近代分析實驗原理：第三課-XRD-1

三、X射線衍射-1

（Xraydiffraction（XRD））近代分析實驗原理（Introductionofmodernanalyticalmethods）12Informationaboutthesizeoftheunitcellofthematerial.Thecrystalstructure.Recreateanimageofthecrystal.DeterminetheCrystalstructures:TheintensitiesofpeaksinXRDpatternαPeakpositionsinXRDpattern薄膜的等傾干涉：3PLDC34E5A1B241.布拉格公式（Bragg’slaw）Althoughthegeometryisidenticaltothatofreflection,thephysicalprocessoccurringisdiffraction。Thepositionofadiffractedbeam,withoutanyreference參考toitsintensity.光程差=AB+BC=dsin+dsin=2dsin滿足衍射的條件為：2dsin=nd為面間距，為Bragg角。這即為Bragg方程。thediffractionangleorBraggangletheorderofthediffractedbeamUsuallymeasuredexperimentally5高次衍射：（thedifferentordersofdiffraction）Forexample:(111)plane(111)(222)6衍射譜：（Thediffractionpattern）Withinacrystalthereareaninfinitenumberofsetsofatomplanes,andBragg’slawappliestoallthese.Thusifacrystalinabeamofradiationisrotated,eachsetofplaneswill,initsturn,diffracttheradiationwhenthevalueofsinbecomesappropriate.Thisistheprinciplebywhichdiffractiondataiscollectedforthewholeofthecrystal.Thearrangementofthediffractedbeams,whentakentogether,isthediffractionpatternofthecrystal.α7θ-2θ粉末X射線衍射：（PowderX-raydiffraction）Bragg’slaw：82.愛瓦德球（theEwaldsphere）Tovisualize形象化thediffractionpatternofacrystal:drawthereciprocallattice倒格子點陣

ofthecrystal,inanorientationequivalenttothatofthecrystalbeingirradiated(a).(ii)drawavectoroflength1/fromtheoriginofthereciprocallatticeinadirectionparallel,andinanoppositedirection,tothebeamofradiation.Herelisthewavelengthoftheradiation,(b).9(iii)fromtheendofthevector,constructasphere,calledtheEwaldsphere,(shownasacircleincross-section),ofradius1/.Eachreciprocallatticepointthatthesphere(circle)touches(ornearlytouches)willgiveadiffractedbeam(c,d).10CuKaradiation,λ=0.15418nmanEwaldsphereradiusof6.486nm-1

thereciprocallatticespacingofcopper,(2.78nm-1)Thismeansthatfarfewerspotsareintercepted截斷,andfewerdiffractedbeamsareproduced.3.Particlesize：11Moreperfectcrystalsdiffractoveraverynarrowrangeofangleswhileverydisorderedcrystalsdiffractoverawiderange.Theapproximaterangeofanglesoverwhichdiffractionoccurs,δ,

centredupontheexactBraggangle,,isgivenby:12Theeffectofcrystalliteshapeontheformofadiffractionspotwellorderedcrystalsharpspotsfuzzyspotsextremelydisorderedorverysmallcrystallite微晶AmorphousmaterialsfewbroadilldefinedringsThesmallerininadirectioninrealspace,thelongerinthereciprocalspaceinnormaldirection.法線方向13BiFeO3/La0.67Sr0.33MnO3bilayeron(111)SrTiO3substrate.14謝樂公式（Scherrerequation）晶粒大小（Particlesize） AlthoughBragg’slawgivesaprecisevalueforthediffractionangle,diffractionactuallyoccursoverasmallrangeofanglesclosetotheidealvaluebecauseofcrystalimperfections.Moreperfectcrystalsdiffractoveraverynarrowrangeofangleswhileverydisorderedcrystalsdiffractoverawiderange.Bragg’slaw：Scherrerequation：Dhklisthecrystalthicknessinthedirectionnormaltothe(hkl)planesdiffractingtheradiation.BraggangleWave

length衍射峰寬（單位為弧度）Theapproximaterangeofanglesoverwhichdiffractionoccurs（inradians）δθ當δθ取積分寬度時，K取1；當δθ取半高寬時，K取0.9K15164.X射線衍射的強度（Theintensitiesofdiffractedbeams）布拉格公式（Bragg’slaw)點陣常數(shù)（thelatticeparameters）謝樂公式（Scherrerequation)晶粒大?。≒articlesize）衍射峰強（Intensity）衍射峰位（Position)衍射峰寬（Width)完整的晶體結構（thecompletecrystalstructure）17影響衍射強度的因素：（dependuponthefollowingfactors）(i)thenatureoftheradiation.(ii)theBraggangleofthediffractedbeam.(iii)thediffractingpoweroftheatomspresent–theatomicscatteringfactor.(iv)thearrangementoftheatomsinthecrystal–thestructurefactor.(v)thethermalvibrationsoftheatoms–thetemperaturefactor.(vi)thepolarisationofthebeamofradiation.(vii)thethickness,shapeandperfectionofthecrystal–theformfactor.(viii)fordiffractionfromapowderratherthanasinglecrystal,thenumberofequivalent(hkl)planespresent–themultiplicity.X射線衍射的運動學近似：InthecaseofX-rays,theinitialbeamisalmostundiminishedonpassingthroughasmallcrystal,andtheintensitiesofthediffractedbeamsareaverysmallpercentageoftheincidentbeamintensity.Forthisreason,itisreasonabletoassumethateachdiffractedX-rayphotonisscatteredonlyonce.Scatteringofdiffractedbeamsbackintotheincidentbeamdirectionisignored.Thisreasonableapproximationisthebasisofthekinematicaltheoryofdiffraction.184.1.原子散射因子（Theatomicscatteringfactor）X-raysarediffractedbytheelectronsoneachatom.ThescatteringoftheX-raybeamincreasesasthenumberofelectrons,orequally,theatomicnumber(protonnumber),oftheatom,Z,increases.ThescatteringpowerofanatomforabeamofX-raysiscalledtheatomicscatteringfactor,fa.原子散射因子：Thenineconstants,ai,biandci,calledtheCromer-Manncoefficients,varyforeachatomorion.Unitsinelectronsin?innmitisdefinedasaratiooftheamplitudeofthewavescatteredfromoneatomtothatscatteredfromoneelectronunderthesamecondition.19Stronglyangledependent:At(sinθ)/λ=zerothescatteringfactorisequaltotheatomicnumber,Z,oftheelementinquestion.Theelementsodium鈉,withZ=11,hasavalueofthescatteringfactorat(sinθ)/λ=zeroof11.ThescatteringfactorforthesodiumionNa+atat(sinθ)/λ=zeroisequaltoZ-1,i.e.10,ratherthan11.Similarly,theatomicscatteringfactorsofK+(Z=18)andCl-(Z=18)areidentical.ComptonscatteringaddstothegeneralbackgroundofscatteredX-rays,andisusuallyignoredinstructuredetermination.204.2.結構因子（Thestructurefactor）Toobtainthetotalintensityofradiationscatteredbyaunitcell,thescatteringofalloftheatomsintheunitcellmustbecombined.Thisiscarriedoutbyaddingtogetherthewavesscatteredfromeachsetof(hkl)planesindependently,toobtainavaluecalledthestructurefactor,F(hkl),foreachhklplane.s0sOArAss0rA=xAa+yAb+zAc光程差：?=λ(rA·s-rA·s0)=rA·(s-s0)λIsI=Is0I=1/λ位相差：（phasedifference）?=rA·R*hklλ=λ(hxA+kyA+lzA)ΦA=2?/λ=2(hxA+kyA+lzA)21thescatteredwaveasacomplexamplitude：Themagnitudeofthescattering（themodulus）phasethetotalscatteringfromaunitcellTheintensityscatteredintothehklbeambyalloftheatomsintheunitcell.22結構因子的計算（Numericalevaluationofstructurefactors）Forexample：TiO2

（rutile）(200)reflectionF(200)=15.513(exp(i20)+exp(i21)+5.326(exp(i23/5)+exp(i28/5)+exp(i27/5)+exp(i22/5))=13.7923I0=F02=024對稱性與衍射強度（Symmetryandreflectionintensities）Crystal（Lackacentreofsymmetry）Diffractionpattern（centrosymmetric）系統(tǒng)消光：（systematicabsences）thesymmetryelementspresentinacrystalForexample：abodycentredlattice0=0（n=1,3,etc.）25反射條件（reflectionconditions）：Theconditionsthatapplyfordiffractiontooccurfrom(hkl)planesintheBravaislattices.對稱中心（acentreofsymmetry）螺旋軸（screwaxes）滑移面（glideplanes）系統(tǒng)消光（systematicabsences）Informationoftheextinction消失ruleThesystematicabsencesthatoccuronadiffractionpatternthusgivedetailedinformationaboutthesymmetryelementspresentinaunitcellandthelatticetypeuponwhichitisbased.左圖為反射條件26SystematicabsencessymmetryconsiderationsStructuralabsencesthescatteringfactorsoftheatomscombineforotherreasonsForexample:NaClKClthe(100)diffractionabsentabsentSystematicabsencesthe(111)reflectionabsentthehalitestructureall-facecentred(F)latticepresentthenumberofelectronsonbothionsis18f=fCl-K+27weightedreciprocallatticeTheareaallocated分配toeachnodeisproportionaltothestructurefactorF(hkl)ofeachreflectionAdiffractionpatternisadirectrepresentationofthereciprocallattice.28ConductionBandValenceBandL2,L3L1KFermiLevel光：IncidentX-ray發(fā)射出的光電子EjectedPhotoelectron1s2s2pComplexscatteringfactor:Thewavelengthatwhichanelectronisexcitedfromoneenergyleveltoanotherforanyatom.ThewavelengthoftheX-raybeamisclosetotheabsorptionedgeofanatominthesample.Friedel’slawbreaksdown0Bijvoetdifference294.3溫度因子（temperaturefactor）Assume：theatomsarestationaryReality:theatomsvibrateCorrections:temperaturefactorFrequency:~1013HzCuKα:λ=1.54?,f=1.951018HzAtomswillbecontinuallydisplacedoutoftheplanebyvaryingamountsonthetimescaleoftheradiation.Smearingout抹掉theelectrondensityofeachatom,anddiminishing衰減eachatomicscatteringfactor.Alargeinterplanar晶面間的spacing,dhkl,whichdiffractatsmallangles,thedisplacements移位areonlyafraction小部分ofdhkl,andtheeffectissmall.Theatomicdisplacementsinplaneswithasmallinterplanarspacing,dhkl,maybeequalorgreaterthandhklitself,causingconsiderablelossindiffractedintensity.30organicmolecularcrystalslowmeltingpointsandlargethermalvibrationsveryweakathighBraggangleInorganic無機crystalshighmeltingpointsharpdiffractionspotsathighBragganglesSampleshouldbemountedonspecialcryogenic冷凍的holders.31theatomicscatteringfactordefinedforstationaryatomsatomictemperaturefactor（alsocalledtheDebye-WallerfactororB-factor）BraggangletheatomicscatteringfactorforathermallyvibratingatomTheeffectofthetemperaturefactoristoreducethevalueoftheatomicscatteringfactorconsiderablyathighervaluesofsinθ/λ.32=thesquareofthemeandisplacementoftheatomfromthenormalequilibriumposition,r0.anisotropictemperaturefactorsOakRidgeThermalEllipsoidProgrampictorialpresentation334.4多重性因子（multiplicity）InPowderX-raydiffraction:SinglecrystalEachhkldiffractionspotwillbecomeconcentricsphericalhklshells34Ingeneral,themultiplicityofequivalent等價reflections,p,willdependupontheunitcelltype.Forany(hkl)reflectionthemultiplicitywillbeatleasttwo.35thepolarisation偏振ofthediffractedbeamincidentbeamofX-raysisunpolarisedScatteredbypowderspartialpolarisationanangledependentreductioninintensityCorrectionfactor:36Braggdiffractiontakesplaceoverarangeofangles,notjustattheexactvalueofθgivenbyBragg’slaw.LorentzfactorCorrectionfactor(inpowderdiffraction):TheLorentzandpolarisationfactorsareusuallycombined,forpowderX-raydiffraction,intoasinglecorrectionterm:(theconstantomitted)TheabsorptionofXray-mustbeincludedageneralformulaapplicabletoallsamplegeometriescannotbegiven.375.晶體結構的確定-位相問題（Structuredetermination-Phaseproblem）MotifSpacegroupCrystalstructureDiffractionpattern?Forexample:onelineofatomsthesquarerootofthemeasuredintensity?38unknownForaone-dimensionalchainavailableexperimentallysolvethephaseproblem39TheprocedureofstructuredeterminationfromasinglecrystalX-raydiffractionexperiment:Obtainanaccuratesetofintensityvalues.Determinetheunitcellandindexthediffractedreflectionsintermsofhkl.Determinethepointgroupandspacegroupofthecrystal,makinguseofsystematicabsences.Constructapossiblemode