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CHAPTER3:
CRYSTAL
STRUCTURES&PROPERTIES
ISSUESTOADDRESS...
Howdoatomsassembleintosolidstructures?(fornow,focusonmetals)
Howdoesthedensityofamaterialdependonitsstructure?
Whendomaterialpropertiesvarywiththesample(i.e.,part)orientation?
1
EnergyandPacking
?
Nondense,randompacking
Energy
typicalneighborbondlength
typicalneighborbondenergy
r
?
Dense,orderedpacking
Energy
typicalneighborbondlength
r
typicalneighbor
bondenergy
2
Dense,orderedpackedstructurestendtohavelowerenergies.
MATERIALSANDPACKING
Crystallinematerials...Long-rangeOrder
atomspackinperiodic,3Darrays
typicalof:
-metals
-manyceramics
-somepolymers
crystallineSiO2
Si
Oxygen
Noncrystallinematerials...
atomshavenoperiodicpacking
occursfor:
-complexstructures
-rapidcooling
"Amorphous"=Noncrystalline
noncrystallineSiO2
3
Lattice+Basis=Crystal
Lattice點(diǎn)陣
Structure
Networkinwhichcrystalstructuresareembedded
Describedby
Specifyingtheatomspositionsinarepeatingunitcell
Referringtheatomstothepointsofintersectionoflines
Basis結(jié)構(gòu)基元
–“Groupofthings”locatedatthelatticepoint
UnitCell晶胞
–Smallestregionthatcompletelydescribesthepattern
Eachlatticepointhasidenticalsurroundingenvironment
CrystalSystems
Unitcell:
smallestrepetitivevolumewhich
containsthecompletelatticepatternofacrystal.
7
crystalsystems
theunitcellgeometry.
crystallattices
thegeometryandatomicarrangements
14
a,b,andcarethelatticeconstants
Fig.3.4,Callister&Rethwisch8e.
5
SevenDistinctsystems
6
Cubic立方
Tetragonal長(zhǎng)方
Hexagonal六方
orthorhombic正交
Rhombohedral斜方
Monoclinic單斜晶系
Triclinic三斜晶系
MetallicCrystalStructures
Mostelementalmetals(about90%)crystallizeintothreedenselypackedcrystalstructures:
?
Body-centeredcubic(BCC
Face-centeredcubic(FCC
體心立方)
面心立方)
–Hexagonalclose-packed(HCP密排六方)
7
BODYCENTEREDCUBICSTRUCTURE(BCC)
?Atomstouchalongbodydiagonal
--Note:Allatomsareidentical;thecenteratomisshadeddifferentlyonlyforeaseofviewing.
Ex:Fe,W,Mo,Nb,Ta,V,Cr
8
Coordination#=
Thenumberofthenearest–neighbouratoms
8
ATOMICPACKINGFACTOR:
BCC
Unitcellcontains:1+8x1/8
=2atoms/unitcellClose-packeddirections:
length=4R
=
3a
atoms
volume
Closepackeddirectionsarebodydiagonals.
unitcell
APF=
atom
volume
a3
unitcell
?
APFforabody-centeredcubicstructure=0.68
9
2
4(3a/4)3
3
APF=Volumeofatomsinunitcell*
Volumeofunitcell
*assumehardspheres
FACECENTEREDCUBICSTRUCTURE(FCC)
Atomstouchalongfacediagonal
Ex:Cu,Ag,Au,Al,Ni,Pt,Fe,Co…
?
Coordination#=12
Thenumberofthenearest
–neighbouratoms
4atoms/unitcell:6facex1/2+8cornersx1/8
10
FACECENTEREDCUBICSTRUCTURE(FCC)
AtomPackingFactor
#ofatoms/cell=8x(1/8)+6x(1/2)=4
=4/3R3
Vcell
=a3
;
Vatom
Relation:
4Ra
2
atoms
volume
unitcell
APF=
atom
volume
a3
unitcell
=0.74
11
4
4(2a/4)3
3
HEXAGONAL
CLOSE-PACKED
STRUCTURE(HCP)
Hexagonal
Ideal:c/a=1.633
a=b≠c
°
°
?
Ex:Mg,Zn,Be,Ti,Co
12
HEXAGONALCLOSE-PACKEDSTRUCTURE(HCP)
12
?
Coordination
#
=
Unitcellcontains:3+1/6X12+1/2X2
=6atoms/cell
APF=0.74
Thisisaclose
Packedstructure
13
FCCStackingSequence
ABCABC...StackingSequence2DProjection
?
?
B
B
C
B
A
B
B
Asites
Bsites
Csites
C
B
C
B
A
?
FCCUnitCell
BC
14
HexagonalClose-PackedStructure(HCP)
ABAB...StackingSequence
?
?
3DProjection
?
2DProjection
Asites
Top
layer
c
Middlelayer
Bsites
Asites
AdaptedfromFig.3.3(a),
Callister&Rethwisch8e.
Bottomlayer
a
15
TheoreticalDensity,
MassofAtomsinUnitCell
TotalVolumeofUnitCell
Density==
nAVCNA
=
where
n=numberofatoms/unitcell
A=atomicweight
VC=Volumeofunitcell=a3forcubic
NA=Avogadro’snumber
=6.022x1023atoms/mol
16
TheoreticalDensity,
?
Ex:Cr(BCC)
A=52.00g/mol
R=0.125nm
n=2atoms/unitcell
R
a=4R/3=0.2887nm
a
Adaptedfrom
Fig.3.2(a),Callister&Rethwisch8e.
atoms
g
unitcell
=volume
theoretical
=7.18g/cm3
=7.19g/cm3
mol
actual
atoms
mol
unitcell
17
a3
6.022x1023
2
52.00
DensitiesofMaterialClasses
Ingeneral
Graphite/Ceramics/Semicond
Metals/Alloys
Composites/fibers
metals>ceramics
>polymers
30
Polymers
Why?
Metalshave...
close-packing
BasedondatainTableB1,Callister
*GFRE,CFRE,&AFREareGlass,Carbon,&AramidFiber-ReinforcedEpoxycomposites(valuesbasedon60%volumefractionofalignedfibers
inanepoxymatrix).
20
10
(metallicbonding)
oftenlargeatomicmasses
Ceramicshave...
lessdensepacking
oftenlighterelements
Polymershave...
lowpackingdensity(oftenamorphous)
lighterelements(C,H,O)
Compositeshave...
intermediatevalues
5
4
3
2
1
0.5
0.4
0.3
DatafromTableB.1,Callister&Rethwisch,8e.
18
(g/cm3)
PlatinumGold,WTantalum
Silver,Mo
Cu,NiSteelsTin,Zinc
Titanium
AluminumMagnesium
Zirconia
AloxideDiamondSinitride
Glass-sodaConcreteSilicon
Graphite
GlassfibersGFRE*
CarbonfibersCFRE*
AramidfibersAFRE*
Wood
PTFE
SiliconePVCPETPC
HDPE,PSPP,LDPE
CrystalsasBuildingBlocks
Someengineeringapplicationsrequiresinglecrystals:
?
--turbineblades
Fig.8.33(c),Callister&Rethwisch8e.(Fig.8.33(c)courtesyofPrattandWhitney).
--diamondsinglecrystalsforabrasives
(CourtesyMartinDeakins,GESuperabrasives,Worthington,OH.Usedwithpermission.)
?
Propertiesofcrystallinematerialsoftenrelatedtocrystalstructure.
--Ex:
Quartzfracturesmoreeasily
alongsomecrystalplanesthanothers.
(CourtesyP.M.Anderson)
19
Polycrystals
Mostengineeringmaterialsarepolycrystals.
Anisotropic
?
AdaptedfromFig.K,colorinsetpagesofCallister5e.
(Fig.KiscourtesyofPaulE.Danielson,TeledyneWahChangAlbany)
1mm
Isotropic
?
?
?
Nb-Hf-Wplatewithanelectronbeamweld.Each"grain"isasinglecrystal.
Ifgrainsarerandomlyoriented,
overallcomponentpropertiesarenotdirectional.
Grainsizestypicallyrangefrom1nmto2cm
(i.e.,fromafewtomillionsofatomiclayers).
?
20
SINGLEVSPOLYCRYSTALS
?
SingleCrystals
-Propertiesvarywithdirection:anisotropic各向異性.
-Example:themodulus
ofelasticity(E)inBCCiron:
Polycrystals
-Propertiesmay/maynotvarywithdirection.
E(diagonal)=273GPa
atomsalongtheedgearemoreseparatedthanalongthefacediagonal.
E(edge)=125GPa
?
200m
-Ifgrainsarerandomlyoriented:isotropic.各向同性
(Epolyiron
=210GPa)
-Ifgrainsaretextured,anisotropic.
21
Polymorphism
Twoormoredistinctcrystalstructuresforthesamematerial(allotropy/polymorphism)
ironsystem
?
titanium
,-Ti
liquid
1538oC
-Fe1394oC
-Fe
912oC
-Fe
BCC
carbon
diamond,graphite
FCC
BCC
22
CrystalPositions,Directions,andPlanes
?Specifyaparticularpointwithinaunitcell,acrystallographicdirection,orsomecrystallographicplaneofatoms.
?Threenumbersorindicesareusedtodesignatepointlocations,directions,andplanes.
?Right-handedcoordinatesystemsituatedatoneofthe
cornersandcoincidingwiththeunitcelledges
23
Crystallographic
Algorithm
Directions
z
Vectorrepositioned(ifnecessary)topassthroughorigin.
Readoffprojectionsintermsofunitcelldimensionsa,b,andc
Adjusttosmallestintegervalues
Encloseinsquarebrackets,nocommas
[uvw]
y
x
ex:1,0,?
-1,1,1
=>
=>
2,0,1
[111]
=>
[201]
whereoverbarrepresentsanegativeindex
24
Crystallographic
directions
[001]
[111]
[011]
<100>cubeedges
[101]
<011>facediagonals
<111>cubediagonals
[010]
[110]
[100]
25
LinearDensity
Numberofatoms
LinearDensityofAtomsLD=
Unitlengthofdirectionvector
[110]
ex:
lineardensityofAlin[110]
direction
a=0.405nm
#atoms
LD
length
a
AdaptedfromFig.3.1(a),
Callister&Rethwisch8e.
26
2a
3.5nm1
2
Crystallographic
Planes
?
MillerIndices:
Reciprocalsofthe(three)axial
interceptsforaplane,clearedoffractions&commonmultiples.AllparallelplaneshavesameMillerindices.
Algorithm
?
1.
Readoffinterceptsofplanewithaxesin
termsofa,b,c
Takereciprocalsofintercepts
Reducetosmallestintegervalues
Encloseinparentheses,nocommasi.e.,(hkl)
27
Crystallographic
Planes
z
c
c
1/
0
0
example
a
1
1/1
1
1
(110)
b
1
1/1
1
1
1.
2.
Intercepts
Reciprocals
y
3.
4.
Reduction
MillerIndices
a
b
x
z
b
1/
0
0
c
1/
0
0
example
a
1/2
1/?
2
2
(100)
1.
2.
Intercepts
Reciprocals
c
3.
4.
Reduction
MillerIndices
y
a
b
x
28
Crystallographic
Planes
z
c
a
1/2
1/?
2
6
(634)
b
1
1/1
1
3
c
3/4
1/?
4/3
4
x
example
InterceptsReciprocals
1.
2.
b
y
a
3.
4.
Reduction
MillerIndices
FamilyofPlanes
{hkl}
Ex:
{100}=(100),(010),(001),(100),(010),(001)
29
FamiliesofPlanes
Z
?Groupsofplanesthatareequivalent
(110)
?Denotedby{hkl}
(101)
(011)
(011)
(101)
Y
(110)
X
{110}
PlanarDensity
PlanarDensityofAtomsPD
Atomscenteredontheplane
=
area
Examples:
Draw(100)and(111)crystallographicplanesforFe(BCC).
Calculatetheplanardensityforeachoftheseplanes.
31
PlanarDensityof(100)Iron
Solution:
AtT<912oCironhastheBCCstructure.
2Drepeatunit
a4 3R
3
(100)
RadiusofironR=0.1241nm
AdaptedfromFig.3.2(c),Callister&Re
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