近代分析实验原理：第三课-XRD-1

三、X射线衍射-1

（Xraydiffraction（XRD））近代分析实验原理（Introductionofmodernanalyticalmethods）12Informationaboutthesizeoftheunitcellofthematerial.Thecrystalstructure.Recreateanimageofthecrystal.DeterminetheCrystalstructures:TheintensitiesofpeaksinXRDpatternαPeakpositionsinXRDpattern薄膜的等倾干涉：3PLDC34E5A1B241.布拉格公式（Bragg’slaw）Althoughthegeometryisidenticaltothatofreflection,thephysicalprocessoccurringisdiffraction。Thepositionofadiffractedbeam,withoutanyreference参考toitsintensity.光程差=AB+BC=dsin+dsin=2dsin满足衍射的条件为：2dsin=nd为面间距，为Bragg角。这即为Bragg方程。thediffractionangleorBraggangletheorderofthediffractedbeamUsuallymeasuredexperimentally5高次衍射：（thedifferentordersofdiffraction）Forexample:(111)plane(111)(222)6衍射谱：（Thediffractionpattern）Withinacrystalthereareaninfinitenumberofsetsofatomplanes,andBragg’slawappliestoallthese.Thusifacrystalinabeamofradiationisrotated,eachsetofplaneswill,initsturn,diffracttheradiationwhenthevalueofsinbecomesappropriate.Thisistheprinciplebywhichdiffractiondataiscollectedforthewholeofthecrystal.Thearrangementofthediffractedbeams,whentakentogether,isthediffractionpatternofthecrystal.α7θ-2θ粉末X射线衍射：（PowderX-raydiffraction）Bragg’slaw：82.爱瓦德球（theEwaldsphere）Tovisualize形象化thediffractionpatternofacrystal:drawthereciprocallattice倒格子点阵

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截断,andfewerdiffractedbeamsareproduced.3.Particlesize：11Moreperfectcrystalsdiffractoveraverynarrowrangeofangleswhileverydisorderedcrystalsdiffractoverawiderange.Theapproximaterangeofanglesoverwhichdiffractionoccurs,δ,

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

length衍射峰宽（单位为弧度）Theapproximaterangeofanglesoverwhichdiffractionoccurs（inradians）δθ当δθ取积分宽度时，K取1；当δθ取半高宽时，K取0.9K15164.X射线衍射的强度（Theintensitiesofdiffractedbeams）布拉格公式（Bragg’slaw)点阵常数（thelatticeparameters）谢乐公式（Scherrerequation)晶粒大小（Particlesize）衍射峰强（Intensity）衍射峰位（Position)衍射峰宽（Width)完整的晶体结构（thecompletecrystalstructure）17影响衍射强度的因素：（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射线衍射的运动学近似：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.UnitsinelectronsinÅinnmitisdefinedasaratiooftheamplitudeofthewavescatteredfromoneatomtothatscatteredfromoneelectronunderthesamecondition.19Stronglyangledependent:At(sinθ)/λ=zerothescatteringfactorisequaltotheatomicnumber,Z,oftheelementinquestion.Theelementsodium钠,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.结构因子（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结构因子的计算（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对称性与衍射强度（Symmetryandreflectionintensities）Crystal（Lackacentreofsymmetry）Diffractionpattern（centrosymmetric）系统消光：（systematicabsences）thesymmetryelementspresentinacrystalForexample：abodycentredlattice0=0（n=1,3,etc.）25反射条件（reflectionconditions）：Theconditionsthatapplyfordiffractiontooccurfrom(hkl)planesintheBravaislattices.对称中心（acentreofsymmetry）螺旋轴（screwaxes）滑移面（glideplanes）系统消光（systematicabsences）Informationoftheextinction消失ruleThesystematicabsencesthatoccuronadiffractionpatternthusgivedetailedinformationaboutthesymmetryelementspresentinaunitcellandthelatticetypeuponwhichitisbased.左图为反射条件26SystematicabsencessymmetryconsiderationsStructuralabsencesthescatteringfactorsoftheatomscombineforotherreasonsForexample:NaClKClthe(100)diffractionabsentabsentSystematicabsencesthe(111)reflectionabsentthehalitestructureall-facecentred(F)latticepresentthenumberofelectronsonbothionsis18f=fCl-K+27weightedreciprocallatticeTheareaallocated分配toeachnodeisproportionaltothestructurefactorF(hkl)ofeachreflectionAdiffractionpatternisadirectrepresentationofthereciprocallattice.28ConductionBandValenceBandL2,L3L1KFermiLevel光：IncidentX-ray发射出的光电子EjectedPhotoelectron1s2s2pComplexscatteringfactor:Thewavelengthatwhichanelectronisexcitedfromoneenergyleveltoanotherforanyatom.ThewavelengthoftheX-raybeamisclosetotheabsorptionedgeofanatominthesample.Friedel’slawbreaksdown0Bijvoetdifference294.3温度因子（temperaturefactor）Assume：theatomsarestationaryReality:theatomsvibrateCorrections:temperaturefactorFrequency:~1013HzCuKα:λ=1.54Å,f=1.951018HzAtomswillbecontinuallydisplacedoutoftheplanebyvaryingamountsonthetimescaleoftheradiation.Smearingout抹掉theelectrondensityofeachatom,anddiminishing衰减eachatomicscatteringfactor.Alargeinterplanar晶面间的spacing,dhkl,whichdiffractatsmallangles,thedisplacements移位areonlyafraction小部分ofdhkl,andtheeffectissmall.Theatomicdisplacementsinplaneswithasmallinterplanarspacing,dhkl,maybeequalorgreaterthandhklitself,causingconsiderablelossindiffractedintensity.30organicmolecularcrystalslowmeltingpointsandlargethermalvibrationsveryweakathighBraggangleInorganic无机crystalshighmeltingpointsharpdiffractionspotsathighBragganglesSampleshouldbemountedonspecialcryogenic冷冻的holders.31theatomicscatteringfactordefinedforstationaryatomsatomictemperaturefactor（alsocalledtheDebye-WallerfactororB-factor）BraggangletheatomicscatteringfactorforathermallyvibratingatomTheeffectofthetemperaturefactoristoreducethevalueoftheatomicscatteringfactorconsiderablyathighervaluesofsinθ/λ.32=thesquareofthemeandisplacementoftheatomfromthenormalequilibriumposition,r0.anisotropictemperaturefactorsOakRidgeThermalEllipsoidProgrampictorialpresentation334.4多重性因子（multiplicity）InPowderX-raydiffraction:SinglecrystalEachhkldiffractionspotwillbecomeconcentricsphericalhklshells34Ingeneral,themultiplicityofequivalent等价reflections,p,willdependupontheunitcelltype.Forany(hkl)reflectionthemultiplicitywillbeatleasttwo.35thepolarisation偏振ofthediffractedbeamincidentbeamofX-raysisunpolarisedScatteredbypowderspartialpolarisationanangledependentreductioninintensityCorrectionfactor:36Braggdiffractiontakesplaceoverarangeofangles,notjustattheexactvalueofθgivenbyBragg’slaw.LorentzfactorCorrectionfactor(inpowderdiffraction):TheLorentzandpolarisationfactorsareusuallycombined,forpowderX-raydiffraction,intoasinglecorrectionterm:(theconstantomitted)TheabsorptionofXray-mustbeincludedageneralformulaapplicabletoallsamplegeometriescannotbegiven.375.晶体结构的确定-位相问题（Structuredetermination-Phaseproblem）MotifSpacegroupCrystalstructureDiffractionpattern?Forexample:onelineofatomsthesquarerootofthemeasuredintensity?38unknownForaone-dimensionalchainavailableexperimentallysolvethephaseproblem39TheprocedureofstructuredeterminationfromasinglecrystalX-raydiffractionexperiment:Obtainanaccuratesetofintensityvalues.Determinetheunitcellandindexthediffractedreflectionsintermsofhkl.Determinethepointgroupandspacegroupofthecrystal,makinguseofsystematicabsences.Constructapossiblemode