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SOLID STATE PHYSICS

发布时间:2013-12-14 14:03:43  

SOLID STATE PHYSICS

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?AIM OF SOLID STATE PHYSICSWHAT IS SOLID STATE PHYSICS AND WHY DO IT?CONTENTREFERENCES

?Solidstatephysics(SSP)explainsthepropertiesof

solidmaterialsasfoundonearth.

?Thepropertiesareexpectedtofollowfrom

Schr?dinger’seqn.foracollectionofatomicnucleiandelectronsinteractingwithelectrostaticforces.Thefundamentallawsgoverningthebehaviourofsolidsareknownandwelltested.?

EP364 SOLID STATE PHYSICS

INTRODUCTION

?Wewilldealwithcrystallinesolids,thatissolids

withanatomicstructurebasedonaregularrepeatedpattern.

Manyimportantsolidsarecrystalline.

Moreprogresshasbeenmadeinunderstandingthebehaviourofcrystallinesolidsthanthatofnon-crystallinematerialssincethecalculationareeasierincrystallinematerials.

EP364 SOLID STATE PHYSICS

INTRODUCTION??

?Solidstatephysics,alsoknownascondensedmatter

physics,isthestudyofthebehaviourofatomswhentheyareplacedincloseproximitytooneanother.

Infact,condensedmatterphysicsisamuchbettername,sincemanyoftheconceptsrelevanttosolidsarealsoappliedtoliquids,forexample.?

EP364 SOLID STATE PHYSICS

INTRODUCTION

?Understandingtheelectricalpropertiesofsolidsis

rightattheheartofand?Theentirecomputerandelectronicsindustryrelies

ontuningofaspecialclassofmaterial,thesemiconductor,whichliesrightattheSolidstatephysicsprovideabackgroundtounderstandwhatgoesoninsemiconductors.

EP364 SOLID STATE PHYSICS

INTRODUCTION

?Newtechnologyforthefuturewillinevitablyinvolvedevelopingandunderstandingnewclassesofmaterials.Bytheendofthiscoursewewillseewhythisisanon-trivialtask.

?So,SSPistheappliedphysicsassociatedwith

EP364 SOLID STATE PHYSICS

INTRODUCTION

?Howcanthisbe?Afterall,theyeachcontainasystemofatomsandespeciallyelectronsofsimilardensity.Andtheplotthickens:graphiteisametal,diamondisaninsulatorandbuckminster-fullereneisasuperconductor.

They are all just carbon!

EP364 SOLID STATE PHYSICS

INTRODUCTION

Chapter 1.Chapter 2.Chapter 3.Chapter 4.Chapter 5.Crystal StructureX-ray CrystallographyInteratomic ForcesCrystal DynamicsFree ElectronTheory

EP364 SOLID STATE PHYSICS INTRODUCTION?????

?Elementary Crystallography?

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?Solid materials (crystalline, polycrystalline, amorphous)CrystallographyCrystal LatticeCrystal StructureTypes of LatticesUnit CellDirections-Planes-Miller Indices in Cubic Unit Cell?

?Typical Crystal Structures Elements of Symmetry

EP364 SOLID STATE PHYSICS

INTRODUCTION(3D–14 Bravais Lattices and the Seven Crystal System)

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?

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?X-ray Diffraction?Bragg equation X-ray diffraction methods?Laue Method?Rotating Crystal Method?Powder MethodNeutron & electron diffraction

EP364 SOLID STATE PHYSICS

INTRODUCTION

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?Energies of Interactions Between AtomsIonic bonding

?NaCl

Comparision of ionic and covalent bonding?Covalent bonding?

?

?

?Metallic bondingVan der waals bondingHydrogen bonding

EP364 SOLID STATE PHYSICS

INTRODUCTION

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?Sound waveLattice vibrations of 1D cystal?Chain of identical atoms?Chain of two types of atomsPhonons ?Heat Capacity?Density of States?Thermal ConductionEnergy of harmonic oscillator?Thermal energy & Lattice Vibrations?Heat Capacity of Lattice vibrations

EP364 SOLID STATE PHYSICS

INTRODUCTION

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?

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?Free electron modelHeat capacity of free electron gasFermi function, Fermi energyFermi dirac distribution functionTransport properties of conduction electrons

EP364 SOLID STATE PHYSICS

INTRODUCTION

?

?Corebook:Solidstatephysics,J.R.HookandH.E.Hall,Secondedition(Wiley)Otherbooksatasimilarlevel:

Solidstatephysics,Kittel(Wiley)

Solidstatephysics,Blakemore(Cambridge)

Fundamentalsofsolidstatephysics,Christman(Wiley)

?Moreadvanced:Solidstatephysics,AshcroftandMermin

EP364 SOLID STATE PHYSICS

INTRODUCTION

CHAPTER 1CRYSTAL STRUCTURE

Elementary CrystallographyTypical Crystal StructuresElements Of Symmetry

By the end of this section you should:?

?

?be able to identify a unit cellin a symmetrical patternknow that there are 7 possible unit cell shapesbe able to define cubic, tetragonal,

orthorhombic and hexagonal unit cell shapes

Crystal Structure18

Crystal Structure

19

?

?Gases have atoms or molecules that do not bond to one another in a range of pressure, temperature and volume. These molecules haven’t any particular order

and move freely within a container.

20

??Similartogases,liquidshaven’tanyatomic/molecularorderandtheyassumetheshapeofthecontainers.Applyinglowlevelsofthermalenergycaneasily

breaktheexistingweakbonds.

Liquidcrystalshavemobile

molecules,butatypeoflong

rangeordercanexist;the

moleculeshaveapermanent

dipole.Applyinganelectricfield

rotatesthedipoleandestablishes

orderwithinthecollectionofCrystal Structuremolecules.21

????Solidsconsistofatomsormoleculesexecutingthermalmotionaboutanequilibriumpositionfixedatapointinspace.Solidscantaketheformofcrystalline,polycrstalline,oramorphousmaterials.Solids(atagiventemperature,pressure,andvolume)havestrongerbondsbetweenmoleculesandatomsthanliquids.Solidsrequiremoreenergytobreakthebonds.

Crystal Structure22

?Single crsytal, polycrystalline, and amorphous, are the

three general types of solids.

Each type is characterized by the size of ordered region within the material.

An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.??

Crystal Structure24

?

whichtheatomsormoleculesarearrangedinadefinite,repeatingpatterninthreedimension.Singlecrystals,ideallyhaveahighdegreeoforder,or

regulargeometricperiodicity,throughouttheentirevolumeofthematerial.

Crystal Structure25

?hasanatomicstructurethatrepeats

periodicallyacrossitswholevolume.Evenatinfinitelengthscales,eachatomisrelatedtoeveryotherequivalentatominthestructurebytranslationalsymmetry

Single Pyrite

Crystal

Amorphous

Solid

Single Crystal

Crystal Structure26

?

?

?

?

?isamaterialmadeupofanaggregateofmanysmallsinglecrystals(alsocalledcrystallitesorgrains).Polycrystallinematerialhaveahighdegreeoforderovermanyatomicormoleculardimensions.Theseorderedregions,orsinglecrytalregions,varyinsizeandorientationwrtoneanother.Theseregionsarecalledasgrains(domain)andareseparatedfromoneanotherbygrainboundaries.Theatomicordercanvaryfromonedomaintothenext.Thegrainsareusually100nm-100micronsindiameter.Polycrystalswithgrains

thatare<10nmindiameterarecallednanocrystalline

PolycrystallinePyrite form(Grain)

Crystal Structure27

??

?

?

?Amorphous(Non-crystalline)Solidiscomposedofrandomlyorientatedatoms,ions,ormoleculesthatdonotformdefinedpatternsorlatticestructures.Amorphousmaterialshaveorderonlywithinafewatomicormoleculardimensions.Amorphousmaterialsdonothaveanylong-rangeorder,buttheyhavevaryingdegreesofshort-rangeorder.Examplestoamorphousmaterialsincludeamorphoussilicon,plastics,andglasses.Amorphoussiliconcanbeusedinsolarcellsandthinfilmtransistors.

Crystal Structure28

?Strictlyspeaking,onecannotprepareaperfectcrystal.For

example,eventhesurfaceofacrystalisakindofimperfectionbecausetheperiodicityisinterruptedthere.?Anotherexampleconcernsthethermalvibrationsoftheatoms

aroundtheirequilibriumpositionsforanytemperatureT>0°K.

?Asathirdexample,actual

crystalalwayscontainssome

foreignatoms,i.e.,impurities.

Theseimpuritiesspoilsthe

perfectcrystalstructure.

29Crystal Structure

What is crystallography?

The branch of science that deals with the geometric description

of crystals and their internal arrangement.

Crystal Structure30

Crystallography is essential for solid state physics?

?

?Symmetryofacrystalcanhaveaprofoundinfluenceonitsproperties.Anycrystalstructureshouldbespecifiedcompletely,conciselyandunambiguously.Structuresshouldbeclassifiedintodifferenttypes

accordingtothesymmetriestheypossess.

Crystal Structure31

?Abasicknowledgeofcrystallographyisessentialforsolidstatephysicists;

?

?tospecifyanycrystalstructureandtoclassifythesolidsintodifferenttypesaccordingtothesymmetriestheypossess.

?Symmetryofacrystalcanhaveaprofoundinfluenceonitsproperties.

Wewillconcerninthiscoursewithsolidswithsimplestructures.

Crystal Structure32?

Whatiscrystal(space)lattice?

Incrystallography,onlythegeometricalpropertiesofthecrystalareofinterest,thereforeonereplaceseachatombyageometricalpointlocatedattheequilibriumpositionofthatatom.

PlatinumPlatinum surface

(scanningtunnelingmicroscope)

Crystal StructureCrystal lattice and structure of Platinum33

y

Aninfinitearrayofpointsinspace,Eachpointhasidenticalsurroundingstoallothers.

Arraysarearrangedexactlyinaperiodicmanner.

Crystal Structure34???

?Crystalstructurecanbeobtainedbyattachingatoms,groupsofatomsormoleculeswhicharecalledbasis(motif)tothelatticesidesofthelatticepoint.

CrystalStructure=CrystalLattice+Basis

Crystal Structure35

?A group of atoms which describe crystal structure

yEa)Situationofatomsattheb)CrystallatticeobtainedbycornersofregularhexagonsCrystal Structureidentifyingalltheatomsin(a)37

??

?Don'tmixupatomslatticepointsLatticepointsinfinitesimalpointsspaceLatticepointsdonecessarilylieatcentreofatomswithareinnotthe

+Basis

38CrystalStructure=CrystalLatticeCrystal Structure

1)

Bravaislatticeisaninfinitearrayofdiscretepointswith

anarrangementandorientationthatappearsexactlythesame,fromwhicheverofthepointsthearrayisviewed.Latticeisinvariantunderatranslation.

Nb film

Crystal Structure40

2) Non-Bravais Lattice

Not only the arrangement but also the orientationmust

appearexactly the same from every point in a bravais lattice.??

?Theredsidehasaneighbourtoitsimmediateleft,theblueoneinsteadhasaneighbourtoitsright.Red(andblue)sidesareequivalentandhavethesameappearanceRedandbluesidesarenotequivalent.Sameappearancecanbeobtainedrotatingblueside180o.

Honeycomb

Crystal Structure41

P

Point D(n1, n2) = (0,2)

Point F (n1, n2) = (0,-1)Athatspacelatticeisasetofpointssuchlatticeatranslationbyavector;fromanypointintheRn=n1a+n2blocatesi.e.asapointanwithexactlytheequivalentpoint,TheP.Thisistranslationalsameenvironmentsymmetry.vectorsvectorsa,bareknownaslatticeintegersand(n1,n2)isapairwhosevaluesof

Crystal Structure42

Theandformasetoflatticevectorsforthelattice.Thechoiceoflatticevectorsisnotunique.ThusonecouldequallywelltaketheCrystal Structure43??

Anidealthreedimensionalcrystalisdescribedby3fundamentaltranslationvectorsa,bandc.Ifthereisalatticepointrepresentedbythepositionvectorr,thereisthenalsoalatticepointrepresentedbythepositionvectorwhereu,vandwarearbitraryintegers.

r’= r + ua + vb + wc(1)

Crystal Structure44

Crystal Structure45

?Thesmallestcomponentofthecrystal(groupofatoms,ionsormolecules),whichwhenstackedtogetherwithpuretranslationalrepetitionreproducesthewholecrystal.bCrystal Structure

46

?Thesmallestcomponentofthecrystal(groupofatoms,ionsormolecules),whichwithpuretranslationalreproducesthewholecrystal.

The choice of

unit cell

is not unique.

baCrystal Structure

47

We define ; these are points with identical environments

Crystal Structure48

Crystal Structure49

Crystal Structure50

Crystal Structure51

Crystal Structure52

Crystal Structure53

Crystal Structure54

Crystal Structure55

Crystal Structure

56

Crystal Structure57

Aunitcelljustfillsspacewhen

translatedthroughasubsetof

Bravaislatticevectors.

Theconventionalunitcellis

chosentobelargerthanthe

primitivecell,butwiththefull

symmetryoftheBravaislattice.

Thesizeoftheconventionalcell

isgivenbythelatticeconstant

a.

Crystal Structure59???

Crystal Structure

60

Primitive Translation Vectors:

Body centered cubic (bcc): conventional ?primitive cell Fractional coordinatesof lattice points in conventional cell: 000,100, 010, 001, 110,101, 011, 111, ? ? ? Simple cubic (sc): primitive cell=conventional cell Fractionalcoordinatesoflatticepoints:000, 100, 010, 001, 110,101, 011, 111

Crystal Structure62

Body centered cubic (bcc): primitive(rombohedron) ?conventional cellFractional coordinates: 000, 100, 101, 110, 110,101, 011, 211, 200 Face centered cubic (fcc): conventional??primitive cell Fractional coordinates: 000,100, 010, 001, 110,101, 011,111, ? ? 0, ? 0 ?, 0 ? ? ,?1 ? , 1 ? ? , ? ? 1

Crystal Structure63

points of primitive cell

Hexagonal close packed cell (hcp):

conventional?primitive cell

Fractional coordinates:

100, 010, 110, 101,011, 111,000, 001

Crystal Structure64

Theunitcelland,consequently,theentirelattice,isuniquelydeterminedbythesixlatticeconstants:a,b,c,α,βandγ.Onlyunit1/8ofeachlatticetothatcellcell.canactuallybepointassignedinaEachassociatedunitcell

point.within8thex1/8figure=1canlattice

beCrystal Structure65???

Aprimitiveunitcellismadeofprimitive

translationvectorsa

nocell1,a

of2,anda

thatthereissmallervolume3such

thatcanbeusedasabuildingblockfor

crystalstructures.

Aprimitiveunitcellwillfillspaceby

repetitionofsuitablecrystaltranslation

vectors.Thisdefinedbytheparallelpiped

a1,a2anda3.Thevolumeofaprimitive

unitcellcanbefoundby

V=a1.(a2xa3)(vectorproducts)

Crystal StructureCubic cell volume = a366???

?

?The primitive unit cellmust have only one lattice point.There can be fora1

P = Primitive Unit Cell

NP = Non-Primitive Unit Cell

Crystal Structure67

AsimplywaytofindtheprimitivecellwhichiscalledWigner-Seitzcellcanbedoneasfollows;

1.Choosealatticepoint.

2.Drawlinestoconnecttheselatticepointtoitsneighbours.

3.Atthemid-pointandnormaltotheselinesdrawnewlines.

The volume enclosed is called as a

Wigner-Seitz cell.

Crystal Structure68

Crystal Structure69

Crystal Structure70

Weasoriginanchooseorigin,onesaylatticethepointpointO.onChoicethelineofeverylatticeiscompletelypointisidentical.arbitrary,sinceThenjoiningwe

pointT.OThistochoose

vectoranypointthelatticevector

canbeonwrittentheline,as;say

R = n1a + n2b + n3c

Tolatticedistinguishalatticefromsquarethe[...]tripleisused.[nina

1n2n3]

[n1n2n3]istheoftheCrystal StructureFig. Shows [111] direction71????

X = 1 , Y = ? , Z = 0X = ? , Y = ? , Z = 1[1 ? 0][2 1 0][1 1 2]

Crystal Structure

72

?direction[n1n2n3]dependontheorigin,negativedirectionscanbewrittenas

[123]Z direction-?R=n1a+n2b+n3cDirection must be

smallest integers.-Z direction

Crystal Structure73

[1 0 0]Crystal Structure74

We can move vector to the origin.X =-1 , Y = 1 , Z = -1/6Crystal Structure75

?

?Withinacrystallatticeitispossibletoidentifysetsofequallyspacedparallelplanes.Thesearecalledlatticeplanes.Inthefiguredensityoflatticepointsoneachplaneofasetis

thesameandalllatticepointsarecontainedoneachsetofplanes

.

The set of

planes in

2D lattice.b

Crystal Structure76

MillerIndicesareasymbolicvectorrepresentationfortheorientationofanatomicplaneinacrystallatticeandaredefinedasthewhichtheplanemakeswiththecrystallographicaxes.

To determine Miller indices of a plane, take the following steps;of the plane along each of the three of the intercepts Crystal Structure77

(1,0,0)Crystal Structure78

(0,1,0)

(1,0,0)Crystal Structure79

(0,0,1)

(0,1,0)(1,0,0)Crystal Structure80

(0,1,0)

(1/2, 0, 0)Crystal Structure81

Crystal Structure

82

Crystal Structure

83

Plane intercepts axes at 3a, 2b, 2c111Reciprocal numbers are: , , 3

22Indices of the plane (Miller): (2,3,3)Indices of the direction: [2,3,3]

(100)

84

(100)Crystal Structure

Crystal Structure86

?Sometimeswhentheunitcellhasrotationalsymmetry,

severalnonparallelplanesmaybeequivalentbyvirtueofthissymmetry,inwhichcaseitisconvenienttolumpalltheseplanesinthesameMillerIndices,butwithcurlybrackets.{100}?(100),(010),(001),(00),(00),(00)

{111}?(111),(11),(11),(11),(),(1),(1),(1)Thusindices{h,k,l}representalltheplanesequivalenttotheplane(hkl)throughrotationalsymmetry.

Crystal Structure87

3D –14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEM?Thereareonlysevendifferentshapesofunitcellwhichcanbestackedtogethertocompletelyfillallspace(in3dimensions)withoutoverlapping.Thisgivesthesevencrystalsystems,inwhichallcrystalstructurescanbeclassified.

CubicCrystalSystem(SC,BCC,FCC)

HexagonalCrystalSystem(S)

TriclinicCrystalSystem(S)

MonoclinicCrystalSystem(S,Base-C)

OrthorhombicCrystalSystem(S,Base-C,BC,FC)

TetragonalCrystalSystem(S,BC)

Trigonal(Rhombohedral)Crystal StructureCrystalSystem(S)???????88

?TheBravaislatticepoints

closesttoagivenpointarethenearestneighbours.

BecausetheBravaislatticeisperiodic,allpointshavethesamenumberofnearestneighboursorcoordinationnumber.Itisapropertyofthelattice.

Asimplecubichascoordinationnumber6;abody-centeredcubiclattice,8;andaface-centeredcubiclattice,12.

Crystal Structure90??

?

AtomicPackingFactor(APF)isdefinedasthevolumeofatomswithintheunitcelldividedbythevolumeoftheunitcell.

a-Simple Cubic (SC)

??

?SimpleCubichasonelatticepointsoitsprimitivecell.Intheunitcellontheleft,theatomsatthecornersarecutbecauseonlyaportion(inthiscase1/8)belongstothatcell.Therestoftheatombelongstoneighboringcells.Coordinatinationnumberofsimplecubicis6.

bc

a

Crystal Structure92

Crystal Structure93

Crystal Structure94

BCChastwolatticepointssoBCC

isanon-primitivecell.

BCChaseightnearestneighbors.

Eachatomisincontactwithits

neighborsonlyalongthebody-

diagonaldirections.

Manymetals(Fe,Li,Na..etc),

includingthealkalisandseveral

transitionelementschoosethe

BCCstructure.

Crystal Structureba95???

a = 4 R

3 APFBCC = Vatoms

V = 0.68

unit cell

2(0,433a)

Crystal Structure96

?

?

?Thereareatomsatthecornersoftheunitcellandatthecenterofeachface.Facecenteredcubichas4atomssoitsnonprimitivecell.Manyofcommonmetals(Cu,Ni,Pb..etc)crystallizeinFCCstructure.

Crystal Structure97

Crystal StructureAtoms are all same.98

4 RVatoms0,74a = APFBCC = = 0.68FCC3Vunit cell

4(0,353a)Crystal Structure99

the unit cellAtomsShared Between:Each atom counts:corner8 cells1/8

face centre2 cells1/2

body centre1 cell1

edge centre2cells1/2

lattice typecell contents

P1 [=8 x 1/8]

I2 [=(8 x 1/8) + (1 x 1)]F4 [=(8 x 1/8) + (6 x 1/2)]C 2 [=(8 x 1/8) + (2 x 1/2)]

Crystal Structure100

Crystal Structure

101

?Acrystalsysteminwhichthreeequalcoplanaraxesintersectatanangleof60,andaperpendiculartotheothers,isofa

differentlength.

Crystal Structure102

Crystal StructureAtoms are all same.103

?Triclinicmineralsaretheleastsymmetrical.Theirthreeaxesarealldifferentlengthsandnoneofthemareperpendiculartoeachother.Thesemineralsarethemostdifficultto

recognize.????? ?????90

oa ??b ??c?= ?= 90o, ? ??90o a ??b ?cCrystal Structure

?= ?= 90o, ? ??90o a ??b ??c,105

Orthorhombic (Simple) ?= ? = ?= 90o

a ??b ??cOrthorhombic (Base-Orthorhombic (BC)centred)?= ? = ?= 90o ?= ? = ?= 90oa ??b ??c

a ??b ??c

Crystal Structure?= ? = ?= 90o a ??b ??c106

Tetragonal (P)

?= ? = ?= 90o

a = b ??cTetragonal (BC) ?= ? = ?= 90o a = b ??c

Crystal Structure107

Rhombohedral (R) or Trigonal(S)

a = b = c, ?= ? = ????90o

Crystal Structure108

Sodium Chloride Structure Na+Cl-Cesium Chloride Structure Cs+Cl-Hexagonal Closed-Packed StructureDiamond Structure

Zinc Blende?????

Crystal Structure109

Sodiumchloridealso

crystallizesinacubiclattice,

butwithadifferentunitcell.

Sodiumchloridestructure

consistsofequalnumbersof

sodiumandchlorineions

placedatalternatepointsofa

simplecubiclattice.

Eachionhassixoftheother

kindofionsasitsnearest

neighbours.

Crystal Structure110???

?IfwetaketheNaClunitcellandremovealltheredClions,we

areleftwithonlytheblueNa.Ifwecomparethiswiththefcc/ccpunitcell,itisclearthattheyareidentical.Thus,theNaisinafcc

sublattice.

Crystal Structure112

Thisstructurecanbe

consideredasaface-

centered-cubicBravaislattice

withabasisconsistingofa

sodiumionat0andachlorine

ionatthecenterofthe

conventionalcell,

a/2(?x??y??z)

LiF,NaBr,KCl,LiI,etc

The lattice constants are in

the order of 4-7 angstroms.???

?Cesiumchloridecrystallizesinacubiclattice.Theunitcellmaybedepictedasshown.(Cs+isteal,Cl-isgold).

Cesiumchlorideconsistsofequalnumbersofcesiumandchlorineions,placedatthepointsofabody-centeredcubiclatticesothateachionhaseightoftheotherkindasitsnearestneighbors.

Crystal Structure114?

?ThetranslationalsymmetryofthisstructureisthatofthesimplecubicBravaislattice,andisdescribedasasimplecubiclatticewithabasisconsistingofacesiumionattheorigin0andachlorineionatthecubecenter

a/2(x?y?z)???

?CsBr,CsIcrystallizeinthisstructure.Thelatticeconstantsareintheorderof4angstroms.

8cell

Thisisanotherstructurethatis

common,particularlyinmetals.

Inadditiontothetwolayersof

atomswhichformthebaseand

theupperfaceofthehexagon,

thereisalsoaninterveninglayer

ofatomsarrangedsuchthat

eachoftheseatomsrestovera

depressionbetweenthreeatoms

inthebase.

Crystal Structure117?

Bravais Lattice : Hexagonal Lattice

He, Be, Mg, Hf, Re (Group II elements)ABABAB Type of Stackinga=b a=120, c=1.633a,basis : (0,0,0) (2/3a ,1/3a,1/2c)

Crystal Structure118

Close packAAAAABACBACBACBAACBACBACBACABACBACBACBASequence ABABAB..-hexagonal close packSequence ABCABCAB..-face centered cubic close pack

Crystal StructureAAABBAAABBAAASequence AAAA…-simple cubicSequence ABAB…-body centeredcubic119

?

?

?Thediamondlatticeisconsistoftwointerpenetratingfacecenteredbravaislattices.Thereareeightatominthestructureofdiamond.Each atom bonds covalently to 4 others equally spread about atom in 3d.

Crystal Structure121

?

?

?The coordination number of diamond structure is 4.The diamond lattice is not a Bravais lattice.Si, Ge and C crystallizes in diamond structure.

Crystal Structure123

?

?Zincblendehasequalnumbersofzincandsulfurionsdistributedonadiamondlatticesothateachhasfouroftheoppositekindasnearestneighbors.Thisstructureisanexampleofalatticewithabasis,whichmustsodescribedbothbecauseofthegeometricalpositionoftheionsandbecausetwotypesofionsoccur.AgI,GaAs,GaSb,InAs,

Zinc

Blende

is

the

name

given

to

the

mineral

ZnS.

It

has

a

cubicclosepacked(facecentred)arrayofSandtheZn(II)sitintetrahedral(1/2occupied)sitesinthelattice.

Crystal Structure126

?Eachoftheunitcellsofthe14Bravaislatticeshasone

ormoretypesofsymmetryproperties,suchasinversion,reflectionor

rotation,etc.

Crystal Structure127

128

?Acenterofsymmetry:Apointatthecenterofthemolecule.

(x,y,z)-->(-x,-y,-z)

Centerofinversioncanonlybeinamolecule.Itisnotnecessarytohaveanatominthecenter(benzene,ethane).Tetrahedral,triangles,pentagonsdon'thaveacenterofinversionsymmetry.AllBravaislatticesareinversionsymmetric.Mo(CO)6?

Crystal Structure129

?

?betweenand

Crystal Structure131

We can not find a lattice that goes into itself under other rotations?A single molecule can have any degree of rotational symmetry, but an infinite periodic lattice –can not.

Crystal Structure132

Crystal Structure

133

Crystal Structure

134

Crystal Structure

135

Can not be combined with translational periodicity!

Crystal Structure

136

?Keplerwonderedwhysnowflakeshave6corners,

never5or7.Byconsideringthepackingofpolygonsin2dimensions,demonstratewhypentagonsandheptagonsshouldn’toccur.

Crystal Structure

138

Crystal Structure139

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