|
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Full Text | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
PAGE 1 MeasurementoftheProtonSpinStructure Functiong1WithDataFromtheEG1-DVCS Experiment By ChristopherE.Pedersen AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida InPartialFulfillmentoftheRequirementsfortheDegree BachelorofArtsinPhysics WrittenUndertheSponsorshipofProfessorDonColladay Sarasota,Florida May,2011 PAGE 2 AcknowledgementsIwouldliketothankDonColladayforsponsoringmythesis;G eorgeRuppeinerandMariana SendovaforsittingonmyBACCcommittee;KeithGrioenandS uchetaJawalkarforthe immenseamountofhelptheygavemewithmythesis;myparents DaveandSuewhohave alwaysbeenthereforme;andmyfriendsBenReinhold,ShaqKa tikala,DolanCochran, RikkiMiller,MikeDexter,TomMcKay,SarahIacobucci,JenZ immermanandmanyothers whohavemademytimeatNewCollegesospecial. i PAGE 3 Contents1Introduction 1 1.1Nucleons......................................11.2TheExperiment.................................2 2Theory 10 2.1Scattering....................................102.2Electron-muonScattering......................... ...14 2.3KinematicsofDeepInelasticScattering............. .......15 2.4StructureFunctions.............................. .18 2.5Therstmomentof g 1 andsumrules.....................21 3DataAnalysis 23 3.1DilutionFactor.................................. 26 3.2BackgroundParticleCorrections................... .....30 3.3Calculating g 1 ..................................33 4ComparisontoOtherDataandConclusions33 ii PAGE 4 ListofFigures 1ThomasJeersonNationalLaboratoryandtheConstantElec tronBeamAcceleratorFacilityfrom[1]........................... ..3 2ASuperconductingradio-frequencycavityfromJLab[2].. .........4 3Thedynamicsofastandingwavefrom[3].Thegureshowsthe amplitudeof thetwowavescreatingthestandingwaveandtheamplitudeof thestanding waveatdierenttimes.Theverticaldashedlinesarepositi onsofantinodes andtheverticaldottedlinesarepositionsofnodes....... ........5 4Theideabehindalinearacceleratorwithanoscillatingel ectromagneticeld from[4].IntheCEBAFthelengthsareallthesamebecausethe particlesvelocityisalreadyveryclosetothespeedoflightandd oesnotchange signicantly.....................................6 5AschematicofthebeammonitoringdevicesinhallBfrom[5] .OneBPM andoneharparelocatedfurtherupthebeamlineandarenotsh own....7 6Ammoniabeadsinsidethetargetinsert................. ....8 7AschematicofthetargetassemblyinsidetheCLASdetector from[6].The dashedlinesrepresentthedetectortorusmagnetpositionp rojectedontothe cutofthetarget..................................9 8TheCLASdetector.Thetorusmagnetsareyellow,thedriftc hambersarein blue,theCherenkovcountersareinmagenta,thescintillat ioncountersare inredandtheelectromagneticcalorimetersareingreen... .........10 9DierentialScatteringCrosssection................. .....11 10Electron-muonscatteringdiagram.Theelectronhasinit ialandnal4momentumkand k 0 andthemuonhasinitialandnal4momentumpand p 0 ..........................................14 11KinematicsofDeepInelasticScattering.............. .......15 iii PAGE 5 12Apolarizedelectroninteractingwithapolarizedproton .Theinitialspins areshownin(i).Theelectronemitsaphotonofspinoneandri psspinsin (ii).Thespincongurationafterthequarkabsorbstheprot onisshownin (iii).........................................20 13Countsforthetoptargetwithbeamandtargetpolarizatio nnegative.The xaxisis Q 2 ,theyaxisisWandthezaxisisnumberofcounts........24 14Materialsinthebeamlinefrom[7].................... ....27 15Dilutionfactorforthetoptargetforarangeof Q 2 binsplottedagainstW forpointswithanuncertaintyoflessthan0.05........... ......29 16ThedynamicsofCherenkovradiationfrom[8]........... ......31 17ACherenkovspectrumwithnumberofphotoelectronsdetec tedonthex axisandnumberofparticlesontheyaxis.Thereddotsarethe electron signal,thecyandotsarethetabove2photoelectrons,theb luedotsare thepionsdetectedusingtheelectromagneticcalorimetera ndothercuts,and thebrownsquaresarethispionsignalscaledtothedierenc ebetweenthe electronsignalandthet.[5]......................... ..37 18Theasymmetrieswiththepolarizationanddilutionfacto rtakenintoaccount areshowninblack.Theasymmetrieswiththepolarization,d ilutionfactor, pairelectroncorrection,andstandardpioncorrectionare showninred.The asymmetrieswiththepolarization,dilutionfactor,paire lectroncorrection, andtotalpioncorrectionareshowninblue.Allofthesearef orthetoptarget.38 19Theenergyofthedetectedelectronforbinswithcounts.. ..........38 20 g 1 forthetoptargetwith A 2 =0inblackand A 2 = p R inblue.......39 21 g 1 with A 2 =0forthetoptargetinblackandforthebottomtargetinred. 39 22 g 1 withthetoptargetinblackandforthebottomtargetinred,an ddata fromEG1bwithapproximatelythesame Q 2 valueplottedingreen.The valuesfortheEG1bexperimentarefrom[9]............... ...40 iv PAGE 6 Abstract TheEG1-DVCSexperimenttookplaceinHallBofThomasJeers onNational AcceleratorFacilityfor128daysin2009.Itscatteredlong itudinallypolarizedelectrons fromalongitudinallypolarized 1 4 NH 3 target.Thedatahasbeencalibratedandis readyforanalysis.Fromthisdatathespinstructurefuncti on g 1 canbecalculated. Thisstructurefunctiongivesawaytoobservetheinternals pinstructureoftheproton. v PAGE 7 1Introduction1.1NucleonsProtonsandneutronsarethetwotypesofparticleswhichmak eupatomicnucleiand arereferredtoasnucleons.Nucleonsarenotthemselvesfun damentalparticles.Theyare hadronswhicharecompositeparticlesconsistingofquarks andgluons.Quarksandgluons arefundamentalparticlesofthestandardmodelofparticle physicsandtheonlyparticles whichinteractthroughthestrongforce.Gluonsarethecarr iersofthestrongforcewhich holdthequarkstogetherintohadrons.Thestrongforceisun iqueinthatthestrengthof theforceincreasesasthequarksaremovedfurtherawayfrom eachother.Thisleadstoa phenomenaknownasconnementinwhichquarksdonotmovefre elyasindividualparticles, butarealwaysfoundinhadrons.Therearetwotypesofhadron s,baryonsandmesons. Valencequarksdeterminethequantumnumberofhadrons.Bar yonsconsistofthreevalence quarksandincludenucleonsandothermoreexoticparticles whichcanbecreatedusing particlesacceleratorsorwhencosmicraysinteractwithth eatmosphere.Mesonsconsistof quarkanti-quarkpairsandarecreatedasvirtualparticles insidenucleiwheretheyholdthe nucleonstogetherandcanalsobecreatedthroughcosmicray interactionsandinparticle accelerators.Thenatureofthestrongforcealsoleadstoas ymptoticfreedomwhereat smalldistancescalesorlargeenergyscalesthequarksinte ractweakly.Thismeansthat ifanucleonisprobedwithalowenergyparticlethequarkint eractionisstrongandthe quarksrespondasacoherentgroup,butathigherenergiesth equarksdonotrespondtothe probecoherentlyandthequarkisknockedoutofthenucleon. Whenthishappensquark anti-quarkpairsarespontaneouslycreatedfromvacuumbec auseittakeslessenergyfor thispairofquarkstobecreatedthantostretchthegluon.Th eoriginalnucleoncapturesa quarkandremainsasabaryonandaquarkanti-quarkryotoget herasamesonandcan createwhatiscalledajetifthereisenoughenergybysplitt ingandcreatingmorehadrons. Nucleonsarespin1/2particlesasarequarks.Theoriginalm odelofthenucleonconsistedofthethreevalencequarksheldtogetherbythegluon swiththespinofthenucleon comingentirelyfromthespinangularmomentumofthevalenc equarks,andnothaving contributionsfromthegluonspinsortheorbitalangularmo mentum.Inexperimentsatthe 1 PAGE 8 StanfordLinearAccelerator(SLAC)andtheEuropeanOrgani zationforNuclearResearch (CERN)itwasshownthatthespinofthevalencequarksactual lycontributesverylittle tothenucleonspin[10].Sincenucleonsarespin1/2conserv ationofangularmomentum requiresthat 1 = 2= S q + G + L z ; (1) where S q isthesumofthequarkspins, G isthesumofthegluonspins,and L z isthe totalorbitalangularmomentum.Ithasbeenshownbyunpolar izedleptonprotonscattering experimentsthatthevalencequarksarenottheonlyquarksw hichcanbeinteractedwith insidethenucleon.Therearealsoquarkanti-quarkpairscr eatedbysplittingofthegluons insidethenucleon.Theyrapidlyannihilatewitheachother ,butstillaectthemomentum ofthenucleon.Thesemayalsohaveanimpactonthespinofthe nucleonbothintheform ofspinandangularmomentum.1.2TheExperimentTheEG-1DVCSexperimentwasconductedattheThomasJeerso nNationalAcceleratorFacility(TJNAF)showningure1usingtheelectronbeam fromtheConstantElectronBeamAcceleratorFacility(CEBAF)andtheCEBAFLargeA cceptanceSpectrometer(CLAS)detectorinHallB.Theexperimentwasconductedo veraperiodof128daysin 2009. Theelectronsourcecreateselectronsusingthephotoelect riceect.Alaserisshown ontoaGaAscrystalcausingexcitationandemissionofelect rons.Thelaseriscircularly polarizedandbychangingthedirectionofpolarizationthe spinoftheelectroncanbe selected.Aftertheelectronsareproducedbytheelectrons ourcetheyareacceleratedupto anenergyof67MeVintheinjector.Usingtheequationforrel ativisticenergy E = mc 2 q 1 v 2 c 2 ; (2) andthatthemassoftheelectronis0.511MeV,thismeansthat whentheelectronsleave theinjectortheyarealreadytravelling99.99%thespeedof light. 2 PAGE 9 Figure1:ThomasJeersonNationalLaboratoryandtheConst antElectronBeamAcceleratorFacilityfrom[1]. Theelectronsarethenfedintotheaccelerator.TheCEBAFis madeupoftwostraight sectionswhicharelinearacceleratorsandtwo180degreetu rnsconsistingof9recirculating beamlines.Theelectronsareguidedthroughtherecirculat ingbeamlinesbymagnets.The electronscirculatethroughtheacceleratoruptovetimes reachinganalbeamenergy ofupto6GeV.Itiseectivelyalinearacceleratorshrunkto 1/10thitslength.Allof theelectronsareatextremelyrelativisticvelocitieswhi chmeansthatgroupsofelectrons ofdierentenergiescanbeacceleratedtogether,butmustb esplitandgothroughdierent beamlinestoberecirculated. Thelinearacceleratorsconsistofsuperconductingradiofrequencycavitiesshownin gure2.Thesearemadeofniobiumwhichiscooledtosupercon ductingtemperatures usingliquidhelium.Theaccelerationisaccomplishedbyan oscillatingelectromagnetic eldwhichresonatesinsidethecavities.Theelectromagne ticeldoscillatesinastanding wavewhichmeanstheamplitudehasastationaryspatialdepe ndence.Standingwaveshave nodesatwhichtheamplitudeisalwayszeroandanti-nodesat whichtheamplitudereaches itsmaximumasseeningure3.Thecavitiesaretunedsuchtha tasagroupofelectronsis passingeachanti-nodeitswitchesfrombeingathighpotent ialtobeingatalowpotential sothattheelectronsareacceleratedtowardsthenextantinodeasshowningure4.The 3 PAGE 10 Figure2:ASuperconductingradio-frequencycavityfromJL ab[2]. cavitiesintheCEBAFacceleratorconsistof5cells,are0.5 mlongandoperateat1497 Mhz[1].Thecavitiesaregroupedintocryomoduleswhicheac hcontain8cavitiesandthere are20cryomodulesineachlinearaccelerator.Thecavities haveanacceleratinggradientof 7.5MV/m.Sincethereare160cavitieswhichare0.5mlongine achlinearacceleratorhas agradientof600MVandthetotalacceleratingpotentialis6 GVafter5passesthrough bothaccelerators.Toprovidethebeamtothethreeseparate hallssimultaneously499Mhz rfderectingcavitiesareused.Thesesendeverythirdbunch ofthebeamtoeachhall. OncethebeamissenttohallBthereareseveraldeviceswithi nthehallusedfor monitoringthebeamasshowningure5.Thebeampositionmon itors(BPMs)constantly measurethepositionandintensityofthebeamat1Hz.Theele ctronbeamprolemonitors calledharpsaremadeupofverythinwiresoftungsten(20and 50 m )andiron(100 m ) [6].Whenmeasurementsaretakenthesewiresaremovedthrou ghthebeamlineand thescatteredelectronsaredetectedinaCherenkovcounter .Thismeasurementdisturbs thebeamandcanonlybedonewhentheexperimentisnottaking data.Tominimize 4 PAGE 11 Figure3:Thedynamicsofastandingwavefrom[3].Thegures howstheamplitudeofthe twowavescreatingthestandingwaveandtheamplitudeofthe standingwaveatdierent times.Theverticaldashedlinesarepositionsofantinodes andtheverticaldottedlinesare positionsofnodes.downtimefromtakingdatathismeasurementisonlydonewhen therearechangestothe beamdeliveredtothehallorifthereseemstobeaproblemwit htheelectronbeam. AMollerpolarimeterisusedtomeasurethebeampolarizatio n.TheMollerpolarimeter measurestheasymmetryinpolarizedelasticelectron-elec tronscattering.Thetargetofthe Mollerpolarimeterconsistofa25 m permendurfoilwhichisanalloyof49%cobalt,49% iron,and2%vandium.Itispositionedata20 angletothebeamlineandtheelectrons inthelmarepolarizedusingamagneticeld.Theelectrons whichscatteroofthislm areseparatedfromthebeambymagnetsanddetectedintwodet ectorsoneithersideofthe beamline.Thepolarizationisthencalculatedusingthekno wnpolarizationoftheelectrons inthelmandthescatteringparameters.Sincemeasurement swiththeMollerPolarimeter 5 PAGE 12 Figure4:Theideabehindalinearacceleratorwithanoscill atingelectromagneticeldfrom [4].IntheCEBAFthelengthsareallthesamebecausethepart iclesvelocityisalready veryclosetothespeedoflightanddoesnotchangesignican tly. alsointerferewiththebeamtheyareonlytakenperiodicall y. Locateddownstreamfromthedetectorandtargetattheendof thebeamlineisthe Faradaycup(FC)whichconsistsof4000kgofleadsupportedb yinsulatingceramicstandos insideavacuumchamber.Itisusedtocollectelectronsandm easurethecurrentrunningto groundandthereforethebeamcurrent.TheFaradaycupsigna lissortedbybeamhelicity andcanbeusedtomeasureandcorrectfordierencesinbeamc urrents. Thetargetusedinthisexperimentwasmadeofsmallammonia( 147 N 11 H 3 )beads.These canbeseeninsidethetargetinsertinFigure6.Thehydrogen atomswhichconsistofone protonarepolarizedusingatechniquecalledDynamicNucle arPolarization(DNP).In thistechniqueunpairedelectronsmustbeinducedintheamm oniabyionizingradiationat temperaturesaround80Kusingeitherthe20MeVelectronbea mattheStanfordUniversity SUN-SHINEfacilityorthe38MeVelectronbeamattheTJNAFFr eeElectronLaser[6]. Theammoniaisstoredinliquidnitrogenuntilitisusedfort hetarget. Topolarizethetargetthetemperatureisdroppedto1Kanda5 Tmagneticeldis applied.Thelargemagneticeldcreatesasignicantdier enceinenergiesbetweenan 6 PAGE 13 Figure5:AschematicofthebeammonitoringdevicesinhallB from[5].OneBPMand oneharparelocatedfurtherupthebeamlineandarenotshown electronwithspinparallelandantiparalleltothemagneti celd.Theratioofthespins canbefoundusingtheMaxwell-Boltzmanndistributionanda ttheverylowtemperature atwhichthetargetiskeptalmostalloftheelectronsareint helowerenergystate.This polarizesthefreeelectronspinsalmostcompletely.Micro waveswithanenergynearthe dierenceintheenergyforthetwoelectronspinstatesareu sedtocausetransitionswhere thespinofbothanelectronandanearbyprotonareripped.Th eelectronreturnstothe lowerenergyspinstateveryrapidlywhiletheprotondoesno tandcanbeusedtoripthe spinofothernearbyprotons.Thisisusedtocreateanoveral lpolarizationintheprotons. Additionalirradiationfromtheelectronbeamoftheaccele ratorfreeshydrogenatomsfrom theammoniawhichdecreasetheabilitytopolarizethetarge t.Annealingthetargetat 80-100Kallowsthesefreehydrogenatomstorecombinewitht heammoniaoroutgas. Afterthisprocessthetargetcanberepolarizedtonearit's originalmaximum,butafter toomanycyclesofannealingthepolarizationcannotbereco vered. Allofthesystemsnecessarytopolarizeandmeasurethepola rizationofthetargetare 7 PAGE 14 Figure6:Ammoniabeadsinsidethetargetinsert. includedinapolarizedtargetsystemwhichisinsertedinto thefrontoftheCLASdetector asshowninFigure7.Thesesystemsincludetherefrigeratio nsystemnecessarytokeep thetargetat1K,the5Tmagnet,themicrowavesource,anNMRs ystemtomeasurethe polarizationofthetarget,andthetargetinsert. TheCLASdetectorcontainsmanydierentlayersofdierent typesofdetectorsas showninFigure8.Thedetectoriscutinto6equivalentsecti onsbythe6superconducting magnets.Thesemagnetsprovideamagneticeldwhichbendst hepathofchargedparticles movingthroughthem.Thisbendinthepathcanthenbeseenbyt heotherdetectorsand usedtodeterminethechargeandmomentumoftheparticle.Th eshapeofthemagnets causesthemagneticeldtobestrongerforforwardangles,w eakerforbackwardangles, andzeroatthecenterofthedetectorinthetargetarea. Thedriftchambersmeasurethetrajectoriesandmomentumof chargedparticles.They containagaswhichisionizedbypassingchargedparticles. Throughthisgaswiresare strungandarekeptatdierentpotentialstocreateanelect riceld.Theelectronsfrom theionizedparticlesinthegasareacceleratedtowardthew ireswithhigherpotentialwith astrongenoughpotentialtocausethemtoionizemorepartic lesinthegas.Thiscascade 8 PAGE 15 Figure7:AschematicofthetargetassemblyinsidetheCLASd etectorfrom[6].Thedashed linesrepresentthedetectortorusmagnetpositionproject edontothecutofthetarget. ofionizedelectronscreatesameasurablecurrentinthewir esclosesttowheretheoriginal chargedparticlepassed.Thepositionsofthewireswhichha dacurrentcanthenbeusedto ndthepositionofthechargedparticleinthechamber.Thed istanceofclosestapproachto eachwirecanalsobecalculatedusingthetotaltimeofright fromthescintillationcounters andthechargedriftvelocity.Thereare3layersofdriftcha mbersinthedetectoratdierent radialdistancesfromthetargetwhichareusedtoreconstru ctthepath.Oncethepathof theparticleinthemagneticeldisknown,themomentumcanb ecalculated. Thescintillatorcountersemitelectromagneticradiation whenachargedparticlepasses throughthem.Theamountofenergydepositedinthescintill atorcountersisdeterminedby theamountoflightproduced.Thiscanbeusedforparticleid entication.Thescintillator countersarealsousedasanaccuratewaytomeasurethetimeo frightofaparticlewhich canbeusedtocalculatethevelocityandthereforethemassp rovidedthemomentumis known.TheCherenkovcountersareusedtodistinguishbetwe enelectronsandnegative pionsandaredescribedinsection3.2.Theelectromagnetic calorimeterconsistsoflayers ofleadandscintillatormaterial.Itmeasurestheenergyde positedbytheparticleandthe depthinthedetectoratwhichtheenergyisdepositedtodete rminetheenergyandtype ofparticle.Itcandistinguishbetweenelectronsandnegat ivepionsbecauseelectronsloose 9 PAGE 16 Figure8:TheCLASdetector.Thetorusmagnetsareyellow,th edriftchambersarein blue,theCherenkovcountersareinmagenta,thescintillat ioncountersareinredandthe electromagneticcalorimetersareingreen.theirenergyprimarilythroughbremsstrahlungradiationa ndnegativepionsprimarilyloose theirenergythroughionization[5].2Theory2.1ScatteringThequantitywhichisusedtodescribescatteringisthedie rentialscatteringcrosssection d d n .Thequantitybiscalledtheimpactparameterandisthedist anceoftheparticle's originalpathfromthecenterofthescattering.Itcanbesee nfromFigure9forclassical 10 PAGE 17 d d d b d n Figure9:DierentialScatteringCrosssection. scatteringifparticlespassthroughanarea d = j bdbd j ; (3) thentheyscatterthroughasolidangle d n= j sin( ) dd j : (4) Thismeansthatthedierentialcrosssectionis d d n = j b sin( ) db d j : (5) Thecrosssectionforscatteringofparticlesaectedbyqua ntummechanicsisdierent thantheclassicalcalculationabovebecausethereareinci dentandoutgoingwavesinstead ofclassicalparticles.Itdependsonbothdynamicinformat ionwhichiscontainedinthe scatteringamplitude( M )andthekinematicsinvolvedwhichiscontainedinthephase spacefactorandincidentrux.Thephasespacefactoristhek inematicinformationinside theintegralandtheincidentruxisthedenominatorofthete rminfrontoftheintegral 11 PAGE 18 inequation(6).Thescatteringcrosssectionforincidentp articles1and2scatteringand formingoutgoingparticles3,4,...,nis[11] = S 4 q ( p 1 p 2 ) 2 ( m 1 m 2 ) 2 Z j M j 2 (2 ) 4 4 ( p 1 + p 2 p 3 p n ) n Y j =3 2 ( p 2j m 2j ) ( p 0j ) d 4 p j (2 ) 4 ; (6) wherePlanck'sconstant h andthespeedoflightchavebeensetto1.Foreachgroup of s i identicalparticleswith i =1 ; 2 ;:::;m andmbeingthenumberofdierentparticles inthenalstate S = 1 ( s 1 !)( s 2 !) ::: ( s m !) isthesymmetryfactor. m j and p j arethemass andfour-momentumofthejthparticle. istheDiracdeltafunctionwhichisequalto 0everywhereexcepttheoriginandintegratesto1.Thefunct ion istheHeavisidestep functionwhichis0if x< 0and1for x> 0. p 0j istherstcomponentofthefourmomentumvector,ortheenergyEoverthespeedoflight,ofth ejthparticles.Thedelta function ( p 2j m 2j c 2 )isequivalenttotherelativisticmassenergyequation p 2j = m 2j c 2 Thisconditioniscalledon-shellandmustalwaysbesatise dforrealparticles.Realor observableparticlesarethosewhichenterorexitanintera ction.Particleswhichpropagate interactionsarevirtualparticleswhichdonothavetobeon -shell.Thestepfunction ( p 0j ) isequivalenttotheenergyofalloftheoutgoingparticlesb eingpositive,andthedelta function 4 ( p 1 + p 2 p 3 p n )implementsenergyandmomentumconservation.The crosssectioncanbeintegratedover p 0j and = S 4 q ( p 1 p 2 ) 2 ( m 1 m 2 ) 2 Z j M j 2 (2 ) 4 4 ( p 1 + p 2 p 3 p n ) n Y j =3 1 2 q ~p 2j m 2j d 3 p j (2 ) 3 ; (7) with p 0j = q ~p 2j m 2j ; (8) where ~p j isthethree-momentumofthejthparticle. Usingtheseformulas,thedierentialcrosssectionforela sticscattering,wheretheinitial andnalparticlesarethesamecanbecalculated.Inthelabf ramewhereparticle2is 12 PAGE 19 initiallyatresttheincidentruxinthelabframecanbewrit tenas q ( p 1 p 2 ) 2 ( m 1 m 2 ) 2 = m 2 j ~p 1 j ; (9) sothat = S 64 2 m 2 j ~p 1 j Z j M j 2 4 ( p 1 + p 2 p 3 p 4 ) p ~p 23 m 23 p ~p 24 m 24 d 3 ~p 3 d 3 ~p 4 : (10) Rewritingthedeltafunctionas 4 ( p 1 + p 2 p 3 p 4 )= E 1 + m 2 p 03 p 04 3 ( ~p 1 ~p 3 ~p 4 ) ; (11) andintegratingover ~p 4 sendsitto ~p 4 = ~p 1 ~p 3 .Usingequation(8) = S 64 2 m 2 j ~p 1 j Z j M j 2 E 1 + m 2 q ~p 3 2 m 23 p ( ~p 1 ~p 3 ) 2 m 23 p ~p 23 m 23 p ( ~p 1 ~p 3 ) 2 m 24 d 3 ~p 3 : (12) Switchingtosphericalcoordinateswhere d 3 ~p 3 = r 2 drd nandintegratingoverrthedierentialcrosssectionis d d n = S 64 2 j M j 2 ~p 23 m 2 j ~p 1 jj ( E 1 + m 2 ) j ~p 3 jj ~p 1 j E 3 cos j : (13) Mottscatteringisthecasewere m 2 E 1 inwhichcasetherecoilofthesecondparticle isnegligibleandthemagnitudeoftheincomingandoutgoing momentumforthemoving particleisthesame.ThecrosssectionforMottscatteringi s d d n = S 64 2 j M j 2 m 22 : (14) Tocalculatethecrosssectionthescatteringamplitudesne edtobecalculatedfortheFeynmandiagramoftheparticularinteractionusingwhatarecal ledFeynmanruleswhichcan befoundin[11]. 13 PAGE 20 k p e k 0 p 0 Figure10:Electron-muonscatteringdiagram.Theelectron hasinitialandnal4momentumkand k 0 andthemuonhasinitialandnal4momentumpand p 0 2.2Electron-muonScatteringElectron-muonscatteringissimilartotheelectron-proto nscatteringexceptthatthemuon isapointparticleunliketheprotonsothescatteringampli tudecanbecalculatedexplicitly. Becausethemuonhasamuchlargermassthantheelectron,rec oilisneglectedandthe crosssectioniswrittenintermsofthescatteringamplitud ebyequation(14).Tocalculate thescatteringamplitudeinthecasewheretheparticlesare notpolarizedandtheirnal spinsarenotdetected,theinitialspinsareaveragedandth enalspinsaresummedover. Usingthismethodthescatteringamplitudecanbewrittenas [11,12] hj M j 2 i = (4 ) 2 q 4 L e L p ; (15) where L e isthetensorassociatedwiththeelectronvertex, L p isthetensorassociated withthemuonvertex,and isthenestructureconstant.Inthecaseofelectron-muon scatteringwheretheelectronmassisneglectedtheseare L e =2( k 0 k + k 0 k ( k 0 k ) g ) ; (16) 14 PAGE 21 and L p =2( p 0 p + p 0 p ( p 0 p M 2 ) g ) : (17) Thescatteringamplitudecannowbecalculatedandis hj M j 2 i = (4 ) 2 q 4 8 ( k 0 p 0 )( k p )+( k 0 p )( k p 0 ) M 2 ( k 0 k ) = (4 ) 2 Q 4 16 M 2 E 0 E cos 2 ( = 2)+ Q 2 2 M 2 sin 2 ( = 2) ; (18) where q = k k 0 isthe4-momentumofthevirtualphoton, Q 2 = q 2 isthepositive 4momentumsquared,and isthelaboratoryscatteringangle.Inthelaboratoryframe wherethemuonisinitiallyatrest,thecrosssectioncannow bewrittenintermsofquantities measuredexperimentallyas[13] d d n = 4 2 E 0 3 EQ 4 cos 2 2 1+ 2 2 Q 2 tan 2 2 ; (19) whereEistheinitialenergyoftheelectron, E 0 isthenalenergyoftheelectron,and = E E 0 istheenergyofthevirtualphoton. 2.3KinematicsofDeepInelasticScattering X k k' q P e e' W Figure11:KinematicsofDeepInelasticScattering. QuantumElectrodynamics(QED)saysthatthelowestorderel ectromagneticinteraction betweenanelectronandanucleonismediatedbyavirtualpho ton.Deepinelasticscattering 15 PAGE 22 allowsprobingoftheinsideofanucleon.Thedeepindeepine lasticscatteringmeansthe virtualphotonhasenoughenergytointeractwithasinglequ arkinsidethenucleon.This isequivalenttothephotonhavingasmallenoughwavelength toresolvetheinsideofthe protonandhappenswhen Q 2 M 2 .Inelasticmeansthatthephotontransfersenough energytocausethenucleontobreakintomultiplehadronswh ichhappenswhen W 2 M 2 Ininclusivemeasurementsonlytheoutgoingelectronisdet ectedandthekinematicsofthe collisionarecalculatedfromtheinitialenergyE,thenal energy E 0 ,angleofderection oftheelectroninthelabframe,andthemassofthenucleonM. Ingure11,kisthe4momentumoftheincomingelectron,Pisthe4-momentumofthe incomingnucleonwhich is(M,0,0,0)inthelabframe,qisthe4-momentumtransfered bythevirtualphoton, k 0 isthe4-momentumoftheelectronaftertheinteraction,and Wisthetotalmassofthe recoilingsystem.Themassoftheelectronisignoredinthef ollowingcalculationsbecause theelectronsareatextremelyhighenergies.Importantqua ntitieswhichcanbecalculated fromexperimentaldataare = P q M = E E 0 ; (20) whichistheelectron'senergylossinthelabframeandtheen ergyofthevirtualphoton, Q 2 = q q =4 EE 0 sin 2 2 ; (21) where isthescatteringangleinthelabframe, W 2 =( P + q ) 2 = M 2 +2 M Q 2 ; (22) and x = Q 2 2 P q = Q 2 2 M (23) inthelabframe.InwhatiscalledtheBreitreferenceframe =0sothevirtualphotoncarriesnoenergy.Thisframecanalwaysbefoundbecause inthelabframe Q 2 = 4 EE 0 sin 2 2 = ~q 2 2 0.Becausethevirtualphotonhasnoenergyifitinteractswi th anasymptoticallyfreequark,themomentumofthequarkmust bethesamebeforeand 16 PAGE 23 aftertheinteractionso p qf ( z )= p qi ( z )+ Q = p qi ( z ) ; (24) where p qi ( z )and p qf ( z )aretheinitialandnalmomentumofthestruckquarkinthez direction.Thisrequiresthat p qi = Q 2 forthequarktointeractwithavirtualphoton.In thisframetheproton4-momentummustbe P i =( M j ~q j Q ; 0 ; 0 ; P q Q )[13].Thismeansthat x = Q 2 2 P q = Qp qi ( z ) P q = p qi ( z ) P i ( z ) ; (25) orthatxisthefractionofthelongitudinalmomentumofthep rotoncarriedbythestruck quarkintheBreitframe. Fromelectron-muonscatteringthecross-sectionforelect ron-nucleonscatteringbysingle photonexchangecanbewrittenas d 2 d n dE 0 = 2 2 Mq 4 E 0 E L W ; (26) where W isthehadronictensorand L istheleptonictensor.Thesetensorscanbe separatedintosymmetric(s)andantisymmetric(a)parts.T heantisymmetricpartofthe hadronictensordependsonthenucleonspin S N andis[14] W ( S N )= W ( s ) + iW ( a ) ( S N ) ; (27) andtheantisymmetricpartoftheleptontensordependsonth eelectronspin S l andis L ( S l )=2[ L ( s ) + iL ( a ) ( S l )] : (28) Togetherequations(26),(27),and(28)give d 2 ( S N ;S l ) d n dE 0 = 2 Mq 4 E 0 E [ L ( s ) W ( s ) L ( a ) ( S l ) W ( a ) ( S N )] : (29) Thesymmetricpartoftheleptonictensorissimplyhalfofth eleptonictensorforthe 17 PAGE 24 unpolarizedcasebecauseitissummedovernalspins,butno tinitialspinsandis L ( s ) = k k 0 + k 0 k g ( k k 0 m 2e )(30) ,where g isthemetrictensor.Theantisymmetricpartoftheleptonic tensoris L ( a ) ( S l )= m e r S l ( k k 0 ) r ; (31) with r beingtheLevi-Civitatensor.Themostgeneralformpossibl eforthesymmetric hadronictensoraftersimplicationis 1 2 M W ( s ) = q q q 2 g W 1 ( x;Q 2 )+ 1 M 2 P P q q 2 q P P q q 2 q W 2 ( x;Q 2 ) ; (32) andtheantisymmetrictermcanbewrittenas 1 2 M W ( a ) ( S N )= r q MS r N G 1 ( x;Q 2 )+[( P q ) S r N ( S N q ) P r ] G 2 ( x;Q 2 ) M ; (33) where W 1 W 2 G 1 ,and G 2 areformsofthestructurefunctions.Theyrelatedtothe standardformsofthestructurefunctionsby F 1 MW 1 ( x;Q 2 ) ; (34) F 2 W 2 ( x;Q 2 ) ; (35) g 1 ( P q ) 2 G 1 ( x;Q 2 ) ; (36) and g 2 ( P q ) G 2 ( x;Q 2 ) : (37) 2.4StructureFunctionsFromtheinterpretationofxintheprevioussectionitiscle arthataquarkandthevirtual photoncanonlyinteractifthemomentumofthequarkintheBr eitframeobeysequation 18 PAGE 25 (24).Ifafunctionf(x)istheprobabilityofaquarkofravor fhavingamomentumbetween xand x + dx anditisassumedthatthequarksdonothaveasubstructureof theirown, theunpolarizedstructurefunction F 1 canbeexpressedas F 1 ( x )= 1 2 X f e 2f f ( x ) dx (38) where e i isthechargeofthequarkravor.Thisisfornotonlytheupand downvalence quarks,butalsofortheseaofup,down,andstrangequarks.T hequarkswhichareheavier thanstrangecanbeneglectedbecauseoftheirhighmasses.I ftheprobabilitydistribution forthemomentumsofthequarksisusedintheequationforela sticscatteringforapoint particleitcanbeshownthatitisthesamefunctionaspresen tedintheprevioussection. Theotherunpolarizedstructurefunctionis F 2 ( x;Q 2 )=2 xF 1 ( x;Q 2 ) : (39) Thecrosssectionintermsofthestructurefunctionsis d dQ 2 d = 4 2 E 0 cos 2 ( = 2) Q 4 E 1 F 2 ( x;Q 2 )+ 1 M tan 2 ( = 2) F 1 ( x;Q 2 ) : (40) Inthepolarizedcasethequarkwhichthevirtualphotoninte ractswithissubjecttothe momentumconstraints,butmustalsohavethecorrectspinto interactwiththephoton. Figure12showsanelectronscatteringfromaprotonwiththe initialspinsparallelandanti parallel.Ifthespinoftheprotonwerecarriedentirelybyt heconstituentquarks,2would havespinparalleltotheprotonand1spinantiparallel.Bec auseoftherequirementsonthe spinofthequarksitwouldbeexpectedthattheelectroncoul dscatteroofeitherofthe twoquarkswiththeirspinparalleltoprotoniftheprotonan delectronspinsareparallel whereasiftheelectronandprotonspinareantiparallelthe electroncouldonlyscatterfrom oneofthequarks.Ifmorespecicmomentumdistributionfun ctions f ( x )and f # ( x )are denedastheprobabilitydistributionfunctionsforaquar kofravorfmomentumxinthe Breitframeandhavingitsspinparallelorantiparalleltot heprotonspinthepolarized 19 PAGE 26 Figure12:Apolarizedelectroninteractingwithapolarize dproton.Theinitialspinsare shownin(i).Theelectronemitsaphotonofspinoneandripss pinsin(ii).Thespin congurationafterthequarkabsorbstheprotonisshownin( iii). structurefunctionscanbewrittenas g 1 ( x;Q 2 )= 1 2 X i e 2i h f i ( x;Q 2 ) f # i ( x;Q 2 ) i = 1 2 X i e 2i f i ( x;Q 2 ) : (41) Thecrosssectioninthelabframecanbecalculatedintermso ftheofthepolarizedstructure functionforspinparallel d "" d n dE 0 = d d n dE 0 2 2 E 0 Q 2 E E + E 0 cos ( ) M g 1 ( x;Q 2 ) 1 M g 2 ( x;Q 2 ) ; (42) where = 2 Q 2 ,andantiparallel d "# d n dE 0 = d d n dE 0 + 2 2 E 0 Q 2 E E + E 0 cos ( ) M g 1 ( x;Q 2 ) 1 M g 2 ( x;Q 2 ) : (43) 20 PAGE 27 Thepolarizedcrosssectionis d d n dE 0 polarized = 1 2 d "# d n dE 0 d "" d n dE 0 (44) andtheunpolarizedcrosssectionis d d n dE 0 unpolarized = 1 2 d "# d n dE 0 + d "" d n dE 0 : (45) Bytakinganasymmetry,theacceptanceofthedetectorcanbe canceled.Thismakesit mucheasiertomakeaccuratemeasurements.Theasymmetryis A k ( x;Q 2 )= d "# d "" d "# + d "" = d polarized d unpolarized : (46) Finally,fortheprotonthepolarizedstructurefunctionca nbeexplicitlycalculatedinterms ofthehelicitydependentquarkmomentumdistributionsfor thedierentquarkravors g 1 ( x;Q 2 )= 1 2 X i e 2i f i ( x;Q 2 )= 4 18 u ( x;Q 2 )+ 1 18 d ( x;Q 2 )+ 1 18 s ( x;Q 2 ) : (47) 2.5Therstmomentof g 1 andsumrules Byintegratingthestructurefunction g 1 ( x;Q 2 )overxthetotalspincontributionforall momentaofthequarkscanbefound.Thisintegraliscallthe rstmomentof g 1 andis p1 ( Q 2 )= Z 1 0 g 1 ( x;Q 2 ) dx = 1 2 Z 1 0 X i e 2i f i ( x;Q 2 )= 4 18 u ( Q 2 )+ 1 18 d ( Q 2 )+ 1 18 s ( Q 2 ) : (48) Isospinsymmetryisanapproximatesymmetrywhichsaysthat thestrongforcetreatsup anddownquarksthesame.Thismeansthatthequarksinproton sandneutronscanbe treatedthesameunderexchangeofupanddownquarks.Anequa tionanalogoustoequation (48)canbewrittenfortheneutronas n1 ( Q 2 )= Z 1 0 g n 1 ( x;Q 2 ) dx = 1 2 Z 1 0 X i e 2i f i ( x;Q 2 )= 1 18 u ( Q 2 )+ 4 18 d ( Q 2 )+ 1 18 s ( Q 2 ) : (49) 21 PAGE 28 TheBjorkensumrulesaysthat[14] p1 ( Q 2 ) n1 ( Q 2 )= 1 6 ( u ( Q 2 ) d ( Q 2 ))= 1 6 g a ; (50) where g a istheaxialcouplingvectorconstant. g a =1 : 26canbecalculatedfrom decay measurementsonbaryonscontainingstrangevalencequarks .Anothersumrulewhichcan bederivedusingtheBjorkensumruleandequation(48)isthe Ellis-Jaesumrule.This saythatassuming s =0, p1 ( Q 2 )= 1 12 g a + 5 36 ( u + d ) : (51) Theassumptionthat s =0comesfromassumingthattheseaquarksareunpolarized. From[15]inthesimplequarkpartonwherethequarksareatre sttheupanddownquark distributionsshouldbe u = 4 3 ; (52) and d = 1 3 : (53) Givingaspinof1/2totheprotoncomingentirelyfromthequa rks.Thisdoesnotsatisfy equation(50).Thisdierencecanbeaccountedforbytherel ativisticconstituentquark modelwhichsaythequarkaremoving.Thismodelsaysthat 75%ofthequarksangular momentumiscomesfromthespinofthequarksandtheother 25%comesfromorbital angularmomentumofthemovingquarks.Thisgives u 1 ; (54) and d 0 : 25 : (55) 22 PAGE 29 From[16] u + d canalsobecalculatedfrom decaymeasurementsandwithQCD correctionstheEllis-Jaesumrulegivesatheoreticalval ueof p1 ( Q 2 =3 GeV 2 )=0 : 165 0 : 016 : (56) ExperimentsdonebytheEuropeanMuonCollaboration(EMC)[ 10]gave p1 =0 : 126 0 : 010( stat: ) 0 : 015( syst: )whichdoesnotagreewiththis.Alaterexperimentbythe E143collaborationatSLACmeasured[16] p1 ( Q 2 =3 GeV 2 )=0 : 133 0 : 010.TheSLAC experimentgivesnalvaluesforthespinsofthequarksof u =0 : 83 0 : 03, d = 0 : 43 0 : 03,and s = 0 : 10 0 : 03.Thisgivesatotalspincontributionfromthequarksof S q =0 : 145 0 : 0045whichmeansonlyabout30%ofthespinoftheprotoncomes from thespinofthequarksandalsothattheseaissignicantlypo larized.Theseresultsarenot expectedbythequarkpartonmodelortherelativisticquark partonmodelandhavebeen thefocusofmuchresearchintomorecomplicatedquarkmodel s.Therehavebeensome successeswiththesemodels,butamodelwhichperfectlyexp lainsthecontributionstothe spinoftheprotonhasnotbeenfoundandcalculatingmoreexa ctexperimentalvaluesfora largerrangeofkinematicsisimportanttohelpwiththecrea tion,testing,andtsofthese models.3DataAnalysisThedatausedwasreceivedwiththeeventsalreadyreconstru ctedandcutsappliedto removedmostofthepioncontaminationwhileleavingasmany electroneventsaspossible. Thesecutsstillleaveatailofthepiondistributionwhichn eedstobecorrectedfor.Thedata wasreceivedasthenumberofcountsbinnedinthevariablesW and Q 2 andwassortedby beamandtargetpolarization,andbytarget.Theexperiment hadatotalof4targets,only3 ofwhichwereuseddirectlyinthisanalysis.Thetopandbott omtargetswerethepolarized targets.TheyarebothAmmonia( NH 3 )inwhichthehydrogenatomshavebeenpolarized. Thereasonforhavingtwotargetswiththesamematerialisth attheabilitytopolarizethe targetdropsasthetargetisexposedtoradiationfromthebe amandmustbeannealed.By 23 PAGE 30 havingtwotargetstheycanbeswitchedwhentheattainablep olarizationofthersttarget dropstoomuchsothetargetsdonothavetobeannealedasofte n.Thethirdtargetwas acarbontargetwhichwasusedtocalculatethedilutionfact or.Thedilutionfactoristhe ratioofscatteringeventsfrompolarizedmaterialtoscatt eringeventsfromallmaterials. Therunnumberrangewasusedtondthetargetandbeampolari zations.Anotherpieceof informationthatwasprovidedwiththedatawastheintegrat edFaradaycupchargewhich isseparatedbybeampolarizationandisnecessarytonormal izethenumberofcountsfor dierentbeamcurrents. Thoughthedatathatthisanalysisstartswithisthenumbero fcountsbinnedinWand Q 2 ,thisisnotwasisactuallymeasuredbythedetector.Thequa ntitieswhichareactually measuredbythedetectoraretheoutgoingenergy E 0 andderectionangle ofthederected electron.Thebeamenergyisanotherexperimentalquantity whichismeasured,though notbythedetector.UsingtheseandtheprotonmassW,and Q 2 canbecalculatedusing equations(22)and(21). ThedataanalysiswasdoneusingRoot.Rootisaprogramandli brarycreatedfor particlephysicsdataanalysis.Rootisextremelyusefulfo rdataanalysisoflargeamounts ofdata.TheprogramitselfincludesaC++scriptandcommand lineinterpreterusedfor preformingthedataanalysis.Alongwiththisithasmanylib rarieswithalargevarietyof commandsusefulfordataanalysis.Italsohasagraphicuser interfacewheretheplotsare displayedandsomechangestotheplotscanbemade. 1 2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 0 100 200 300 400 500 600 700 3 10 q2vsww00 Entries 3.12219e+07Mean x 1.945Mean y 2.546 RMS x 0.7782RMS y 0.3953 q2vsww00 Entries 3.12219e+07Mean x 1.945Mean y 2.546 RMS x 0.7782RMS y 0.3953 vs WW Inclusive (ptarg = neg, beam = neg) 2 Q Figure13:Countsforthetoptargetwithbeamandtargetpola rizationnegative.Thex axisis Q 2 ,theyaxisisWandthezaxisisnumberofcounts. 24 PAGE 31 Anexampleofoneoftheplotsoftherawcountsisshowningur e13.Thestatistical uncertaintyonthesecountsis p N andthestatisticaluncertaintyisconsideredtobethe dominantsourceofuncertaintyintheexperiment,sothesys tematicuncertaintyisignored. Fromthenumberofcountsforthedierentpolarizations( N tb )wherethearrowinplace oftrepresentsthetargetpolarizationandthearrowinplac eofbrepresentsthebeam polarization,therawasymmetriesaresimply A raw = N ## ( "" ) ( Fc ) N #" ( "# ) N ## ( "" ) +( Fc ) N #" ( "# ) ; (57) whereFcistheratiooftheFaradaycupcurrentforbeampolar izationparalleltotarget polarizationoveranti-paralleltotargetpolarization.T hearrowsnotinparenthesisarefor onetargetpolarizationandthearrowsinparenthesisarefo rtheother.Theelectromagnetic interactionisnonparityviolating.Thismeanswhenthespi nofboththeelectronandproton areswitched,thecrosssectionisthesameandtheresultsfo rbothtargetpolarizationscan becombinedaftertheasymmetryiscalculated. Thenumberofcountsusedtocalculatetherawasymmetryisdi lutedbyseveralthings. Firstthebeamandtargetmaterialarenotcompletelypolari zed.Thebeampolarization P b =84 : 55wascalculatedusing[17]andtherunnumbersforthedata. Thiswasdoneby assumingthepolarizationwasconstantbetweenmeasuremen tsandweightingtheaverage bythenumberofruns.Thebeampolarizationwasassumedtobe constantbetweenruns becausethemeasurementsweremadewhentherewasreasontob elievethatthebeam polarizationhadchangedanditisassumedthatitwasstable betweenthesechanges.The targetpolarizationwascalculatedusing[18]whichhadtob esortedbytargetandby polarizationdirectionandthenaveragedovertherunrange used.Thesenumbersarethe averagepolarizationforthetargetovertherunperiod.The polarizationofthetargetis measuredregularlythroughouttherunandthetargetisswit chedwhenthepolarization decreasestoacertainvalue. 25 PAGE 32 3.1DilutionFactorThemajorcauseoftheasymmetrybeingdilutedisunpolarize dmaterialsinthebeamline. Thesematerialsincludealuminum"banjowindows",theKapt ontargetcup,thehelium surroundingthetarget,andthenitrogenatomintheammonia molecule.Thesematerials areunpolarizedandparticlesscatteredfromthemdonotcon tributetothenumerator oftheasymmetry,buttheydocontributesignicantlytothe denominator.Thenumber ofcurrentnormalizedcountsforbeamandtargetpolarizati onsparalleliswrittenas n + Beamandtargetpolarizationsanti-parallelarewrittenas n .Thenormalizednumberof countsfromthebackgroundiswrittenas n B andthesecountsareincludedin n + and n Thenormalizednumberofcountsfromallmaterialsduringam moniarunsiswrittenasas n a = n + + n ,andthecurrentnormalizedcountsfromjustthepolarizedt argetmaterials iswrittenas n Ppol .Theundilutedasymmetry A undil is A undil = n + n n + + n n B = A raw f d ; (58) where f d isthedilutionfactorandisequalto f d = n Ppol n A = n A n B n A =1 n B n A : (59) Theonlypolarizedmaterialinthebeamlinearethe3hydroge n( 11 H )ontheammonia. ji X meansanelementXwithiprotons,atomicweightj,andtheref orj-ineutrons.The crosssectionforthepolarizedmaterialwillbekeptsepara teandwillbewrittenas Ppol Forcalculatingthescattingfromthebackground,itwouldb eidealifrunswithanitrogen targetcouldbeused,buttherearedicultieswithnitrogen targets.Instead,acarbon ( 126 C )targetisused.Thenitrogen( 147 N )crosssection( N )canbeapproximatedasthe crosssection( D )of7deuterons( 21 H )(oneprotonandoneneutron)andsimilarlythe carbon( 126 C )crosssection( C )as6deuteronsandtheHelium( 42 He )crosssection( He ) as2deuterons.Thealuminum( 2713 Al )iscomposedof13protonsand14neutrons,but thecrosssectionoftheprotonandneutronwillbeassumedto beapproximatelythesame and = D 2 = P + Neu 2 P Neu isusedasanapproximation,where P isthecross 26 PAGE 33 sectionoftheprotonand Neu isthecrosssectionoftheneutron.Nowthecrosssection ofaluminumis Al 27 andthekapton( 126 C 22 11 H 10 147 N 2 168 O 5 )targetcupcrosssection K 382 .Sincetheammonia( 147 N 11 H 3 )targetmaterialconsistsofoneunpolarized nitrogenand3polarizedprotonsthetotalammoniacrosssec tionis A =14 +3 Ppol Figure14showsthematerialsinthebeamlinefordierentta rgetcongurations.The normalizednumberofcountsfromtheallmaterialinthebeam lineforaspecictarget isproportionaltothecrosssectionforeachmaterialinvol vedmultipliedbythenumberof molesofthatmaterialpercubiccentimetermultipliedbyth elengthofthatmaterialinthe beamline.Forammoniathismeans Figure14:Materialsinthebeamlinefrom[7]. n A / 0A l A A +( L l A ) 0He He + l Al 0Al Al + l K 0K K / [ l A 14 17 A +( L l A ) He + l Al Al + l K K ] + 3 17 l A A Ppol ; (60) 27 PAGE 34 where i isthemassdensityofthematerialiin gm cm 3 0i = i # nucleons where# nucleons is thenumberofnucleonspermoleculeofthatmaterialin # mols cm 3 l i isthelengthofmaterial ithatthebeampassesthrough,andListhelengthoftheheliu mthatthebeampasses throughwithnotargetmaterial.Forcarbonthetotalnormal izednumberofcountsfrom allmaterialinthebeamline( n C )is n C / 0C l C A +( L l C ) 0He He + l Al 0Al Al + l K 0K K / [ l C C +( L l C ) He + l Al Al + l K K ] : (61) Fortheammoniatarget,thenormalizednumberofcountsfrom thebackgroundis n B / [ l A 14 17 A +( L l A ) He + l Al Al + l K K ] ; (62) andthenormalizednumberofcountsfromthepolarizedmater ialsis n Ppol / 3 17 l A A Ppol : (63) Nowthedilutionfactoris f d =1 C n C n A ; (64) where C = l A 14 17 A +( L l A ) He + l Al Al + l K K l C C +( L l C ) He + l Al Al + l K K ; (65) becausethenormalizednumberofcountsforthesedierentm aterialsareproportionalto thecrosssectionwiththesameconstantofproportionality ,sothat n B = Cn c .Thelengths ofthetargetmaterialsotherthan l a andLcanbemeasuredstraightforwardlybyphysical measurements,buttheammoniaisnotonesolidpiece.Itisin theformofgranulesor beadswhichlltheentire1.5cmtargetcup.Thelength l A istheeectivelengthifthe granuleswereconsolidatedintoasolidpieceofmaterialan dtheratio l A 1 : 5 cm iscalledthe packingfraction.Therestofthetargetcupislledwithliq uidhelium.Thedilutionfactor cannowbecomputedusingthisequationandinformationfrom [7]shownintable1.The dilutionfactorforthetoptargetforarangeof Q 2 binscanbeseeningure15.Points 28 PAGE 35 Table1:Constantsforcalculatingthedilutionfactorfrom [7]. PropertyHeliumCarbonAluminumKaptonAmmoniaTopBottom Density( gm cm 3 )0.1452.1932.7001.4300.8660.866 Length(cm)L=2.010.3980.01660.00660.8600.910 Molsofnucleons cm 2 0.2910.8730.04480.00940.7450.788 C----0.7190.743 withuncertaintiesofmorethan0.05arenotshown.Pointswi thlargeuncertaintiesarenot shownatmanypointsthroughoutthisworkbecausethequanti tiesbeingcalculatedhave beencalculatedforotherexperimentsinsimilarkinematic ranges.Thedataproducedin thisworkisonlyusefulforareasinwhichithasasmalluncer taintyandmayimproveupon previousdata.Pointswithlargeruncertaintiesalsomakei tmorediculttoseetrendsin thedatawithsmalleruncertaintiesandsincethesevaluesa restillcalculatedandaresimply notplotted,theycanbeaddedatanytime.Afunctionwasmade forthispurposeandthe maximumuncertaintyplottedcanbechangedeasily. topdilu2 Entries 70919Mean 2.849RMS 0.1438 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 topdilu2 Entries 70919Mean 2.849RMS 0.1438 Top Dilution Factor Q^2 = 1.125 topdilu3 Entries 102877Mean 2.665RMS 0.2157 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 topdilu3 Entries 102877Mean 2.665RMS 0.2157 Top Dilution Factor Q^2 = 1.406 topdilu4 Entries 131427Mean 2.457RMS 0.3006 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 topdilu4 Entries 131427Mean 2.457RMS 0.3006 Top Dilution Factor Q^2 = 1.757 topdilu5 Entries 147981Mean 2.081RMS 0.4423 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 topdilu5 Entries 147981Mean 2.081RMS 0.4423 Top Dilution Factor Q^2 = 2.196 topdilu6 Entries 142300Mean 1.894RMS 0.5177 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 topdilu6 Entries 142300Mean 1.894RMS 0.5177 Top Dilution Factor Q^2 = 2.745 topdilu7 Entries 100994Mean 1.845RMS 0.4869 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 topdilu7 Entries 100994Mean 1.845RMS 0.4869 Top Dilution Factor Q^2 = 3.465 topdilu8 Entries 34314Mean 1.752RMS 0.4155 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 topdilu8 Entries 34314Mean 1.752RMS 0.4155 Top Dilution Factor Q^2 = 4.322 topdilu9 Entries 12716Mean 1.595RMS 0.3237 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 topdilu9 Entries 12716Mean 1.595RMS 0.3237 Top Dilution Factor Q^2 = 5.358 topdilu10 Entries 3132Mean 1.295RMS 0.1528 W 11.522.53 Ratio 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 topdilu10 Entries 3132Mean 1.295RMS 0.1528 Top Dilution Factor Q^2 = 6.697 Figure15:Dilutionfactorforthetoptargetforarangeof Q 2 binsplottedagainstWfor pointswithanuncertaintyoflessthan0.05. 29 PAGE 36 3.2BackgroundParticleCorrectionsMostofthebackgroundparticlescanbedistinguishedfrome lectronsbyapplyingcutsto thedata,butthedataisstillcontaminatedbyhighenergypi onsmisidentiedaselectrons andelectronsfromelectron-positronpairproductionfrom piondecays.Writingthecounts fromacontaminationas N ## ( #" ) c ,thecorrectedasymmetrycanbewrittenas A corr = ( N ## N ## c ) ( N #" N #" c ) ( N ## N ## c )+( N #" N #" c ) = A raw RA c 1 R ; (66) where R = N c N N c = N ## c + N #" c ,and A c = N ## c + N #" c N c Assaidbefore,themajorityofthepionsareremovedfromthe datausingcuts.The mainwaytoseparatepionsandelectronsinthedetectoristo usetheCherenkovcounter. Whenachargedparticleismovingthroughadielectricmediu mfasterthanthespeedof lightinthatmedium,itcreatesashockwavesimilartohowso methingmovingfasterthan thespeedofsoundcreatesasonicboom.Inamaterialwithani ndexofrefraction R ,the speedoflightis v c = c R andinatimetthephotonsmoveadistance x c = c R t isdened astheratiooftheparticlesvelocitytothespeedoflightc, = v p c ,where v p istheparticle velocity.Fromgure16itcanbeseenthattheangleofthecon eoflightfromthepath obeystherelation cos ( )= 1 R = c v p R ; (67) andtheparticleonlyemitsradiationwhen > 1 R .Bychoosingamaterialwithan appropriateindexofrefraction,particlesofacertainene rgycanbedistinguished.Inthe CLASdetector,perruorobutane( C 4 F 10 )withanindexofrefractionof1.00153isused.The energyoftheparticlewithmass m p is E p = m p c 2 p 1 2 ,Cherenkovradiationisonlyemitted when E p > R q 2 R 1 m p =18 : 1 m p : (68) Foranegativepionwhichhasmass0.1396GeVthisis2.52GeV, butforanelectronwhich hasmass0.000511GeVthisis0.00925GeV.Thismeansthatthe Cherenkovcountercanvery eectivelylteroutpionsfromelectronswhentheenergyof theparticleislessthan2.5GeV. 30 PAGE 37 Figure16:ThedynamicsofCherenkovradiationfrom[8]. TheCherenkovcounterusesphotomultipliertubestodetect thelightfromtheCherenkov radiationandmeasurethenumberofphotoelectronscreated inthephotomultipliertubes. Forthecontaminationcorrections,theanalysisfromasimi larpreviousexperimentin hallbofJLab(calledtheEG1bexperiment)[5]wasused.Figu re17showsanexampleof thespectrumfromtheCherenkovcounter. Thettotheelectronsabove2photoelectronsisthebestapp roximationoftheactual numberofelectrons.Thepionsignalscaledtothedierence betweentheelectrontand signalisthebestapproximationofthenumberofpionsconta minatingtheelectronsignal. ThecutontheCherenkovcounterforenergiesoflessthan3Ge Vis2photoelectronsso thesenumbersaresummedforpointsabove2photoelectronsa ndtheirratioisusedas thestandardcontaminationwhichistheratioofpionstoele ctronsforenergieslessthan 2.5GeV.Forgreaterthan3GeV,theCherenkovcounterisnote ectivefordistinguishing pionsfromelectronsandtheCherenkovcutisnotapplied.To calculatethecontamination inthisregion,theelectronandpiontaresummedovertheen tirespectrumandtheratio iscalledthetotalcontamination.Between2.5and3GeVthec utisstillapplied,butthe eciencydecreasessothecontaminationisbetweenthestan dardandtotalcontamination. 31 PAGE 38 Inthisanalysisthestandardcontaminationisusedandthed ierencebetweenthestandard andtotalcontaminationisaddedtotheuncertaintyforthep ointswithenergiesabove 2.5GeV.Forbothmethodsthe,ratiocanbewrittenas[5] R = e a + b + cp + dp ; (69) where istheangleofderectionoftheparticle,pisthemomentumof theparticle,and a,b,c,anddareconstantswhichcanbefoundin[5]andaredi erentforthestandardand totalcontaminationsandfordierentkinematicranges.Al so,sincethisratioissmall,as istheasymmetryofthecontamination, RA c istakentobezero. Thereisnowaytodirectlydierentiatescatteredelectron sfromelectronscreatedby electron-positronpaircreation.Toestimatethenumberof electronsfromthissource,the numberofpositronsneedstobefound,becauseeachelectron shouldhaveacorresponding positronwiththesamekinematics.Thisiscomplicatedbyth efactthattheacceptanceof thedetectorforparticlesofdierentchargesisdierentd uetothemagneticeldinthe detector.IntheEG1-DVCSexperimenttherewerenotveryman yrunsinwhichthecurrent ofthemagnetwasreversed,socalculationsofthisratiowou ldhavelargeuncertainties,but intheEG1bexperimentthereweremanymorerunswithopposit emagnetcurrents.The positronshaveallofthesamecutsappliedastheelectronse xceptofcoursethecharge, butthepioncontaminationisfarhigher.Thisisremovedina similarmethodtothe pioncorrectionforelectrons.Oncetheuncontaminatedpos itronspectrumisfound,the ratioistakenbetweenthepositronspectrumandtheelectro nspectrumintheopposite toruscurrentnormalizedforbeamcurrents.Thisisusedast hecontaminationduetopair symmetricelectronsintheelectronspectrum.Thiscontami nationalsofollowstheformof equation(69)andtheconstantscanbefoundin[5]. Theasymmetrieswiththepolarizationanddilutionfactort akenintoaccount,using noradiativecorrections,thestandardpionandpairelectr oncorrection,andthetotalpion correctionandpairelectroncorrectionareshowningure1 8.Byplottingtheenergiesfor eachbincalculatedusingWand Q 2 ,showningure19itcanbeseenwhereeachcorrection applies.Thetotalcorrectiondoesnotapplyformostpoints andiscloseto,butnotthe 32 PAGE 39 sameasthestandardcorrectionwhereitapplies.Inthisana lysisthedierencebetweenthe correctedasymmetryfortwocorrectionswasaddedtotheunc ertaintyintheareawhere thetotalcorrectionappliesandthestandardcorrectionwa susedeverywhere. 3.3Calculating g 1 Nowthatthecorrectedasymmetries A k havebeencalculated,theycanbeusedtocalculate thestructurefunction g 1 ( W;Q 2 ).Thestructurefunction g 1 ( W;Q 2 )intermsofthevirtual photonabsorptionasymmetries A 1 and A 2 is[5]and[16] g 1 ( W;Q 2 )= F 1 ( W;Q 2 ) 1+ r 2 ( A 1 + rA 2 ) ; (70) where r = p Q 2 and A k canbewrittenintermsofthevirtualphotonabsorptionasym metriesas A k = D ( A 1 + A 2 ) ; (71) where D = 1 E 0 E 1+ R =(1+2(1+ 1 r 2 )tan 2 ( 2 )) 1 ,and R 0 : 18istheratiooflongitudinal totransversevirtualphotonabsorptioncrosssections.Us ingthese,thestructurefunction intermsof A k is g 1 ( W;Q 2 )= F 1 ( W;Q 2 ) 1+ r 2 ( A k D +( r ) A 2 ) : (72) Thetermdependingon A 2 issmallbecausethe r issmalland A 2 isassumedtobe closetozero[16]. A 2 alsocanbeshowntohaveanupperboundof p R ,andthoughin thekinematicregionofthisexperimentitshouldbemuchsma ller,thiscanbeusedasa verygenerousestimateoftheeect A 2 willhaveonthestructurefunction.Ascanbeseen ingure20thisapproximationfor A 2 alters g 1 signicantly,butassuming A 2 isactually muchsmallerthanthisthestructurefunctionsforbothtarg etsareshowningure21. 4ComparisontoOtherDataandConclusionsThenaldatafor g 1 F 1 canbecomparedtodatafromotherexperiments.ThemostcomparableexperimentisanearlierexperimentatJLabcalledt heEG1bexperiment.This 33 PAGE 40 Table2:Comparisonof Q 2 valuesusedingure22. Experiment Q 2 1( GeV 2 ) Q 2 2 Q 2 3 Q 2 4 Q 2 5 Q 2 6 Q 2 7 Q 2 8 EG1-DVCS 1.1251.4061.7572.1962.7453.4564.3225.358 EG1B 1.21.441.712.052.923.484.164.96 experimentmeasuredawiderrangeofWand Q 2 values,butwithmoreuncertainty.A comparisonofthevalueobtainedbythetwoexperimentsispl ottedingure22.Thisgure showsthatthattheuncertaintyforthevaluescalculatedin thisthesisimproveconsiderably onthepreviousuncertainties. ThedatadoesnotagreeexactlywiththevaluesfromEG1b.One sourceofthisdierence isthatthe Q 2 valuesfortheEG1bdataarenotexactlythesameasforthedat ainthis thesis.Ascanbeseenfromthedatathevalueof g 1 F 1 risesas Q 2 rises.The Q 2 valuesfor thedatafromthisthesisandtheEG1bexperimentarecompare dintable2. Thedatainthisthesisshouldbeusefulformoreprecisecomp utationsofthemoments ofthestructurefunctionandpropertiesoftheprotoningen eral.Inordertocalculatethe momentsofthestructurefunctiontheentirerangeofxfrom0 to1wouldneedtobepart ofthedataortheendsoftherangewouldneedtobeapproximat ed.Unfortunatelythatis beyondthescopeofthisthesisatthistime.References [1]C.W.Leemann,D.R.Douglas,andG.A.Krat.TheContinuo usElectronBeam AcceleratorFacility:CEBAFattheJeersonLaboratory. Ann.Rev.Nucl.Part.Sci. 51:413{450,2001. [2]ThomasJeersonNationalAcceleratorFacilityOceofS cienceEducation.Cebaf center-cavitydisplay. http://education.jlab.org/sitetour/ccentercavity.l. html ,April2011. [3]File:standingwaves.svg. http://commons.wikimedia.org/wiki/File: Standingwaves.svg ,April2010. 34 PAGE 41 [4]File:lineaeracceleratoren.svg-wikimediacommons. http://commons.wikimedia. org/wiki/File:Lineaer_accelerator_en.svg ,February2010. [5]NevzatGuler. SpinStructureOfTheDeuteron .PhDthesis,OldDominionUniversity, December2009. [6]C.D.Keithetal.ApolarizedtargetfortheCLASdetector Nucl.Instrum.Meth. A501:327{339,2003. [7]SuchetaJawalkar,NicholasKvaltine,PeterBosted,and YelenaProk.Calculating lengthbetweenthebanjowindowsandeectivelengthof nh 3 .Technicalreport,EG1DVCS,December2010. [8]File:cherenkov.svg-wikipedia,thefreeencyclopedia http://en.wikipedia.org/ wiki/File:Cherenkov.svg ,March2006. [9]Clasdatabase= > measuremente93m8. http://depni.sinp.msu.ru/cgi-bin/jlab/ msm.cgi?eid=93&mid=8&data=on ,April2011. [10]J.Ashmanetal.Aninvestigationofthespinstructureo ftheprotonindeepinelastic scatteringofpolarizedmuonsonpolarizedprotons. Nucl.Phys. ,B328:1,1989. [11]DavidGriths. IntroductiontoElementaryParticles .Wiley-VCH,2ndedition,2008. [12]FrancisHalzenandAlanMartin. QuarksandLeptons:Anintroductorycoursein modernparticlephysics .JohnWileyandSons,NewYork,USA,1984. [13]RobertG.Fersch. MeasurementofInclusiveProtonDouble-SpinAsymmetriesa nd PolarizedStructureFunctions .PhDthesis,CollegeofWilliamandMary,August2008. [14]E.LeaderandE.Predazzi.AnIntroductiontogaugetheo riesandmodernparticle physics.Vol.1:Electroweakinteractions,thenewparticl esandthepartonmodel. Camb.Monogr.Part.Phys.Nucl.Phys.Cosmol. ,3:1,1996. [15]B.W.FilipponeandXiang-DongJi.Thespinstructureof thenucleon. Adv.Nucl. Phys. ,26:1,2001. 35 PAGE 42 [16]K.Abeetal.Measurementsoftheprotonanddeuteronspi nstructurefunctionsg1 andg2. Phys.Rev. ,D58:112003,1998. [17]Beampolarization-eg1-dvcs-wiki. http://clasweb.jlab.org/rungroups/ eg1-dvcs/wiki/index.php/Beam_Polarization ,March2011. [18]Targettable-eg1-dvcs-wiki. http://clasweb.jlab.org/rungroups/eg1-dvcs/ wiki/index.php/Target_Table ,October2009. 36 PAGE 43 Figure17:ACherenkovspectrumwithnumberofphotoelectro nsdetectedonthexaxisand numberofparticlesontheyaxis.Thereddotsaretheelectro nsignal,thecyandotsarethe tabove2photoelectrons,thebluedotsarethepionsdetect edusingtheelectromagnetic calorimeterandothercuts,andthebrownsquaresarethispi onsignalscaledtothedierence betweentheelectronsignalandthet.[5]. 37 PAGE 44 topaerrCtot2 Entries 3116Mean 2.833RMS 0.1346 W 11.522.53 Asymmetry 0 0.05 0.1 0.15 0.2 0.25 0.3 topaerrCtot2 Entries 3116Mean 2.833RMS 0.1346 topaerr2 Entries 3115Mean 2.833RMS 0.1443 Top Electromagnetic Asymmetry Q^2 = 1.125 topaerrCtot3 Entries 6914Mean 2.697RMS 0.1894 W 11.522.53 Asymmetry 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 topaerrCtot3 Entries 6914Mean 2.697RMS 0.1894 topaerr3 Entries 6913Mean 2.691RMS 0.197 Top Electromagnetic Asymmetry Q^2 = 1.406 topaerrCtot4 Entries 12310Mean 2.508RMS 0.2775 W 11.522.53 Asymmetry 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 topaerrCtot4 Entries 12310Mean 2.508RMS 0.2775 topaerr4 Entries 12309Mean 2.49RMS 0.2809 Top Electromagnetic Asymmetry Q^2 = 1.757 topaerrCtot5 Entries 18954Mean 2.233RMS 0.4368 W 11.522.53 Asymmetry 0 0.1 0.2 0.3 0.4 0.5 0.6 topaerrCtot5 Entries 18954Mean 2.233RMS 0.4368 topaerr5 Entries 18954Mean 2.168RMS 0.4176 Top Electromagnetic Asymmetry Q^2 = 2.196 topaerrCtot6 Entries 20093Mean 2.048RMS 0.522 W 11.522.53 Asymmetry 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 topaerrCtot6 Entries 20093Mean 2.048RMS 0.522 topaerr6 Entries 20093Mean 1.969RMS 0.5017 Top Electromagnetic Asymmetry Q^2 = 2.745 topaerrCtot7 Entries 15440Mean 1.924RMS 0.4521 W 11.522.53 Asymmetry 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 topaerrCtot7 Entries 15440Mean 1.924RMS 0.4521 topaerr7 Entries 15438Mean 1.92RMS 0.465 Top Electromagnetic Asymmetry Q^2 = 3.465 topaerrCtot8 Entries 5987Mean 1.854RMS 0.273 W 11.522.53 Asymmetry 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 topaerrCtot8 Entries 5987Mean 1.854RMS 0.273 topaerr8 Entries 5985Mean 1.883RMS 0.2975 Top Electromagnetic Asymmetry Q^2 = 4.322 topaerrCtot9 Entries 1656Mean 0RMS 0 W 11.522.53 Asymmetry 0 0.2 0.4 0.6 0.8 1 topaerrCtot9 Entries 1656Mean 0RMS 0 topaerr9 Entries 1655Mean 1.925RMS 1.557e-08 Top Electromagnetic Asymmetry Q^2 = 5.358 Figure18:Theasymmetrieswiththepolarizationanddiluti onfactortakenintoaccount areshowninblack.Theasymmetrieswiththepolarization,d ilutionfactor,pairelectron correction,andstandardpioncorrectionareshowninred.T heasymmetrieswiththe polarization,dilutionfactor,pairelectroncorrection, andtotalpioncorrectionareshown inblue.Alloftheseareforthetoptarget. W 11.522.53 Energy (Gev) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 E2bins2 Entries 8869978Mean 2.808RMS 0.137 Energy of outgoing electron Q^2 = 1.125 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 E2bins3 Entries 1.258287e+07Mean 2.603RMS 0.2025 Energy of outgoing electron Q^2 = 1.406 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 3 3.5 E2bins4 Entries 1.461438e+07Mean 2.364RMS 0.2789 Energy of outgoing electron Q^2 = 1.757 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 E2bins5 Entries 1.417955e+07Mean 1.938RMS 0.4284 Energy of outgoing electron Q^2 = 2.196 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 E2bins6 Entries 9981004Mean 1.72 RMS 0.4803 Energy of outgoing electron Q^2 = 2.745 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 3 3.5 4 E2bins7 Entries 5019617Mean 1.678RMS 0.4531 Energy of outgoing electron Q^2 = 3.465 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 3 3.5 E2bins8 Entries 1321555Mean 1.605RMS 0.3999 Energy of outgoing electron Q^2 = 4.322 W 11.522.53 Energy (Gev) 0 0.5 1 1.5 2 2.5 3 E2bins9 Entries 289268Mean 1.505RMS 0.3286 Energy of outgoing electron Q^2 = 5.358 Figure19:Theenergyofthedetectedelectronforbinswithc ounts. 38 PAGE 45 topgerrCstanA22 Entries 6859Mean 2.839RMS 0.1455 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 topgerrCstanA22 Entries 6859Mean 2.839RMS 0.1455 topgerrCstan2 Entries 3115Mean 2.838RMS 0.1464 Q^2 = 1.125 topgerrCstanA23 Entries 14221 Mean 2.69RMS 0.2004 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 topgerrCstanA23 Entries 14221 Mean 2.69RMS 0.2004 topgerrCstan3 Entries 6832Mean 2.693RMS 0.2 Q^2 = 1.406 topgerrCstanA24 Entries 22528Mean 2.483RMS 0.2863 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 topgerrCstanA24 Entries 22528Mean 2.483RMS 0.2863 topgerrCstan4 Entries 11186Mean 2.489RMS 0.2857 Q^2 = 1.757 topgerrCstanA25 Entries 29708Mean 2.178RMS 0.4034 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 topgerrCstanA25 Entries 29708Mean 2.178RMS 0.4034 topgerrCstan5 Entries 15264Mean 2.186RMS 0.4043 Q^2 = 2.196 topgerrCstanA26 Entries 30744Mean 1.934RMS 0.5074 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 topgerrCstanA26 Entries 30744Mean 1.934RMS 0.5074 topgerrCstan6 Entries 16583Mean 1.968RMS 0.5016 Q^2 = 2.745 topgerrCstanA27 Entries 22318Mean 1.899RMS 0.4759 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 topgerrCstanA27 Entries 22318Mean 1.899RMS 0.4759 topgerrCstan7 Entries 12864Mean 1.93RMS 0.4683 Q^2 = 3.465 topgerrCstanA28 Entries 8858Mean 1.88RMS 0.299 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 topgerrCstanA28 Entries 8858Mean 1.88RMS 0.299 topgerrCstan8 Entries 5334Mean 1.885RMS 0.298 Q^2 = 4.322 topgerrCstanA29 Entries 2621Mean 1.769RMS 0.09814 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 topgerrCstanA29 Entries 2621Mean 1.769RMS 0.09814 topgerrCstan9 Entries 1627Mean 1.768RMS 0.09756 Q^2 = 5.358 Figure20: g 1 forthetoptargetwith A 2 =0inblackand A 2 = p R inblue. topgerrCstan2 Entries 3115Mean 2.838RMS 0.1464 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 topgerrCstan2 Entries 3115Mean 2.838RMS 0.1464 Q^2 = 1.125 topgerrCstan3 Entries 6832Mean 2.693RMS 0.2 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 topgerrCstan3 Entries 6832Mean 2.693RMS 0.2 Q^2 = 1.406 topgerrCstan4 Entries 11186Mean 2.489RMS 0.2857 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 topgerrCstan4 Entries 11186Mean 2.489RMS 0.2857 Q^2 = 1.757 topgerrCstan5 Entries 15264Mean 2.186RMS 0.4043 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 topgerrCstan5 Entries 15264Mean 2.186RMS 0.4043 Q^2 = 2.196 topgerrCstan6 Entries 16583Mean 1.968RMS 0.5016 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 topgerrCstan6 Entries 16583Mean 1.968RMS 0.5016 Q^2 = 2.745 topgerrCstan7 Entries 12864Mean 1.93RMS 0.4683 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 topgerrCstan7 Entries 12864Mean 1.93RMS 0.4683 Q^2 = 3.465 topgerrCstan8 Entries 5334Mean 1.885RMS 0.298 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 topgerrCstan8 Entries 5334Mean 1.885RMS 0.298 Q^2 = 4.322 topgerrCstan9 Entries 1627Mean 1.768RMS 0.09756 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 topgerrCstan9 Entries 1627Mean 1.768RMS 0.09756 Q^2 = 5.358 Figure21: g 1 with A 2 =0forthetoptargetinblackandforthebottomtargetinred. 39 PAGE 46 topgerrCstan2 Entries 3115Mean 2.838RMS 0.1464 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 topgerrCstan2 Entries 3115Mean 2.838RMS 0.1464 Q^2 1 topgerrCstan3 Entries 6831Mean 2.692RMS 0.2009 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 topgerrCstan3 Entries 6831Mean 2.692RMS 0.2009 Q^2 2 topgerrCstan4 Entries 11185Mean 2.466RMS 0.2986 W 11.522.53 g1/F1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 topgerrCstan4 Entries 11185Mean 2.466RMS 0.2986 Q^2 3 topgerrCstan5 Entries 15261Mean 2.115RMS 0.4436 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 topgerrCstan5 Entries 15261Mean 2.115RMS 0.4436 Q^2 4 topgerrCstan6 Entries 16582Mean 2 RMS 0.5185 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 topgerrCstan6 Entries 16582Mean 2 RMS 0.5185 Q^2 5 topgerrCstan7 Entries 12864Mean 1.93RMS 0.4683 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 topgerrCstan7 Entries 12864Mean 1.93RMS 0.4683 Q^2 6 topgerrCstan8 Entries 5326Mean 1.775RMS 0.4009 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 topgerrCstan8 Entries 5326Mean 1.775RMS 0.4009 Q^2 7 topgerrCstan9 Entries 1611Mean 1.617RMS 0.3195 W 11.522.53 g1/F1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 topgerrCstan9 Entries 1611Mean 1.617RMS 0.3195 Q^2 8 Figure22: g 1 withthetoptargetinblackandforthebottomtargetinred,a nddatafrom EG1bwithapproximatelythesame Q 2 valueplottedingreen.ThevaluesfortheEG1b experimentarefrom[9]. 40 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||