showerseaccelerator

AAppendixInthemainbodyofthisbookwehavedescribedparticleandnuclearphysicsandtheunderlyinginteractionsconciselyandincontext.
Wehavehereandthereelucidatedthebasicprinciplesandmethodsoftheexperimentsthathaveledustothisknowledge.
Wenowwanttobrieflydescribetheindivid-ualtoolsofexperimentalphysics–theparticleacceleratorsanddetectors–whoseinventionanddevelopmenthaveoftenbeenasinequanonforthediscoveriesdiscussedhere.
Moredetaileddiscussionsmaybefoundintheliterature[Kl92a,Le94,Wi93].
A.
1AcceleratorsParticleacceleratorsprovideuswithdierenttypesofparticlebeamswhoseenergies(atthetimeofwriting)canbeanythinguptoaTeV(106MeV).
Thesebeamsserveontheonehandas"sources"ofenergywhichifusedtobombardnucleicangenerateavarietyofexcitedstatesorindeednewparticles.
Ontheotherhandtheycanactas"probes"withwhichwemayinvestigatethestructureofthetargetparticle.
Themostimportantquantity,whetherwewanttogeneratenewparticlesorexciteasystemintoahigherstate,isthecentreofmassenergy√softhereactionunderinvestigation.
InthereactionofabeamparticleawithtotalenergyEawithatargetparticlebwhichisatrestthisis√s=2Eambc2+(m2a+m2b)c4.
(A.
1)Inhighenergyexperimentswheretheparticlemassesmaybeneglectedincomparisontothebeamenergythissimpliesto√s=2Eambc2.
(A.
2)Thecentreofmassenergyforastationarytargetonly,wesee,growswiththesquarerootofthebeamparticle'senergy.
Ifabeamparticlewithmomentumpisusedtoinvestigatethestructureofastationarytarget,thenthebestpossibleresolutionischaracterisedbyitsreduceddeBrogliewavelengthλ–=/p.
ThisisrelatedtotheenergyEthrough(4.
1).
342AAppendixAllacceleratorsessentiallyconsistofthefollowing:aparticlesource,astructuretoactuallydotheacceleratingandanevacuatedbeampipe.
Itshouldalsobepossibletofocusanddeecttheparticlebeam.
Theaccelerat-ingprincipleisalwaysthesame:chargedparticlesareacceleratediftheyareexposedtoanelectriceld.
AparticlewithchargeZewhichtraversesapo-tentialdierenceUreceivesanamountofenergy,E=ZeU.
Inthefollowingwewishtobrieflypresentthethreemostimportanttypesofaccelerators.
Electrostaticaccelerators.
IntheseacceleratorstherelationE=ZeUisdirectlyexploited.
Themaincomponentsofanelectrostaticacceleratorareahighvoltagegenerator,aterminalandanevacuatedbeampipe.
Inthemostcommonsort,theVandeGraaaccelerator,theterminalisusuallyametallicspherewhichactsasacapacitorwithcapacitanceC.
Theterminalischargedbyarotating,insulatedbandandthiscreatesahighelectriceld.
Fromanearthedpotentialpositivechargesarebroughtontothebandandthenstrippedoontotheterminal.
Theentiresetupisplacedinsideanearthedtankwhichislledwithaninsulatinggas(e.
g.
,SF6)topreventprematuredischarge.
ThevoltageU=Q/Cwhichmaybebuiltupinthiswaycanbeasmuchas15MV.
Positiveions,producedinanionsource,attheterminalpotentialnowtraverseinsidethebeampipetheentirepotentialdierencebetweentheterminalandthetank.
Protonscaninthiswayreachkineticenergiesupto15MeV.
EnergiestwiceashighmaybeattainedintandemVandeGraaacceler-ators(Fig.
A.
1).
Heretheacceleratingpotentialisusedtwiceover.
Negativeionsarerstproducedatearthpotentialandthenacceleratedalongabeampipetowardstheterminal.
Athinfoil,orsimilar,placedtherestripssomeoftheelectronsotheionsandleavesthempositivelycharged.
Theaccelerat-ingvoltagenowentersthegameagainandprotonsmayinthiswayattainkineticenergiesofupto30MeV.
Heavyionsmayloseseveralelectronsatonceandconsequentlyreachevenhigherkineticenergies.
VandeGraaacceleratorscanprovidereliable,continuousparticlebeamswithcurrentsofupto100μA.
Theyareveryimportantworkhorsesfornuclearphysics.
Protonsandbothlightandheavyionsmaybeacceleratedinthemuptoenergiesatwhichnuclearreactionsandnuclearspectroscopymaybesystematicallyinvestigated.
Linearaccelerators.
GeV-typeenergiesmayonlybeattainedbyrepeat-edlyacceleratingtheparticle.
Linearaccelerators,whicharebaseduponthisprinciple,aremadeupofmanyacceleratingtubeslaidoutinastraightlineandtheparticlesprogressalongtheircentralaxis.
Everypairofneighbour-ingtubeshaveoppositelyarrangedpotentialssuchthattheparticlesbetweenthemareaccelerated,whiletheinteriorofthetubesisessentiallyeldfree(Wider¨oetype).
Ahighfrequencygeneratorchangesthepotentialswithaperiodsuchthattheparticlesbetweenthetubesalwaysfeelanacceleratingforce.
AfterpassingthroughntubestheparticleswillhavekineticenergyA.
1Accelerators343MagnetChargingbeltTerminalTankNegative-ionsourceMagnetStrippingcanalAcceleratortubewithintermediateelectrodesBeamofpositiveIonsFig.
A.
1.
SketchofatandemVandeGraaaccelerator.
Negativeionsareacceler-atedfromthelefttowardstheterminalwheresomeoftheirelectronsarestrippedoandtheybecomepositivelycharged.
Thiscausesthemtonowbeacceleratedawayfromtheterminalandthepotentialdierencebetweentheterminalandthetankistraversedforasecondtime.
E=nZeU.
Suchacceleratorscannotproducecontinuousparticlebeams;theyacceleratepacketsofparticleswhichareinphasewiththegeneratorfrequency.
Sincethegeneratorfrequencyisxed,thelengthsofthevariousstagesneedtobeadjustedtotthespeedoftheparticlesasitpassesthrough(Fig.
A.
2).
Ifwehaveanelectronbeamthislastsubtletyisonlyrelevantfortherstfewaccelerationsteps,sincethesmallelectronmassmeansthattheirvelocityisverysoonnearlyequaltothespeedoflight.
OntheotherhandDrifttubesRfgeneratorIonsourceFig.
A.
2.
Sketchofthefundamentalsofa(Wider¨oetype)linearaccelerator.
Thepotentialsofthetubesshownareforoneparticularmomentintime.
Theparticlesareacceleratedfromthesourcetotherstdrifttube.
ThelengthsLiofthetubesandthegeneratorfrequencyωmustbeadjustedtoeachothersothatwehaveLi=viπ/ωwhereviistheparticlevelocityattheithtube.
Thisdependsbothuponthegeneratorvoltageandthetypeofparticlebeingaccelerated.
344AAppendixthetubelengthsgenerallyneedtobecontinuallyalteredalongtheentirelengthofprotonlinearaccelerators.
Thenalenergyofalinearacceleratorisdeterminedbythenumberoftubesandthemaximalpotentialdierencebetweenthem.
Atpresentthelargestlinearacceleratorintheworld,wheremanyimpor-tantexperimentsondeepinelasticscatteringonucleonshavebeencarriedout,istheroughly3kmlongelectronlinearacceleratorattheStanfordLin-earAcceleratorCenter(SLAC).
Hereelectronspassthrougharound100000acceleratingstagestoreachenergiesofabout50GeV.
Synchrotrons.
Whileparticlespassthrougheachstageofalinearaccel-eratorjustonce,synchrotrons,whichhaveacircularform,maybeusedtoaccelerateparticlestohighenergiesbypassingthemmanytimesthroughthesameacceleratingstructures.
Theparticlesarekeptontheircircularorbitsbymagneticelds.
Theacceleratingstagesaremostlyonlyplacedatafewpositionsuponthecircuit.
TheprincipleofthesynchrotronistosynchronouslychangethegeneratorfrequencyωoftheacceleratingstagestogetherwiththemagneticeldBinsuchawaythattheparticles,whoseorbitalfrequenciesandmomentapareincreasingasaresultoftheacceleration,alwaysfeelanacceleratingforceandaresimultaneouslykeptontheirassignedorbitsinsidethevacuumpipe.
Thismeansthatthefollowingconstraintsmustbesimultaneouslyfullled:ω=n·cR·pcEn=positiveinteger(A.
3)B=pZeR,(A.
4)whereRistheradiusofcurvatureofthesynchrotronring.
Technicallimita-tionsupontheBandωavailablemeanthatonehastoinjectpreacceleratedparticlesintosynchrotronringswhereupontheycanbebroughtuptotheirpreassignednalenergy.
Linearacceleratorsorsmallersynchrotronsareusedinthepreaccelerationstage.
Synchrotronsalsoonlyproducepacketsofpar-ticlesanddonotdelivercontinuousbeams.
Highparticleintensitiesrequirewellfocusedbeamsclosetotheidealor-bit.
Focusingisalsoofgreatimportanceinthetransportofthebeamfromthepreacceleratortothemainstageandfromtheretotheexperiment(injec-tionandextraction).
Magneticlenses,madefromquadrupolemagnets,areusedtofocusthebeaminhighenergyaccelerators.
Theeldofaquadrupolemagnetfocuseschargedparticlesinoneplaneonitscentralaxisanddefo-cusesthemontheotherplaneperpendiculartoit.
Anoverallfocusinginbothplanesmaybeachievedbyputtingasecondquadrupolemagnet,whosepolesarerotatedrelativetothoseoftherstonethrough90,aftertherstmagnet.
Thisprincipleofstrongfocusingissimilartotheopticalcombina-tionofthindivergingandconverginglenseswhichalwayseectivelyfocuses.
A.
1Accelerators345KQFDQDDQFDQDDEBFig.
A.
3.
Section(toscale)ofasynchrotronfromabove.
Theessentialacceleratingandmagneticstructuresareshowntogetherwiththebeampipe(continuousline).
Highfrequencyacceleratortubes(K)areusuallyonlyplacedatafewpositionsaroundthesynchrotron.
Theeldsofthedipolemagnets(D),whichkeeptheparticlesontheircircularpaths,areperpendiculartothepage.
Pairsofquadrupolemagnetsformdoubletswhichfocusthebeam.
Thisisindicatedbythedottedlineswhich(exaggeratedly)showtheshapeofthebeamenvelope.
ThequadrupolesmarkedQFhaveafocusingeectintheplaneofthepageandtheQDquadrupolesadefocusingeect.
FigureA.
3depictstheessentialsofasynchrotronandthefocusingeectsofsuchquadrupoledoublets.
Particlesacceleratedinsynchrotronslosesomeoftheirenergytosyn-chrotronradiation.
Thisreferstotheemissionofphotonsbyanychargedparticlewhichisforcedontoacircularpathandisthusradiallyaccelerated.
Theenergylosttosynchrotronradiationmustbecompensatedbytheaccel-eratingstages.
ThislossisforhighlyrelativisticparticlesΔE=4παc3Rβ3γ4whereβ=vc≈1andγ=Emc2,(A.
5)perorbit–itincreasesinotherwordswiththefourthpoweroftheparti-cleenergyE.
Themassdependencemeansthatthisrateofenergylossisabout1013timeslargerforelectronsthanforprotonsofthesameenergy.
Themaximalenergyinmodernelectronsynchrotronsisthusabout100GeV.
Synchrotronradiationdoesnotplayanimportantroleforprotonbeams.
Thelimitontheirnalenergyissetbytheavailableeldstrengthsofthedipolemagnetswhichkeeptheprotonsintheorbit.
ProtonenergiesuptoaTeVmaybeachievedwithsuperconductingmagnets.
Therearetwotypesofexperimentwhichuseparticlesacceleratedinsyn-chrotrons.
Thebeammay,afterithasreacheditsnalenergy,bedeectedoutoftheringandledotowardsastationarytarget.
Alternativelythebeammaybestoredinthesynchrotronuntilitiseitherlooseduponathin,internaltargetorcollidedwithanotherbeam.
346AAppendixStoragerings.
Thecentreofmassenergyofareactioninvolvingastationarytargetonlygrowswiththesquarerootofthebeamenergy(A.
2).
Muchhighercentreofmassenergiesmaybeobtainedforthesamebeamenergiesifweemploycollidingparticlebeams.
ThecentreofmassenergyforaheadoncollisionoftwoparticlebeamswithenergyEis√s=2E–i.
e.
,itincreaseslinearlywiththebeamenergy.
Theparticledensityinparticlebeams,andhencethereactionrateforthecollisionoftwobeams,isverytiny;thustheyneedtoberepeatedlycollidedinanyexperimentwithreasonableeventrates.
Highcollisionratesmay,e.
g.
,beobtainedbycontinuouslyoperatingtwolinearacceleratorsandcollidingtheparticlebeamstheyproduce.
Anotherpossibilityistostoreparticlebeams,whichwereacceleratedinasynchrotron,attheirnalenergyandattheacceleratingstagesjusttopuptheenergytheylosetosynchrotronradiation.
Thesestoredparticlebeamsmaybethenusedforcollisionexperiments.
ConsiderasanexampletheHERAringattheDeutscheElektronen-Synchrotron(GermanElectronSynchrotron,DESY)inHamburg.
Thisismadeupoftwoseparatestorageringsofthesamediameterwhichrunpar-alleltoeachotheratabout1mseparation.
Electronsareaccelerateduptoabout30GeVandprotonstoabout920GeVbeforestorage.
Thebeamtubescometogetherattwopoints,wherethedetectorsarepositioned,andtheoppositelycirclingbeamsareallowedtocollidethere.
Constructionisrathersimplerifonewantstocollideparticleswiththeirantiparticles(e.
g.
,electronsandpositronsorprotonsandantiprotons).
Insuchcasesonlyonestorageringisneededandtheseequalmassbutoppo-sitelychargedparticlescansimultaneouslyrunaroundtheringinoppositedirectionsandmaybebroughttocollisionatvariousinteractionpoints.
Ex-amplesofthesearetheLEPring(LargeElectronPositronRing)atCERNwhere86GeVelectronsandpositronscollideandtheSppS(SuperProtonAn-tiprotonSynchrotron)where310GeVprotonsandantiprotonsarebroughtviolentlytogether.
BothofthesemachinesaretobefoundattheEuropeanNuclearResearchCentreCERNjustoutsideGeneva.
AnexampleofaresearchcomplexofacceleratorsisshowninFig.
A.
4;thatofDESY.
AtotalofsevenpreacceleratorsservicetheDORISandHERAstorageringswhereexperimentswithelectrons,positronsandprotonstakeplace.
Twopreacceleratorstagesareneededfortheelectron-positronringDORISwherethebeamseachhaveamaximalenergyof5.
6GeV.
Threesuchstagesarerequiredfortheelectron-protonringHERA(30GeVelectronsand820GeVprotons).
DORISalsoservesasansourceofintensivesynchrotronradiationandisusedasaresearchinstrumentinsurfacephysics,chemistry,biologyandmedicine.
A.
1Accelerators347ElectronsPositronsProtonsSynchrotonradiationHERAWestHallPETRAHASYLABHERADesyIIDesyIIIDORISARGUSPIALINACILINACIIILINACIIFig.
A.
4.
TheacceleratorcomplexattheGermanElectronSynchrotron,DESY,inHamburg.
TheDORISandHERAstorageringsareservicedbyachainofpreac-celerators.
Electronsareacceleratedupto450MeVintheLINACIorLINACIIlinearacceleratorsbeforebeinginjectedintotheDESYIIsynchrotron,wheretheymayreachupto9GeV.
ThencetheyeitherpassintoDORISorthePETRAsyn-chrotron.
PETRAactsasanalpreacceleratorforHERAandelectronenergiesofupto14GeVmaybeattainedthere.
BeforeHERAwascommissionedPETRAworkedasanelectron-positronstorageringwithabeamenergyofupto23.
5GeV.
PositronsareproducedwiththehelpofelectronsacceleratedinLINACIIandarethenaccumulatedinthePIAstorageringbeforetheirinjectionintoDESYIIwheretheyarefurtheracceleratedandthenledotoDORIS.
ProtonsareacceleratedinLINACIIIupto50MeVandthenpreacceleratedintheprotonsynchrotronDESYIIIupto7.
5GeVbeforebeinginjectedintoPETRA.
Theretheyattain40GeVbeforebeinginjectedintoHERA.
TheHERAring,whichisonlypartiallyshownhere,hasacircumferenceof6336m,whilethecircumferenceofPETRAis2300mandthatofDESYII(III)isaround300m.
(CourtesyofDESY)348AAppendixA.
2DetectorsTheconstructionanddevelopmentofdetectorsforparticleandnuclearphysicshas,aswithacceleratorphysics,developedintoanalmostindepen-dentbranchofscience.
Thedemandsuponthequalityandcomplexityofthesedetectorsincreasewiththeeverhigherparticleenergiesandcurrentsinvolved.
Thishasnecessarilyledtoastrongspecialisationamongthedetec-tors.
Therearenowdetectorstomeasuretimes,particlepositions,momentaandenergiesandtoidentifytheparticlesinvolved.
Theprinciplesunderly-ingthedetectorsaremostlybasedupontheelectromagneticinteractionsofparticleswithmatter,e.
g.
,ionisationprocesses.
Wewillthereforerstbrieflydelineatetheseprocessesbeforeshowinghowtheyareappliedintheindivid-ualdetectors.
Interactionofparticleswithmatter.
Ifchargedparticlespassthroughmattertheyloseenergythroughcollisionswiththemedium.
Alargepartofthiscorrespondstointeractionswiththeatomicelectroncloudswhichleadtotheatomsbeingexcitedorionised.
TheenergylosttoionisationisdescribedbytheBethe-Blochformula[Be30,Bl33].
Approximatelywehave[PD94]dEdx=4πmec2nz2β2e24πε02ln2mec2β2I·(1β2)β2(A.
6)whereβ=v/c,zeandvarethechargeandspeedoftheparticle,nistheelectrondensityandIistheaverageexcitationpotentialoftheatomsH2CPb0.
1110100110100p/mc1dEρdx[MeV/gcm--2]Fig.
A.
5.
Roughsketchoftheaverageenergylossofchargedparticlestoionisationprocessesinhydrogen,carbonandlead.
Theenergylossdividedbythedensityofthematerialisplottedagainstp/mc=βγfortheparticleinalog-logplot.
Thespecicenergylossisgreaterforlighterelementsthanforheavyones.
A.
2Detectors349(typically16eV·Z0.
9fornuclearchargenumbersZ>1).
Theenergylossthusdependsuponthechargeandspeedoftheparticle(Fig.
A.
5)butnotuponitsmass.
Itdecreasesforsmallvelocitiesas1/v2,reachesaminimumaroundp/m0c≈4andthenincreasesonlylogarithmicallyforrelativisticvelocities.
Theenergylosstoionisationperlengthdxtraversednormalisedtothedensityofthematterattheionisationminimum,andalsoforhigherparticleenergies,isroughly1/·dE/dx≈2MeV/(gcm2).
Electronsandpositronsloseenergynotjusttoionisationbutalsotoafurtherimportantprocess:bremsstrahlung.
Electronsbrakingintheeldofanucleusradiateenergyintheformofphotons.
Thisprocessstronglydependsuponthematerialandtheenergy:itincreasesroughlylinearlywithenergyandquadraticallywiththechargenumberZofthemedium.
Aboveacrit-icalenergyEc,whichmaybecoarselyparameterisedbyEc≈600MeV/Z,bremsstrahlungenergylossismoreimportantforelectronsthanisionisation.
Forsuchhighenergyelectronsanimportantmaterialparameteristheradi-ationlengthX0.
Thisdescribesthedistanceoverwhichtheelectronenergydecreasesduetobremsstrahlungbyafactorofe.
HighenergyelectronsarebestabsorbedinmaterialswithhighchargenumbersZ;e.
g.
,lead,wheretheradiationlengthisjust0.
56cm.
Whilechargedparticlestraversingmatterloseenergyslowlytoelectro-magneticinteractionsbeforenallybeingabsorbed,theinteractionofapho-tonwithmattertakesplaceatapoint.
TheintensityIofaphotonbeamthereforedecreasesexponentiallywiththethicknessofthemattertraversed:I=I0·eμ.
(A.
7)Theabsorptioncoecientμdependsuponthephotonenergyandthetypeofmatter.
Theinteractionofphotonswithmatteressentiallytakesplaceviaoneofthreeprocesses:thephotoelectriceect,theComptoneectandpairpro-duction.
Theseprocessesdependstronglyuponthemediumandtheenergyinvolved.
ThephotoelectriceectdominatesatlowenergiesinthekeVrange,theComptoneectforenergiesfromseveral100keVtoafewMeVwhileinhighenergyexperimentsonlypairproductionisofanyimportance.
Herethephotonisconvertedinsidethenucleareldtoanelectron-positronpair.
ThisisthedominantprocessaboveseveralMeV.
InthisenergyrangethephotoncanalsobedescribedbytheradiationlengthX0:theconversionlengthλofahighenergyphotonisλ=9/7·X0.
TheenergydependenceofthesethreeprocessesinleadisillustratedinFig.
A.
6.
Wewishtobrieflymentiontwofurtherprocesseswhichareusefulinparticleidentication:theradiationofCherenkovlightandnuclearreactions.
Cherenkovradiationisphotonemissionfromchargedparticlesthatcrossthroughamediumwithavelocitygreaterthanthespeedoflightinthatmedium.
Thesephotonsareradiatedinaconewithangle350AAppendix10.
10.
010.
010.
1110100μ/ρ[cm2/g]PhotoeffectTotalPairproductionEγ[MeV]ComptoneffectFig.
A.
6.
Thephotonabsorptioncoecientμinleaddividedbythedensityplottedagainstthephotonenergy.
Thedashedlinesarethecontributionsoftheindividualprocesses;thephotoelectriceect,theComptoneectandpairproduction.
AboveafewMeVpairproductionplaysthedominantrole.
θ=arccos1βn(A.
8)aroundthepathofthechargedparticle(nistherefractiveindexofthemedium).
TheenergylosstoCherenkovradiationissmallcomparedtothatthroughionisation.
Nuclearreactionsareimportantfordetectingneutralhadronssuchasneutronsthatdonotparticipateinanyoftheaboveprocesses.
Possiblere-actionsarenuclearssionandneutroncapture(eV–keVrange),elasticandinelasticscattering(MeVrange)andhadronproduction(highenergies).
Measuringpositions.
Theabilitytomeasurethepositionsandmomentaofparticlesisimportantinordertoreconstructthekinematicsofreactions.
Themostcommondetectorsofthepathsofparticlesexploittheenergylostbychargedparticlestoionisation.
Bubblechambers,sparkchambers,andstreamerchambersshowuswhereparticlespassthroughbymakingtheirtracksvisiblesothattheymaybepho-tographed.
Thesepictureshaveahighillustrativevalueandpossessacertainaestheticappeal.
Manynewparticleswerediscoveredinbubblechambersinparticularinthe1950'sand1960's.
Thesedetectorsarenowadaysonlyusedforspecialapplications.
A.
2Detectors351xyxionisingparticleEAnodewiresFig.
A.
7.
Groupofthreeproportionalchambers.
Theanodewiresofthelayersmarkedxpointintothepage,whilethoseoftheylayerrunatrightanglestothese(dashedline).
Thecathodesaretheedgesofthechambers.
Apositivevoltageappliedtotheanodewiresgeneratesaeldliketheonesketchedintheupperlefthandcorner.
Aparticlecrossingthroughthechamberionisesthegasinitspathandtheelectronsdriftalongtheeldlinestotheanodewire.
Intheexampleshownasignalwouldbeobtainedfromonewireintheupperxplaneandfromtwointhelowerxlayer.
Proportionalcountersconsistofat,gas-lledformsinwhichmanythin,parallelwires(r≈10μm)arearranged.
Thewiresaremaintainedataposi-tivepotentialofafewkVandaretypicallyarrangedatseparationsofabout2mm.
Chargedparticlespassingthroughthegasionisethegasatomsintheirpathsandtheso-releasedelectronsdriftototheanodewires(Fig.
A.
7).
Theelectriceldstrengthsaroundthethinwiresareveryhighandsotheprimaryelectronsareacceleratedandreachkineticenergiessuchthattheythemselvesstarttoionisethegasatoms.
Achargeavalancheisletloosewhichleadstoameasurablevoltagepulseonthewire.
Thearrivaltimeandamplitudeofthepulseareregisteredelectronically.
Theknownpositionofthewiretellsuswheretheparticlepassedby.
Thespatialresolutioninthedirectionperpen-diculartothewiresisoftheorderofhalfthewireseparation.
Animprovedresolutionandareconstructionofthepathinallthreespatialcoordinatesisinpracticeobtainedbyusingseverallayersofproportionalcounterswiththewirespointingindierentdirections.
Driftchambersfunctionsimilarlytoproportionalchambers.
Thewiresare,however,atafewcentimetresseparation.
Thepositionoftheparticle'spathxisnowobtainedfromthetimeofthevoltagepulsetwireonthewirerelativetothetimet0thattheparticlecrossedthroughthedetector.
Thislattertimehastobemeasuredinanotherdetector.
Ideallyweshouldhavethelinearrelationx=xwire+vdrift·(twiret0),(A.
9)352AAppendixiftheelectriceldduetoadditionalelectrodes,andhencethedriftveloc-ityvdriftofthereleasedelectronsinthegas,areveryhomogeneous.
Driftchambers'spatialresolutioncanbeasgoodas50μm.
Severallayersareagainrequiredforathreedimensionalreconstruction.
Wirechambersareveryuse-fulforreconstructingpathsoverlargeareas.
Theymaybemadetocoverseveralsquaremetres.
Siliconstripdetectorsaremadeoutofsiliconcrystalswithverythinelectrodesattachedtothematseparationsofabout,e.
g.
,20μm.
Achargedparticlecrossingthewaferproduceselectron-holepairs,insiliconthisonlyrequires3.
6eVperpair.
Anexternalvoltagecollectsthechargeattheelec-trodeswhereitisregistered.
Spatialresolutionslessthan10μmmaybereachedinthisway.
Measuringmomenta.
Themomentaofchargedparticlesmaybedeter-minedwiththehelpofstrongmagneticelds.
TheLorentzforcecausestheseparticlestofollowcircularorbitswhichmaythenbe,e.
g.
,measuredinbubblechamberphotographsorreconstructedfromseveralplanesofwirechambers.
A"ruleofthumb"forthemomentumcomponentp⊥perpendiculartothemagneticeldmaybeobtainedfromthemeasuredradiusofcurvatureoftheparticlepathRandtheknown,homogeneousmagneticeldB:p⊥≈0.
3·B·RGeV/cTm.
(A.
10)Magneticspectrometersareusedtoindirectlydeterminetheradiusofcurvaturefromtheanglewhichtheparticleisdeectedthroughinthemag-neticeld;onemeasurestheparticle'spathbeforeandafterthemagnets.
Thismethodofmeasuringthemomentaactuallyhassmallererrorsthanadirectdeterminationoftheradiusofcurvaturewouldhave.
Therelativeac-curacyofthesemeasurementstypicallydecreaseswithincreasingmomentaasδ(p)/p∝p.
Thisisbecausetheparticlepathbecomesstraighterathighmomenta.
Measuringenergies.
Ameasurementoftheenergyofaparticleusuallyrequirestheparticletobecompletelyabsorbedbysomemedium.
Theab-sorbedenergyistransformedintoionisation,atomicexcitationsorperhapsCherenkovlight.
Thissignalwhichmay,withthehelpofsuitabledevices,betransformedintoameasurableoneisproportionaltotheoriginalenergyoftheparticle.
Theenergyresolutiondependsuponthestatisticaluctuationsofthetransformationprocess.
Semiconductordetectorsareofgreatimportanceinnuclearphysics.
Electron-holepairscreatedbychargedparticlesareseparatedbyanexter-nalvoltageandthendetectedasvoltagepulses.
Ingermaniumonly2.
8eVisrequiredtoproduceanelectron-holepair.
Insilicon3.
6eVisneeded.
Semi-conductordetectorsaretypicallyafewmillimetresthickandcanabsorblightnucleiwithenergiesuptoafewtensofMeV.
PhotonenergiesaredeterminedA.
2Detectors353throughthephotoelectriceect–onemeasuresthesignaloftheabsorbedphotoelectron.
ThelargenumberNofelectron-holepairsthatareproducedmeansthattheenergyresolutionofsuchsemiconductorcountersisexcellent,δE/E∝√N/N.
For1MeVparticlesitisbetween103and104.
Electromagneticcalorimetersmaybeusedtomeasuretheenergiesofelectrons,positronsandphotonsaboveabout100MeV.
Oneexploitsthecascadeofsecondaryparticlesthattheseparticlesproduceviarepeatedbremsstrahlungandpairproductionprocessesinsidethematerialofthecalorimeter.
Theproductionofsuchameasurableionisationorvisiblesig-nalisillustratedinFig.
A.
8.
Thecompleteabsorptionofsuchashowerinacalorimetertakesplace,dependingupontheenergyinvolved,overadistanceofabout15–25timestheradiationlengthX0.
WewillconsidertheexampleofhomogeneouscalorimetersmadeofNaI(Tl)crystalsorleadglass.
NaIdopedwithsmallamountsofthalliumisaninorganicscintillatorinwhichchargedparticlesproducevisiblewavelengthphotons.
Thesephotonsmaythenbeconvertedintoanelectricpulsewiththehelpofphotomultipliers.
CalorimetersaremadefromlargecrystalsofNaI(Tl)withphotomultipliersattachedtotheirbacks(seeFig.
13.
4).
TherelativeenergyresolutiontypicallyhasvaluesoftheorderofδE/E≈12%/4E[GeV].
NaI(Tl)isalsoofgreatimportancefornuclear-gammaspectroscopy,andhenceforenergies100.
Transi-tionradiationisproducedwhenchargedparticlescrossfromonematerialtoanotherwhichhasadierentdielectricconstant.
Theintensityoftheradiationdependsuponγ.
Thusanintensitymeasurementcanenableustodistinguishbetweendierenthadronswiththesamemomenta.
Thisisinfacttheonlywaytoidentifysuchparticlesiftheenergyofthehadronisabove100GeV.
Transitionradiationmayalsobeemployedtodistinguishbetweenelectronsandpions.
Thetinymassoftheelectronsmeansthatthisisalreadypossibleforenergiesaround1GeV.
356AAppendix–Neutrondetectionisaspecialcase.
(n,α)and(n,p)nuclearreactionsareusedtoidentifyneutrons—fromthosewiththermalenergiestothosewithmomentauptoaround20MeV/c.
Thechargedreactionproductshavexedkineticenergiesandthesemaybemeasuredinscintillationcountersorgasionisationcounters.
Formomentabetween20MeV/cand1GeV/conelooksforprotonsfromelasticneutron-protonscattering.
Theprotontargetisgenerallypartofthematerialofthedetectoritself(plasticscintillator,countergas).
Athighermomentaonlyhadroncalorimetermeasurementsareavailabletous.
Theidenticationisthen,however,asarulenotun-ambiguous.
Adetectorsystem.
WewishtopresentasanexampleofasystemofdetectorstheZEUSdetectorattheHERAstoragering.
Thisdetectormea-suresthereactionproductsinhighenergyelectron-protoncollisionswithcentreofmassenergiesuptoabout314GeV(Fig.
A.
9).
Itissoarrangedthatapartfromthebeampiperegionthereactionzoneishermeticallycov-ered.
Manydierentdetectors,chosentooptimisethemeasurementofen-ergyandmomentumandtheidenticationofthereactionproducts,makeupthewhole.
Themostimportantcomponentsarethewirechambers,whicharearrangeddirectlyaroundthereactionpoint,and,justoutsidethese,auranium-scintillatorcalorimeterwheretheenergiesofelectronsandhadronsaremeasuredtoahighprecision.
A.
2Detectors35788226m54321010,38mProtons93755654Electrons11958Fig.
A.
9.
TheZEUSdetectorattheHERAstorageringinDESY.
Theelectronsandprotonsarefocusedwiththehelpofmagneticlenses(9)beforetheyaremadetocollideattheinteractionpointinthecentreofthedetector.
Thetracksofchargedreactionproductsareregisteredinthevertexchamber(3)whichsurroundsthereactionpointandalsointhecentraltrackchamber(4).
Thesedriftchambersaresurroundedbyasuperconductingcoilwhichproducesamagneticeldofupto1.
8T.
Theinuenceofthismagneticeldontheelectronbeamwhichpassesthroughitmustbecompensatedbyadditionalmagnets(6).
Thenextlayerisauranium-scintillatorcalorimeter(1)wheretheenergiesofelectrons,photonsandalsoofhadronsmaybemeasuredtoagreataccuracy.
Theironyokeofthedetector(2),intowhichthemagneticuxofthecentralsolenoidreturns,alsoactsasanabsorberforthebackwardscalorimeter,wheretheenergyofthosehighenergyparticleshowersthatarenotfullyabsorbedinthecentraluraniumcalorimetermaybemeasured.
Largeareawirechambers(5),positionedbehindtheironyoke,surroundthewholedetectorandareusedtobetraythepassageofanymuons.
Thesechambersmaybeusedtomeasurethemuons'momentasincetheyareinsideeitherthemagneticeldoftheironyokeoranadditional1.
7Ttoroidaleld(7).
Finallyathickreinforcedconcretewall(8)screensotheexperimentalhallasfarasispossiblefromtheradiationproducedinthereactions.
(CourtesyofDESY)358AAppendixA.
3CombiningAngularMomentaThecombinationoftwoangularmomenta|j1m1and|j2m2toformatotalangularmomentum|JMmustobeythefollowingselectionrules:|j1j2|≤J≤j1+j2,(A.
11)M=m1+m2,(A.
12)J≥|M|.
(A.
13)ThecoupledstatesmaybeexpandedwiththehelpoftheClebsch-Gordancoecients(CGC)(j1j2m1m2|JM)inthe|j1j2JMbasis:|j1m1|j2m2=J=j1+j2J=|j1j2|M=m1+m2(j1j2m1m2|JM)·|j1j2JM.
(A.
14)Theprobabilitythatthecombinationoftwoangularmomenta|j1m1and|j2m2producesasystemwithtotalangularmomentum|JMisthusthesquareofthecorrespondingCGC's.
Thecorollary|j1j2JM=m1=+j1m1=j1m2=Mm1(j1j2m1m2|JM)·|j1m1|j2m2,(A.
15)alsoholds.
Forasystem|JM,whichhasbeenproducedfromacombina-tionoftwoangularmomentaj1andj2,thesquareoftheCGC'sgivestheprobabilitythattheindividualangularmomentamaybefoundinthestates|j1m1and|j2m2.
Equations(A.
14)and(A.
15)mayalsobeappliedtoisospin.
Consider,forexample,theΔ+baryon(I=3/2,I3=+1/2)whichcandecayintop+π0orn+π+.
ThebranchingratiocanbefoundtobeB(Δ+→p+π0)B(Δ+→n+π+)=(121+120|32+12)2(12112+1|32+12)2=232132=2.
(A.
16)TheCGC'sarelistedforcombinationsoflowangularmomenta.
Thevaluesforj1=1/2andj2=1maybefoundwiththehelpofthegeneralphaserelation(j2j1m2m1|JM)=(1)j1+j2J·(j1j2m1m2|JM).
(A.
17)A.
3CombiningAngularMomenta359j1=1/2j2=1/2m1m2JMCGC1/21/211+11/21/210+1/21/21/200+1/21/21/210+1/21/21/2001/21/21/211+1j1=1j2=1/2m1m2JMCGC11/23/23/2+111/23/21/2+1/311/21/21/2+2/301/23/21/2+2/301/21/21/21/301/23/21/2+2/301/21/21/2+1/311/23/21/2+1/311/21/21/22/311/23/23/2+1j1=1j2=1m1m2JMCGC1122+11021+1/21011+1/21120+1/61110+1/21100+1/30121+1/201111/20020+2/30010000001/30121+1/20111+1/21120+1/611101/211001/31021+1/210111/21122+1360AAppendixA.
4PhysicalConstantsTableA.
1.
Physicalconstants[Co87,La95,PD98].
Thenumbersinbracketssignifytheuncertaintyinthelastdecimalplaces.
Thesizesofc,μ0(andhenceε0)aredenedbytheunits"metre"and"ampere"[Pe83].
Theseconstantsarethereforeerrorfree.
ConstantsSymbolValueSpeedoflightc2.
99792458·108ms1Planck'sconstanth6.
6260755(40)·1034Js=h/2π1.
05457266(63)·1034Js=6.
5821220(20)·1022MeVsc197.
327053(59)MeVfm(c)20.
38937966(23)GeV2mbarnAtomicmassunitu=M12C/12931.
49432(28)MeV/c2MassoftheprotonMp938.
27231(28)MeV/c2MassoftheneutronMn939.
56563(28)MeV/c2Massoftheelectronme0.
51099906(15)MeV/c2Elementarychargee1.
60217733(49)·1019AsDielectricconstantε0=1/μ0c28.
854187817·1012As/VmPermeabilityofvacuumμ04π·107Vs/AmFinestructureconstantα=e2/4πε0c1/137.
0359895(61)Class.
electronradiusre=αc/mec22.
81794092(38)·1015mComptonwavelengthλ–e=re/α3.
86159323(35)·1013mBohrradiusa0=re/α25.
29177249(24)·1011mBohrmagnetonμB=e/2me5.
78838263(52)·1011MeVT1NuclearmagnetonμN=e/2mp3.
15245166(28)·1014MeVT1Magneticmomentμe1.
001159652193(10)μBμp2.
792847386(63)μNμn1.
91304275(45)μNAvogadro'snumberNA6.
0221367(36)·1023mol1Boltzmann'sconstantk1.
380658(12)·1023JK1=8.
617385(73)·105eVK1GravitationalconstantG6.
67259(85)·1011Nm2kg2G/c6.
70711(86)·1039(GeV/c2)2FermiconstantGF/(c)31.
16639(1)·105GeV2Weinberganglesin2θW0.
23124(24)MassoftheW±MW80.
41(10)GeV/c2MassoftheZ0MZ91.
187(7)GeV/c2Strongcouplingconst.
αs(M2Zc2)0.
119(2)SolutionstotheProblemsChapter21.
Protonrepulsionin3He:VC=cαR=(M3HeM3H)·c2(MnMp)·c2=Emaxβ(MnMpme)·c2.
ThisyieldsR=1.
88fm.
Theβ-decayrecoilandthedierencebetweentheatomicbindingenergiesmaybeneglected.
Chapter31.
a)AtSaturnwehavet/τ=4yrs/127yrsandwerequireN01τet/τ·5.
49MeV·0.
055=395Wpowertobeavailable.
ThisimpliesN0=3.
4·1025nuclei,whichmeans13.
4kg238Pu.
b)AtNeptune(after12years)371Wwouldbeavailable.
c)Thepoweravailablefromradiationdecreasesas1/r2.
HenceatSaturn395Wpowerwouldrequireanareaof2.
5·103m2and371WatNeptunecouldbeproducedbyanareaof2.
3·104m2.
Thiswouldpresumablyleadtoconstructionandweightproblems.
2.
a)ApplyingtheformulaN=N0eλttobothuraniumisotopesleadsto99.
280.
72=eλ238teλ235twhichyields:t=5.
9·109years.
Uraniumisotopes,likeallheavy(A>56)elements,areproducedinsupernovaexplosions.
Thematerialwhichissoejectedisusedtobuildupnewstars.
Theisotopicanalysisofmeteoritesleadstotheageofthesolarsystembeing4.
55·109years.
b)After2.
5·109years,(1eλt)ofthenucleiwillhavedecayed.
Thisis32%.
362SolutionstoChapter3c)Equation(2.
8)yieldsthatatotalof51MeVisreleasedinthe238U→206Pbdecaychain.
Inspontaneousssion190MeVissetfree.
3.
a)A2(t)=N0,1·λ1·λ2λ2λ1eλ1teλ2tforlargetimest,becauseofλ1λ2:A2(t)=N0,1·λ1.
b)Theconcentrationof238UinconcretecanthusbefoundtoberoomvolumeV:400m3e.
concretevolumeVB:5.
4m3=U=V·AVB·λ238=1.
5·1021atomsm3.
4.
NuclearmassesforxedAdependquadraticallyuponZ.
Fromthedeni-tionsin(3.
6)theminimumoftheparabolaisatZ0=β/2γ.
Theconstantaainβandγispartoftheasymmetryterminthemassformula(2.
8)and,accordingto(17.
12),doesnotdependupontheelectromagneticcouplingconstantα.
The"constant"ac,whichdescribestheCoulombrepulsionandentersthedenitionofγ,isontheotherhandproportionaltoαandmaybewrittenas:ac=κα.
InsertingthisintoZ0=β/2γyieldsZ0=β2aa/A+κα/A1/3=1α=2κAZ0A1/3(Aβ2aaZ0).
AssumingthattheminimumofthemassformulaisexactlyatthegivenZonends1/αvaluesof128,238and522forthe18674W,18682Pband18688Ranuclides.
Stable18694Pucannotbeobtainedjustby"twiddling"α.
5.
TheenergyEreleasedinAZX→A4Z2Y+αisE=B(α)δBwhereδB=B(X)B(Y).
Notethatwehavehereneglectedthedierenceintheatomicbindingen-ergies.
Ifwefurtherignorethepairingenergy,whichonlyslightlychanges,weobtainE=B(α)BZδZBAδA=B(α)2BZ4BA=B(α)4av+83as13A1/3+4acZA1/31Z3Aaa12ZA2.
PuttingintheparametersyieldsE>0ifA>150.
Naturalα-activityisonlysignicantforA>200,sincethelifetimeisextremelylongforsmallermassnumbers.
6.
Themothernucleusandtheαparticleareboth0+systemswhichimpliesthatthespinJandparityPofadaughternucleuswithorbitalangularmomentumLandspatialwavefunctionparity(1)Lmustcombineto0+.
ThismeansthatJP=0+,1,2+,3,···areallowed.
SolutionstoChapter4363Chapter41.
a)Inanalogyto(4.
5)thereactionratemustobey˙N=σ˙Ndnt,where˙Ndsigniesthedeuteronparticlecurrentandntistheparticlearealdensityofthetritiumtarget.
TheneutronratefoundinanysolidangleelementdΩmustthenobeyd˙N=dσdΩdΩ˙Ndnt=dσdΩFR2IdeμtmtNA,whereeistheelementaryelectriccharge,mtisthemolarmassoftritiumandNAisAvogadro'snumber.
Insertingthenumbersyieldsd˙N=1444neutrons/s.
b)Rotatingthetargetawayfromtheorthogonalincreasestheeectiveparticleareadensity"seen"bythebeambyafactorof1/cosθ.
Arotationthrough10thusincreasesthereactionrateby1.
5%.
2.
ThenumberNofbeamparticlesdecreasesaccordingto(4.
5)withthedistancexcoveredasex/λwhereλ=1/σnistheabsorbtionlength.
a)Thermalneutronsincadmium:WehavenCd=CdNAACdwheretheatomicmassofcadmiumisgivenbyACd=112.
40gmol1.
Wethusobtainλn,Cd=9μm.
b)Forhighlyenergeticphotonsinleadonemayndinananalogousmanner(APb=207.
19gmol1)λγ,Pb=2.
0cm.
c)Antineutrinospredominantlyreactwiththeelectronsintheearth.
Theirdensityisne,earth=earthZAearthNA.
Wethereforeobtainλν/earth=6.
7·1016m,whichisabout5·109timesthediameteroftheplanet.
Note:thenumberofbeamparticlesonlydecreasesexponentiallywithdistanceifonereactionleadstothebeamparticlesbeingabsorbed;acriterionwhichisfullledintheaboveexamples.
Thesituationisdierentifk1reactionsareneeded(e.
g.
,αparticlesinair).
InsuchcasestherangeisalmostconstantL=k/σn.
364SolutionstoChapter5Chapter51.
a)FromQ2=(pp)2and(5.
13)onendsQ2=2M(EE),withMthemassoftheheavynucleus.
ThisimpliesthatQ2islargestatthesmallestvalueofE,i.
e.
,θ=180.
Themaximalmomentumtransferisthenfrom(5.
15)Q2max=4E2MMc2+2E,b)From(5.
15)wendforθ=180thattheenergytransferν=EEisν=E111+2EMc2=2E2Mc2+2E.
TheenergyofthebackwardlyscatterednucleusisthenEnucleus=Mc2+ν=Mc2+2E2Mc2+2EanditsmomentumisP=Q2max+ν2c2=4ME2Mc2+2E+4E4c2(Mc2+2E)2.
c)ThenuclearComptoneectmaybecalculatedwiththehelpofΔλ=hMc(1cosθ).
Thesameresultasforelectronscatteringisobtainedsincewehaveneglectedtheelectronrestmassina)andb)above.
2.
Thoseαparticleswhichdirectlyimpingeuponthe56Fenucleusareab-sorbed.
Elasticallyscatteredαparticlescorrespondtoa"shadowscatter-ing"whichmaybedescribedasFraunhoferdiractionuponadisc.
ThediameterDofthediscisfoundtobeD=2(3√4+3√56)·0.
94fm≈10fm.
IntheliteratureDismostlyparameterisedbytheformulaD=23√A·1.
3fm,whichgivesthesameresult.
Thewavelengthoftheαparticlesisλ=h/p,wherepistobeunderstoodasthatinthecentreofmasssystemofthereaction.
Usingpc=840MeVonendsλ=1.
5fm.
Therstminimumisatθ=1.
22λ/D≈0.
18≈10.
2.
Theintensitydis-tributionofthediractionisgivenbytheBesselfunctionj0.
ThefurtherminimacorrespondtothenodesofthisBesselfunction.
Thescatteringangleought,however,tobegiveninthelaboratoryframeandisgivenbyθlab≈9.
6.
SolutionstoChapter53653.
Thesmallestseparationoftheαparticlesfromthenucleusiss(θ)=a+asinθ/2forthescatteringangleθ.
Theparameteraisobtainedfrom180scattering,sincethekineticenergyisthenequaltothepotentialenergy:Ekin=zZe2c4πε0c2a.
For6MeVαscatteringogold,wehavea=19fmands=38fm.
FordeviationsfromRutherfordscatteringtooccur,theα-particlesmustmanagetogetclosetothenuclearforces,whichcanrsthappenataseparationR=Rα+RAu≈9fm.
AmoredetaileddiscussionisgiveninSec.
18.
4.
SincesRnonuclearreactionsarepossiblebetween6MeVαparticlesandgoldandnodeviationfromtheRutherfordcross-sectionshouldthereforebeexpected.
Thiswouldonlybepossibleformuchlighternuclei.
4.
Thekineticenergyoftheelectronsmaybefoundasfollows:2MαEkinα≈λ–α!
=λ–e≈cEkine=Ekine≈2Mαc2Ekinα=211MeV.
Themomentumtransferismaximalforscatteringthrough180.
Neglect-ingtherecoilwehave|q|max=2|pe|=2λ–e≈22MαEkinα=423MeV/c,andthevariableαinTable5.
1maybefoundwiththehelpof(5.
56)tobeαmax=|q|maxR=423MeV·1.
21·3√197fm197MeVfm=15.
1.
Thebehaviourofthefunction3α3(sinααcosα)fromTable5.
1issuchthatithas4zeropointsintherange00.
003.
370SolutionstoChapter9b)Theaveragenumberofresolvedpartonsisgivenbytheintegraloverthepartondistributionsfromxminto1.
Thenormalisationconstant,A,hastobechosensuchthatthenumberofvalencequarksisexactly3.
Onends:SeaquarksGluonsx>0.
30.
0050.
12x>0.
030.
44.
9Chapter91.
a)Therelationbetweentheeventrate˙N,thecross-sectionσandtheluminosityLisfrom(4.
13):˙N=σ·L.
Thereforeusing(9.
5)˙Nμ+μ=4πα22c23·4E2·L=0.
14/s.
Atthiscentreofmassenergy,√s=8GeV,itispossibletoproducepairsofu,d,sandc-quarks.
TheratioRdenedin(9.
10)canthereforebecalculatedusing(9.
11)andwesoobtainR=10/3.
Thisimplies˙Nhadrons=103·˙Nμ+μ=0.
46/s.
b)At√s=500GeVpaircreationofall6quarkavoursispossible.
TheratioisthusR=5.
Toreachastatisticalaccuracyof10%onewouldneedtodetect100eventswithhadronicnalstates.
FromNhadrons=5·σμ+μ·L·tweobtainL=8·1033cm2s1.
Sincethecross-sectionfallsosharplywithincreasingcentreofmassenergies,futuree+e-acceleratorswillneedtohaveluminositiesofanorder100timeslargerthanpresentdaystoragerings.
2.
a)FromthesuppliedparametersweobtainδE=1.
9MeVandthusδW=√2δE=2.
7MeV.
AssumingthatthenaturaldecaywidthoftheΥissmallerthanδW,themeasureddecaywidth,i.
e.
,theenergydependenceofthecross-section,merelyreectstheuncertaintyinthebeamenergy(andthedetectorresolution).
Thisisthecasehere.
b)Usingλ–=/|p|≈(c)/Ewemayre-express(9.
8)asσf(W)=3π2c2Γe+eΓf4E2[(WMΥc2)2+Γ2/4].
Intheneighbourhoodofthe(sharp)resonancewehave4E2≈M2Υc4.
Fromthisweobtainσf(W)dW=6π2c2Γe+eΓfM2Υc4Γ.
SolutionstoChapter10371Themeasuredquantitywasσf(W)dWforΓf=Γhad.
UsingΓhad=Γ3Γ+=0.
925ΓwendΓ=0.
051MeVforthetotalnaturaldecaywidthoftheΥ.
Thetrueheightoftheresonanceoughtthereforetobeσ(W=MΥ)≈4100nb(withΓf=Γ).
Theexperimentallyobservedpeakwas,asaresultoftheuncertaintyinthebeamenergy,lessthanthisbyafactorofover100(seeParta).
Chapter101.
p+p→.
.
.
stronginteraction.
p+K→.
.
.
stronginteraction.
p+π→.
.
.
baryonnumbernotconserved,soreactionimpossible.
νμ+p→.
.
.
weakinteraction,sinceneutrinoparticipates.
νe+p→.
.
.
Lenotconserved,soreactionimpossibleΣ0→.
.
.
electromagneticinteraction,sincephotonradiatedo.
2.
a)C|γ=1|γ.
Thephotonisitsownantiparticle.
ItsC-parityis1sinceitcouplestoelectricchargeswhichchangetheirsignundertheC-paritytransformation.
C|π0=+1|π0,sinceπ0→2γandC-parityisconservedintheelectromagneticinteraction.
C|π+=|π,notaC-eigenstate.
C|π=|π+,notaC-eigenstate.
C(|π+|π)=(|π|π+)=1(|π|π+),C-eigenstate.
C|νe=|νe,notaC-eigenstate.
C|Σ0=|Σ0,notaC-eigenstate.
b)Pr=rPp=pPL=LsinceL=r*pPσ=σ,sinceσisalsoangularmomentum;PE=E,positiveandnegativechargesare(spatially)ippedbyPtheeldvectorthuschangesitsdirection;PB=B,magneticeldsarecreatedbymovingcharges,thesignofthedirectionofmotionandofthepositionvectorarebothipped(cf.
Biot-Savartlaw:B∝qr*v/|r|3).
P(σ·E)=σ·EP(σ·B)=σ·BP(σ·p)=σ·pP(σ·(p1*p2))=σ·(p1*p2)3.
a)Sincepionshavespin0,thespinofthef2-mesonmustbetransferredintoorbitalangularmomentumforthepions,i.
e.
,=2.
SinceP=(1),theparityofthef2-mesonisP=(1)2P2π=+1.
SincetheparityandC-paritytransformationsofthef2-decaybothleadtothe372SolutionstoChapter10samestate(spatialexchangeofπ+/πandexchangeoftheπ-chargestates)wehaveC=P=+1forthef2-meson.
b)AdecayisonlypossibleifPandCareconservedbyit.
SinceC|π0|π0=+1|π0|π0andtheangularmomentumargumentofa)remainsvalid(=2→P=+1),thedecayf2→π0π0isallowed.
Forthedecayintotwophotonswehave:C|γ|γ=+1.
Thetotalspinofthetwophotonsmustbe2andthez-componentSz=±2.
There-foreoneofthetwophotonsmustbelefthandedandtheotherrighthanded.
(Sketchthedecayinthecentreofmasssystemanddrawinthemomentaandspinsofthephotons!
)OnlyalinearcombinationofSz=+2andSz=2canfullltherequirementofparityconservation,e.
g.
,thestate(|Sz=+2+|Sz=2).
Applyingtheparityoperatortothisstateyieldstheeigenvalue+1.
Thismeansthatthedecayintotwophotonsisalsopossible.
4.
a)Thepiondecaysinthecentreofmassframeintoachargedleptonwithmomentumpandaneutrinowithmomentump.
Energyconservationsuppliesmπc2=m2c4+|p|2c2+|p|c.
ForthechargedleptonwehaveE2=m2c4+|p|2c2.
Takingv/c=|p|c/Eoneobtainsfromtheaboverelations1vc=2m2m2π+m2=0.
73forμ+0.
27·104fore+.
b)Theratioofthesquaredmatrixelementsis|Mπe|2|Mπμ|2=1ve/c1vμ/c=m2em2μm2π+m2μm2π+m2e=0.
37·104.
c)Weneedtocalculate(E0)=dn/dE0=dn/d|p|·d|p|/dE0∝|p|2d|p|/dE0.
Fromtheenergyconservationequation(seeParta)wendd|p|/dE0=1+v/c=2m2π/(m2π+m2)and|p|=c(m2πm2)/(2mπ).
Puttingittogetherwegete(E0)μ(E0)=(m2πm2e)2(m2πm2μ)2(m2π+m2e)2(m2π+m2μ)2=3.
49.
Thereforethephasespacefactorforthedecayintothepositronislarger.
d)TheratioofthepartialdecaywidthsnowonlydependsuponthemassesoftheparticlesinvolvedandturnsouttobeΓ(π+→e+ν)Γ(π+→μ+ν)=m2em2μ(m2πm2e)2(m2πm2μ)2=1.
28·104.
Thisvalueisingoodagreementwiththeexperimentalresult.
SolutionstoChapter13373Chapter111.
ThetotalwidthΓtotofZ0maybewrittenasΓtot=Γhad+3Γ+NνΓνandΓν/Γ=1.
99(seetext).
From(11.
9)itfollowsthatσmaxhad=12π(c)2M2ZΓeΓhadΓtot.
SolvingforΓtotandinsertingitintotheaboveformulayieldsfromtheexperimentalresultsNν=2.
96.
VaryingtheexperimentalresultsinsidetheerrorsonlychangesthecalculatedvalueofNνbyabout±0.
1.
Chapter131.
Thereducedmassofpositroniumisme/2.
From(13.
4)wethusndthegroundstate(n=1)radiustobea0=2αmec=1.
1·1010m.
TherangeoftheweakforcemaybeestimatedfromHeisenberg'suncer-taintyrelation:R≈MWc=2.
5·103fm.
Atthisseparationtheweakandelectromagneticcouplingsareofthesameorderofmagnitude.
Themassesofthetwoparticles,whoseboundstatewouldhavetheBohrradiusR,wouldthenbeM≈2αRc≈2·104GeV/c2.
Thisisequivalenttothemassof4·107electronsor2·104protons.
Thisvividlyshowsjusthowweaktheweakforceis.
2.
From(18.
1)thetransitionprobabilityobeys1/τ∝E3γ|rfi|2.
Ifmisthereducedmassoftheatomicsystem,wehave|rfi|∝1/mandEγ∝m.
1/τ=m/me·1/τHimpliesτ=τH/940forprotonium.
3.
Thetransitionfrequencyinpositroniumfe+eisgivenbyfe+efH=74gegpmpme|ψ(0)|2e+e|ψ(0)|2HUsing(13.
4)onends|ψ(0)|2∝m3red=[(m1·m2)/(m1+m2)]3.
Onesoobtainsfe+e=204.
5GHz.
Onecananalogouslyndfμ+e=4.
579GHz.
374SolutionstoChapter14Thedeviationsfromthemeasuredvalues(0.
5%and2.
6%respectively)areduetohigherorderQEDcorrectionstothelevelsplitting.
Thesearesuppressedbyafactoroftheorderα≈0.
007.
4.
a)Theaveragedecaylengthiss=vτlab=cβγτwhereγ=EB/mBc2=0.
5mΥ/mBandβγ=γ21.
Onethusobtainss=0.
028mm.
b)From0.
2mm=cβγτ=τ·|pB|/mBweobtain|pB|=2.
3GeV/c.
c)Fromtheassumption,mB=5.
29GeV/c2=mΥ/2,theB-mesonsdonothaveanymomentuminthecentreofmassframe.
Inthelaboratoryframe,|pB|=2.
3GeV/candthus|pΥ|=2|pB|.
WeobtainfromthisEΥ=m2Υc4+p2Υc4=11.
6GeV.
d)Fromfour-momentumconservationpΥ=pe++peweobtain(settingme=0)EΥ=Ee++EeandpΥc=Ee++EefromthiswegetEe+=8.
12GeVandEe=3.
44GeV(orvice-versa).
Chapter141.
Angularmomentumconservationrequires=1,sincepionsarespin0.
Inthe(=1)state,thewavefunctionisantisymmetric,buttwoidenticalbosonsmusthaveatotallysymmetricwavefunction.
2.
ThebranchinthedenominatorisCabibbo-suppressedandfrom(10.
21)wethusexpect:R≈20.
3.
a)FromthedecaylawN(t)=N0et/τweobtainthefractionofthedecayingparticlestobeF=(N0N)/N0=1et/τ.
Inthelaboratoryframewehavetlab=d/(βc)andτlab=γτ,whereτistheusuallifetimeintherestframeoftheparticle.
WethusobtainF=1expdβcγτ=1expd1m2c4E2cEmc2τ,andfromthiswendFπ=0.
9%undFK=6.
7%.
b)Fromfourmomentumconservationweobtain,e.
g.
,forpiondecayp2μ=(pπpν)2anduponsolvingfortheneutrinoenergygetEν=m2πc4m2μc42(Eπ|pπ|ccosθ).
Atcosθ=1wehavemaximalEν,whileforcosθ=1itisminimal.
WecansoobtainEmaxν≈87.
5GeVandEminν≈0GeV(moreprecisely:11keV)inpiondecay.
Inthecaseofkaondecay,weobtainEmaxν≈191GeVandEminν≈0GeV(moreprecisely:291keV).
SolutionstoChapter15375Chapter151.
b)Alloftheneutralmesonsmadeoutofu-andd-quarks(andsimilarlythess(φ)meson)areveryshortlived;cτ<100nm.
Thedilatationfactorγthattheywouldneedtohaveinordertotraverseadistanceofseveralcentimetresinthelaboratoryframeissimplynotavailableatthesebeamenergies.
Sincemesonswithheavyquarks(c,b)cannotbeproduced,asnotenoughenergyisavailable,theonlypossiblemesonicdecaycandidateistheK0S.
SimilarlytheonlybaryonsthatcomeintoquestionaretheΛ0andtheΛ0.
TheprimarydecaymodesoftheseparticlesareK0S→π+π,Λ0→pπandΛ0→pπ+.
c)WehaveforthemassMXofthedecayedparticlefrom(15.
1)M2X=m2++m2+2p2+/c2+m2+p2/c2+m22c2|p+||p|cos<)(p+,p)wherethemassesandmomentaofthedecayproductsaredenotedbym±andp±respectively.
Considertherstpairofdecayproducts:thehypothesisthatwehaveaK0S→π+π(m±=mπ±)decayleadstoMX=0.
32GeV/c2whichisinconsistentwiththetrueK0mass(0.
498GeV/c2).
ThehypothesisΛ0→pπ(m+=mp,m=mπ)leadstoMX=1.
11GeV/c2whichisinverygoodagreementwiththemassoftheΛ0.
TheΛ0possibilitycan,aswiththeK0hypothesis,becondentlyexcluded.
Consideringthesecondpairofdecayparticleswesimilarlynd:K0hypothesis,MX=0.
49GeV/c2;Λ0hypothesis,MX=2.
0GeV/c2;theΛ0hypothesisalsoleadstoacontradiction.
InthiscasewearedealingwiththedecayofaK0.
d)ConservationofstrangenessinthestronginteractionmeansthataswellastheΛ0,whichismadeupofaudsquarkcombination,afur-therhadronwithans-quarkmustbeproduced.
TheobservedK0SdecaymeansthatthiswasaK0(sd).
1Chargeandbaryonnumberconserva-tionnowcombinetoimplythatthemostlikelytotalreactionwasp+p→K0+Λ0+p+π+.
Wecannot,however,excludeadditional,unobservedneutralparticlesorveryshortlivedintermediatestates(suchasaΔ++).
2.
LetusconsiderthepositivelychargedΣparticles|Σ+=|u↑u↑s↓and|Σ+=|u↑u↑s↑.
Sincethespinsofthetwou-quarksareparallel,wehave3i,j=1i1BoththeK0andtheK0candecayasK0S(cf.
Sec.
14.
4).
376SolutionstoChapter15Werstinspect2σu·σs=3i,j=1iWealreadyknowthersttermonther.
h.
s.
from(15.
10).
Itis3forS=1/2baryonsand+3forS=3/2baryons.
Thesecondtermis+1.
ThisyieldsΔMss=493c3παs|ψ(0)|21m2u,d4mu,dmsfortheΣstates,493c3παs|ψ(0)|21m2u,d+2mu,dmsfortheΣstates.
TheaveragemassdierencebetweentheΣandΣbaryonsisabout200MeV/c2.
Withthemassformula(15.
12)wehaveMΣMΣ=ΔMss(Σ)ΔMss(Σ)=493c3παs|ψ(0)|26mu,dms≈200MeV/c2,whereweassumethatψ(0)isthesameforbothstates.
Wethusobtain(mu,d=363MeV/c2,ms=538MeV/c2)αs|ψ(0)|2=0.
61fm3.
Insertingahydrogenatom-typewavefunction,|ψ(0)|2=3/4πr3,andαs≈1,yieldsaroughapproximationfortheaverageseparationrofthequarksinsuchbaryons:r≈0.
8fm.
3.
TheΛisanisospinsinglet(I=0).
Toarstapproximationthedecayisjustthequarktransitions→u,whichchangestheisospinby1/2.
Thusthepion-nucleonsystemmustbeaI=1/2state.
ChargeconservationimpliesthatthethirdcomponentisIN3+Iπ3=1/2.
ThematrixelementsofthedecayoftheΛ0areproportionaltothesquaresoftheClebsch-Gordancoecients:σ(Λ0→π+p)σ(Λ0→π0+n)=(1121+12|1212)2(112012|1212)2=(√2/3)2(√1/3)2=2.
4.
Theprobabilitythatamuonbecapturedfroma1sstateintoa12Cnucleusis1τμC=2π12BeipνrigAσiI12Cψμ(r)(r=0)2p2νdpνdΩ(2π)3dEν.
SolutionstoChapter15377SincecarbonhasJP=0+andboronJP=1+,thisisapurelyaxialvectortransition.
Wefurtherhavedpν/dEν=1/c,dΩ=4πand|ψμ(r=0)|2=3/(4πr3μ).
Theradiusofthe12Cmuonicatomicisfoundtoberμ=aBohrmeZmμ=42.
3fm,andtheenergyisEν=mμc213.
3MeV≈90MeV.
Thisyieldstheabsorptionprobability1τμC=2πc4πcE2ν(2π)3(c)312BigAσiI12C2|ψ(0)|2.
Theseareallknownquantitiesexceptforthematrixelement.
Thismaybeextractedfromtheknownlifetimeofthe12B→12C+e+νedecay:1τ12B=12π37c612CigAσiI+12B2E5max.
Wethusnallyobtain1τμC≈1.
5·104s1.
Thetotaldecayprobabilityofthemuondecayin12Cisthesumoftheprobabilitiesofthefreemuondecayingandofitsbeingcapturedbythenucleus:1τ=1τμ+1τμC.
5.
Thesebranchingratiosdependprimarilyupontwothings:a)thephasespaceandb)thefactthatthestrangenesschangesintherstcase(Cabibbosuppression)butnotinthelatter.
Aroughestimatemaybeobtainedbyassumingthatthematrixelementsare,apartfromCabibbosuppression,identical.
From(10.
21)and(15.
49)onendsW(Σ→n)W(Σ→Λ0)≈sin2θCcos2θC·E1E25=120·257MeV81MeV5≈16.
Thisagreementisnotbadatall,consideringthecoarsenessofourapprox-imation.
InthedecayΣ+→n+e++νewewouldneedtwoquarkstochangetheiravours;(suu)→(ddu).
378SolutionstoChapter166.
a)Baryonnumberconservationmeansthatbaryonscanneitherbeanni-hilatednorcreatedbutratheronlytransformedintoeachother.
Henceonlytherelativeparitiesofthebaryonshaveanyphysicalmeaning.
b)Thedeuteronisagroundstatep-nsystem,i.
e.
,=0.
Itsparityisthereforeηd=ηpηn(1)0=+1.
Sincequarkshavezeroorbitalangularmomentuminnucleons,thequarkintrinsicparitiesmustbepositive.
c)ThedownwardscascadeofpionsintothegroundstatemaybeseenfromthecharacteristicXrays.
d)Sincethedeuteronhasspin1,thed-πsystemisinastatewithtotalangularmomentumJ=1.
Thetwonalstateneutronsareidenticalfermionsandsomusthaveanantisymmetricspin-orbitwavefunction.
Only3P1ofthefourpossiblestateswithJ=1,3S1,1P1,3P1and3D1fulllsthisrequirement.
e)Fromnn=1,weseethatthepionparitymustbeηπ=η2n(1)1/ηd=1.
f)Thenumberofquarksofeachindividualavour(NqNq)issepa-ratelyconservedinparityconservinginteractions.
Thequarkpari-tiescanthereforebeseparatelychosen.
Onecouldthuschoose,e.
g.
,ηu=1,ηd=+1,givingtheprotonapositiveandtheneutronaneg-ativeparity.
Thedeuteronwouldthenhaveanegativeparityandthechargedpionsapositiveone.
Theπ0asauu/ddmixedstatewouldthoughkeepitsnegativeparity.
Particleslike(π+,π0,π)or(p,n)althoughinsidethesameisospinmultipletswouldthenhavedistinctparities–aratherunhelpfulconvention.
Forηn=ηp=1,ontheotherhand,isospinsymmetrywouldbefullled.
Theparitiesofnucleonsandoddnucleiwouldthenbetheoppositeofthestandardconvention,whilethoseofmesonsandevennucleiwouldbeunchanged.
TheΛandΛcparitiesarejustthoseofthes-andc-quarksandmaybechosentobepositive.
Chapter161.
Theranges,λ≈c/mc2,are:1.
4fm(1π),0.
7fm(2π),0.
3fm(,ω).
Twopionexchangewithvacuumquantumnumbers,JP=0+,I=0,generatesascalarpotentialwhichisresponsiblefornuclearbinding.
Becauseofitsnegativeparity,thepionisemittedwithanangularmomentum,=1.
Thespindependenceofthiscomponentofthenuclearforceisdeterminedbythis.
Similarpropertiesholdfortheandω.
Theisospindependenceisdeterminedbytheisospinoftheexchangeparticle;I=1fortheπandandI=0fortheω.
Sinceisospinisconservedinthestronginterac-tion,theisospinofinteractingparticlesiscoupled,justasisthecasewithangularmomentum.
2.
Taking(16.
1),(16.
2)and(16.
6)intoaccountweobtainSolutionstoChapter17379σ=4πsinkbk2.
Atlowenergies,wherethe=0partialwavedominates,weobtaininthek→0limit,thetotalcross-section,σ=4πb2.
Chapter171.
AtconstantentropySthepressureobeysp=UVS,whereVisthevolumeandUistheinternalenergyofthesystem.
IntheFermigasmodelwehavefrom(17.
9):U=35AEFandhencep=35AEFV.
From(17.
3)wendforN=Z=A/2:A=2Vp3F3π23=2V(2MEF)3/23π23=EFV=2EF3V.
TheFermipressureisthenp=2A5VEF=25NEF,whereNisthenucleondensity.
ThisimpliesforN=0.
17nucleons/fm3andEF≈33MeVp=2.
2MeV/fm3=3.
6·1027bar.
2.
a)Weonlyconsidertheoddnucleons.
Theevenonesareallpairedointhegroundstate.
Therstexcitedstateisproducedeitherby(I)theexcitationoftheunpairednucleonintothenextsubshellor(II)bythepairingofthisnucleonwithanotherwhichisexcitedfromalowerlyingsubshell.
73Li2311Na3316S4121Sc8336Kr9341NbGroundstate1p13/21d35/21d13/21f17/21g39/21g19/2Excited(I)1p11/22s11/2(1f17/2)(2p13/2)(1g17/2)(1g17/2)Excited(II)(1s11/2)1p11/22s11/21d13/22p11/22p11/2JP0experiment3/23/2+3/2+7/29/2+9/2+JP0model3/25/2+3/2+7/29/2+9/2+JP1experiment1/25/2+1/2+3/2+7/2+1/2JP1case(I)1/21/2+(7/2)(3/2)(7/2+)(7/2+)JP1case(II)(1/2+)1/21/2+3/2+1/21/2380SolutionstoChapter17Thosestateswhoseexcitationwouldbebeyonda"magic"boundaryareshownhereinbrackets.
Thisrequiresalotofenergyandsoisonlytobeexpectedforhigherexcitations.
Asonesees,thepredictivepowersoftheshellmodelaregoodforthosenucleiwheretheunlledsubshellisonlyoccupiedbyasinglenucleon.
b)The(p-1p13/2;n-1p13/2)in63LiimpliesJP=0+,1+,2+,3+.
4019Khasfrom(p-1d13/2;n-1f17/2)apossiblecouplingto2,3,4,5.
3.
a)An17Onucleusmaybeviewedasbeingan16Onucleuswithanad-ditionalneutroninthe1f5/2shell.
TheenergyofthislevelisthusB(16O)B(17O).
The1p1/2shelliscorrespondinglyatB(15O)B(16O).
ThegapbetweentheshellsisthusE(1f5/2)E(1p1/2)=2B(16O)B(15O)B(17O)=11.
5MeV.
b)Onewouldexpectthelowestexcitationlevelwiththe"right"quantumnumberstobeproducedbyexcitinganucleonfromthetopmost,occu-piedshellintotheoneabove.
For16OthiswouldbetheJP=3state,whichisat6.
13MeV,andcouldbeinterpretedas(1p11/2,1d5/2).
Theexcitationenergyis,however,signicantlysmallerthanthetheoreticalresultof11.
5MeV.
Itseemsthatcollectiveeects(statemixing)aremakingthemselvesfelt.
Thisisconrmedbytheoctupoleradiationtransitionprobability,whichisanorderofmagnitudeabovewhatonewouldexpectforasingleparticleexcitation.
c)The1/2+quantumnumbersmakeitnaturaltointerprettherstex-citedstateof17Oas2s1/2.
Theexcitationenergyisthenthegapbe-tweentheshells.
d)Assuming(morethanalittlenaively)thatthenucleiarehomogenoussphereswithidenticalradii,onendsfrom(2.
11)thatthedierenceinthebindingenergiesimpliestheradiusis(3/5)·16αc/3.
54MeV=3.
90fm,whichismuchlargerthanthevalueof3.
1fm,whichfollowsfrom(5.
56).
Intheshellmodelonemayinterpreteachofthesenucleiasan16Onucleuswithanadditionalnucleon.
Thevalencenucleoninthed5/2shellthushasalargerradiusthanonewouldexpectfromtheabovesimpleformulawhichdoesnottakeshelleectsintoaccount.
e)ThelargerCoulombrepulsionmeansthatthepotentialwellfeltbytheprotonsin17Fisshallowerthanthatoftheneutronsin17O.
Asaresultthewavefunctionoftheexcited,"additional"protonin17Fismorespreadoutthanthatoftheequivalent"additional"neutronin17Oandthenuclearforcefeltbytheneutronisstrongerthanthatactingupontheproton.
Thisdierenceisnegligibleforthegroundstatesincethenucleonismorestronglybound.
4.
Attheupperedgeoftheclosedshellswhichcorrespondtothemagicnumbers50and82wendthecloselyadjacent2p1/2,1g9/2andthe2d3/2,1h11/2,3s1/2levelsrespectively.
ItisthusnaturalthatfornucleiwithSolutionstoChapter17381nucleonnumbersjustbelow50or82thetransitionbetweenthegroundstateandtherstexcitedstateisasingleparticletransition(g9/2p1/2andrh11/2d3/2,s1/2respectively).
Suchprocessesare5thorder(M4orE5)andhenceextremelyunlikely[Go51].
5.
a)Thespinofthestateisgivenbythecombinationoftheunpairednu-cleonswhichareinthe(p-1f7/2,n-1f7/2)state.
b)Thenuclearmagneticmomentisjustthesumofthemagneticmomentsoftheneutroninthef7/2shell1.
91μNandoftheprotoninthef7/2shell+5.
58μN.
From(17.
36)wewouldexpectagfactorof1.
1.
6.
a)Inthede-excitationi→fofanSmnucleusatresttheatomreceivesarecoilenergyofp2Sm/2Mwhere|pSm|=|pγ|≈(EiEf)/c.
Inthecaseathandthisis3.
3eV.
ThesameamountofenergyislostwhenthephotonisabsorbedbyanotherSmnucleus.
b)Ifwesetthematrixelementin(18.
1)toone,thisimpliesalifetimeofτ=0.
008ps,whichisequivalenttoΓ=80meV.
Inactualmea-surementsonendsτ=0.
03ps,i.
e.
,Γ=20meV[Le78],whichisofasimilarsize.
Sincethewidthofthestateismuchsmallerthantheenergyshiftof2·3.
3eV,noabsorptioncantakeplace.
Thermalmotionwillchange|pSm|byroughly±√M·kT.
Atroomtemperaturethiscorrespondstosmearingtheenergyby±0.
35eV,whichisalsoinsucient.
c)IftheSmatomemitsaneutrinobeforethedeexcitation,then|pSm|ischangedby±|pν|=±Eν/c.
Iftheemissiondirectionsoftheneutrinoandofthephotonareoppositetoeachother,thentheenergyoftheradiatedphotonis3.
12eVlargerthantheexcitationenergyEiEf.
ThiscorrespondstotheclassicalDopplereect.
Inthiscaseresonantuorescenceispossiblefortheγradiation.
Themomentumdirectionoftheneutrinocanbedeterminedinthisfashion(fordetails,see[Bo72]).
7.
Thethreelowestprotonshellsinthe14Onucleus,the1s1/2,1p3/2and1p1/2,arefullyoccupiedasarethetwolowestneutronshells.
The1p1/2shellis,however,empty(sketchedonp.
273).
Thusoneofthetwovalencenucleons(oneoftheprotonsintheir1p1/2shell)cantransformintoaneutronattheequivalentlevelandwiththesamewavefunction(superallowedβ-decay).
Wethushaveψnψp=1.
Thisisa0+→0+transition,i.
e.
,apureFermidecay.
Eachofthetwoprotonscontributesatermtothematrixelementequaltothevectorpartof(15.
39).
Thetotalistherefore|Mfi|2=2g2V/V2.
Equation(15.
47)nowbecomesln2t1/2=1τ=m5ec42π37·2g2V·f(E0).
Usingthevectorialcoupling(15.
56)onendsthehalf-lifeis70.
373s–whichisremarkablyclosetotheexperimentalvalue.
Note:thequantumnumbersanddeniteshellstructureheremeansthatthisisoneofthefew382SolutionstoChapter18caseswhereanuclearβ-decaycanbecalculatedexactly.
Inpracticethisdecayisusedtodeterminethestrengthofthevectorialcoupling.
Chapter181.
a)InthecollectivemodelofgiantresonancesweconsiderZprotonsandNneutronswhosemutualvibrationsaredescribedbyaharmonicos-cillator.
TheHamiltonianmaybewrittenasH=p22m+mω22x2whereω=80A1/3MeV,andm=A/2MNisthereducedmass.
ThesolutionoftheSchr¨odingerequationyieldsthelowestlyingoscillatorstates[Sc95]ψ0=14√π√x0·e(x/x0)2/2wherex0=/mω,ψ1=14√π√x0·√2xx0e(x/x0)2/2.
Theaveragedeviationisx01:=ψ0|x|ψ1=√2√πx0xx02e(x/x0)2dxx0=√2√πx0.
For40Cawehavex0=0.
3fmandx01=0.
24fm.
b)ThematrixelementisZx01.
Itssquareistherefore23fm2.
c)Thesingleparticleexcitationshaveabouthalftheenergyofthegiantresonance,i.
e.
,ω≈40A1/3MeV.
Thereducedmassinthiscaseisapproximatelythenucleonmass,sincethenucleonmovesinthemeaneldoftheheavynucleus.
Thisincreasesx0,andthusx01,byafactorof√40.
The24nucleonsintheoutermostshelleachcontributetothesquareofthematrixelementwithaneectivechargee/2.
Thesquareofthematrixelementissoseentobe27.
6fm2.
Theagreementwiththeresultofb),i.
e.
,themodelwheretheprotonsandneutronsoscillatecollectively,isverygood.
2.
SeeSec.
17.
4(17.
38)and(18.
49f).
Fromδ=abR,R=ab21/3.
andthenucleondensity,whichinthenucleusisroughlyN≈0.
17nucleons/fm3,itfollowsthata≈8fm,b≈6fm.
SolutionstoChapter193833.
TheFermivelocityisvF=pF/M2N+p2F/c2=0.
26c.
Theangularve-locityisω=|L|Θ≈60AMN(a2+b2)2/5=0.
95·1021s1,wherea=2b=3√4R,andwehaveemployedthevalueofRfrom(5.
56).
Thespeedisv=a·ωandisabout0.
03coraround12%oftheFermivelocity.
ThehighrotationalvelocitycausesaCoriolisforcewhichisre-sponsibleforbreakingupthenucleonpairs.
Chapter191.
a)Inthereaction4p→α+e++2νe,26.
72MeVofenergyisreleased.
Theneutrinoscarryo0.
52MeV,andso26.
2MeVremainstoheatupthesun.
Thenumberofhydrogenatomswhichareconvertedintoheliumeverysecondis:˙Np=4·4·1026W26.
2*1.
6·1013Ws≈0.
4·1039atoms/s.
b)0.
4·1010kg/sc)≈7%d)≈130terrestrialmassese)Nuclearreactionstakeplaceintheinteriorofthesun,primarilyatradiirByburningo7%ofthehydrogentheheliumconcen-trationintheinteriorofthesunisincreasedbyabout50%.
Doublingthisconcentrationmeansthathydrogenburningisnolongerecient:heliumburningstartsupandthesunswellsintoaredgiant.
2.
a)Twoneutrinosareproducedforevery4HenucleuscreatedΦν=˙Nν4πa2=2·˙EBHe·4πa2=5.
9·1010cm2s1.
b)Thenumberof71GanucleiisfoundtobeN71Ga=totalmassofgalliumaveragemassperatom·proportionof71Ga=3·104kg(0.
40·71+0.
60·69)·931.
5·1.
6·1013J/c2·0.
40=1.
0·1029,andfromthiswecanndthereactionrate:˙NReaktion=N71Ga·σνGe·Φν·ε=1.
0·1029·2.
5·1045cm2·5.
9·1010cm2s1·0.
5=0.
7/day.
384SolutionstoChapter19SinceN(t)=˙Nreactionτ(1et/τ)weexpect8Geatomsafter3weeksandafteraverylongtimewewouldexpecttohave11atoms.
Note:Thecross-sectiondependsstronglyupontheenergy.
Thevaluequotedisanaverageweightedaccordingtotheenergyspectrumofsolarneutrinos.
3.
a)ThenumberofneutronsintheneutronstarisNn=1.
8·1057.
TheenergyreleasedbyfusingNnprotonsinto56Feis2.
6·1045J.
b)Weneglectthegravitationalenergyoftheironcoreintheoriginalstar,(sinceR10km).
Thustheenergyreleasedduringtheimplosionisthegravitationalenergyoftheneutronstarminustheenergyneededtotransformtheironintofreeneutrons(thislastistheenergywhichwasoriginallyreleasedduringthefusionofhydrogenintoiron):EImplosion≈3GM25R2.
6·1045J=3.
3·1046J.
Theenergyreleasedviatheimplosionduringthesupernovaexplosionismorethantentimeslargerthanthefusionenergy.
Althoughonlyabout20.
.
.
50%ofthematteroftheoriginalstarendsupintheneutronstar,thefusionenergyreleasedduringtheentirelifetimeofthestarisslightlylessthantheenergyreleasedinthesupernovaexplosion.
c)Mostoftheenergyisradiatedoasneutrinoemission:e++e→νe+νe,νμ+νμ,ντ+ντ.
Thepositronsinthisprocessaregeneratedinthereaction:p+νe→e++n.
Neutrinoscan,however,alsobedirectlyproducedin:p+e→n+νe.
Thelasttwoprocessesareresponsibleforthetransformationoftheprotonsin56Fe.
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Math.
Soc.
Japan17(1935)48IndexAbsorptioncoecient,349Activity,26α-decay,26,31Antiproton,152Asymmetry–energy,20,248–term,20Asymptoticfreedom,108,178Atomicnumber,13Axialvectorcurrent,145,223,272Barn,46Baryonnumber,102Baryons,102,201–225β-decay,25–29,218–220,224,271–273–allowed,273–double,28,279–forbidden,273–inverse,133,248–neutrinolessdouble,280–superallowed,273Bethe-Blochformula,348Bhabhascattering,116Bindingenergy,18–20,25,237,246–248,250,252–ofdeuteron,17,234–237–ofhypernuclei,251Bjorkenscalingvariable,86Bohrmagneton,360Bohrradius,171,172,360Bornapproximation,56,266Bosons–Higgs,162–W,132–134–W,mass,153–Z0,113,122,132,152–Z0,mass,154Bottonium,181,182–decays,185Breitframe,90Breit-Wignerformula,119,155Bremsstrahlung,349Bubblechamber,350C-Parity,5,145Cabibboangle,139,219Cabibbo-Kobayashi-Maskawamatrix,seeCKMmatrix,140Callan-Grossrelation,88Calorimeter,353Carboncycle,332Centralpotential,234,242,253CERN,41,152,346Chaos,337,338Chargeconjugation,5,145Chargeradius,67–ofkaons,81–ofneutron,77–ofnuclei,68–ofpions,81–ofproton,77Chargedcurrents,134Charmonium,174–177Cherenkovcounter,355Cherenkovradiation,349Chromomagneticinteraction,180,181,210,211,238CKMmatrix,140Clebsch-Gordancoecients,209,358,359Colour,103,104,123,124,139Colourwavefunction,207Compoundnucleus,316Comptoneect,349Connement,105,108,178,239,240Conservationlaws,166394IndexConstituentquarks,101,181,189,225,338Cooperpairs,161,162,307Coriolisforce,383Coulomb–barrier,31–excitation,301–303–potential,31,171–term,20,33–threshold,301Couplingconstant–electromagnetic,7,171–strong,104,110,125–weak,135,139,158Covalentbond,239,339CPviolation,141,146,199Criticaldensity,323Cross-section,44–dierential,48–geometric,44–total,46Crystalball,175Current–charged,132–134–neutral,132Currentquarks,101de-Brogliewave-length,42Decay–constant,26–width,119Deformation,34,261–263,307,308Deformationparameter,261,306Delayedneutronemission,276Δresonance,84,201,202,208DESY,41,346,347Deuteron,234–bindingenergy,17,234–237–dipolemoment,235–magneticmoment,234–quadrupolemoment,234Dipole–t,77–formfactor,75–oscillator,289–transition,electric,36,176,288–transition,magnetic,36,176,177Diquarks,239Diracparticle,74,213Directnuclearreaction,265Dopplershift,322Doubleβ-decay,279–neutrinoless,280Driftchamber,351DWBA,268Electron–charge,11,13–discovery,11–magneticmoment,74,360–mass,11Electronvolt,6EMCeect,93,238Exclusivemeasurement,43Fermi–constant,135,136,219–decay,218,219,272–energy,246–function,271–gas,80,245,246–gas,degenerate,245–momentum,80,246,249,250–motion,79,91,92–pressure,249Feynmandiagrams,50Finestructure,172Finestructureconstant,7Fission–barrier,33,34–spontaneous,26Flavour,98,132Flux,44FNAL,41,122Formfactor,58,62–64–Dipole,75–electric,75–magnetic,75–ofkaons,80–ofnuclei,62,64,66–ofnucleons,75,77–ofpions,80Formationexperiments,204Four-momentum,53,54–transfer,73,74Four-vector,53Friedmanmodel,323ftvalue,272Index395Fusionreactions,304g-factor–ofDiracparticle,74–ofelectron,74–ofneutron,75–ofnuclei,260–ofproton,75G-parity,195γ-decay,36–38Gamowfactor,32Gamow-Tellerdecay,219,272–274Giantdipoleresonance,38,285,291,297Giantresonance,290Gluons,104,106,125,126Goldenrule,49,135,218,265Goldhaberexperiment,277GravitationalDopplershift,248Hadronicmatter,319,321Hadronisation,123,126,180Hadrons,102Halflife,26Hardcore,233,249Helicity,60,144–146HERA,41,346,347Hubbleconstant,323Hundrule,263Hypernestructure,172,260Hypernuclei,250Hyperons,201,204,250–magneticmoment,215–semileptonicdecays,223,224Impulseapproximation,78,90Inclusivemeasurement,43Internalconversion,37Invariantmass,204,205Ionisation,348–350,353Isobars,14Isomers,37Isospin,5,21,190,230–strong,5,219–symmetry,99–weak,5,217,219Isotones,14Isotopes,14Jacobianpeak,153Jets,125Kcapture,29,133,274,275,277Kaons,80,120,195–chargeradius,81Klein-Gordonequation,241Kurieplot,276LargeHadronCollider,162Larmorfrequency,216LEP,41,151,346Leptonicavormixingmatrix,142Leptonicprocesses,133Leptons,114,116,131–families,131–familynumber,131–number,131–universality,117Lifetime,26Linearaccelerator,342Liquiddropmodel,20log-ftvalue,272Luminosity,47Magicnumbers,254,255Magneticmoment,74–inthequarkmodel,213–ofhyperons,215–ofdeuteron,234–ofelectron,74–ofnuclei,259,260–ofnucleons,74,213,214Magneticspectrometer,62,352Mass–defect,13–formula,19,248–invariant,54–number,14Meissnereect,160–162Mesons,102,103,189,190–J/ψ,121,122,175–J/ψ,decay,183–186–Υ,122,181,184–B,185–D,184–exchangeof,240–pseudoscalar,189,192Mirrornuclei,21,257,258396IndexMomentofinertia,300,301,304–308Momentumtransfer,57,59,60,63,74Monopoleformfactor,80,81Moseley'slaw,13Mottcross-section,60Muons,128–decay,128Neutralcurrents,137Neutrinolessdoubleβ-decay,280Neutrinos,128–129,132–helicity,145,277–279–mass,129,276,277–oscillations,129–131–scattering,147,148Neutron–capture,17,35,38–chargeradius,77–decay,218,222–lifetime,222–magneticmoment,75,214,360–mass,14–scattering,38Neutronstars,248–250Nilssonmodel,264Non-leptonicprocesses,134Nuclear–force,229–231–magneton,75,213,360–matter,311–photoelectriceect,38,289–spinresonance,260–temperature,312Nuclearssion–induced,35–spontaneous,33nuclearshadowing,94Nucleonresonances,83,201Nuclides,14Pairproduction,349Pairing–force,263,307–term,20Parity,4,255,273–intrinsic,4,189–ofnuclearlevels,255–ofquarkonia,176–violation,144–147Partialwaves,230,231Partialwidth,119Partonmodel,89–91Phasespace,48Phasetransition–electroweak,327Photoelectriceect,349Pickupreaction,269Pions,80,102,195,240,242,250–chargeradius,81–decay,129,146,147–parity,196Pointinteraction,180Point-likeinteraction,135,137Poissonequation,242Positronium,171–173Potentialwell,246,247Principalquantumnumber,171,174Probabilityamplitude,48Productionexperiments,204,223Propagator,52,60,134Proportionalcounter,351Proton–Anti-,152–charge,13–chargeradius,77–decay,102–magneticmoment,75,213,360–mass,14Proton-protoncycle,332Quadrupolemoment,261,262–ofdeuteron,234,235Quadrupoleoscillations,297Quadrupoletransition,36,288Quantumchaos,312Quantumchromodynamics,50,104,105,107,178Quantumelectrodynamics,50Quarkmatter,249Quark-gluonplasma,321,327,328Quarkonia,171Quarks,97–99–mass,122–top,122Quasi-particles,337,338r-process,335Radiationlength,349Index397Renormalisation,158Resonances,118,119,204Resonantabsorption,278Rosenbluthformula,75Rotationbands,300,302–307Rutherfordscattering,12–cross-section,56s-process,335s-waves,231Samplingcalorimeter,353Sargent'srule,222,223Scalingviolations,108–110Scattering–angle,55–deepinelastic,88,89,147,148–elastic,42,56,57,59–experiments,41–inelastic,43,69,83,84–neutrino-electron,158–neutrino-nucleon,92,98,99–ofneutrinos,147,148–ofneutrons,315–phase,230–232–quasi-elastic,78,339–Rutherford,12Scintillator,353,354,356Seaquarks,91,92,98,100,149,152Selfsimilarity,111Semiconductordetector,352Semileptonicprocesses,133Separationenergy,38Shellmodel,245,253,255,339Siliconstripdetector,352SLAC,75SLC,151Sparkchamber,350Spindependence–ofnuclearforce,234,242Spin-orbitinteraction,172,234,261Spin-spininteraction,193,210,211,238Spontaneoussymmetrybreaking,160,161Standardmodel,165–167Storagering,346Strangeparticles,120Strangeness,120,190,201Streamerchamber,350Strippingreaction,265,266Structurefunctions,85,91Superconduction,307Superconductivity,160–162Superuidity,307Surfaceterm,20,33Surfacethickness,68Synchrotron,344,345–radiation,345Tauleptons,116,128–decay,128,133,138Tensorforce,234,235Thomsonmodel,11Topquarks,182Transition–matrixelement,48Transitionradiationdetector,355Transmission,31,32Transversemomentum,153Tunneleect,31,33Universe–closed,323–criticaldensity,323–open,323VAtheory,145,218Valencenucleon,259,270Valencequarks,91,92,98,149,152VandeGraaaccelerator,342VanderWaalsforce,239,339Vectorcurrent,145,223,272Vectormesons,119,189–192Vertex,50Virtualparticles,50Volumeterm,19–20Weakisospin,156Weinbergangle,157,158Weizs¨ackerformula,19Whitedwarves,250Woods-Saxonpotential,255Yukawapotential,241,242ZEUSdetector,356Zweigrule,121,184,196