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Takeji Hashimoto Principles and Applications of X-ray, Light and Neutron Scattering Principles and Applications of X-ray, Light and Neutron Scattering Takeji Hashimoto Principles and Applications of X-ray, Light and Neutron Scattering TakejiHashimoto KyotoUniversity Kyoto,Japan ISBN978-981-16-1644-0 ISBN978-981-16-1645-7 (eBook) https://doi.org/10.1007/978-981-16-1645-7 TranslationfromtheJapaneselanguageedition:X-sen,Hikari,ChuseishiSanrannoGenritoOyoby TakejiHashimoto,©KodanshaScientificLtd.2017.PublishedbyKodanshaScientificLtd..AllRights Reserved. ©SpringerNatureSingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsofreprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageand retrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknown orhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface This book presents basic principles and theories underlying elastic scattering of X-ray,light,andneutron.Thebookalsopresentsapplicationsofthemtotheexplo- ration of internal structures of soft matters in the length scale ranging from sub- nanometerstoafewten-micronmeters.Theapplicationsarebasedonthefollowing fundamentalconcepts.Spatialvariationsofphysicalparametersrelevanttointernal structuresofthematters,suchaslocalmassdensity,canbegenerallydecomposed intoFouriermodeswithwavevectorqandamplitudeofη(q):Thescatteredintensity I(q)fromthematteratthescatteringvectorqisequaltothespectralintensityη(q)2 oftheFouriermodewiththewavevectorq.Thus,I(q)isafundamentalquantityin physicalsciencewhichinvestigatestherelationshipbetweenstructuresandphysical propertiesorfunctionsofthematters.Themattersincludenotonlyvariousmaterials studiedinmaterialsciencesbutalsocells,basicunitsoflivingcreaturesstudiedin biology,andlifesciences. Inrecentyears,highfluxandhighbrillianceX-raybeamsandhighfluxpulsedand continuousneutronbeams,bothofwhicharecoupledwithadvancedscatteringappa- ratusesand2Ddetectors,havebeenmadeavailablewithoutbigdifficultiesinmany nationalfacilitiesinmanycountries.Combinedapplicationsofthesetwoscattering methodsinconcertwithalaserlightscatteringmethodprovidethemostpowerful methodologyforinsituandreal-timecharacterizationsoftheinternalstructuresof various soft matters formed in thermal equilibrium or evolving in nonequilibrium processes, regardless of their optical clarity and opacity which come into play in caseofthescatteringofvisiblelight. Thescatteringmethodhasbeenwidelyappliedinvariousfieldssuchasphysics, chemistry,biology, materials science, and engineering sciences, and their interdis- ciplinary fields, as well. This is especially true for wide-angle X-ray and neutron scatteringanddiffractionmethodsforthecharacterizationofmattersinmicroscopic (atomic and molecular) scale. This book specifically focuses on small-angle scattering of X-ray, light, and neutron for the characterization of mesoscopic scale structuresofmattersinthelengthscalespanningfrom∼10nmto∼10μm. v vi Preface This book has its inception in a series of lectures entitled “Theories and Applications of X-ray and Light Scattering from Polymers” for graduate students atDepartmentofPolymerChemistry,KyotoUniversity,fortheindustrialresearch laboratoryinToyobo,Co.Ltd.,Japan,andinmanyotherindustrialresearchlabora- toriesinJapan.Aseriesoflecturesentitled“PrinciplesandApplicationsofElastic Scattering for Characterization of Supermolecular Assemblies of Polymers” was deliveredinDepartmentofAppliedPhysics,NagoyaUniversity,Japan;aseriesof lectures entitled “Physics of Elastic Scattering of Light, X-ray, and Neutron and ApplicationstoSoft-MaterPhysicalScienceandMaterialsScience”wasdelivered inFrontierSoftmaterialBeamLineConsortium(inSPring8),Hyogo,Japan.Based ontheselecture notes,theJapanese versionof thisbook was firstpublished under thesametitleinAugust2017fromKodanshyaScientific,Tokyo,Japan.Thepresent bookisarevisedversionoftheJapanesebookbyaddingsomemorechaptersand sections. Theauthortakespleasureinacknowledgingthegenerousassistanceofmanyof hiscolleagues.HeisespeciallyindebtedtoProf.KazuoSakurai,KitakyushyuCity University,andProf.YuyaShinohara,TokyoUniversity,forreadingandcriticizing themanuscript,andtoDr.ShinichiKoizumi,SeniorEditor,SpringerJapan,forthe editorialassistance.TheauthorisindebtedtoDr.ZhaoYue,TakasakiResearchLabo- ratory,JapanAtomicEnergyResearchAgency,Japan,fornumericalcalculationsof someofthetheoreticalscatteringprofilesandpatternspresentedinthisbook. Kyoto,Japan TakejiHashimoto Contents 1 GeneralIntroduction .......................................... 1 1.1 Basic Concepts Underlying Relationship Between theScatteringandStructures .............................. 1 1.2 VariousStructuresCharacterizedbyScatteringMethods ...... 3 References .................................................... 8 PartI FundamentalsofX-ray,LightandNeutronScattering 2 ScatteringMechanismofX-ray,LightandNeutronScattering .... 11 2.1 PrincipleofScatteringExperiments ........................ 11 2.2 X-rayandLight ......................................... 11 2.3 Neutrons ............................................... 12 2.4 ComparisonofScatteringMechanism ...................... 13 2.4.1 X-rayScatteringandLightScattering .............. 13 2.4.2 NeutronScattering .............................. 16 References .................................................... 18 3 ScatteringofSchrödingerWave:InterferenceofScattered WavesandBornApproximation ................................ 19 3.1 FieldTheory(OpticalTheory)ofScattering ................. 19 3.2 Schrödinger’sWaveEquation ............................. 20 3.3 WaveEquationinFreeSpace:PlaneWaveandSpherical Wave .................................................. 21 3.4 Wave Equation in Potential Field: Born Hypothesis inScatteringofSchrödingerWave ......................... 24 3.5 CriteriafortheBornHypothesis ........................... 29 3.6 DifferentialScatteringCross-SectionofSchrödingerWave .... 30 3.7 NotesofThisChapter .................................... 32 References .................................................... 37 4 ScatteringofX-rayandVisibleLight ........................... 39 4.1 Maxwell’sEquationsofElectromagneticWave .............. 39 4.2 FundamentalsofElectromagneticWave .................... 41 vii viii Contents 4.3 FundamentalEquationsinScatteringofElectromagnetic Wave .................................................. 42 4.4 BornHypothesisinScatteringofElectromagneticWave: ScatteringfromOscillatingDipoles ........................ 45 4.5 NotesofThisChapter .................................... 49 FurtherReading ................................................ 52 5 ResponseofElectronstoX-rayandVisibleLight:Thomson ScatteringandRayleighScattering ............................. 53 5.1 FrequencyResponseofElectrons .......................... 53 5.2 X-rayScattering:ScatteringfromFreeElectrons ............. 57 5.3 Scattering of Visible Light: Scattering from Bound Electrons ............................................... 62 5.4 Differential Scattering Cross-Section and Scattering Length ................................................. 67 5.4.1 DifferentialScatteringCross-SectionforX-ray, LightandNeutronScattering ..................... 67 5.4.2 Scattering Length and Scattering Length Density ....................................... 69 5.5 NotesofThisChapter .................................... 70 FurtherReading ................................................ 72 6 PolarizationofScatteredElectromagneticWaveandLight ScatteringfromOpticallyAnisotropicScatteringElements ........ 73 6.1 ScatteringwithPolarizedIncidentWave .................... 73 6.2 ScatteringwithUnpolarizedIncidentWave ................. 74 6.3 LightScatteringfromOpticallyAnisotropicScattering ElementsandDepolarizedLightScattering ................. 75 6.3.1 Induced Dipole Moment of Optically AnisotropicScatteringElements .................. 76 6.3.2 OpticallyIsotropicScatteringandOptically AnisotropicScattering ........................... 78 7 Rayleigh–Gans–Born–DebyeScatteringfromLargeParticles ..... 81 7.1 PhysicalBasisofRayleigh–Gans–Born–DebyeScattering .... 81 7.2 InterferenceofScatteredWavesfromScatteringElements .... 82 7.3 ScatteringStructureAmplitudeandStructure:Reciprocal Space(FourierSpace)andRealSpace ...................... 86 7.4 Scattering Structure Factor and Autocorrelation Function:MathematicsofFourierTransformandPhysics ofReciprocalPhenomena ................................ 87 7.5 IntraparticleInterferenceandInterparticleInterference ....... 90 FurtherReading ................................................ 92 Contents ix 8 PhysicsofReciprocalPhenomena ............................... 93 8.1 Introduction ............................................ 93 8.2 ScatteringfromVerySmallParticles:p(z)=δ(z) ............ 94 8.3 Scattering from Homogeneous Matters: p(z) = p (Constant) ............................................ 95 0 8.4 ScatteringfromMatterswithFiniteSize .................... 96 8.5 Scattering from Periodic Structure: Diffraction fromCrystalLattice ..................................... 98 8.6 Shape Anisotropy of Structures and Anisotropy ofScatteringPatterns .................................... 101 8.7 NotesofThisChapter .................................... 104 9 ScatteringStructureFactor:SpectralIntensityDistribution ofFourierModesofStructure .................................. 107 9.1 GeneralConcept ........................................ 107 9.2 Example: Scattering Structure Factor of Lamellar MicrodomainsinBlockCopolymers ....................... 110 9.3 CoupledInternalStructuresofMattersinr-SpaceAre Decoupledinq-Space .................................... 113 9.3.1 Dilute,IsolatedSystems ......................... 114 9.3.2 SystemswithFiniteParticleConcentrations ........ 116 References .................................................... 118 PartII X-rayandNeutronScattering:IsotropicScattering 10 ScatteringfromIsolatedParticles ............................... 121 10.1 ScatteringfromaSphericallySymmetricalParticle ........... 121 10.2 ScatteringfromaSphere ................................. 122 10.2.1 General Scattering Equation and Its Characteristics ................................. 122 10.2.2 FormFactorMaxima ............................ 123 10.2.3 AsymptoticBehavioratq→0:Guinier’sLaw ....... 125 10.2.4 AsymptoticBehavioratq→∞:Porod’sLaw ....... 126 10.3 ScatteringfromanEllipsoidofRevolution .................. 127 10.3.1 PerfectOrientation .............................. 127 10.3.2 RandomOrientation ............................ 134 10.4 ScatteringfromaCylinder ................................ 140 10.4.1 PerfectOrientation .............................. 140 10.4.2 RandomOrientation ............................ 143 10.4.3 RandomlyOrientedInfinitesimallyThinRod ....... 143 10.4.4 RandomlyOrientedInfinitesimallyThinDisk ....... 145 10.5 ScatteringfromaGaussianChain .......................... 148 10.5.1 DebyeScatteringFunction ....................... 148 10.5.2 AsymptoticBehavioratq→0:Guinier’sLaw ....... 151 10.5.3 AsymptoticBehavioratq→∞:Porod’sLaw ....... 152 x Contents 10.6 Scattering from Particles with Large Aspect Ratios: Asymptotic Behaviors, Radius of Gyration ofCross-Section,andLorentzFactors ...................... 152 10.6.1 LongCylinder ................................. 152 10.6.2 ThinDisk ..................................... 156 10.6.3 Summary of Asymptotic Behaviors and Crossover Behaviors in q-Dependence ofScatteringIntensityDistribution ................ 159 10.6.4 LorentzFactor ................................. 160 10.7 NotesofThisChapter .................................... 163 References .................................................... 169 FurtherReading ................................................ 169 11 ScatteringfromFluctuations:StatisticalTheoryofScattering ..... 171 11.1 Debye–Bueche Statistical Theory of Scattering fromHeterogeneousMatters .............................. 171 11.1.1 GeneralTheory ................................ 171 11.1.2 Spread of Correlation Function in r-Space andSpreadofScatteringFunctioninq-Space ....... 174 11.2 ExperimentalEvaluationofStatisticalParameters ............ 177 11.2.1 InverseFourierTransformation ................... 177 11.2.2 Debye–BuechePlot ............................. 178 11.3 Relationships Between Structures and Characteristic StatisticalParameters .................................... 181 11.3.1 MeanSquareFluctuations ....................... 181 11.3.2 CorrelationFunction ............................ 183 11.3.3 IntegralParametersandDifferentialParameters ofCorrelationFunction .......................... 185 11.4 SpatialCorrelationFunctionforAssemblyofAsymmetric Particles ............................................... 189 11.4.1 3DIsotropicSystemsHaving3DCorrelation Functions ...................................... 189 11.4.2 3DIsotropicSystemsHaving1DCorrelation Function ...................................... 190 11.4.3 3DIsotropicSystemsHaving2DCorrelation Function ...................................... 194 11.5 RadiusofGyration ...................................... 197 11.5.1 Guinier’sLaw .................................. 197 11.5.2 RadiusofGyrationforParticlesHavingVarious Shapes ........................................ 200 11.6 ScatteringfromDilutePolymerSolutions ................... 201 11.6.1 DescriptionofScatteringBasedonFluctuation Theory and Excess Scattering Due toConcentrationFluctuations .................... 203