Crystal Structure Slides

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Chapter 3 - 1

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

Chapter 3 - 2

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

-metals-many ceramics-some polymers

atoms have no periodic packingNoncrystalline materials...

-complex structures-rapid cooling

crystalline SiO2

noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.23(b),Callister & Rethwisch 8e.

Adapted from Fig. 3.23(a),Callister & Rethwisch 8e.

Materials and Packing

Si Oxygen

typical of:

occurs for:

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Chapter 3 - 3

Metallic Crystal Structures How can we stack metal atoms to minimize

empty space?2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

Chapter 3 -

Polycrystals

4

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Chapter 3 -

Crystal Structure

5

Chapter 3 -

Bravais Lattice

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Chapter 3 - 7

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

Coordination # = 6(# nearest neighbors)

Simple Cubic Structure (SC)

Click once on image to start animation(Courtesy P.M. Anderson)

Chapter 3 - 8

Body Centered Cubic Structure (BCC)

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

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Chapter 3 - 9

Face Centered Cubic Structure (FCC)

Chapter 3 - 10

A sites

B B

B

BB

B B

C sites

C C

CA BB sites

ABCABC... Stacking Sequence 2D Projection

FCC Unit Cell

FCC Stacking Sequence

B B

B

BB

B BB sites

C C

CAC C

CA

AB

C

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Chapter 3 - 11

ABAB... Stacking Sequence 3D Projection 2D Projection

Adapted from Fig. 3.3(a),Callister & Rethwisch 8e.

Hexagonal Close-Packed Structure (HCP)

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

Chapter 3 - 12

Theoretical Density,

where n = number of atoms/unit cellA = atomic weight VC = Volume of unit cell = a3 for cubicNA = Avogadros number

= 6.022 x 1023 atoms/mol

Density = =

VCNAn A =

CellUnitofVolumeTotalCellUnitinAtomsofMass

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Chapter 3 - 13

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

theoretical

a = 4R/ 3 = 0.2887 nm

actual

aR

= a3

52.002atoms

unit cellmolg

unit cellvolume atoms

mol

6.022x 1023

Theoretical Density,

= 7.18 g/cm3

= 7.19 g/cm3

Adapted from Fig. 3.2(a), Callister & Rethwisch 8e.

Chapter 3 - 14

Densities of Material Classesmetals > ceramics > polymers

Why?

Data from Table B.1, Callister & Rethwisch, 8e.

(g/

cm )3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

1

2

20

30Based on data in Table B1, Callister

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% volume fraction of aligned fibers

in an epoxy matrix).10

345

0.30.40.5

Magnesium

Aluminum

Steels

Titanium

Cu,NiTin, Zinc

Silver, Mo

TantalumGold, WPlatinum

GraphiteSiliconGlass -sodaConcreteSi nitrideDiamondAl oxide

Zirconia

HDPE, PSPP, LDPEPC

PTFE

PETPVCSilicone

Wood

AFRE*CFRE*GFRE*Glass fibers

Carbon fibersAramid fibers

Metals have... close-packing

(metallic bonding) often large atomic masses

Ceramics have... less dense packing often lighter elements

Polymers have... low packing density

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

Composites have... intermediate values

In general

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Chapter 3 - 15

Some engineering applications require single crystals:

Properties of crystalline materials often related to crystal structure.

(Courtesy P.M. Anderson)

-- Ex: Quartz fractures more easily along some crystal planes than others.

-- diamond singlecrystals for abrasives

-- turbine bladesFig. 8.33(c), Callister & Rethwisch 8e. (Fig. 8.33(c) courtesy of Pratt and Whitney).

(Courtesy Martin Deakins,GE Superabrasives, Worthington, OH. Used with permission.)

Crystals as Building Blocks

Chapter 3 - 16

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).

Adapted from Fig. K, color inset pages of Callister 5e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

Polycrystals

Isotropic

Anisotropic

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Chapter 3 - 17

Single Crystals-Properties vary withdirection: anisotropic.

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

Data from Table 3.3, Callister & Rethwisch 8e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Polycrystals-Properties may/may notvary with direction.

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

-If grains are textured,anisotropic.

200 m Adapted from Fig. 4.14(b), Callister & Rethwisch 8e.(Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

Single vs PolycrystalsE (diagonal) = 273 GPa

E (edge) = 125 GPa

Chapter 3 - 18

Polymorphism Two or more distinct crystal structures for the same

material (allotropy/polymorphism)

titanium, -Ti

carbondiamond, graphite

BCC

FCC

BCC

1538C

1394C

912C

-Fe

-Fe

-Fe

liquidiron system

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Chapter 3 - 19Fig. 3.4, Callister & Rethwisch 8e.

Crystal Systems

7 crystal systems

14 crystal lattices

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

a, b, and c are the lattice constants

Chapter 3 - 20

Point CoordinatesPoint coordinates for unit cell

center area/2, b/2, c/2

Point coordinates for unit cell corner are 111

Translation: integer multiple of lattice constants identical position in another unit cell

z

x

ya b

c

000

111

y

z

2c

b

b

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Chapter 3 - 21

Crystallographic Directions

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 values4. Enclose in square brackets, no commas

[uvw]

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

families of directions

z

x

Algorithm

where overbar represents a negative index

[111 ]=>

y

Chapter 3 - 22

ex: linear density of Al in [110] direction

a = 0.405 nm

Linear Density Linear Density of Atoms LD =

a

[110]Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD ==Adapted fromFig. 3.1(a),Callister & Rethwisch 8e.

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Chapter 3 - 23

HCP Crystallographic Directions

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

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]

[ 1120 ]ex: , , -1, 0 =>

Adapted from Fig. 3.8(a), Callister & Rethwisch 8e.

dashed red lines indicate projections onto a1 and a2 axes a1

a2

a3

-a32

a2

2a1

-

a3

a1

a2

z

Algorithm

Chapter 3 - 24

HCP Crystallographic Directions Hexagonal Crystals

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

=

=

=

'ww

t

v

u

)vu( +-)'u'v2(

31

-

)'v'u2(31

-=

]uvtw[]'w'v'u[

Fig. 3.8(a), Callister & Rethwisch 8e.

-

a3

a1

a2

z

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Chapter 3 - 25

Crystallographic Planes

Adapted from Fig. 3.10, Callister & Rethwisch 8e.

Chapter 3 - 26

Crystallographic Planes 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, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no

commas i.e., (hkl)

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Chapter 3 - 27

Crystallographic Planesz

x

ya b

c

4. Miller Indices (110)

example a b c z

x

ya b

c


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

1 1 03. Reduction 1 1 0

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

2 0 03. Reduction 2 0 0

example a b c

Chapter 3 - 28

Crystallographic Planesz

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

(001)(010),

Family of Planes {hkl}

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

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Chapter 3 - 29

Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used

example a1 a2 a3 c

4. Miller-Bravais Indices (1011)

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

1 0 -1-1

11

3. Reduction 1 0 -1 1a2

a3

a1

z

Adapted from Fig. 3.8(b), Callister & Rethwisch 8e.

Chapter 3 - 30

Crystallographic Planes We want to examine the atomic packing of

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 - 31

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

(100)

Radius of iron R = 0.1241 nm

R3

34a =

Adapted from Fig. 3.2(c), Callister & Rethwisch 8e.

2D repeat unit

= Planar Density =a2

1atoms

2D repeat unit=

nm2atoms12.1

m2atoms

= 1.2 x 10191

2R

334area

2D repeat unit

Chapter 3 - 32

Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell

333 22

R3

16R3

42a3ah2area =

===

atoms in planeatoms above planeatoms below plane

ah23

=

a2

1= =

nm2atoms7.0

m2atoms0.70 x 1019

3 2R3

16Planar Density =

atoms2D repeat unit

area

2D repeat unit

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Chapter 3 - 33

X-Ray Diffraction

Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.

Cant resolve spacings < Spacing is the distance between parallel planes of

atoms.

Chapter 3 - 34

X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

c

d = n2 sin c

Measurement of critical angle, c, allows computation of planar spacing, d.

Incoming X-rays diffract from crystal planes.


reflections must be in phase for a detectable signal

spacing between planes

d

extra distance travelled by wave 2

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Chapter 3 - 35

X-Ray Diffraction Pattern

Adapted from Fig. 3.22, Callister 8e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2

Diffraction pattern for polycrystalline -iron (BCC)

Inte

nsi

ty (re

lativ

e)z

x

ya b

c

z

x

ya b

c

Chapter 3 -

Octahedral and Tetrahedral Voids

36

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Chapter 3 -

Voids in FCC

37

Chapter 3 -

Octahedral Voids in HCP

38

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Chapter 3 -

Tetrahedral Voids in HCP

39

Chapter 3 -

Voids in BCC

40

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Chapter 3 - 41

Adapted from Fig. 2.7, Callister & Rethwisch 8e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 byCornell University.)

Degree of ionic character may be large or small:

Atomic Bonding in Ceramics

SiC: smallCaF2: large

Chapter 3 - 42

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.--Reflected in chemical

formula:

CaF2: Ca2+

cationF-

F-

anions+

AmXpm, p values to achieve charge neutrality

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Chapter 3 - 43

Coordination # and Ionic Radii

Centre of Triangle

Centre of Tetrahedron

Centre of Octahedron

Centre of Cube

Chapter 3 - 44

Rock Salt StructureSame concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure

rNa = 0.102 nm

rNa/rCl = 0.564

cations (Na+) prefer octahedral sites


rCl = 0.181 nm

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Chapter 3 - 45

AX Crystal Structures

939.0181.0170.0

Cl

Cs==

+

r

r


Cesium Chloride structure:

Since 0.732 < 0.939 < 1.0, cubic sites preferred

So each Cs+ has 8 neighbor Cl-

Chapter 3 -

Zinc Blende Structure

46

402.0184.0074.0Zn

==

Sr

r

Tetrahedral sites

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Chapter 3 - 47

AmXp Crystal Structures

Calcium Fluorite (CaF2) Cations in cubic sites

UO2, ThO2, ZrO2, CeO2


Fluorite structure 752.0133.0100.0

Cl

Ca==

+

r

r

Chapter 3 - 48

AmBnXp Crystal Structures


Perovskite structure

Ex: complex oxideBarium Titanate- BaTiO3

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Crystal Structure Slides