Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • ENGR-45_Lec-03_Crystal_Structure.ppt1

Bruce Mayer, PE Engineering-45: Materials of Engineering

Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engineering 45

CrystallineCrystallineMicroStructMicroStruct

ureure



Learning GoalsLearning Goals

Learn How atoms assemble into solid structures• Use metals as Prototypical Example

Determine Relationship Between Material density and material MicroStructure

Understand how material properties vary with the sample (i.e., part) orientation



Properties of Solid MaterialsProperties of Solid Materials

Mechanical: Characteristics of materials displayed when Forces and/or Moments are applied to them.

Physical: Characteristics of materials that relate to the interaction of materials with various forms of Energy.

Chemical: Material characteristics that relate to the e− structure of a material.

Dimensional: Size, shape, and finish



Material PropertiesMaterial Properties Chemical Physical Mechanical Dimensional

Composition Melting Point Tensile properties Standard Shapes

Microstructure Thermal Toughness Standard Sizes

Phases Magnetic Ductility Surface Texture

Grain Size Electrical Fatigue Stability

Corrosion Optical Hardness Mfg. Tolerances

Crystallinity Acoustic Creep

Molecular Wt Gravimetric Compression

Flammability



Energy and Atomic PackingEnergy and Atomic Packing NonDense, Random Packing

Dense Regular Packing

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

Regular Structures Tend to have LOWER Energy → Energetically Favored



Materials and Atomic-PackingMaterials and Atomic-Packing

Si Oxygen

crystalline SiO2

noncrystalline SiO2

CRYSTALLINE Materials• atoms pack in periodic, 3D arrays

• typical of: Metals, many Ceramics, and some Polymers

NONcrystalline Materials• atoms have no periodic packing

• occurs for:– Complex structures

– Rapid Cooling

"Amorphous" Noncrystalline



Crystal BasicsCrystal Basics Crystalline Material

• atomic arrangement in solid:– periodic and repeating

3D array

– Long Range Order

lattice

Unit Cell Smallest Repeating Entity Within a Lattice• Geometry

– Lattice Constants: a, b, c

– interaxial angles:



Crystal StructureCrystal Structure

Crystal Structure geometry + atom-positions• i.e.; the spatial arrangement of Atoms,

Ions, or Molecules

Some Types

polymer

Crystal Type Typical Metal Typical CeramicsFace Centered Cubic (FCC) Cu, Al NaCl

Body Centered Cubic (CCC) Cr, W CsCl

Hexagonal Close Packed (HCP Ti, Zn ZnSB



Example Crystal SystemsExample Crystal Systems Tab 3.2 In Text

Shows the 7 Most Common Crystal Systems• Some Examples

Cubic• a = b = c = = = 90°

Hexagonal• a = b c = = 90°, =

120°



Simple Cubic Stucture (SC)Simple Cubic Stucture (SC) Rare due to poor

packing (only Po has this structure)

Close-packed directions are cube edges.

Coordination No. = 6• CoOrd No. (CN) = the

No. of Nearest Neighbors

Ro

tate



Polonium, Z = 84, SC StructurePolonium, Z = 84, SC Structure HardSphere Model

a (pm) b (pm) c (pm) 335.9 335.9 335.9 90 90 90

Lattice Constants InterAxial 's

Reduced Sphere Mod

Note: 100 PicoMeters = 1 Å http://www.webelements.com/webelements/index.html

http://www.webelements.com/webelements/elements/text/Po/key.html

http://www.webelements.com/webelements/elements/text/Po/xtal.html



Atomic Packing Factor, APFAtomic Packing Factor, APF Assuming HardSphere Model

APF For Simple Cubic Structure = 52%

Cell Unit of Volume TOTAL

Cell Unit aIn ATOMS of VolumeAPF

close-packed directions

a

R=0.5a

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

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume



Body Centered Cubic (BCC)Body Centered Cubic (BCC) HardSphere Model

Atoms per Unit Cell = 1 + (8 x 1/8) = 2

CoOrd No. = 8• 8 Atoms Touch the

“Center” Atom Rotate



Atomic Packing Factor: BCC Atomic Packing Factor: BCC Find Radius, r, in

terms of Lattice Const, a

Thus r in Terms of a

Atoms Touch On Cube Diagonal, L• L = a3 = L = 4r

aarra 433.04

343

And Vsphere = (4/3)R3

So the BCC APF

cellma

atmacellat

APF33

3

3

43

34

2

%02.68BCCAPF



Face Centered Cubic (FCC)Face Centered Cubic (FCC) HardSphere Model

Atoms per Unit Cell = 6x½ + (8 x 1/8) = 4

CoOrd No. = 12• CN = 4top + 4bot + 4midRotate



Atomic Packing Factor: FCC Atomic Packing Factor: FCC Find Radius, r, in

terms of Lattice Const, a

Thus r in Terms of a

Atoms Touch on Face Diagonal, f• f = a2 = 4r

aarra 3536.022

142

And Vsphere = (4/3)R3

So the FCC APF

cellma

atmacellat

APF33

3

3

22

134

4

%05.74FCCAPF



FCC Stacking SequenceFCC Stacking Sequence

An ABCABC... Stacking Sequence• The 2D

projection

A sites

B B

B

BB

B B

C sites

C C

CA

B

B sites B B

B

BB

B B

B sitesC C

CA

C C

CA

The FCCUnit Cell

AB

C



Hexagonal Close Packed (HCP) Hexagonal Close Packed (HCP) Built Up in an A-B

Stacking Pattern Exhibits NonCubic

Symmetry on a & c axes• c:a Ratio 1.633

Atoms per Cell = 6• (12 x 1/6)corners +

(2 x ½)top/bot + 3mid = 6

APF and CN are the SAME as the FCC Structure• APF = 74.05%

– Close Packeda

c



HCP CoOrdination NumberHCP CoOrdination Number For 3 Stacking

Planes Below, Consider the CENTER Atom

Observe The Center Atom’s Nearest Neighbors• 6 Surrounding in the

Center Plane

• 3 Touching From Below

• 3 Touching From Above

Thus

12363 HCPCN



Structure of Compounds Structure of Compounds Compounds: Often

have similar close-packed structures

NaCl Structure• Ionic Radii

– Na+ = 116 pm

– Cl– = 167 pm

Expand Na+ ions to Reveal Close-Packed X-tal Structure



Theoretical Density, Theoretical Density, Atomic Radii for Crystals are Measured by

X-Ray Diffraction • Can use The Data From XRD Measurements to

Calc Density for Crystals

On the Macro Scale Densities are Calc’d by Weight & Measure of Chunks of Crystals

n AVcNA

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

Volume/unit cell

(cm3/unit cell)Avogadro's number (6.023 x 1023 atoms/mol)

Fcn of Ratom



Theoretical Density Example Theoretical Density Example For Copper

• Ratom = 128 pm

• Acu = 63.54 g/mol

• Xtal Structure = FCC

Recall From FCC APF Calc

FCC Cu has 4 atoms per Unit Cell; so

And The VC = a3

3330

233

kg/m8893g/pm10893.8

10023.6pm128216

54.634

AC

Cu

NV

nA

macro = 8940 kg•m-3

0.53% HIGHER than Theoretical Value

RaaR 2222

1

333 22822 RRa


Bruce Mayer, PE Engineering-45: Materials of Engineering 15

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 Some Elements at 20 °CCharacteristics of Some Elements at 20 °C



16

(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-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

Graphite Silicon

Glass -soda Concrete

Si nitride Diamond Al oxide

Zirconia

HDPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE *

CFRE *

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Why? Metals have... • close-packing (metallic bonding) • large atomic mass Ceramics have... • less dense packing (covalent bonding) • often lighter elements Polymers have... • poor packing (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values

Data from Table B1, Callister 6e.

Densities Of Material ClassesDensities Of Material Classesmetals > ceramics > polymers



SINGLE Crystal Applications SINGLE Crystal Applications Most Crystalline

Materials Are Composed of many Small Crystals• i.e., they are

POLYcrystalline and exhibit a GRAIN Structure

Grain Structure Introduces Weakness into the Material

But Making LARGE Single Crystals is Very Difficult; i.e., Expensive

Single Xtal Examples• SemiConductor

wafers

• Turbine Blades



1-Material; Several Crystal Types1-Material; Several Crystal Types Polymorphism More Than One

Crystal Structure• Often Found in Compounds

Allotropy Polymorphism in ELEMENTAL Solids

Examples

• Carbon Allotropes– Graphite

– Diamond

– Bucky Balls/Tubes

• Iron Allotropes– BCC Ferrite (RT)

– FCC Austenite (> 912 °C)

– BCC Delta (~TMelt)



Cabon Cabon Allotropes Allotropes Diamond - tetrahedral,

covalent bonds, Single-Element Form; the ZincBlende Crystal structure

Graphite – Layers of Hexagonally Bonded C-atoms

C60 Fullerenes



WhiteBoard WorkWhiteBoard Work Example

similar to P3.15• Uranium

• Orthorhombic

• Lattice Constants– a = 286 pm

– b = 587 pm

– c = 495 pm

• ratom = 138.5 pm

= 19050 kg/m3

• AU = 238.03 g/mol

→ FIND APF

SR Case



Uran

ium

Un

it Cell (B

CR

)U

raniu

m U

nit C

ell (BC

R)



Uran

ium

Un

it Cell (B

C)

Uran

ium

Un

it Cell (B

C)



Uran

ium

Un

it Cell (S

R)

Uran

ium

Un

it Cell (S

R)



All Done for TodayAll Done for Today

Uranium hasthe Highest ZOf Any Natural

Element



P3-15P3-15

Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

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