms intersected in selected 2D planein unit cell

Selected 2D area

222 aaa

r p 2287.02

2=

2

13

2

1072.12.17

mmnm

atoms


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ex: linear density of Al in [110]direction

a = 0.405 nm

Linear Atomic Density

Linear Density of Atoms LD =

a

[110]

Unit length of direction vector

Number of atoms diameter

# atoms

length

13.5 nma 2

2LD -


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Polymorphism Some materials can have more than one crystal structure. This is

referred to as polymorphism (or allotropy ). Temperature and pressure cause metals to change crystal structure.

titanium, -Ti

carbondiamond, graphite BCC

FCC

BCC

1538C

1394C

912C

-Fe

-Fe

-Fe

liquid

iron system


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Polymorphic Forms of CarbonDiamond tetrahedral bonding of

carbon hardest material known very high thermal

conductivity large single crystals gem stones

small crystals used togrind/cut other materials

diamond thin films hard surface coatings

used for cutting tools,medical devices, etc.


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Polymorphic Forms of Carbon (cont)Graphite layered structure parallel hexagonal arrays of

carbon atoms

weak van der Waals forces between layers planes slide easily over one another -- good

lubricant


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Polymorphic Forms of CarbonFullerenes and Nanotubes

Fullerenes spherical cluster of 60 carbon atoms, C 60 Like a soccer ball Carbon nanotubes sheet of graphite rolled into a tube

Ends capped with fullerene hemispheres

C l B ildi Bl k


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Properties of crystalline materials often related to their crystal structure. Single crystal the repeated unit cell is perfect throughout the

material. Some engineering applications require single crystals:-- diamond single

crystals for abrasives or turbine blade.

However, most crystalline solids are made up of small crystals ( grains ). Thisis called polycrystalline materials.


Crystals as Building Blocks


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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 typically range from 1 nm to 2 cm.

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

1 mm

Polycrystals

Isotropic

Anisotropic-Properties varywith direction:anisotropic .

-Properties do

not vary withdirection:isotropic .


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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 .

-If grains are textured ,anisotropic.

200 mm

Single vs PolycrystalsE (diagonal) = 273 GPa

E (edge) = 125 GPa

C i C l S


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Ceramic Crystal Structures

Oxide structures oxygen anions larger than metal cations close packed oxygen in a lattice (usually FCC) cations fit into interstitial sites among oxygen ions


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Factors that Determine Crystal Structure1. Relative sizes of ions Formation of stable structures:

--maximize the # of oppositely charged ion neighbors.


- -

- -+

unstable

- -

- -+

stable

- -

- -+

stable2. Maintenance of

Charge Neutrality :--Net charge in ceramic

should be zero.--

CaF 2 : Ca2+

cationF-

F-

anions+


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X-Ray Diffraction

Diffraction gratings must have spacings comparable tothe wavelength of diffracted radiation. Cant resolve spacings Spacing is the distance between parallel planes of

atoms.


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X-Ray Diffraction Crystal planes of target metal act as mirrors reflecting

X-ray beam. If rays leaving a set of planes

are out of phase (as in case ofarbitrary angle of incidence)

no reinforced beam isproduced.

If rays leaving are in phase,

reinforced beams areproduced.

3-36 Figure 3.28


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X-Rays to Determine Crystal Structure

X-ray

intensity(fromdetector)

q

qc

d n

2 sin qc

Measurement of

critical angle, qc,allows computation ofplanar spacing, d .

Incoming X-rays diffract from crystal planes.

reflections mustbe in phase fora detectable signal

spacingbetweenplanes

d q q

extradistance

travelledby wave 2


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X-Ray Diffraction Pattern

Adapted from Fig. 3.20, Callister 5e.

(110)

(200)

(211)

z

x

y a b

c

Diffraction angle 2 q

Diffraction pattern for polycrystalline -iron (BCC)

I n t e

n s

i t y

( r e

l a t i

v e

)

z

x

y a b

c z

x

y a b

c


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Quartz is crystallineSiO 2:

Basic Unit: Glass is noncrystalline ( amorphous) Fused silica is SiO2 to which no

impurities have been added Other common glasses contain

impurity ions such as Na +, Ca 2+,

Al3+

, and B3+

(soda glass)Adapted from Fig. 3.41,Callister & Rethwisch 3e .

Glass Structure

Si0 4 tetrahedron4-

Si 4+

O2-

Si4+Na +

O2-


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Layered Silicates Layered silicates (e.g., clays, mica, talc)

SiO 4 tetrahedra connectedtogether to form 2-D plane

A net negative charge is associatedwith each (Si 2O5)2- unit

Negative charge balanced byadjacent plane rich in positivelycharged cations

Adapted from Fig.3.13, Callister &Rethwisch 3e.


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Atoms may assemble intocrystalline or amorphous structures.

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

SUMMARY

Common metallic crystal structures areFCC , BCC , and HCP.Coordination number and atomic packing factor are the samefor both FCC and HCP crystal structures.

Crystallographic points , directions and planes are specified interms of indexing schemes. Crystallographic directions andplanes are related to atomic linear densities and planar densities .

Ceramic crystal structures are based on:-- maintaining charge neutrality

-- cation-anion radii ratios.

Interatomic bonding in ceramics is ionic and/or covalent.


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Some materials can have more than one crystalstructure. This is referred to as polymorphism (orallotropy ).

SUMMARY

Materials can be single crystals or polycrystalline .Material properties generally vary with single crystalorientation (i.e., they are anisotropic ), but are generallynon-directional (i.e., they are isotropic ) in polycrystalswith randomly oriented grains.

X-ray diffraction is used for crystal structure andinterplanar spacing determinations.

crystal and amorphous structure in materials

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