2fermionsAssociatedwiththeelectronsisaconservedquantityexpressedasthequantumnumberknownastheleptonnumberTheleptonnumberofthenegatronisbyconvention1Theleptonnumberofthepositronalsotheantipartic ID: 937642
Download Pdf The PPT/PDF document "Chapter15DecayNotetostudentsandotherread..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Chapter15 DecayNotetostudentsandotherreaders:ThisChapterisintendedtosupplementChapter9ofKrane'sexcellentbook,"IntroductoryNuclearPhysics".KindlyreadtherelevantsectionsinKrane'sbook rst.Thisreadingissupplementarytothat,andthesubsectionorderingwillmirrorthatofKrane's,atleastuntilfurthernotice. -particle'sareeitherelectrons1orpositronsthatareemittedthroughacertainclassofnucleardecayassociatedwiththe\weakinteraction".ThediscovererofelectronswasHenriBecquerel,whonoticedthatphotographicplates,coveredinblackpaper,storednearra-dioactivesources,becamefogged.Theblackpaper(meanttokeeptheplatesunexposed)wasthickenoughtostop -particles,andBecquerelconcludedthatfoggingwascausedbyanewformofradiation,onemorepenetratingthan -particles.Thename\ ",followednaturallyasthenextletterintheGreekalphabetafter , -particleshavingalreadybeendiscoveredandnamedbyRutherford.Sincethatdiscovery,wehavelearnedthat -particlesareabout100timesmorepenetratingthan -particles,andarespin-1 2fermions.Associatedwiththeelectronsisaconservedquantity,expressedasthequantumnumberknownastheleptonnumber.Theleptonnumberofthenegatronis,byconvention+1.Theleptonnumberofthepositron,alsotheanti-particleofthenegatron,is-1.Thus,inanegatron-positronannihilationevent,th
enextleptonnumberiszero.Onlyleptonscancarryleptonnumber.(Moreonthissoon.)Recall,fromChapter13(Chapter6inKrane),ourdiscussionofthevariousdecaymodesthatareassociatedwith decay: 1Technically,theword\electron"canrepresenteitheranegatron(afancywordfore )orapositron(e+).I'lluse\electron"interchangeablywiththismeaning,andalsoe .Usuallythecontextdeterminesthemeaning.1 2CHAPTER15. DECAYAZXN !AZ+1X0N 1+e + e decayAZXN !AZ 1X0N+1+e++e +decayAZXN !AZ 1X0N+1+eelectroncapture(")(15.1)Weseefromtheaboveprocessesthatthereareotherparticlescalledneutrinos.Neutrinosarealsospin-1 2leptons(partofthelargerfermionfamily).Theyareverynearlymassless(butproventohavemass2).Theelectronneutrinoisgiventhesymbole,andhasleptonnumber+1.Theantineutrino,the e,hasleptonnumber-1.AsketchoftheorganizationoffundamentalparticlesisgiveninFigure15.1.Figure15.1:Theparticlephysicsclassi cationofbosonsandfermions,withthesub-classi cationsofbaryonsandfermionsshown.Threeviewsof decayTherearethreewaysofviewing decay.The rstisthe\radiologicalphysicsview"expressedby(15.1).Thenextisthe\nuclearphysicsview",wherewerecognizethatthedecaysofthenucleiareactuallycausedbytransformationsofthenucleonconstituents,asexpressedin(15.2).n !p+e + e decayp !n+e++e +deca
yp+e !n+eelectroncapture(")(15.2)Afreeneutronwilldecaywithameanlife, =885:7(8)s,about11minutes.Afreeprotonisbasicallystable.Oncethesenucleonsareboundinanucleus,however,conservationofenergy,withtheavailabilityoflowerenergystates,dictateswhetherornottheseprocessesarefreetoproceed.Then,thereisthemoremicroscopicview,the\particlephysicsview"expressedin(15.3),d !u+e + e decayu !d+e++e +decayu+e !d+eelectroncapture(")(15.3) 2Adirectmeasurementofneutrinomasssuggeststhatitsupperlimitisme2:2eV.Indirectmeasure-mentoftheneutrinomasssuggestthat0:04eVme0:3eV.Forthemoremassiveleptonfamilygroups,m180keV,andm 15:5MeV. 15.1.ENERGYRELEASEIN DECAY3thatrepresentsthetransitionsofnucleons,asreallytransitionsbetweentheup(u)anddown(d)quarks.Aparticlephysicspictureof -decayisgiveninFigure15.2.Figure15.2:Theparticlephysicsviewof -decay.Inthiscase,theweakforceiscarriedbytheintermediatevectorboson,theW .Inthecaseof -decay,theweakforceiscarriedbytheintermediatevectorboson,theW+,theantiparticletotheW .Thereisalsoaneutralintermediatevectorboson,Z0,thatisresponsibleforsuchthingsasscattering.Consequencesof -decay's3-body nalstate -decayhas3\bodies"inthe nalstate:therecoildaughternucleus,thee,andaneutrino.Typically,thedaught
ernucleus(eveninthecaseoffreeneutrondecay,ismuchmoremassivethantheleptons,therefore,theleptonscarryo mostoftheenergy.(Evenintheworstpossiblecase,thatoffreeneutrondecay,therecoilprotoncanatmostabout0.4keV,orabout0.05%ofthereactionQ-value.)Consequently,ifonemeasuresthekineticenergyoftheresultantelectron,onemeasuresadistributionofenergies,that(generally)peaksatsmallenergies,andreachesan\end-point"energy,theso-called -endpoint.This -endpointrepresentsthecasewherethe'senergyapproacheszero.SeeFigure15.3.Figure15.3:Atypicalelectronenergyspectrumthatismeasuredina decay.Theendpointenergyisthemaximumenergythatcanbegiventotheelectron,andthatiscloselyrelatedtothereactionQ-value(smallrecoilcorrection).Atlesserenergies,thecarrieso someoftheavailablekineticenergythatQprovides.Thisleadsnaturallytoadiscussionof...15.1Energyreleasein decayNeutrondecayn !p+e + emnc2=mpc2+me c2+m ec2+QnQn=mnc2 mpc2 me c2 m ec2Qn=(939:565580(81) 938:272013(23) 0:5110999(0))[MeV] m ec2Qn=0:782568(84)[MeV] m ec2(15.4)Since410 8m ec22:210 6[MeV],wecansafelyignoretheneutrinorestmassenergy,withintheexperimentaluncertainofthereactionQ, 4CHAPTER15. DECAYQn=0:782568(84)[MeV](15.5)Accountingforprotonrecoil,theexactrelationshipbetweentheelectrone
ndpointenergyandQ,isgivenby:Tmaxe=(mp+me)c2" 1+s 1+2Qnmpc2 [(mp+me)c2]2#TmaxeQn 1+me=mp:(15.6)Puttinginnumericalvalues,wascalculateTmaxe=0:782142(84)[MeV],whichagreeswiththedirectmeasurementofTmaxe=0:782(13)[MeV].Wecancalculatetheproton'srecoilenergybyusingConservationofEnergy:Tmaxp=Qn TmaxeTmaxeQn1 1 1+me=mpTmaxeQn(me=mp):(15.7)ThisevaluatesnumericallytoTmaxp0:426(84)[keV].Qfor -decayFor -decayAZXN !AZ+1X0N 1+e + e(15.8)Goingbacktothede nitionofQintermsofnuclearmasses,andignoring,henceforth,themassoftheneutrino:Q =mN(AZXN) mN(AZ+1X0N 1) mec2;(15.9)wherethesubscript\N"connotesnuclear(notatomic)masses.Therelationshipbetweenthenuclear(nosubscript\N")andatomicmassis: 15.1.ENERGYRELEASEIN DECAY5m(AZXN)c2=mN(AZXN)c2+Zmec2 ZXi=1Bi;(15.10)whereBiisthebindingenergyofthei'thatomicelectron.Substituting(15.10)in(15.9),toeliminatethe(lesswellknown)nuclearmassesresultsin:Q =m(AZXN) Zmec2 m(AZ+1X0N 1) (Z+1)mec2 mec2+"ZXi=1Bi Z+1Xi=1B0i#=m(AZXN) m(AZ+1X0N 1)c2+"ZXi=1Bi Z+1Xi=1B0i#;=m(AZXN) m(AZ+1X0N 1)c2+"ZXi=1(Bi B0i) B0Z+1#;(15.11)notingthattheelectronmasseshavecanceledinthiscase.ThefactorZXi=1Bi Z+1Xi=1B0i=ZXi=1(Bi B0i) B0Z+1isthedi erenceintheenergyoftheelectronicorbitalcon gurationoftheparentanddau
gh-ternuclei.Generally,thisdi erencecanbeignored.However,inthecaseoflargeZnuclei,itcanamounttoabout10keV.ForaccuratedeterminationsofQ,thedi erenceinatomicelectronbindingenergymustbeaccountedfor.Qfor +-decaySimilarconsiderationsfor +decayleadto:Q +=m(AZXN) m(AZ 1X0N+1) 2mec2+"ZXi=1Bi Z 1Xi=1B0i#:(15.12)Herewenotethattheelectronrest-massenergiesdonotcompletelycancel.However,thediscussionregardingtheelectronbindingenergyremainsthesame.QforelectroncaptureForelectroncapture: 6CHAPTER15. DECAYQ"=m(AZXN) m(AZ 1X0N+1)c2 Bn+"ZXi=1Bi Z 1Xi=1B0i#:(15.13)Thelattertermrelatedtoelectronbindingenergy,ZXi=1Bi Z 1Xi=1B0iisgenerallyignored,forthereasonscitedabove.However,thethebindingenergyofthecapturedelectron,Bncanapproach100keVforlarge-Znuclei,andcannotbeignored.Discussionpoint:Freeneutrondecay,revisitedFromourcurrentunderstandingoftheweakinteraction,theelectroniscreatedwhenadownquarkchangesintoanupquark.TheQvalueforthisreactionis0.782MeV.Letusseeifwecanapplysomereasoningfromclassicalphysicstosaysomethingabouttheobservationofsuchadecay.Iftheelectronwerea\point"particle,anditwascreatedsomewhereinsidetheneutronatradiusr,iswouldfeelanattraction:V(r)= e2 40((Rp r) Rp"3 2 1 2r Rp2#+(r Rp) r);whereRpistheradiusoftheprot
on.Weareassumingthatthequarksaremovingsofastinsidetheproton,thatalltheelectronseesisacontinuousblurofchargeaddinguptooneunitofcharge.So,ifRp1:2fm(fromRN=1:22 1:25[fm]A1=3),wecanconcludethatthekineticenergythattheelectronisrequiredtohavetoescapedthenucleusfallsintherange:e2 40RpTe3 2e2 40Rp1:2[MeV]Te1:8[MeV]inotherwords,itcannothappen.Thisisincontradictionwiththeobservationthatitdoesdecay,withameanlifeofabout11minutes.Classdiscussion:Canyouexplainthis? 15.2.FERMI'STHEORYOF DECAY715.2Fermi'stheoryof decayFermi'stheoryof decaystartswithastatementofFermi'sGoldenRule#2fortransitionrate,:=2 ~jVifj2(Eif);(15.14)whereVisapotentialthatcausesthetransitionfromaninitialquantumstate i(theparentnucleusinthethiscase)toa nalone, f,thatincludeswavefunctionsofthedaughternucleus,theelectronanditsneutrino.Vifh fjVj iiisthetransitionamplitude.ThederivationofFermi'sGoldenRule#2isgenerallyreservedforgraduatecoursesinQuantumMechanics,butaversionofthederivationisavailableinChapter13,foryourinterest.Whatconcernsusnow,istocalculatethedensityof nalstates,(Eif),forthe -transition.Thisderivation guressoprominentlyinthe -spectrum,andtheendpointenergy.StartinginChapter13,thedensityofstatesisderivedfornon-relativisticparticles
withmass,relativisticparticleswithmass(theelectroninthiscase),andmasslessparticles(theneutrinointhiscase).Westartwith(13.21).Thenumberofstates,N,ofaparticleinthe nalstatewithenergyEisgivenby:dN= 2n2dn:(15.15)wheren=p n2x+n2y+n2z,and(nx;ny;nz)arethequantumnumbersofafreeparticleinnin niteboxpotential,withsideL.themomentumandthen'sarerelatedby:pi=ni~=L:(15.16)Putting(15.16)into(15.15)gives:dN=1 22L3 ~3p2dp:(15.17)Or,dividingbydE,dN dE=1 22L3 ~3p2dp dE:(15.18) 8CHAPTER15. DECAYWeshouldpointoutthat(15.18)isvalidforallparticles,massless,relativisticandnon-relativistic,since(15.16)isuniversal.Allweneeddonowisrelatemomentumtoenergytocomputethedensityfactors.Fortheneutrino,whichwearenowtreatingasmassless,p=E=cdp=dE=cdN dE=1 22L3 ~3c3E2(15.19)Fortheelectron,thatmustbetreatedrelativistically,pe=p E2e (mec2)2=cdpe=[Ee=(cp E2e (mec2)2)]dEedNe dEe=1 22L3 ~3c3p E2e (mec2)2EedNe dTe=1 22L3 ~3c3p Te(Te+2mec2)(Te+mec2)(15.20)For decaywehavetwoparticlesinthe nalstate,sowecanexpresstherateofdecaytoproduceanelectronwithmomentumpas:d dp=2 ~jVifj2dNe dpdN dEif;(15.21)Ifqisthemomentumoftheneutrino,Eif=Te+cqdEif=c(dq)(Te xed):(15.22)Thus,d dp=2 ~cjVifj21 22L3 ~3p21 22L3 ~3q2(Eif [Te+T]):(15.23)Wherethe-funct
ionaccountsspeci callyfortheconservationofenergy. 15.2.FERMI'STHEORYOF DECAY9RecallthatthefreeelectronandneutrinowavefunctionshavetheformL 3=2exp(i~px=~)andL 3=2exp(i~qx=~),respectively.Thus,theLforthesideoftheboxdisappearsfromthecalculation.Wealsoreplaceq=(Q Te)=c,ignoringtherecoilofthedaughternucleus.Finally,integratingoverallpossibleneutrinoenergies,weobtain:d dp=jMifj2 23~7c3p2(Q Te)2ord dp=jMifj2 23~7cp2q2(15.24)whereMif=L3Vif.ThuswehavederivedFermi'scelebratedequation.Justabriefnoteondimensions:jVifj2hasunits[E2]becauseallthewavefunctionsinsidearenormalized.GettingridofalltheL'sresultsinMifhavingunits[length3energy].(15.24)iscorrectdimensionally.AllowedtransitionsNowweexaminetheformofthe\matrixelement"Mif.Thishaschangedformseveraltimesduringthederivation,andwillagain,toconformwithKrane'sbook.WenowrewriteMif=gMifMif=h(ei~px=~)(ei~qx=~) X0jO j Xi;(15.25)wheregisthe\strength"ofthe transition.Thisisascalarquantitythatplaystheroleofe,theelectriccharge,forelectromagnetictransitions.Theunnormalizedelectronwavefunctionisexp(i~px=~),andtheunnormalizedneutrinowavefunctionisexp(i~qx=~). X0isthewavefunctionofthedaughternucleus,while Xisthewavefunctionoftheparentnucleus.Finally,O istheweakinteractionope
rator,thecauseofthetransition.Werecallfromtheclassdiscussions,thattheelectronandneutrinowavefunctionshavewavelengthsthataremanytimesthesizeofthenucleus.So,itseemsreasonabletoexpandthesewavefunctionsinaTaylorseriesexpansion,toseehowfarweget.Namely,exp(i~px=~)=1+i~px ~ ~px ~2+ 10CHAPTER15. DECAYexp(i~qx=~)=1+i~qx ~ ~qx ~2+(15.26)Thustheleading-ordertermof(15.25)is:M0if=h X0jO j Xi:(15.27)IfM0if=0,the decayiscalledan\allowed"transition,andtherateisrelativelyprompt.IfM0if=0,thenwemustgotohigherordertermsin(15.26).Thesearecalled\forbidden"transitions,andoccur,butatmuchslowerrates.(Moreonthistopiclater.)Kranelikestoadoptthefollowingshorthand.Forallowedtransitions,weseethat:d0 dp=g2jM0ifj2 23~7cp2q2:(15.28)IfwehaveN(t) -emittersinasample,themomentumspectrumofelectronsthatmaybemeasuredis:N0(p)dp=N(t)d0 =g2N(t)jM0ifj2 23~7c5p2q2dp:(15.29)IfN(t)changeslittleoverthecourseofthemeasurementofthespectrum(theusualcase):N0(p)dp=C(0)p2q2dp;(15.30)wherewehavegatheredallconstantswithinsidethelargeparenthesesin(15.29)intoaglobalconstantC(0),thatisdeterminedexperimentally.Itcanbedeterminedthroughtheanormalizationcondition,ZdpN0(p)1:Conventionalforms:N0(p),N0(Te)N0(p)expressedin(15.30)containspandq,thatarere
latedbyconservationofenergy.Intermsofsinglemomentumvariable,N0(p)dp=C(0) c2p2hQ p (cp)2+(mec2)2+mec2i2dp;(15.31) 15.2.FERMI'STHEORYOF DECAY11usingrelativistickinematicrelationships.Themaximumpossiblepoccurswhentheneutrinocomponentdropstozero.Thisiseasilyfoundtobe:pmax=1 cp Q2+2Qmcc2:(15.32)AnevenmorecommonexpressionistoshowN0intermsofTe.We ndthisbysaying:N0(Te)dTe=N0(p)dp=N0(p)dp dTedTe;(15.33)Applyingrelativistickinematicrelationships,we nd:N0(Te)dTe=C(0) c5p T2e+2Temec2(Te+mec2)(Q Te)2dTe:(15.34)Herethe -endpointatQ=Teisevident.Accountingof"forbiddeness"andnuclearCoulombe ect.Therearetwootherattributesof -spectrawemusttakeaccountof,beforewestartusingthetheoreticalspectralshapetoassistinanalyzingdata.The rstofthesehastodowiththeinteractionofthedaughter'sCoulombchargewiththeresultantelectronorpositroninthe nalstate.Thisnuclearchargehasnoe ect,ofcourse,ontheneutralneutrino.Goingbackto(15.25),wewrotetheelectronwavefunctionasafreeplanewave.Inactualfact,thatwasafairlycrudeapproximation.Theseplanewavesaredistortedsigni cantlybytheattractionthe wouldfeel,andtherepulsionthatthepositronwouldfeel.Incidentally,thereisnoe ectonourconclusionsregarding\allowed"or\forbidden".Accountingforthisisquiteinvolved,butnotbeyond
ourcapabilities.Wewouldhavetogobackto(15.25)andwritetheelectronwavefunctionsintermsoffreeparticlesolutionstotheCoulombpotential.(InNERS311welearnalotaboutboundstatesoftheCoulombpotential.)Ihaveneverseendetaileddiscussionofthisinevengraduate-leveltexts,andinterestedstudentsareusuallytoldtoseekoutthepapersintheliterature.Theresultis,however,thatthe -spectraaremultipliedbyacorrectionfactor,theFermifunction,thatdependsonthechargeofthedaughternucleus,Z0,andtheelectronmomentumandsign,F(Z0;p).Thee ectithascouldhavebeenanticipatedfromclassicalconsiderations.Theelectronspectraisdraggedbacktowardlesservalues,whilethepositronspectraarepushedtowardhighervalues.SeeFigure(9.3)inKrane. 12CHAPTER15. DECAYThe\forbiddeness"ofthedecayalsoa ectstheshapeofthespectrum.Thisisalsoamultiplicativecorrectiontothe -spectrum.Therearedi erenceshapesdependingonthelevelof\forbiddeness",andthatisdeterminedbytheamountoforbitalangularmomentum,L,carriedawaybytheelectron-neutrinopair,aswellastheirmomenta.ExamplesoftheseshapefactorsaregiveninTable15.2,forthe\uniqueforbiddentransitions"3. L SL(p;q) 0 1 Allowed1 (p2+q2)=(mec)2 Unique rstforbidden2 (p4+10 3p2q2+q4)=(mec)4 Uniquesecondforbidden3 (p6+7p4q2+7p2q4+q6)=(mec)6 Uniquethirdforbidden... ...
... Table15.1:Shapefactorsforthe rstthreeuniqueforbiddentransitions.The -spectrumrevealedWithallthesevariousfactorsa ectingthespectralshapeanddecayratesfor decay,wewritedownthe nalformthatisemployedfordataanalysis:N(p)/jMLifj2p2(Q Te)2SL(p;q)F(Z0;p);(15.35)where,1.MLifisthenuclearmatrixelementassociatewiththetransition.Itcandependonpandq,aswellasthealignmentofspinandangularmomentumvectors.Itexhibitsaverystrongdependenceontheangularmomentum,L,carriedo bytheleptonpair.MLifalsodependsstronglyonthe\closeness"oftheinitialand nalnuclearquantumwavefunctions.Theclosertheinitialand nalnuclearquantumstatesare,thelargertheiroverlap,resultinginalargerMLif.2.p2(Q Te)2isthe\statisticalfactor"associatedwiththedensityof nalstates.3.F(Z0;p),theFermifunction.Itaccountsforthedistortionofthespectralshapeduetoattraction/repulsionoftheelectron/positron.4.SL(p;q)accountsforspectralshapedi erences.Itdependsonthetotalorbitalangularmomentumcarriedo bytheelectron-neutrinopair,~L,theirtotalspinvalue,~S,andtheirorientationwithrespecttoeachother. 3Relativisticquantummechanicsallowsustocalculatetheseinthespecialcaseofuniquetransitions.Thesetransitionsareonesinwhichtheangularmomentumvectorandthetwoleptonspinsareallaligned. 15.3.EXPERIMEN
TALTESTSOFFERMI'STHEORY1315.3ExperimentaltestsofFermi'stheoryKurieplots:Shapeofthe spectrumToemploy(15.35)toanalyze spectra,oneplots:s N(p) SL(p;q)F(Z0;p)vs:Te;(15.36)usingtheinitialassumptionthatL=0,sothatSL(p;q)=1.Ifthedatapointsfallanastraightline(statisticaltestsmaybenecessary),oncecaneasilyobtaintheQ-valuefromthex-intercept.ThistypeofplotiscalledaKurieplot(namedafterFranzKurie.)onehasalsoidenti ed,fromtheshape,thatthisisanallowedtransition.IftheKurieplotisnotstraight,onemustsuccessivelytestshapefactorsuntilastraightlinematchisobtained.Oncetheshapefactorisdetermined,thelevelofforbiddenessisdetermined,andtheQ-valuemaybeextrapolatedfromthedataunambiguously.Totaldecayrate:Theft1=2,log10ftvaluesPuttingintheCoulombandshapefactorsinto(15.28)allowsustodeterminethetotaldecayratefora -decayprocess, =g2jMLifj2 23~7cZpmax0dpSL(p;q)F(Z0;p)p2q2=g2m5ec4jMLifj2 23~71 (mec)5Zpmax0dpSL(p;q)F(Z0;p)p2q2g2m5ec4jMLifj2 23~7fL(Z0;Q);(15.37)wherethedimensionlessintegralinlargesquarebrackets,isatheoreticalfactorthatmaybepre-computedandemployedinthedataanalysis.Thisisconventionallywrittenintermsofhal\rife,t1=2=log(2)= .Thus,fL(Z0;Q)t1=2ft1=2=loge(2)23~7 g2m5ec4jMLifj2:(15.38) 14CHAPTER15. DECAYThisisknowncolloquially
astheftvalue.(Pronouncede tee.)Theft'scanbequitelarge,andsometimesthe\logft"valueisquoted.(Pronouncedloge tee.)Theprecisede nitionislog10(ft1=2).MassoftheneutrinoOurapplicationsof -decayignoretheneutrinomass,buttheyturnouttobecriticallyimportantforcosmology.Thereisoneimportantfact:theydo havemass,butitisverysmall.Thetablebelowshowsthecurrentstateofthemassdeterminationsofthethreegenerationsofleptons,e,,and .lepton\ravorneutrinosymbolmass(eV) ee0:04 !2:21:70105 1:5510715.4AngularmomentumandparityselectionrulesClassi cationoftransitionsin decayTheeandtheinthe nalstatesofa decayeachhaveintrinsicspin-1 2.Conservationoftotalangularmomentumrequiresthat:~IX=~IX0+~L+~S;(15.39)where~IX,~IX0arethetotalangularmomentaoftheparentanddaughter,respectively,and~L,~Sarethetotalorbitalandtotalspinangularmomentum,respectively,oftheepair.Therefore,theIcanbeL.orjL1j.IfL=0,thenI=1.Thereareonlytwocasesforleptonspinalignment.S=0,whentheeintrinsicspinsanti-align,iscalledaFermitransition.S=1,whentheeintrinsicspinsalign,iscalledaGamow-Tellertransition.Generally,asL",#,t1=2",becausethereismuchlessoverlapoftheewavefunctionswiththenucleus.TheentirecharacterizationschemeisgiveninTable15.4.Nomenclaturealert! 15.4.ANGULARM
OMENTUMANDPARITYSELECTIONRULES15 TypeofTransition SelectionRules Le ? ft superallowed I=0;1 0 no 1103{1104allowed I=0;1 0 no 2103{106 1stforbidden I=0;1 1 yes 106{108unique1stforbidden I=2 1 yes 108{109 2ndforbidden I=1;2 2 no 21010{21013unique2ndforbidden I=3 2 no 1012 3rdforbidden I=2;3 3 yes 1018unique3rdforbidden I=4 3 yes 41015 4thforbidden I=3;4 4 no 1023unique4thforbidden I=5 4 no 1019 Table15.2:Classi cationoftransitionsin decay.Notes:()0+!0+canonlyoccurviaFermidecay.()UniquetransitionsareGamow-Tellertransitionswhere~Land~Sarealigned.Theshapefactorshaveverysimpleformsinthiscase.()Forthen2forbiddentransitions,theI=(n 1)transitionisoftenassociatedwiththen 2forbiddentransition,beingindistinguishableinthemeasurementsoftheseprocesses. Nomenclature Meaning ~L;L Totalorbitalangularmomentumoftheepair~S;S TotalspinangularmomentumoftheepairFermi(F)transition eintrinsicspinsanti-align,S=0Gamow-Teller(GT)transition eintrinsicspinsalign,S=1Superallowed Thenucleonthatchangedform,didnotchangeshell-modelorbital.Allowed L=0transition.M0if=0.See(15.27).nthforbidden Theepaircarryo nunitsoforbitalangularmomentumUnique ~Land~Sarealigned. 16CHAPTER15. DECAYExampleso
fallowed decaysThisisstraightoutofKrane.14O(0+)!14N(0+)mustbeapureFermidecaysinceitis0+!0+.Otherexamplesare34Cl!34S,and10C!10B.6He(0+)!6Li(1+),a0+!1+transition.ThismustbeapureGamow-Tellerdecay.Othersimilarexamplesare13B(3 2 )!13C(1 2 ),and230Pa(2 )!230Th(3 ).n(1 2+)!p(1 2+)Thisisamixedtransition.TheFtransitionpreservesthenucleonspindirection,theGTtransition\ripsthenucleonspin.(Showdrawing.) decaycaneitherbeoftheFtype,theGTtypeoramixtureofboth.Wemaygeneralizethematrixelementandcouplingconstantasfollows,foralloweddecays:gM0=gFM0F+gGTM0GT=gFh X0j1j Xi+gGTh X0jO"#j Xi;(15.40)whereO"#symbolizesanoperatorthat\ripsthenucleonspinfortheGTtransition.TheoperatorfortheFtransitionissimply1,(i.e.unity),andjustmeasurestheoverlapbetweentheinitialand nalnuclearstates.ThefractionofFtransitionsis:fF=g2FjM0Fj2 g2FjM0Fj2+g2GTjM0GTj2=y2 1+y2;(15.41)where,ygFM0F gGTM0GT:(15.42)TablesofyvaluesaregiveninKraneonpage290.15.4.1MatrixelementsforcertainspecialcasesThissectionismeanttoexplainseveralthingsgivenwithoutexplanationinKrane'sChap-ter9. 15.5.COMPARATIVEHALF-LIVESANDFORBIDDENDECAYS17Mif=p 2,forsuperallowed0+!0+transitions ThiswasstatednearthetopofthetextonKrane'sp.284.Weknowthata0+!0+allowedtransition(superorregular),mustbeanF
transition.Inthecasethatitisalsoasuperallowedtransition,wecanwriteexplicitly:Mif=D X0(0+)1 p 2[e(")(#)+e(#)(")] 1 X(0+)E;(15.43)wheretheintrinsicspinsoftheepairareshownexplicitly.Thisspinwavefunctionisproperlynormalizedwiththep 2asshown.Separatingthespinspart,andthespacepart,Mif=1 p 2h X0j Xih(e(")(#)+e(#)("))j~0i=p 2;(15.44)sinceh X0j Xi=1forsuperallowedtransitions,andhe(")(#)j~0i=he(#)("))j~0i=1.Usingthisknowledge,onecanmeasuredirectly,gFfrom0+!0+superallowedtransitions.Adapting(15.38)forsuperallowedtransitions,g2F=loge(2)3~7 m5ec41 ft1=2meas;(15.45)givingadirectmeasurementofgFviameasuringft.Table9.2inKrane(page285)showshowremarkableconstantftisfor0+!0+superallowedtransitions.ThispermitsustoestablishthevalueforgFtobe:gF=0:8810 4MeVfm3:(15.46)15.5Comparativehalf-livesandforbiddendecaysNotcoveredinNERS312.15.6NeutrinophysicsNotcoveredinNERS312. 18CHAPTER15. DECAY15.7Double- DecayNotcoveredinNERS312.15.8 -delayedelectronemissionNotcoveredinNERS312.15.9Non-conservationofparityNotcoveredinNERS312.15.10 spectroscopyNotcoveredinNERS312.Mif=1,forneutron decay,n!p+e +~e ThiswasstatednearthetopofthetextonKrane'sp.290.Inthiscase,foranFtransition:Mif=D X0(0+)1 p 2[e(")(#)+e(#)(")] 1 X(0+)E;