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Methodsofx-rayMethodsofx-ray➢AcrystalisathreedimensionaldiffractionThelatticeperiodicityofthecrystaldeterminesthesamplingregionsofthediffractionpattern➢7.3.1Thesource7.3.1ThesourceandpropertyofX-Röntgen1901b.d.X-rayThewavelengthsofX-rayareThewavelengthsofX-rayareintherangeof100-0.01Å(103~••hardx-100～1Å：softx-1～0.05Å:usedinmedicalperspective,detectionofmaterialsChracteristicX-raysChracteristicX-raysproducedbyelectronictransitionbetweenatomicenergylevelsK1,K2,KandNiK1,K2,KandNi8.058.03Cu••••IK12IK2=1.541870K=1.39225•IK•(infactKisaclosedoublet,associatedwiththetwospinstatesof2pelectrons)K1,K2,K1,K2,KandNiCu••••IK12IK2=1.541870K=1.39225•IK•(absorptionedgeforNifitlerisat1.4881(ThethicknessofNifoil:0.021CommercialCommercialx-raytubetargetwavelengthsandsuitableWavelengthsv.sTargetWavelengthsv.sTargetWSLACBNLCERNSLACBNLCERNFNALKEKDESYLaueequationandBragg’sLaueequationandBragg’sLaueLauefirstmathematicallydescribeddiffractionfromconsiderX-raysscatteredfromeveryatomineveryunitcellinthecrystalandhowtheyinterferewitheachtogetadiffractionspotyoumusthaveconstructive•MaxVon•TheNobelPrizeinPhysics1914“forhisdiscoveryofthediffractionofX-raysbycrystals”ThederivationoftheLaueThederivationoftheLaueInterferenceAD-CB=a·s-a·s0a·(s-s0)=a(cos-cos0)=ha—lattice0—anglewhichamakeswith—anglewhichamakeswitha00=0=0≠a(cos-cos0)=LaueInthediffractiondirection,thedifferencebetweentheincidenta·(s-s)LaueInthediffractiondirection,thedifferencebetweentheincidenta·(s-s)=a(cos-cos)=andthediffractedbeamthroughanytwolatticepointsmustbeanintegralnumberofwavelengths.00b·(s-s0)=b(cos-cos0)=c·(s-s0)=c(cos-cos0)=la,b,c—lattice0,0,0—anglewhichmakewiths0,,—anglewhicha,makeswithh,k,l—indicesofdiffraction,Thevectorform(000)to=ma+nbThedifferencesin=Tmnp·(s-=(ma+nb+pc)·(s-ࢉࢉࢉ·+(-0+ࢉࢽ·(=mh+cequalto2.TheBragg’sThe2.TheBragg’sTheNobelPrizeinPhysics"fortheirservicesintheanalysisofcrystalstructurebymeansofX-rays"WilliamLawrence1890-1971,SirWilliamHenry1862-1942,•BraggdiscoveredthatyoucouldconsiderthediffractiontohavearisenfromreflectionfromlatticeplanesTheThederivationofBragg’sThederivationofBragg’s PD=ThederivationofBragg’s PD=Conditionfor2d(hkl)sinn=(n=1,2,3,…theangleoftheorderofthe2d(hklnnd(hkl(nh,nk,nlnd(hkl)d(nh,nk,nl(dHKLnd(hkl)d(nh,nk,nl(dHKLn2d==ThederivationofBragg’s PD=ThederivationofBragg’s PD=Conditionfor2d(hkl)sinn=(n=1,2,3,…theangleoftheorderofthe2d(hklnd(hkl(nh,nk,nln2dsin=where,hkl—reflection3.The3.ThereciprocallatticeandEwaldAdiffractionpatternisnotadirectrepresentationofthecrystalThediffractionpatternisarepresentationofthereciprocalReciprocala*bVb*ccbVc*aVaccaa b∗∗Reciprocala*bVb*ccbVc*aVaccaa b∗∗∗∗∗ 倒易空间矢量∗∗∗RealspaceandRealspaceandreciprocal•Imaginaryreciprocallatticesarecreatedtohelpusunderstanddiffractiongeometrya*=bcsin b*= c*= Ewalddiffraction12OBEwalddiffraction12OBS球心O’到倒易点阵点B的连线方向SEwaldsphereandlimited12Ewaldsphereandlimited121/dhkl2.02Å，1.43Å，1.17Å，1.01Å，0.90Å，0.83Å，0.76Å……当用波长为kα的铁靶照射时，因λα2=097Å，只有四个d大于它，靶进行照射，因λkα/2=0.77Å，故前IntensityofX-ray•Supposeaunitcelldefinedbyvectorsa,b,c,containingNatomswithatomcoordinates(xj,yj,zj)forjatom,itsscatteringfactorisfj.ࢉࢉIntensityofX-ray•Supposeaunitcelldefinedbyvectorsa,b,c,containingNatomswithatomcoordinates(xj,yj,zj)forjatom,itsscatteringfactorisfj.ࢉࢉൌࢉࢉࢉࢉ••Thedifferenceinpathઢࢉൌࢉࢉࢉࢉ·ࢉൌࢉࢠࢉࢉࢉࢠࢉ•ThephaseIntensityofX-rayThestructureIntensityofX-rayThestructuref1exp[i1]f2exp[i2]f3exp[i3]...fnexp[infjexp[in]fjexp[2i(hxjkyjlzjNNNNfjcos2kyjlzj)fjisin2(hxjkyjlzjjeiei2cosisineiei(ei eiei2CalculationstructureMetallicsodiumatomcoordinates(0,0,0)and(1/2,1/2,fCalculationstructureMetallicsodiumatomcoordinates(0,0,0)and(1/2,1/2,fNa[cos2(h0k0l0)isin2(h0k0lcos2(h1k1l1)isin2(h1k1l222222fNa[1cos(hk(=systematic2.Unitcellhasa21screwaxisalongthecaxisatEquivalentpositionandx,,N/N/22.Unitcellhasa21screwaxisalongthecaxisatEquivalentpositionandx,,N/N/2kylz)]kyl(zFfjjjjjjjjN/N/12i(lz)]FffjjjjN/fjexp2i(lzj)(1exp2i 2(x,,N/(lfjexp2πi(lzj=0systematicalongzThedirectionofX-rayLauea·(s-s0)=ThedirectionofX-rayLauea·(s-s0)=a(cos-cos0)=b·(s-s0)=b(cos-cos0)=c·(s-s0)=c(cos-cos0)=Bragg’s2dsin=TheintensityofX-rayNfjexp[2i(hxkyjlzjNNfjcos2(hxjkyjlzj)fjisin2(hxjkyjlzjjsystematicsystematicCrystalstructurewhichcontaincentering,glideplaneandscrewaxiswillhavesystematicabsences.Namely,somereflectionswillbesystematicallysystematicabsenceandConditionsforCauseofIn-centredI A-systematicabsenceandConditionsforCauseofIn-centredI A-h,k,lnotallevenandnotF-h+k+lnotmultiplesofR- k+lnotmultiplesof dl 21,42,lnotmultiplesoflnotmultiplesoflnotmultiplesof Screw 31,32,62,41,61,RocksaltCrystalRocksaltCrystalstructureofNa+(0,0,0),(1/2,1/2,0),(1/2,0,1/2),Cl-(1/2,0,0),(0,1/2,0),(0,0,1/2),(1/2,1/2,FNa+=fNa+[(1+ei(h+k)+ei(h+l)+ei(k+l)FCl-=fCl-eih[(1+ei(h+k)+ei(h+l)+ei(k+l)F=FNa++FCl-=(fNa++fCl-eih)[(1+ei(h+k)+ei(h+l)+ei(k+l)F=fNa++fCl-=F=fNa++fCl-=(fNa++fCl-eih)[(1+ei(h+k)+ei(h+l)+ei(k+l)Whenhklnotallevenandnotall(1+ei(h+k)+ei(h+l)+ei(k+l)FNaC(systematic(1+ei(h+k)+ei(h+l)+ei(k+l))=WhenhklalloddorallFNaCl=4(fNa++fCl-eih)WhenhklallWhenhklalloddXRDXRDOfRocksalt7.3.4MethodsofX-ray7.3.4MethodsofX-rayMonochromaticcameramethod--MonochromaticX-Rotation,Oscillation,WeissenbergLaue--whiteX---MonochromaticX-1.Singlecrystal1.Singlecrystalthecrystalcanrotatearoundanaxisinthehorizontalplane,andthescatteredX-raysarerecordedinanelectronicdetectorwhichcanmovearoundthecrystal.Structure=•SolvingStructure=•Solvingacrystalstructureinvolvesdeterminingunknownphases and(x,y,FDimeric[CuI(ophen)22F(hkl)fjexp2i(hxjkyjlzj)A(hkl)(xyz)(1/V)F(hkl)exp2i(hxjkyjlzjCrystalstructureCrystalsystemandCellparametersCrystalstructureCrystalsystemandCellparameters(x,y,z)V1F(hkl)e2i(hxkylz) F(hkl)(x,y, K RadiationX-rayRadiationX-raySynchrotron•-poorsensitivity,highbackground,lowdynamicScintillationgoodsensitivity,lowbackground,highdynamicImaginggoodsensitivity,lowbackground,gooddynamicrange,veryefficientdatacollectionCCDsandMultiwirefastreadout,goodsensitivity,lowbackground,gooddynamicrange,veryefficientdatacollection•••LaueorareadetectorFixedLaueorareadetectorFixed2.Powdera.Camera--Debye2.Powdera.Camera--DebyeScherrer1OB2IfthespacingbetweenaIfthespacingbetweenapairofdiffractionlinesis4=2L/R(radian)=(2L/R)=(L/2R)×(180/)=2R==LR==L/100mmb.PowderMonochromaticX-POb.PowderMonochromaticX-POsampleDiffractionIncidentPowderDiffractedbeamX-ray IncidentbeamAutomateddiffractometerAutomateddiffractometerpowderCrystalstructuresdetermination–pattern=WeknowBragg'sandtheequationforinterplanarCrystalstructuresdetermination–pattern=WeknowBragg'sandtheequationforinterplanarspacing,d,forcubiccrystalsisgivenby:adk2lWhereaisthelatticeThis4d2sin2sin2k2lsin l2222systematicabsenceand ConditionsforextinctionCauseofsystematicabsenceand ConditionsforextinctionCauseof h,k,lnotallevenandnotall-h+k+lnotmultiplesofA-R- sin l4a2sin2h2k2lCharacteristiclinesequencesinthecubicSimpleBody-centeredcubic:Face-centeredcubic:Diamondcubic:1,2,CharacteristiclinesequencesinthecubicSimpleBody-centeredcubic:Face-centeredcubic:Diamondcubic:1,2,3,4,5,6,8,9,10,11,12,13,14,16,2,4,6,8,10,12,14,16,18,3,4,8,11,12,16,19,20,24,27,32,3,4,11,16,19,24,27,32,(h,k,lallevenorall(h,k,lalloddorallevenwhensin(hl2222IndexingofthecubicCharacteristiclinesequenceIndexingofthecubicCharacteristiclinesequenceincubic(hkl)100,110,111,200,210,211,220,221,222,300,（h2＋k2＋l21,2,3,4,5,6,8,9,10,11,12,13,(hkl)110,200,211,220,310,222,321,400,2,4,6,8,10,12,14,16…→(1:2:3:4:5:6:（h2＋k2＋l2(hkl)111,200,220,311,222,400,331,420,3,8,11,12,16,19,20（h2＋k2＋l2ExampleforpowercrystalExampleforpowercrystalMeasurerelativeMeasurethedistancesoflinepair2L1,2L2,2L3,…Calculateinaccordingtotheformulae4= =2L/4R=Sample:Condition:CuK,=1.5418Å,R=50mm4)Calculate4)CalculateCalculatesin21:sin22:sin23:sin24:…=3:4:8:11:…IdentifyBravaislattice→facecubiccalculateh2+k2+l2and=2L/4R=h,k,lallevenorall123456789=2L/4R=h,k,lallevenorall123456789(h2k2l2sin2(h2k2l2sin2(h2k2l2asin24a=5.628Z2.165(5.628108236.0224sin2Whyusehighangle l2222(h24sin2Whyusehighangle l2222(h2k2l2asin24°a=5.6282dsinθ=➢Highanglevaluesgivesmoreaccurated-Least-squaremethod,plotmethod,highanglevalues,…Peak•••••CrystalSpacegroupsymmetryPeak•••••