固体物理+晶体结构.ppt

SOLIDSTATE,CrystalsStructures1.UnitCell,Crystalstructurebasicsunitcellssymmetrylattices,Someimportantcrystalstructuresandpropertiesclosepackedstructuresoctahedralandtetrahedralholesbasicstructuresferroelectricity,Objectives,Bytheendofthissectionyoushould:beabletoidentifyaunitcellinasymmetricalpatternknowthatthereare7possibleunitcellshapesbeabletodefinecubic,tetragonal,orthorhombicandhexagonalunitcellshapes,WhySolids?,mostelementssolidatroomtemperatureatomsinfixedposition“simple”case-crystallinesolidCrystalStructure,Whystudycrystalstructures?descriptionofsolidcomparisonwithothersimilarmaterials-classificationcorrelationwithphysicalproperties,Earlyideas,Crystalsaresolid-butsolidsarenotnecessarilycrystallineCrystalshavesymmetry(Kepler)andlongrangeorderSpheresandsmallshapescanbepackedtoproducesregularshapes(Hooke,Hauy),?,Groupdiscussion,Keplerwonderedwhysnowflakeshave6corners,never5or7.Byconsideringthepackingofpolygonsin2dimensions,demonstratewhypentagonsandheptagonsshouldntoccur.,Emptyspacenotallowed,Definitions1.Theunitcell,“Thesmallestrepeatunitofacrystalstructure,in3D,whichshowsthefullsymmetryofthestructure”,Theunitcellisaboxwith:3sides-a,b,c3angles-,Sevenunitcellshapes,Cubica=b=c=90Tetragonala=bc=90Orthorhombicabc=90Monoclinicabc=90,90Triclinicabc90Hexagonala=bc=90,=120Rhombohedrala=b=c=90,2Dexample-rocksalt(sodiumchloride,NaCl),Wedefinelatticepoints;thesearepointswithidenticalenvironments,Choiceoforiginisarbitrary-latticepointsneednotbeatoms-butunitcellsizeshouldalwaysbethesame.,Thisisalsoaunitcell-itdoesntmatterifyoustartfromNaorCl,-orifyoudontstartfromanatom,ThisisNOTaunitcelleventhoughtheyareallthesame-emptyspaceisnotallowed!,In2D,thisISaunitcellIn3D,itisNOT,AllM.C.Escherworks(c)CordonArt-Baarn-theNetherlands.Allrightsreserved.,Summary,Unitcellsmustlinkup-cannothavegapsbetweenadjacentcellsAllunitcellsmustbeidenticalUnitcellsmustshowthefullsymmetryofthestructurenextsection,CrystalsStructures2:Symmetry,Bytheendofthissectionyoushould:beabletorecognizerotationalsymmetryandmirrorplanesknowaboutcentresofsymmetrybeabletoidentifythebasicsymmetryelementsincubic,tetragonalandorthorhombicshapesunderstandcentringandrecognizeface-centred,body-centredandprimitiveunitcells.Knowsomesimplestructures(Fe,Cu,NaCl,CsCl),Symmetry“Somethingpossessessymmetryifitlooksthesamefrom1orientation”,RotationalsymmetryCanrotateby120abouttheC-Clbondandthemoleculelooksidentical-theHatomsareindistinguishable,Thisiscalledarotationaxis-inparticular,athreefoldrotationaxis,asrotateby120(=360/3)toreachanidenticalconfiguration,Ingeneral:n-foldrotationaxis=rotationby(360/n),Wetalkaboutthesymmetryoperation(rotation)aboutasymmetryelement(rotationaxis),?Thinkofexamplesforn=2,3,4,5,6,MirrorPlaneSymmetry“Ariseswhenonehalfofanobjectisthemirrorimageoftheotherhalf”,Thismoleculehastwomirrorplanes,Oneishorizontal,intheplaneofthepaper-bisectstheH-C-HbondsOtherisvertical,perpendiculartotheplaneofthepaperandbisectstheCl-C-Clbonds,CentreofSymmetry“presentifyoucandrawastraightlinefromanypoint,throughthecentre,toanequaldistancetheotherside,andarriveatanidenticalpoint”(phew!),CentreofsymmetryatS,Nocentreofsymmetry,AllM.C.Escherworks(c)CordonArt-Baarn-theNetherlands.Allrightsreserved.,Unitcellsymmetries-cubic,4foldrotationaxes(passingthroughpairsofoppositefacecentres,paralleltocellaxes)TOTAL=3,Unitcellsymmetries-cubic,4foldrotationaxesTOTAL=3,3-foldrotationaxes(passingthroughcubebodydiagonals)TOTAL=4,Unitcellsymmetries-cubic,4foldrotationaxesTOTAL=3,3-foldrotationaxesTOTAL=4,2-foldrotationaxes(passingthroughdiagonaledgecentres)TOTAL=6,Mirrorplanes-cubic,3equivalentplanesinacube,6equivalentplanesinacube,TetragonalUnitCella=bc;=90,ca,b,elongated/squashedcube,Reductioninsymmetry,CubicTetragonalThree4-axesOne4-axisTwo2-axesFour3-axesNo3-axesSix2-axesTwo2-axesNinemirrorsFivemirrorsSeetutorial,Example,CaC2-hasarocksalt-likestructurebutwithnon-sphericalcarbides,2-,C,C,Carbideionsarealignedparalleltocca,btetragonalsymmetry,CubicUnitCell,a=b=c,=90,a,c,b,Manyexamplesofcubicunitcells:e.g.NaCl,CsCl,ZnS,CaF2,BaTiO3,Allhavedifferentarrangementsofatomswithinthecell.Sotodescribeacrystalstructureweneedtoknow:theunitcellshapeanddimensionstheatomiccoordinatesinsidethecell,PrimitiveandCentredLattices,Coppermetalisface-centredcubicIdenticalatomsatcornersandatfacecentresLatticetypeFalsoAg,Au,Al,Ni.,PrimitiveandCentredLattices,-Ironisbody-centredcubicIdenticalatomsatcornersandbodycentre(nothingatfacecentres)LatticetypeIAlsoNb,Ta,Ba,Mo.,PrimitiveandCentredLattices,CaesiumChloride(CsCl)isprimitivecubicDifferentatomsatcornersandbodycentre.NOTbodycentred,therefore.LatticetypePAlsoCuZn,CsBr,LiAg,PrimitiveandCentredLattices,SodiumChloride(NaCl)-NaismuchsmallerthanCsFaceCentredCubicRocksaltstructureLatticetypeFAlsoNaFl,KBr,MgO.,Anothertypeofcentring,Sidecentredunitcell,Notation:A-centredifatominbcplaneB-centredifatominacplaneC-centredifatominabplane,UnitcellcontentsCountingthenumberofatomswithintheunitcell,Manyatomsaresharedbetweenunitcells,AtomsSharedBetween:Eachatomcounts:corner8cells1/8facecentre2cells1/2bodycentre1cell1edgecentre4cells1/4,latticetypecellcontentsP1=8x1/8I2=(8x1/8)+(1x1)F4=(8x1/8)+(6x1/2)C?,e.g.NaClNaatcorners:(81/8)=1Naatfacecentres(61/2)=3Clatedgecentres(121/4)=3Clatbodycentre=1Unitcellcontentsare4(Na+Cl-),2=(8x1/8)+(2x1/2),Summary,CrystalshavesymmetryEachunitcellshapehasitsownessentialsymmetryInadditiontothebasicprimitivelattice,centredlatticesalsoexist.Examplesarebodycentred(I)andfacecentred(F),CrystalsStructures3:MillerIndex,Bytheendofthissectionyoushould:understandtheconceptofplanesincrystalsknowthatplanesareidentifiedbytheirMillerIndexandtheirseparation,dbeabletocalculateMillerIndicesforplanesknowthed-spacingequationfororthogonalcrystalsbeabletousethed-spacingequation,LatticePlanesandMillerIndices,Imaginerepresentingacrystalstructureonagrid(lattice)whichisa3Darrayofpoints(latticepoints).Canimaginedividingthegridintosetsof“planes”indifferentorientations,AllplanesinasetareidenticalTheplanesare“imaginary”Theperpendiculardistancebetweenpairsofadjacentplanesisthed-spacingNeedtolabelplanestobeabletoidentifythem,Findinterceptsona,b,c:1/4,2/3,1/2Takereciprocals4,3/2,2Multiplyuptointegers:(834)ifnecessary,Exercise-WhatistheMillerindexoftheplanebelow?,Findinterceptsona,b,c:1/2,1,1/2Takereciprocals2,1,2Multiplyuptointegers:(2,1,2),Planeperpendiculartoycutsat,1,(010)plane,Generallabelis(hkl)whichintersectsata/h,b/k,c/l(hkl)istheMILLERINDEXofthatplane.,Thisdiagonalcutsat1,1,(110)plane,NBanindex0meansthattheplaneisparalleltothataxis,UsingthesamesetofaxesdrawtheplaneswiththefollowingMillerindices:,(001),(111),d-spacingformula,Fororthogonalcrystalsystems(i.e.=90):-,Forcubiccrystals(specialcaseoforthogonal)a=b=c:-,e.g.for(100)d=a(200)d=a/2(110)d=a/2etc.,Atetragonalcrystalhasa=4.7,c=3.4.Calculatetheseparationofthe:(100)(001)(111)planes,Acubiccrystalhasa=5.2(=0.52nm).Calculatethed-spacingofthe(110)plane,4.73.42.4,Summary,WecanimagineplaneswithinacrystalEachsetofplanesisuniquelyidentifiedbyitsMillerindex(hkl)Wecancalculatetheseparation,d,foreachsetofplanes(hkl),CrystalsStructures4:ClosedPackedStructures,Bytheendofthissectionyoushould:understandtheconceptofclosepackingknowthedifferencebetweenhexagonalandcubicclosepackingknowthedifferenttypesofinterstitialsitesinaclosepackedstructurerecogniseanddemonstratethatcubicclosepackingisequivalenttoafacecentredcubicunitcell,InorganicCrystalStructures,AllcrystalstructuresmaybedescribedintermsoftheunitcellandatomiccoordinatesofthecontentsManyinorganicstructuresmaybedescribedasarraysofspacefillingpolyhedra-tetrahedra,octahedra,etc.Manystructures-ionic,metallic,covalent-maybedescribedasclosepackedstructures,Closepackedstructures-metals,Mostefficientwayofpackingequalsizedspheres.In2D,haveclosepackedlayers,Coordinationnumber(CN)=6.Thisisthemaximumpossiblefor2Dpacking.,Canstackclosepacked(c.p.)togive3Dstructures.,Twomainstackingsequences:,Ifwestartwithonecplayer,twopossiblewaysofaddingasecondlayer(canhaveoneorother,butnotamixture):,Twomainstackingsequences:,Ifwestartwithonecplayer,twopossiblewaysofaddingasecondlayer(canhaveoneorother,butnotamixture):,LetsassumethesecondlayerisB(red).Whataboutthethirdlayer?Twopossibilities:(1)CanhaveApositionagain(blue).ThisleadstotheregularsequenceABABABA.Hexagonalclosepacking(hcp)(2)CanhavelayerinCposition,followedbythesamerepeat,togiveABCABCABCCubicclosepacking(ccp),Hexagonalclosepacked,Cubicclosepacked,Nomatterwhattypeofpacking,thecoordinationnumberofeachequalsizesphereisalways12,Wewillseethatothercoordinationnumbersarepossiblefornon-equalsizespheres,ThesewillbestudiedfurtherinthelabThereasonswhyaparticularmetalprefersaparticularstructurearestillnotwellunderstood,Metalsusuallyhaveoneofthreestructuretypes:ccp(=fcc,seenextslide),hcporbodycentredcubic(bcc),Closepackedionicstructures,Ionicstructures-cations(+ne)andanions(-ne)Inmanyionicstructures,theanions,whicharelargerthanthecations,formac.p.arrayandthecationsoccupyinterstitialholeswithinthisanionarray.Twomaintypesofinterstitialsite:TETRAHEDRAL:CN=4OCTAHEDRAL:CN=6,TetrahedralT+,TetrahedralT-,OctahedralO,ccp=fcc?,Buildupccplayers(ABCpacking),Addconstructionlines-canseefccunitcell,c.playersareorientedperpendiculartothebodydiagonalofthecube-,Hexagonalclosepackedstructures(hcp),hcp,bcc,Summary,ClosepackingoccursinavarietyofmetalsWecanenvisagelayers,positionsdenotedbyA,BandC,sothathexagonalclosepackingisrepresentedbyABABAandcubicclosepackingisrepresentedbyABCABCAccpisequivalenttoface-centredcubicSmallionscanoccupyinterstitialsitesinaclosepackedstructure-bothtetrahedral(4)andoctahedral(6)sitesexist,CrystalsStructures-5:InterstitialSites,Bytheendofthissectionyoushould:knowhowatompositionsaredenotedbyfractionalcoordinatesbeabletocalculatebondlengthsforoctahedralandtetrahedralsitesinacubebeabletocalculatethesizeofinterstitialsitesinacube,Objectives,Bytheendofthissectionyoushould:knowhowatompositionsaredenotedbyfractionalcoordinatesbeabletocalculatebondlengthsforoctahedralandtetrahedralsitesinacubebeabletocalculatethesizeofinterstitialsitesinacube,Fractionalcoordinates,Usedtolocateatomswithinunitcell,0,0,0,0,0,0,Note:atomsareincontactalongfacediagonals(closepacked),1.2.3.4.,OctahedralSites,Coordinate,Distance=a/2,Coordinate0,0=1,0Distance=a/2,Inafacecentredcubicanionarray,cationoctahedralsitesat:,00,00,00,Tetrahedralsites,Relationofatetrahedrontoacube:,i.e.acubewithalternatecornersmissingandthetetrahedralsiteatthebodycentre,Candividethef.c.c.unitcellinto8minicubesbybisectingeachedge;inthecentreofeachminicubeisatetrahedralsite,So8tetrahedralsitesinafcc,Bondlengthsimportantdimensionsinacube,Facediagonal,fd(fd)=(a2+a2)=a2,Bodydiagonal,bd(bd)=(2a2+a2)=a3,Octahedral:halfcelledge,a/2Tetrahedral:quarterofbodydiagonal,1/4of3aAnion-anion:halffacediagonal,1/2of2a,Bondlengths:,Sizesofinterstitials,fcc/ccp,Spheresareincontactalongfacediagonalsoctahedralsite,bonddistance=a/2radiusofoctahedralsite=(a/2)-rtetrahedralsite,bonddistance=a3/4radiusoftetrahedralsite=(a3/4)-r,Summaryf.c.c./c.c.panions,4anionsperunitcellat:0000004octahedralsitesat:0000004tetrahedralT+sitesat:4tetrahedralT-sitesat:,AvarietyofdifferentstructuresformbyoccupyingT+T-andOsitestodifferingamounts:theycanbeempty,partfullorfull.Wewilllookatsomeoftheselater.Canalsovarytheanionstackingsequence-ccporhcp,Summary,ByunderstandingthebasicgeometryofacubeanduseofPythagorastheorem,wecancalculatetheoctahedralbondlength(a/2)andtetrahedralbondlength(3a/4)inafccstructureAsaconsequence,wecancalculatetheradiusoftheoctahedralinterstice=(a/2)-randofthetetrahedralinterstice=(a3/4)-r,whereristheradiusofthepackingion.,CrystalsStructures6:SomeExamples,Bytheendofthissectionyoushould:beabletodrawasimplecrystalstructureprojectionappreciatethatavarietyofimportantcrystalstructurescanbedescribedbyclose-packingbeabletocompareandcontrastsimilarstructures,CrystalStructureProjections,AnotherwayofdescribingstructuresMakedrawingsofastructureprojecteddownoneaxisontoaunitcellface,b,a,ORIGIN,Example1-Rocksalt,Example2-ZincBlende(Sphalerite),Example3-Fluoritestructure,DescriptionsofStructures,Withccpanionarray:Rocksalt,NaClOoccupiedZincBlende,ZnST+(orT-)occupied*Antifluorite,Na2OT+andT-occupiedWithhcpanionarray:Wurtzite,ZnST+(orT-)occupiedWithccpcationarray:Fluorite,ZrO2T+andT-occupied*majorstructureinsemiconductor/microelectronicsindustry,RocksaltstructureMostalkalihalides(LiF,KCl,etc.)Manydivalentmetalchalcogenides(CaO,BaS,etc.)Somecarbides,nitrides(TiC,LaN,etc.)SphaleriteorZincBlendeStructureManychalcogenides(ZnSe,HgS,etc.)Manypnictides(BN,GaAs,etc.)Somehalides(CuX)Diamond(C,Si)Majorstructureinthesemiconductor/microelectronicsindustryFluorite/AntifluoritestructureSomedivalenthalides(CaF2)Sometetravalentoxides(PbO2)Somemonovalentchalcogenides(Li2O,Na2S),Sphalerite(ZnS)vsDiamondstructure,Ballandstickshowsusthe4-foldcoordinationinbothstructures,Lookingattetrahedrainthestructurehelpsusseethe“diamondshape”,Fluoritestructure,Canbethoughtofasa3Darrayofalternatingemptyandoccupiedcubes,CadmiumChloride,CdCl2structure,ccpClions,halfofOoccupiedbyCd,T+T-sitesempty.Layeredstructure,LayersofCdionssandwichedbetweenchloridelayers.AdjacentsandwichesarelinkedbyCl-ClvanderWaalsbonds.,Inthecloselyrelatedcadmiumiodidestructure(CdI2),layersaresimilarbutwithhcp(ABABAB)packingofiodideions.Thesestructuresarecommonindivalenttransitionmetalhalides(generallynotfluorides,whicharetooionicandadoptalternativestructures)Bondingmusthaveconsiderablecovalentcharacter-ifweconsidercoordinationofCl-threenearestneighbourCdionstooneside,withtwelvenext-nearest-neighbourClions.Theanti-CdCl2structureoccursinCs2O,NickelArsenide(NiAs)structure,h.c.p.analogueofrocksaltstructureh.c.p.AswithoctahedralNi,cpointingtowardsus,cpointingupwards,Inthec-direction,theNi-Nidistanceisrathershort.Overlapof3dorbitalsgivesrisetometallicbonding.TheNiAsstructureisacommonstructureinmetalliccompoundsmadefrom(a)transitionmetalswith(b)heavyp-blockelementssuchasAs,Sb,Bi,S,Se.,CoordinationofAsisalso6butasatrigonalprism:,Summary(seelab/webpages),SummaryofAXstructures,wurtziteZnSCN=4sphalerite,NaCl,NiAsCN=6,CsClCN=8,Generaltrendistogethighercoordinationnumberswithlarger(heavier)cations.ThisisseenalsowithAX2structures,SummaryofAX2structures,SiO2,BeF2silicastructureCN=4:2,TiO2,MgF2rutilestructureCN=6:3,CdCl2,CdI2layerstructureCN=6:3,Note:rutilestructurewillbeintroducedinthelab,PbO2,CaF2fluoritestructureCN=8:4,Compare:1)Be,Mg,Cafluorides2)Si,Ti,Pbdioxides,CrystalsStructures7:Understanding,Bytheendofthissectionyoushould:understandtheconceptoftheradiusofanatomorionknowthetrendsinionicradiuswithcoordinationnumber,oxidationstate,groupknowabouttheradiusratioandbeabletocalculatethisforoctahedraland8-foldcoordination,Ionicradiiandbonddistances,Ionicradiicannotbe“measured”-estimatedfromtrendsinknownstructures(reference-Shannon,ActaCryst.(1976)A32751),Oxideion:r0takenas1.26,Refs:Krugetal.Zeit.PhysChem.Frankfurt436(1955)Krebs,FundamentalsofInorganicCrystalChemistry,(1968),Notes:IonradiiforgivenelementincreasewithCN,Notes:IonradiiforgivenelementincreasewithCNIonradiiforgivenelementdecreasewithincreasingoxidationstate/positivecharge,Notes:IonradiiforgivenelementincreasewithCNIonradiiforgivenelementdecreasewithincreasingoxidationstate/positivechargeRadiiincreasegoingdownagroup,Anionsoftenbiggerthancations,Radiusratiorules,Rationalisationforoctahedralcoordination:R=radiusoflargeion,r=radiusofsmallion,Ifr/R0.414,theanionsarepushedapartIfr/R0.414,coordinationchanges:,Asimplepredictiontool,butbeware-itdoesntalwayswork!,CoordinationMinimumr/RLinear,2-Trigonal,30.155Tetrahedral,40.225Octahedral,60.414Cubic,80.732Closepacked,121.000,Radiusratiorules,Rationalisationfor8-foldcoordination:,Radiusratiorules,Rationalisationfor8-foldcoordination:,Unitcelledgea=2RAtomstouchalongdiagonal(ifsmallionfitsperfectlyintospace)soa3=2(R+r)Divide:3=(R+r)/RMultiplyout3R=R+rR(3-1)=rr/R=3-1=0.732,Otherwaysofclassifyingstructures,1)StructureFieldMapse.g.forAxByOzcompounds,plotradiusofAagainstradiusofBandnotetrendsofstructureasrAandrBchange.2)Mooser-PearsonplotsFocusesonthecovalentcharacterofbonds.Plotofdifferenceinelectronegativityversusaverageprincipalquantumnumberofatomsinvolved.,Summary,IonradiiforgivenelementincreasewithincreasingCNandwithdecreasingoxidationstateIonicradiiincreasegoingdownagroupItispossibletocalculatetheradiusratioswhichgiveanindicationofthelikelycoordinationofagivenion,CrystalsStructures8:Perovskite,Bytheendofthissectionyoushould:beabletoidentifyanddrawtheperovskitestructureunderstandhowtheperovskitestructurecanbecomepolarisableknowthebasicpropertiesofbariumtitanateknowthebasicpropertiesofYBa2Cu3O7understandhowpropertiesaremodifiedbyappropriatesubstitutions,Perovskite-anInorganicChameleon,ABX3-threecompositionalvariables,A,BandX,CaTiO3-dielectricBaTiO3-ferroelectricPb(Mg1/3Nb2/3)O3-relaxorferroelectricPb(Zr1-xTix)O3-piezoelectric(Ba1-xLax)TiO3-semiconductor(Y1/3Ba2/3)CuO3-x-superconductor,NaxWO3-mixedconductor;electrochromicSrCeO3-H-protonicconductorRECoO3-x-mixedconductor(Li0.5-3xLa0.5+x)TiO3-lithiumionconductorLaMnO3-x-Giantmagneto-resistance,PerovskiteStructure,ABO3e.g.KNbO3SrTiO3LaMnO3,SrTiO3cubic,a=3.91,InSrTiO3,Ti-O=a/2=1.955Sr-O=a2/2=2.765,CNofA=12,CNofB=6,OR,Thefractionalcoordinatesforcubicperovskiteare:A=(,)A=(0,0,0)B=(0,0,0)ORB=(,)X=(,0,0)(0,0)(0,0,)X=(,0)(,0,)(0,)Drawoneoftheseasaprojection.,InSrTiO3,Ti-O1.95atypicalbondlengthforTi-O;stableasacubicstructure,InBaTiO3,Ti-Oisstretched,2.0Toolongforastablestructure.Tidisplacesoffitscentralpositiontowardsoneoxygensquarepyramidalcoordination,larger,Thiscreatesanetdipolemoment:,Displacementby5-10%Ti-Obondlength,Randomdipoleorientations,paraelectric,Aligneddipo