南京大学-晶体生长课件-chapter-7-晶体结构

第七章晶体结构及其表征,CrystalStructuresandTheirCharacterizations,Lattice,Basis,andtheUnitCellCommonCrystalStructuresMillerIndicesforCrystalDirectionsandPlanesTheReciprocalLatticeMethodsforCrystalStructuralCharacterization,HowdoatomsARRANGEthemselvestoformsolids?,AtomsinaCrystal,Lattice,Basis,andUnitCell,Anidealcrystallinesolidisaninfiniterepetitionofidenticalstructuralunitsinspace.Therepeatedunitmaybeasingleatomoragroupofatoms.,Animportantconcept:,crystalstructure=lattice（格点）+basis（基元）,primitiveCell:simplestcell,containonelatticepointNotnecessaryhavethecrystalsymmetry,unitcell:thesmallestcomponentofthecrystal,whichwhenstackedtogetherwithpuretranslationalrepetitionreproducesthewholecrystal,lattice:aperiodicarrayofpointsinspace.Theenvironmentsurroundingeachlatticepointisidentical.,basis:theatomorgroupofatoms“attached”toeachlatticepointinordergeneratethecrystalstructure.,Thetranslationalsymmetryofalatticeisgivenbythebasevectorsorlatticevectors.Usuallythesevectorsarechoseneither:tobetheshortestpossiblevectors,ortocorrespondtoahighsymmetryunitcell,Example:a2-Dlattice,Thesetwochoicesoflatticevectorsillustratetwotypesofunitcells:,Conventional(crystallographic)unitcell:largerthanprimitivecell;chosentodisplayhighsymmetryunitcell,Primitiveunitcell:hasminimumvolumeandcontainsonlyonelatticepoint,体心立方(BodyCenteredCubic)含有81/8+1=2个原子固体物理学原胞只要求含有1个原子。a1=(a/2)i+(a/2)j+(a/2)k=a/2(i+j+k)同理：a2=a/2(ij+k)a3=a/2(i+jk),体心立方结构，固体物理学原胞的体积是晶体学原胞的体积的1/2.,Alatticetranslationvectorconnectstwopointsinthelatticethathaveidenticalsymmetry:,Inour2-Dlattice:,CommonCrystalStructures,2-Donly5distinctpointlatticesthatcanfillallspace,3-Donly14distinctpointlattices(Bravaislattices),The14Bravaislatticescanbesubdividedinto7different“crystalclasses”,basedonourchoiceofconventionalunitcells.,AugusteBravais(1811-1863),Bravais,agraduateofthecolePolytechniqueandaprofessorofphysics,workedoutamathematicaltheoryofcrystalsymmetrybasedontheconceptofthecrystallattice,ofwhichtherewere14.,BravaisLattices,Attachingabasisofatomstoeachlatticepointintroducesnewtypesofsymmetry(reflection,rotation,inversion,etc.)basedonthearrangementofthebasisatoms.Wheneachofthese“pointgroups”iscombinedwiththe14possibleBravaislattices,thereareatotalof230differentpossible“spacegroups”in3-D.Wewillfocusonthefewthatarecommonformetals,semiconductors,andsimplecompounds.,CrystalStructureDiagrams,NaClCsCl,(c)fluorite(d)perovskite,(e)Lavesphase(f)A15,CrystalStructureDiagrams(continued),diamondstructure(C,Si,Ge),hexagonalclosepacked(Be,Mg,Zn),AnalysisofCommonCrystalStructures,1.ThethreeBravaislattices,basis:atomat000,basis:atomsat000,basis:atomsat0000,0,0,2.NaClstructure(manyionicsolids),lattice:face-centeredcubic(fcc)basis:Naat000,Clat,Naatcorners:(8x1/8)=1Naatfacecentres(6x1/2)=3Clatedgecentres(12x1/4)=3Clatbodycentre=1Unitcellcontentsare4(Na+Cl-),3.CsClstructure(someionicsolidsandintermetallicalloys),lattice:simplecubic(sc)basis:Csat000,Clat,4.diamondstructure(C,Si,Ge),lattice:face-centered-cubic(fcc)basis:000,5.Zincblende(闪锌矿）structure(cubicZnS,GaAs,InP,InSb,SiC),lattice:face-centered-cubic(fcc)basis:Znat000,Sat,Latticeparameters,Interstitialvoidspaces,Ifwelookdownontopoftwolayersofclose-packedspheres,wecanpickouttwoclassesofvoidspaceswhichwecalltetrahedralandoctahedralholes,Eachsphereinaclose-packedlatticeisassociatedwithoneoctahedralsite,whereasthereareonlyhalfasmanytetrahedralsites.ThiscanbeseeninthisdiagramthatshowsthecentralatomintheBlayerinalignmentwiththehollowsintheCandAlayersaboveandbelow.,6.hexagonal-close-packed(ex:Cd,Mg,Ti,Zn),lattice:hexagonalbasis:000,1/32/31/2,7.Wurtzite(纤锌矿）structure(hexagonalZnS),Unitcellsfor(a)hexagonalwurtzitestructure,and(b)thecorrespondingcrystallinelayerstacks.,8.Fluorite(萤石）structure(CaF2),Fluorite,CaF2,havingtwiceasmanyionsoffluorideasofcalcium,makesuseofalleighttetrahedralholesintheCPPlatticeofcalciumions(orange)depictedhere.ThecalciumionatAissurroundedbyeightfluorideions,andthisisofcoursethecaseforallofthecalciumsites.Sinceeachfluorideionhasfournearest-neighborcalciumions,thecoordinationinthisstructureisdescribedas(8:4).,9.Perovskitestructure(ABO3),OI,OII,OIII连接成等边三角形氧八面体：每个原胞有8个这样的三角形面，围成八面体Ti在八面体的中心，Ba在八面体的间隙里。,10.Spinelstructure(AB2O4),Thespinelstructure(sometimescalledgarnetstructure)isnamedafterthemineralspinel(MgAl2O4);thegeneralcompositionisAB2O4.Itisessentiallycubic,withtheO-ionsformingafcclattice.Thecations(usuallymetals)occupy1/8ofthetetrahedralsitesand1/2oftheoctahedralsitesandthereare32O-ionsintheunitcell.,ItsimplyhastwotypesofcubicbuildingunitsinsideabigfccO-ionlattice,fillingall8octants.,Thespinelstructureisveryflexiblewithrespecttothecationsitcanincorporate;thereareover100knowncompounds.Inparticular,theAandBcationscanmix!Inotherwords,thecompositionwithrespecttooneunitcellcanbe(A8)(B16)O32,orA8(B8A8)O32=A(AB)O4inregularchemicalspelling,or(A8/3B16/3)(A16/3B32/3)O32andsoon,withtheatomsinthebracketsoccupyingtherespectivesiteatrandom.,ThespinelmineralshavethegenericformulaAB2O4,whereAisacationwith+2chargeandBisacationwith+3.Themostcommonmembersinclude:,Thespinelstructureisalsointerestingbecauseitmaycontainvacanciesasregularpartofthecrystal.Forexample,ifmagnetiteisslowlyoxidizedbylyingaroundacoupleofbillionyears,orwhenrockscool,Fe2+willturnintoFe3+(oxidation,inchemicalterms,meansyoutakeelectronsaway).IfallFe2+isconvertedintoFe3+,chargebalancerequiresanetformulaofFe21,67O32perunitcellandthismeansthat2,33sitesmustbevacant-wehavewhatiscalledadefectspinel.Inaway,thecompositionisnowFe21,67Vac2,33O3;havinglotsofvacanciesasanintegralpartofthestructure.,(a)Viewingalong001direction,(b)Viewingalong111direction,Octahedralcationsareinyellowandgreen,tetrahedralinpurple.Betweenoxygenlayerswefindhexagonalpatternsofoctahedralcations,andhexagonalpatternsoftetrahedralcationswithanoctahedralcationatthecenter.,3layersofoxygenatomsareshowninalternatingshadesofblue.Atomsinoctahedralsitesareshowninshadesofgreenandyellow.Atomsintetrahedralsitesareshowninpurple.Contrarytoexpectationsbasedonionicradius,inspinelthealuminumatomsareinoctahedralsitesandthemagnesiumsinthetetrahedral.,(a)Viewingalong001direction,(b)Viewingalong111direction,PolyhedralRepresentation,Filledoctahedraformcriss-crossrows,withalternatinglayersofparallelrowsoffsetasshownontherightsideofthediagram.Thesquareholesenclosedbytherowsofoctahedraarefilledwithtetrahedra.,Octahedralcationsareinyellowandgreen,tetrahedralinpurple.Betweenoxygenlayerswefindhexagonalpatternsofoctahedralcations,andhexagonalpatternsoftetrahedralcationswithanoctahedralcationatthecenter.,NormalandInverseSpinelsSincetherearetwiceasmanyfilledoctahedraastetrahedra,andtheformulaforspinelisAB2O4,itispossibletofillalltheoctahedrawithB(trivalent)atomsandthetetrahedrawithA(divalent)atoms.WecantreversetherolesofAandBandfillallthesites.Ifweweretotry,wecouldonlyfillhalftheoctahedrawithA,andthetetrahedrawouldonlytakeuphalftheBatoms.Theremainderwouldfillthestill-vacantoctahedra.Suchastructureiscalledaninversespinel.Spinelandchromitearenormalspinels,magnetiteisaninversespinel.Inreality,mostspinelsaresomewherebetweenthetwoendstates.,ConceptMap,Dataforsomeinorganiccrystalstructures,MillerIndicesforCrystalDirectionshkl=afamilyofsymmetry-equivalentplanes,CrystalPlanesandDirections,Crystaldirectionsarespecifiedhklasthecoordinatesofthelatticepointclosesttotheoriginalongthedesireddirection:,Notethatforcubiclattices,thedirectionhklisperpendiculartothe(hkl)plane,Note:hkl=aspecificdirection;=afamilyofsymmetry-equivalentdirections,MillerIndicesSiliconWafers,Flatsarecutoutofwaferstoindicatecrystalorientationofwafersurface.EssentialtoknowsurfaceorientationofSiwaferifyouwanttofabricatecircuitsonit.,DirectionsinHexagonalCrystals,Directionsarewrittengenerallyasuvwandareenclosedinsquarebrackets.Notethatthesymboluvwincludesallparalleldirections,justas(hkl)specifiesasetofparallelplanes.,AswithMillerindices,directionsinhexagonalcrystalsaresometimesspecifiedintermsofafour-indexsystem,uvtwcalledWeberindices.,3-indexsetuvw,4-indexsetuvtw,HexagonallatticesandMiller-Bravaisindices,Miller-Bravaisindicesinhexagonallattices.ThethreesetsofidenticalplanesmarkedhavedifferentMillerindicesbutsimilarMiller-Bravaisindices,Miller-Bravaisindices(hkil)andareonlyusedinthehexagonalsystem.Theindexiisgivenby:,DiffractionsandCrystalStructures,TheReciprocalLatticeZoneandZoneAxisDiffractionfromCrystalsElectronDiffractionandCrystallography,three-dimensional“diffractiongrating”,X-ray,Lauespots,TheEwaldsphereisageometricconstructusedinelectron,neutron,andX-raycrystallographywhichdemonstratestherelationshipbetween:thewavevectoroftheincidentanddiffractedx-raybeams,thediffractionangleforagivenreflection,thereciprocallatticeofthecrystal,ThinkinginReciprocalSpace:Ewaldssphere,TheReciprocalLattice,Thereciprocallatticeiscomposedofallpointslyingatpositionsfromtheorigin,sothatthereisonepointinthereciprocallatticeforeachsetofplanes(hkl)inthereal-spacelattice.,Thisseemslikeanunnecessaryabstraction.Whatisthepayofffordefiningsuchareciprocallattice?,Thereciprocallatticesimplifiestheinterpretationofx-rayandelectrondiffractionsfromcrystalsThereciprocallatticefacilitatesthecalculationofwavepropagationincrystals(latticevibrations,electronwaves,etc.),TheReciprocalLattice,Crystalplanes(hkl)inthereal-spaceordirectlatticearecharacterizedbythenormalvectorandtheinterplanarspacing:,LongpracticehasshownCMphysiciststheusefulnessofdefiningadifferentlatticeinreciprocalspacewhosepointslieatpositionsgivenbythevectors,Thisvectorisparalleltothehkldirectionbuthasmagnitude2/dhkl,whichisareciprocaldistance,TheReciprocalLattice:AnAnalogy,Wavesoflatticevibrationsorelectronwavesmovingthroughacrystalwithaperiodicityspecifiedbybasevectorscanlikewisebedecomposedintoasumofplanewaves:,Intheanalysisofelectricalsignalsthatareperiodicintime,weuseFourieranalysistoexpresssuchasignalinthefrequencydomain:,Iff(t)hasperiodT,thenthecoefficientCisnonzeroonlyforfrequenciesgivenby,n=integer,Here,thecoefficientCkisnonzeroonlywhenthevectorkisareciprocallatticetranslationvector:,A,B,andCarethebasevectorsofthereciprocallattice(somebooksusea*,b*,c*),DefinitionofReciprocalLatticeBaseVectors,Thesereciprocallatticebasevectorsaredefined:,Whichhavethesimpledotproductswiththedirect-spacelatticevectors:,Socompare,forexample:,frequencytime,reciprocallatticedirectlattice,ZoneandZoneAxis,DiffractionfromCrystals,IntensitiesofdiffractionmaximacanvarymoreinformationaboutdetailedstructureSymmetryofthecrystalstructureisreflectedinthediffractionpattern,X-rayDiffraction(XRD),X-RayDiffraction:DiscoveryofX-Rays,Nov.,1895:Wm.Rntgendiscoveredthatwhencertainsubstancesareexposedtothebeamofacathoderaytube,anewkindofpenetratingraycapableoffoggingphotographicplatesevenwhenshieldedwasemitted-calleditx-rays.Thesex-raysalsoionizedgasesthroughwhichtheypassed-1stNobelPrizeinphysics(1901).Wavenatureofx-rays(transverse)establishedbyCharlesGloverBarklain1906althoughtherecontinuedtobecontroversyaboutthis.,X-RayDiffraction:Ludwig-MaximiliansUniversityofMunichGroupin1912,Rntgen,directorofthephysicslaboratory.ArnoldSommerfeld,DirectoroftheInstituteforTheoreticalPhysics.Experimentalworkonwave-nature(andwavelength)ofx-rays.PaulvonGroth,professorofmineralogy,worldrenownedauthorityoncrystallographyandmineralogy.Interestedinatomic/molecularmeaningofcrystalstructure.PaulPeterEwald,studentofSommerfeld,workingonpropagationofx-raysinsinglecrystals.MaxvonLaue,ProvatdozentinSommerfeldsInstitute.Photos:RntgenSommerfeld,vonGroth,Ewald,vonLaue.Hofgartencaf.http:/www.munich-info.de/portrait/p_hofgarten_en.html,X-RayDiffraction:April,1912,MaxvonLauejoinedSommerfeldsgroupasaprivatelecturerin1909,andhewasimmediatelystruckbytheatmospherethatwassaturatedwithquestionsforthenatureofX-rays.“ManyinstitutesinMunichUniversityhadmathematicalmodelsoftheseproposedspace-latticestructures,mainlythankstotheenthusiasticsupportofthetheorybythecrystallographerPaulvonGroth,butnoonehadyetprovedthatcrystalshavethisstructure.vonGrothwasanotherfrequentparticipantoftheHofgartencafcircle,andthankstohimvonLauequicklylearnedaboutcrystaloptics,andsoonbecameknownasalocalspecialistinthesubject.,X-RayDiffraction:April,1912,OneeveninginFebruary1912,thephysicistPeterPaulEwaldsoughtvonLauesadviceaboutsomedifficultieshewashavingwithhisdoctoralthesisonthebehavioroflongelectromagneticwavesinthehypotheticalspacelatticesofcrystals.VonLauecouldntanswerEwaldsquestion,buthismindbegantowander.Suddenly,aconnectionclickedinhismind.Ifdiffractionandinterferenceoccurswhenthewavelengthoflightisasimilarsizetothewidthoftheslitofanopticalgrating,andifX-rayswereindeedwavesthathaveawavelengthatleasttenthousandtimesshorterthanvisiblelight,thenintheorythespacesbetweentheatomsinacrystalmightbejusttherightsizetodiffractX-rays.Ifallthisweretrue,vonLauethought,abeamofX-rayspassingthroughacrystalwillbediffracted,formingacharacteristicinterferencepatternofbrightspotsonaphotographicplate./nobel_prizes/physics/laureates/1914/perspectives.html,X-RayDiffraction:April,1912,VonLauedesignedanexperimentinwhichheplacedacoppersulphatecrystalbetweenanX-raytubeandaphotographicplate.Hisassistants,WaltherFriedrichandPaulKnipping,carriedouttheexperiment.Afterafewinitialfailures,theymetwithsuccesson23April,1912.X-rayspassingthroughthecrystalformedthepatternofbrightspotsthatprovedthehypothesiswascorrect.”/nobel_prizes/physics/laureates/1914/perspectives.html,ZnS,InstrumentalTechnology:X-RayDiffraction:SetupofLaue,FriedrichandKnipping,ThesourceofRntgensradiationisseparatedfromthecrystalunderinvestigationbyaleadscreen,S,piercedatB1,andaseriesofever-finerleaddiaphragmsB2(intheleadchamberK),B3andB4.AroundthecrystalKrphotographicplatesmaybeplacedatvariouspositionsP15.TheextensionRisaddedtotrapthestraightforwardlypassingraysandobviatedisturbingsecondaryraysofthewall.ForprecisionmeasurementsthereisadiaphragmAbforthepinholeB1inscreenS(Friedrichetal.,1912).Kubbinga,“CrystallographyfromHaytoLaue,”p.27.Zincsulfide.,p.28,fig.18.,VonLaue-Braggs,“Regardingtheexplanation,Lauethinksitisduetothediffractionoftherntgenraysbytheregularstructureofthecrystal.Heis,however,atpresentunabletoexplainthephenomenoninitsdetail.*OncebackinCambridge,WillieW.L.BraggcontinuedtopourovertheLaueresults,andrecalledthecrystalstructuretheoriesofWilliamPopeandWilliamBarlow.HebecameconvincedthattheeffectwasopticalandvisualizedanexplanationintermsofthesimplereflectionofX-raysfromtheplanesofatomsinthecrystal.HetherebydevisedBraggsLaw.,n=2dsin.”*Letter,LarsVegardW.H.Bragg,June26,1912.JohnJenkins,“AUniquePartnership:WilliamandLawrenceBraggandthe1915NobelPrizeinPhysics,”Minerva,2001,Vol.39,No.4,pp.380-381.,X-raydiffraction:BraggsLaw,SpecularReflection,Rays1and2interfereconstructivelyifTotalPathDifferenceisintegralmultipleofthewavelength,Totalp.d.=AB+BC,OABandOCBareequivalent.AB=BC=dhklsin,Diffractionconditionis:2dhklsin=n,W.H.BottomWlliamLawrenceBragg(1890-1971)SwedishpostagestampwithBraggs,W.H.spotsgrowarfadeinintensity.Ontheotherhand,thepositionsoftheKikuchilinesareextremelysensitivetothetiltofthespecimen.,(a)Geometryofcrystalrotation,andpositionofKikuchilineswithrespecttodiffractionspots.,(b)TheKikuchilineintersectstheg-diffractionspotwhenthespecimenisattheexactBraggorientation(left),butisdisplacedbyxwhenofftheexactorientation(right).,(c)Relationshipbetweenthedeviationvector,s,androtationofcrystalbyangle,asinthetop(right),andforexactBraggorientation(left).,Thedeviationparameter,scanbedeterminedbys=g2x/(kr),=s/g,s/g=x/L,r/L=2,s=g.=g.x/L=g.x.2/r2dsin=,2/g=1/k,s=gx/r.g/k=g2.x/(rk),Wesaythats0iftheexcessKikuchilineliesoutsideitscorrespondingdiffractionspotg,asshowninFig.b.Inthiscase,thereciprocallatticepointliesinsidetheEwalssphere.,菊池线对的中线，即(hkl)面与荧光屏的截线。两条中线的交点，即两个对应的平面所属的晶带轴与荧光屏的截点，称为菊池极。菊池衍射谱中，参与衍射的晶带很多。菊池极也有多个，而斑点衍射谱中一般仅一、两个晶带参与衍射。,Nimetal,SomeindexedKikuchilinesfora110bccdiffractionpattern,TheKikuchibandsneartheforwardbeamshowthecorrect3-foldsymmetryofthecrystal,andnotthefalse6-foldsymmetryofdiffractionspotsinthezoneaxisofSi.,DiffractionfromCrystals,DiffractionfromaPrimitiveLattice,Definitions:(1)Selectionrules:thestructureofthecrystalimposescertainselectionrules以决定哪些diffractedbeams是被允许的。(2)Weights(weightingfactor):每个reciprocallattice的points有一个weightingfactors(与scatteringfactors类似)。(3)Structurefactor:theunit-cellequivalentsoftheatomicscatteringamplitude,f().(每个晶胞的scatteringamplitude)。,StructureFactors:TheIdea,由structurefactor的概，可以解为何某些特定的平面在特定crystalstructure內是能存在的(forbiddenspots)。而某些特殊晶格的reciprocallattice:,fcc的reciprocallattice为bcc,bcc的reciprocallattice为fcc,theatomswithintheunitcellallscatterwithaphasedifferencegivenby,此式为structurefactor的keyequation，可应用于任何unitcell，无论何种晶体，含有多少atoms，位置等等。,SomeImportantStructures:bcc,fcc,andhcp,(1)Thereciprocallatticeforbccandfccarethemselvesspeciallattices.(2)Allreciprocallatticesofcubicmaterialsaresimplecubic,butsomeofthelatticepointshaveazerostructurefactor.,bcc-crystals:,沒有其他可能性,故其reciprocallattice象一个fcc。(所有平面在reciprocallattice上均为整数!),fcc-crystals:有4个原子perunitcell,故其reciprocallattice象一个bcc。,hcp-crystals:比较复杂,(1)除(0001)外，thepatternscanbedifferentforeverymaterial因为c/aratioisdifferent。(2)Weuse3-indexnotationtoderivethestructure-factorrules.(3)Weuse4-indexMiller-BravaisnotationtoindexthelatticeplanesandthustheDPs.,(paper:Frank,F.C.(1965)Actacryst.18.P.862)forhcpindexing!,为(hkil)和(defg)两平面之间的夾角!,Extendingfccandhcptoincludeabasis,下三种材以fcc为始再延伸加上basis:NaCl,GaAs,andSi,(1)NaCl:foreveryNaatom,thereisaClatomrelatedtoitbythevector1/2,0,0.Toemphasizethecubicsymmetry,thealternativebasisvector1/2,1/2,1/2isselected.,由于元素的f同，故在hklallodd的点，会有4(fNa-fcl)不同表现,sensitivetothechemistryofcompound!称之为“chemicallysensitivereflections”!,(2)GaAs:可视为Ga(000)及As(1/4,1/4,1/4)(tetrahedron),(3)Si:Si,Ge,及Diamond均相同,唯一同处,当,(200)reflection在Si,Diamond均为零,在原有的hcp结构上加上另外一个hcplattice,(4)Wurtzite:是hcp的变种，包括(BeO,ZnO,AlN),就象GaAs(zincblende)对fcc的变化,Applyingthebccandfccanalysistosimplecubic,(1)ExtendingbcctoNiAl,Ni(000),Al(1/2,1/2,1/2),其实NiAl是simplecubic，