7.2SmallAngleNeutronScattering..............................227.3Re ec - PDF document

7.2SmallAngleNeutronScattering..............................227.3Re ectometry........................................228Acknowledgements229References229.1ScatteringandOptics...................................239.2Re ectometry........................................23ARadiusofGyrationofSomeHomogeneousBodies241IntroductionTheneutronisaspin1/2sub-atomicparticlewithmassequivalentto1839electrons(1.67492810�27kg),amagneticmomentof-1.9130427n(-9.649178310�27JT�1)andalifetimeof15minutes(885.9s).Quantummechanicstellsusthat,whilstitiscertainlyparticulate,theneutronalsohasawavenatureandassuchcandisplaythegamutofwavebehaviorsincludingre ection,refractionanddiraction.Thisintroductioncoversbrie ythetheoryofneutronscatteringandthatoftwotechniquesthatmakeuseofthewavepropertiesofneutronstoprobethestructureofmaterials,namelysmallangleneutronscattering(diraction)andneutronre ectometry(re ectionandrefraction).Sincethisintroductionisexactlythat,thereaderisencouragedtolooktotheextensiveliteratureonthesubjectandarecommendedreadinglistisprovidedattheend.Muchofthematerialpresentedherehasbeentakenfromthosereferences.2NeutronScattering2.1Neutron-nucleusinteractionThescatteringofneutronsoccursintwoways,eitherthroughinteractionwiththenucleus(nuclearscattering)orthroughinteractionofunpairedelectrons(andhencetheresultantmagneticmoment)withthemagneticmomentoftheneutron(magneticscattering).Itistheformerofthesethatthisintroductionwilladdress.Letusconsidertheelasticscatteringofabeamofneutronsfromasinglenucleus.Inthiscasewetreatthenucleusasbeingrigidlyxedattheoriginofcoordinatesandthereisnoexchangeofenergy(Figure(1)).ThescatteringwilldependupontheinteractionpotentialV(r)betweentheneutronandthenucleus,separatedbyr.Thispotentialisveryshortrangeandfallsrapidlytozeroatadistanceoftheorderof10�15m.Thisisamuchshorterdistancethanthewavelength2 bevariedbyreplacinghydrogenwithdeuteriumandpotentiallybemadetomatchthatofsomeothercomponentinthesystem.Thistechniqueofcontrastvariationisoneofthekeyadvantagesofneutronscatteringoverx-raysandlight.Asmentionedabove,theneutroncanalsointeractwiththemagneticmomentofanatom.Thismagneticinteractionhasaseparatemagneticscatteringlengththatisofthesameorderofmagni-tude,butindependentfrom,thenuclearscatteringlength.Thus,forexample,onecanusecontrastvariationtoremovethenuclearcomponentofthescatteringandleaveonlythemagnetic.Themagneticinteractionisspin-dependent,soitisalsopossibletoextractinformationaboutthemag-netizationthroughtheuseofpolarizedneutrons.Theseadvancedusesarebeyondthescopeofthisintroduction,butmoreinformationcanbefoundinthereferencemateriallistedattheend.Havingtreatedthecaseofasinglenucleus,ifwenowconsiderathree-dimensionalassemblyofnucleiwhilstmaintainingtheassumptionofelasticscatteringtheresultantscatteredwavewillthenbe s=�Xibi reikreiq
r(3)whereq=ki�ksandisknownasthescatteringvectorwithkiandksbeingthewavevectorsoftheincomingandscatteredneutronsrespectively.2.2ScatteringCrossSectionThescatteringcrosssectionisameasureofhow\big"thenucleusappearstotheneutronandthushowstronglyneutronswillbescatteredfromit. Figure2:Thegeometryofascatteringexperiment(afterSquires)ImagineaneutronscatteringexperimentwhereabeamofneutronsofagivenenergyEisincident4 onageneralcollectionofatoms(yoursample-itcouldbeacrystal,asolutionofpolymers,apieceofrock,etc)(Figure2).Ifweagainassumeelasticscattering(suchthattheenergyoftheneutronsdoesnotchange)wecansetupaneutrondetectortosimplycountalltheneutronsscatteredintothesolidangled inthedirection;.Thedierentialcrosssectionisdenedbyd d =numberofneutronsscatteredpersecondintod indirection; d (4)whereisthenumberofincidentneutronsperunitareapersecond,referredtoastheincident ux.Thename\crosssection"suggeststhatthisrepresentsanareaandindeed,wecanseethatthedimensionsof uxare[area�1time�1]andthoseofthenumeratorinequation(4)are[time�1]resultingindimensionsof[area]forthecrosssection.Thetotalscatteringcrosssectionisdenedbytheequations=totalnumberofneutronsscatteredbysecond (5)andisrelatedtothedierentialscatteringcrosssectionbys=Zd d d (6)Thecrosssectionisthequantitythatisactuallymeasuredinascatteringexperimentandthebasicproblemistoderivetheoreticalexpressionsthatdescribeitforgivensystemsofscatterers.Exper-imentallythecrosssectionsareusuallyquotedperatomorpermoleculeandthusthedenitionsabovearethendividedbythenumberofatomsormoleculesinthescatteringsystem.Wecancalculatethecrosssectiond=d forscatteringfromasinglexednucleususingtheexpressionsgivenabove.Denotingthevelocityoftheneutronsasvandagaintreatingelasticscattering,thenumberofscatteredneutronspassingthroughanareadSpersecondisvdSj sj2=vdSb2 r2=vb2d (7)Theincidentneutron uxis=vj ij2=v(8)Fromequation(4)d d =vb2d d =b2(9)andthenintegratingoverallspace(4steradians)weobtaintot=4b2(10)Wecanperformasimilarcalculationfortheassemblyofnucleiwhosewavefunctionwasgiveninequation(3)aboveandobtainthedierentialcrosssectiond d (q)=1 NNXibieiq
r2(11)whichwecannowseeisafunctionofthescatteringvector,q.5 Figure3:Scatteringlengthdensityofwaterasafunctionofdistancefromagivenoxygenatom(afterKline)So,wecannowmakethereplacementofthesumind d (q)=1 NNXibieiq
r2(18)bytheintegralofthescatteringlengthdensitydistributionacrossthewholesampleandnormalizebythesamplevolumed d (q)=N Vd d (q)=1 VZV(r)eiq
rdr2(19)Thisresultisknownasthe\Rayleigh-GansEquation"andshowsusthatsmallanglescatteringarisesasaresultofinhomogeneitiesinscatteringlengthdensity((r)).==Visknownasthemacroscopiccrosssection.TheintegraltermistheFouriertransformofthescatteringlengthdensitydistributionandthedierentialcrosssectionisproportionaltothesquareofitsamplitude.ThislatterfactmeansthatallphaseinformationislostandwecannotsimplyperformtheinverseFouriertransformtogetfromthemacroscopiccrosssectionbacktothescatteringlengthdensitydistribution.Asdiscussedpreviously,thedierentialcrosssectiond=d isthedirectlymeasuredquantityinascatteringexperiment.Inthecaseofsmallanglescatteringtheresultsareusuallynormalizedbythesamplevolumetoobtaintheresultonan\absolute"scaleasthispermitsstraightforwardcomparisonofscatteringfromdierentsamples.ThusthedierentialmacroscopiccrosssectionisusedasdenedbytheRayleigh-Gansequationabove.Aswiththeatomiccrosssection,themacroscopiccrosssectionhasthreecomponentsd d (q)=dcoh d (q)+dinc d +dabs d (20)Informationaboutthedistributionofmatterinthesampleiscontainedinthecoherentcomponent,whilsttheincoherentcomponentisnotq-dependentandcontributesonlytothenoiselevel.Theabsorptioncomponentisusuallysmallandsimplyreducestheoverallsignal.7 Whilstdierenttypesofsystemhavedierentnaturalbasesforthedistributionofscatteringlengthdensity,allarefundamentallyequivalent-wejustusedierentwaystodescribethem.Inthecaseofparticulatesystemswherewehave\countable"unitsthatmakeupthescattering,wecanthinkaboutthespatialdistributionofthoseunitssuchthatZVf(r)dr2�!NXiNXjf(ri�rj)(21)Inpolymerstheunitsmightbethemonomersinthechain,inproteinswemightconsiderpolypep-tidesubunitsandinageneralparticulatesystemtheindividualparticles(betheymoleculesoroildroplets)mightbeused.Innon-particulatesystems(forexamplemetalalloysorbicontinuousmicroemulsions)astatisticaldescriptionmaybeappropriatewhereby(r)isdescribedbyacorrelationfunction (r).3.1GeneralTwoPhaseSystem Figure4:Asystemcontainingtwophaseswithscatteringlengthdensities1and2Sowhatisthepracticalresultoftheabovediscussion?Letusimagineageneraltwophasesystemsuchasthatpresentedingure4.Itconsistsoftwoincompressiblephasesofdierentscatteringlengthdensities1and2.ThusV=V1+V2(22)(r)=(1inV12inV2(23)TakingtheRayleigh-Gansequation(equation(19))andbreakingthetotalvolumeintotwosub8 volumesd d (q)=1 VZV11eiq
rdr1+ZV22eiq
rdr22(24)d d (q)=1 VZV11eiq
rdr1+2(ZVeiq
rdr�ZV1eiq
rdr1)2(25)(26)Soatnon-zeroqvaluesd d (q)=1 V(1�2)2ZV1eiq
rdr12(27)wherethedierenceinscatteringlengthdensitiesencapsulatesbothmaterialproperties(density,composition)andradiationproperties(scatteringlengths),whilsttheintegraltermdescribesthespatialarrangementofthematerial. Figure5:TwosystemswherethestructureisthesamebutthescatteringlengthdensitiesarereversedTheaboveequationleadsto\Babinet'sPrinciple"thattwostructures,suchasthoseshowningure5,whichareidenticalotherthanfortheinterchangeoftheirscatteringlengthdensitiesgivethesamecoherentscattering(theincoherenttermmaybedierent).Thisisaresultofthelossofphaseinformationmentionedpreviously-thereisnoway(fromasinglemeasurement)todetermineif1isgreaterthan2orviceversa.Thusitisimportantwhendesigningsmallanglescatteringexperimentstoconsidertheappropriateuseofcontrastvariation-usuallybysubstitutionofhydrogenfordeuterium-inordertobeabletosolvethestructure.9 4AnalysisofSmallAngleScatteringDataOncethevariousinstrumentaleectshavebeenremovedandthescatteringispresentedasd=d (q)itisthennecessarytoperformsomesortofanalysistoextractusefulinformation.Unlessthereissomespecicorientationofscatteringobjectswithinthesample,thescatteringcanbeaveragedtogivethemacroscopiccrosssectionasafunctionofthemagnitudeofq.Itisthisthatismostcommonlypresentedandisknownasthe1-Dsmallanglescatteringpattern.Thereareessentiallytwoclassesofanalysis:model-dependentandmodel-independent.Theformerconsistsofbuildingamathematicalmodelofthescatteringlengthdensitydistribution,whilstthelatterconsistofdirectmanipulationsofthescatteringdatatoyieldusefulinformation.4.1ModelIndependentAnalysis4.1.1TheScatteringInvariantPorodshowedthatthetotalsmallanglescatteringfromasampleisaconstant(i.e.invariant)irrespectiveofthewaythesampledensityisdistributed(gure6). Figure6:Twosystemswherethecontrastandvolumefractionarethesame,butthedistributionofmatterisdierent.Bothare10%blackand90%white.IntegratethedierentialcrosssectionwithrespecttoQQ=Zd d (q)dq(28)=(2)3((r)� )2(29)andforanincompressibletwo-phasesystemQ 4=Q=221(1�1)(2�1)2(30)10 Thus,intheory,thisanalysisallowsforthecalculationofthevolumefractionofeachcomponentinatwo-phasesystemgiventhecontrast,orthecontrastgiventhevolumefractions.HoweverinpracticeitisdiculttomeasurethescatteringinawideenoughQrangetobeabletocalculateQ.4.1.2PorodScatteringAlsoduetoPorodisalawforscatteringathighvaluesofQ(Q1/D,whereDisthesizeofthescatteringobject),iftherearesharpboundariesbetweenthephasesofthesystem.ThelawstatesthatatlargeQI(q)/q�4(31)andthus Q
limq��1(I(q)
q4)=S V(32)whereQisthescatteringinvariantmentionedpreviouslyandS=Visthespecicsurfaceofthesample.Ifweconsiderthesystemsshowningure6wecanseethatthespecicsurfaceofthelefthandsamplewillbelargerthanthatoftherighthandone,buttheyhavethesamescatteringinvariant.4.1.3GuinierAnalysisWherethePorodapproximationconsidersthehigh-Qlimitofscattering,thelowQlimitcanbedescribedusinganapproximationduetoGuinier.TheGuinierapproximationisformulatedasI(Q)=I(0)e�(QRg)2 3(33)ln(I(Q))=ln(I(0))�R2g 3Q2(34)andthustheradiusofgyrationofthescatteringobject,Rg,canbeextractedfromtheslopeofaplotofln(I(Q))vsQ2,bearinginmindthatthevalidityoftheapproximationislimitedtovaluesofQRg1.TheradiusofgyrationofasphereisgivenbyR2g=3 5R2(35)andtheequationsforotherbodiesaregiveninAppendixA.11 4.2.1TheFormFactorforSpheresForasphereofradiusrP(q)="3(sin(qr)�qrcos(qr)) (qr)3#2(38) Figure7:FormFactorspheresofradius3A.Rg=23A13 Figure10:Formfactorofspheresofradius30Awithadistributionofradii.Thepolydispersitiesarep==Rmeanquotedasapercentage.Sizedistributionsforthenon-zeropolydispersitesareinset.5NeutronRe ectometryAsmentionedintheintroduction,neutronsobeythesamelawsaselectromagneticwavesandassuchdisplayre ectionandrefractiononpassingfromonemediumtoanother.Whilstthere ectionofneutronsasaphenomenonhasbeenknownsincerstreportedbyFermiandco-workersinthelate1940'sitisonlyrelativelyrecently(1980's)thatapplicationofthetechniqueasastructuralprobewasseriouslyconsidered.Asweshallsee,there ectionofneutronsisapowerfultechniquefortheprobingofsurfacestructuresandburiedinterfacesonthenanometerlengthscale.There ectionofneutronsfromsurfacesis,however,verydierentfrommostneutronscattering.Inthediscussionofscatteringtheoryandsmallanglescatteringweassumedthatwecouldcalculatethescatteringcrosssectionbyaddingthescatteringfromeachnucleusinthesample.ThisinvolvedtheassumptionthataneutronisonlyscatteredonceonpassingthroughthesampleandiscalledtheBornapproximation.Weignoredmultiplescatteringbecausetheseeectsareusuallyveryweak.However,inthecaseofre ectionclosetoandbelowthecriticalangle(whereneutronsaretotallyre ectedfromasmooth16 surface)wearenolongerconsideringweakscatteringandtheBornapproximationnolongerholds.Thankfullywecanusetheconstructsofclassicalopticstodiscussthebehaviorofneutronsunderthesecircumstances.5.1SpecularRe ection Figure11:(a)Theinterfacebetweentwobulkmediaofrefractiveindicesn0andn1showingincidentandre ectedwavesatangle0andthetransmittedwaveatangle1.(b)Athinlmofthicknessdandrefractiveindexn1betweentwobulkmediaofrefractiveindicesn0,n2.Specularre ectionisdenedasre ectioninwhichtheangleofre ectionequalstheangleofincidence.Considertheneutronbeamwithwavevectork0ingure11(a)incidentonaplanarboundarybetweenmedia1and2.Therefractiveindexattheboundarybetweentwomediaisdenedasusualn=k1 k0(43)wherek1;k0aretheneutronwavevectorsinsideandoutsidethemedium.Therefractiveindexofamaterialagainstvacuumiscommonlywrittenasni=1�2i 2(44)whereiisthescatteringlengthdensityofmediumi(aspreviouslydenedforSANS)andabsorp-tionhasbeenignored.Sinceusuallyn1,neutronsareexternallyre ectedfrommostmaterialswithSnell'slawgivingthecriticalanglebelowwhichtotalre ectionoccurscosc=n2 n1(45)Forneutronsincidentonthesurfaceofamaterial(e.g.waterorsilicon)fromair(whichhasarefractiveindexverycloseto1)wecanobtainasimplerelationshipbetweenthecriticalangle,neutronwavelengthandscatteringlengthdensityofthematerialc=r  (46)17 Itisimportanttonotethattheonlychangeinwavevectorisinthezdirection(perpendiculartotheinterface)andhenceaspecularre ectivityexperimentmeasuresthescatteredintensityasafunctionofqz=2kz.Assuch,there ectometryexperimentprovidesinformationaboutstructureperpendiculartotheinterface.Itispossibletomeasurethere ectionatnon-specularanglestoextractinformationaboutthein-planestructureofthesample.Sucho-specularre ectionisbeyondthescopeofthisintroductionsothereaderisencouragedtoconsulttheliteratureformoreinformation.5.1.1ClassicalOpticsIthasbeenshownthatthesamelawsapplyforthere ectionandrefractionofneutronsasforanelectromagneticwavewithitselectricvectorperpendiculartotheplaneofincidence(s-wave).Thusthere ectivityisgivenbyFresnel'slawwhereforcthere ectivityR=1andforcR=jrj2=n0sin0�n1sin1 n0sin0+n1sin1(47)TheFresnelcalculationcanbeextendedtothecaseofathinlmattheinterfacegure11(b).Abeamincidentonsuchasystemwillbemultiplyre ectedandrefractedattheinterfacesbetweenthelayers.Takingintoaccountthephasechangesthatoccur,there ectionandrefractioncoecientsforeachpairofadjoiningmediamaybecalculatedbyaninnitesumofamplitudesofthere ectedandrefractedrays.ForasinglethinlmofthicknessdthisleadstoanexactequationfortheinterferencefromthelmR=jrj2=r01+r12e2i1 1+r01r12e2i12(48)whererijistheFresnelre ectioncoecientatinterfaceijgivenbyrij=pi�pj pi+pj(49)withpj=njsinjandj=(2=)njdsinj(theopticalpathlengthinthelm).Thisapproachcanbeextendedeasilytothreeorfourdiscretelayers,butbeyondthatlevelofcomplexityamoregeneralsolutionisrequired.OnesuchstandardmethodisthatdescribedbyBornandWolfwhere,onapplyingtheconditionthatthewavefunctionsandtheirgradientsbecontinuousateachboundary,acharacteristicmatrixforeachlayercanbederivedsuchthatforthejthlayerMj=cosj�(1=pj)sinj�pjsinjcosj(50)Theresultingre ectivityisthenobtainedfromtheproductofthecharacteristicmatricesMR=[M1][M2]:::[Mn]byR=(M11+M12ps)pa�(M21+M22)ps (M11+M12ps)pa+(M21+M22)ps(51)whereMijarethecomponentsofthe22matrixMR.18 alternativeistodescribetheroughnessordiusenessofaninterfacebyusingalargenumberofverythinlayerswithaGaussiandistributionofscatteringlengthdensityinequations(50)and(51).Thetreatmentinequation(53)waspreferredasitwasmuchlesscomputationallyintensive.However,giventheincreaseincomputingpowerinrecentyears,itisprobablynowpreferredtousethe\exact"solution.5.1.3Kinematic(Born)ApproximationItwasstatedintheintroductiontothissectionthattheBornapproximationdoesnotholdbecausewearenotinaweakscatteringregimewhencriticalre ectionisconcerned.However,ifwelookathighenoughvaluesofq,thenthescatteringisweakandtheBornapproximationholds.Asforsmallanglescattering,thedierentialcrosssectionisgivenbytheFouriertransformofthescatteringlengthdensitydistributionoverthewholesample.Takingintoaccountthespeculargeometrysuchthatonlyqzvarieswecanobtainanexpressionforthere ectivityR(qz)=162 q2zj^(qz)j2(57)where^(qz)istheone-dimensionalFouriertransformofthescatteringlengthdensityprolenormaltotheinterface.ThiscanalsobeexpressedintermsofthescatteringlengthdensitygradientthusR(qz)=162 q4zj^0(qz)j2(58)Theaboveexpressioncanthenbeusedtocalculatethere ectivityeasilyfromagivenscatteringlengthdensityprole.Whilstitislimitedtotheweakscatteringregime,therearemanysystemswherethisisvalid.Inparticulartheadsorptionofmolecules(e.g.surfactants,polymers)attheair-waterinterfacewherethescatteringlengthdensityofthewatercanbesettomatchthatofair.Thevaliditycanbeextendedtonon-zerobulkcontrastsystemsusingtheDistortedWaveBornApproximationorsimilartechniques.6AnalysisofRe ectometryDataTheanalysisofre ectometrydataconsistsofgeneratingascatteringlengthdensityprole,calcu-latingthere ectivityusingoneofthemethodsdescribedpreviouslyandcomparingtheresulttothere ectivityproleobtainedfromexperiment.Anexampleofcalculatedre ectivityprolesisshowningure12AsaresultofthelossofphaseinformationontakingthemagnitudeoftheFouriertransform,agivenre ectivityprolecannotbeuniquelydescribedbyasinglescatteringlengthdensityprole.Howeverthisdeciencycanbeovercomethroughtheuseofmultiplecontrasts.Indeed,ifonecanmeasurethesamesystemwhilstchangingonlythescatteringlengthdensityofeitherbulksideoftheinterfacialthenthephaseinformationcanberecoveredandthedatainvertedtogettheSLDprole.Thiscanbeachievedbychangingthesolvent(e.g.byexchangingH2OforD2O)aslongas20 7RecommendedReading7.1NeutronScattering\NeutronScattering-APrimer"byRogerPynn(LAUR-95-3840LosAlamosScience,Vol.19,1990.)http://www.mrl.ucsb.edu/~pynn/primer.pdf\IntroductiontoThermalNeutronScattering"byG.L.Squires(CambridgeUniversityPress,1978)Thisisanexcellentbookifyouwantthenitty-grittyofscatteringtheory.ItisnowavailablefromDoverPublicationsandatthetimeofwritingisonly$12fromAmazon.com7.2SmallAngleNeutronScattering\TheSANSToolbox"byBoualemHammouda-availableasaPDFfromtheNCNRwebsite.http://www.ncnr.nist.gov/staff/hammouda/the_SANS_toolbox.pdfTheNCNRSANSwebsitecontainstutorialsandtoolsrelatingtoSANSaswellasinformationabouttheNCNRSANSinstruments.http://www.ncnr.nist.gov/programs/sans/7.3Re ectometryTheNCNRRe ectometrywebsitehastutorialsandtoolsrelatingtore ectometryalongwithinformationabouttheNCNRre ectometryinstruments.http://www.ncnr.nist.gov/programs/reflect/8AcknowledgementsThisintroductionisanamalgamofmaterialfromanumberofsources.ThesectiononneutronscatteringwasbasedheavilyonsectionsinSquiresandBaconandthosetwobooks(Squiresinparticular)willrewardthededicatedreader.ThesectiononsmallangleneutronscatteringwasbasedonasetofpowerpointslidespresentedbyStevenKlineoftheNISTCenterforNeutronResearchatprevioussummerschools.Thesectiononneutronre ectometrywasbasedona1990reviewarticlebyBobThomasofOxfordUniversityandJePenfoldoftheISISneutronfacility.9ReferencesThesearevariousbooksandpapersthatrelatetothematerialpresentedhereandfallintothe\extendedreading"category.Thisisbynomeansanexhaustivelistandthereaderisencouraged22 ARadiusofGyrationofSomeHomogeneousBodiesSphereofRadiusRR2g=3 5R2SphericalshellwithradiiR1�R2R2g=3 5R51�R52 R31�R32EllipsewithsemiaxesaandbR2g=a2+b2 4Ellipsoidwithsemiaxesa,b,cR2g=a2+b2+c2 5PrismwithedgesA,B,CR2g=A2+B2+C2 12CylinderwithradiusRandlengthlR2g=R2 2+l2 12EllipticalcylinderwithsemiaxesaandbandheighthR2g=a2+b2 4+h2 12HollowcircularcylinderwithradiiR1�R2andheighthR2g=R21+R22 2+h2 1224