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PAGE 1 LorentzSymmetryViolatingExtensionsofthe NuclearShellModel by STEVENWILCOX AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts Undertheco-sponsorshipofPatrickMcDonaldandDonaldColladay Sarasota,FL May,2013 PAGE 2 Abstract TheinvarianceofphysicallawswithrespecttoLorentzsymmetryisoneofthemostwell supportednotionsinallofphysics.Nevertheless,asphysicistssearchforcluesforunderstandingthenatureofgravityatquantummechanicalscalesandforresolvingunexplained phenomenasuchastheobservedantisymmetryofmatterandantimatterintheuniverse, theviolationofLorentzcovariancehasbecomeanincreasinglyattractiveindicatorfornew discoveriesandunconventionalphysics.Themotivationforthissearchcomesfromtheoreticalpredictionsofleadingtheoriesbeyondthestandardmodel,suchasstringtheory andloopquantumgravity,aswellasfromthefactthatmostotherknownsymmetriesbesidesCPTandLorentzsymmetryarebrokeninsomephysicalprocesses.Toaccommodate thissearch,aframeworkcalledtheStandardModelExtensionSMEhasbeendeveloped andextensivelystudied[ColladayandKostelecky].TheStandardModelExtension includesallrstorderLorentzviolatingcouplingsofStandardModeleldsandhasbeen showntopreservemanyoftheimportantfeaturesoftheStandardModel.Inthisthesis, arstordernon-relativisticexpansionoftheLorentzviolatingperturbationsgivenbythe SMEareappliedtothesingleparticlenuclearshellmodel,andtheimplicationsofthese resultsarediscussed. PatrickMcDonald DonaldColladay DivisionofNaturalSciences 5/16/2013 ii PAGE 3 Acknowledgements IwouldliketoacknowledgemythesissponsorsDr.DonaldColladayandDr.Patrick McDonaldfortheirwillingnesstooverseethisresearchprojectandtoentrustmewith itsexecution.Furthermore,Iwouldliketothankthemfortheirpatienceandguidance throughoutmyundergraduatecareer.TheyweretheprinciplereasonIchoseNewCollege ratherthananotherinstitution.Furthermore,Iwouldliketothankmyparentsandmy sisterfortheirloveandsupport.Itishardtooverestimatethevaluethatyourunconditional kindnesshasaddedtomylife. iii PAGE 4 Contents Abstract ii Acknowledgementsiii Tables vi ListofFiguresvii 1Foreward1 2TheHistoricalDevelopmentoftheNuclearShellModel3 3TheSingleParticleShellModel9 3.1SimpleHarmonicOscillator..........................15 3.2InniteSphericalWell.............................21 4LorentzSymmetryViolatingPerturbationsoftheSPSM25 4.1TheStandardModelExtension........................25 4.2PerturbationsbyaStaticBackgroundField.................28 4.3MomentumPerturbations...........................36 4.4ConclusionsandFurtherWork.........................45 AClebsch-GordonCoecients47 iv PAGE 5 BExperimentalDataSupportingtheNuclearShellModel:Figuresand Discussion51 Bibliography58 v PAGE 6 ListofTables 3.1HarmonicOscillatorSpectrum.........................20 3.2ZerosoftheSphericalBesselFunctions, j l z .................23 3.3Energyspectrumfortherstseveninnitewellstates............24 4.1Clebsch-GordanCoecients..........................29 A.1Clebsch-GordonCoecients..........................50 vi PAGE 7 ListofFigures 3.1Heliumatomwithanuclearpointchargeattheorigin...........11 3.2SphericalBesselFunctions j 0 red, j 1 magenta, y 0 green, y 1 blue...22 B.1IonizationEnergyofNeutralAtomsasafunctionofAtomicNumber[Heyde]52 B.2Numberofstableisotopesandisotonesforvariousprotonandneutronnumbers[Heyde]...............................53 B.3Energyof -EmissionVersusNeutronNumber[Heyde1994]........54 B.4SingleParticleHarmonicOscillatorSpectrumwithSpin-OrbitCoupling[Heyde1994]56 B.5UnperturbedSingleParticleHarmonicOscillatorSpectrum[Heyde].57 vii PAGE 8 Chapter1 Foreward Theevolutionoffundamentalphysicsistodayasexcitingandvibrantasever.Within thelastonehundredyears,mankindhasmadeunprecedentedprogressinthesciences andmathematics,andfromthisimpetuswehavewitnessedthetechnologicalrevolutionof thelastcentury.Nevertheless,attheturnofthe20 th century,researchersbeganprobing phenomenawhichprogressivelyescapedourcommonplaceexperiences,andthecomplexity andcounterintuitivenatureoftheseobservationsbecameincreasinglyevident.Withthe adventofquantummechanics,scientistswereforcedtoabandoncommonplacenotionsof observationinordertoassemblepiecesofdisparateinsightintoanewfundamentaltheory.Withsteadyeortandburstsofrevelation,theyforged,redacted,andreinterpreted theirpropositionsinordertoprovidethecohesivephysicalandmathematicalframework forquantummechanics.Thisworkledtothedevelopmentofquantumeldtheoryand theformulationoftheStandardModelofparticlephysics,themostcomprehensivephysicaltheorytodate.Nevertheless,asresearchersandinquisitivemindsmoveonwardto theunansweredquestionsofthiscentury,therearemanychallengesinstore.Likeour predecessors,today'sphysicistsmusttakeupthetorchofdevelopingnewandinnovative explanationsforourobservationsaswellascreatingthemethodsandtechnologytotest thesehypotheses.Asthesehurdlesareconfronted,itissatisfyingtoseehowfarmankind 1 PAGE 9 hascomeaswellasitspotentialforchange,innovation,adaptation,andunderstanding. Itisthislegacywhichservesasalamppostfortheinquisitivemindsofthisage,bothfor thepurposeofinspirationandforperseverance.Inthisway,thevalueofthishistoryisas muchacatalystforusastheresultsareintheircontinuedapplicationtoscienticresearch. 2 PAGE 10 Chapter2 TheHistoricalDevelopmentofthe NuclearShellModel Althoughphilosophiesregardingthefundamentalconstituentsofmatterdatebacktothe timeoftheGreeks,theunderstandingofatomicandnuclearstructurereallybeganinthe 20thcentury.OneoftherstattemptstodescribetheatomwasmadebyJ.J.Thompson in1897soonafterhediscoveredtheelectron.Hispropositionwasthatatomsconsistedof electronswhichwerelodgedinaconglomerateofpositivechargesimilartoplumsinapudding.In1911,ErnestRutherfordsoughttotestthismodelbyringastreamof particles atathingoldfoiltomeasurehowtheparticleswerescattered.Whilemostoftheparticles simplypassedthroughthefoilundeected,asmallfractionoftheparticleswerescattered atlargeangles.Fromthisdata,Rutherfordconcludedthatatomspossessadensecore,or nucleus,whichisasmallfractionoftheatom'svolume,andheclaimedthattheinfrequent interactionswiththeatom'scenterwereresponsibleforthelargescatteringangles.This precipitatedRutherford'splanetarymodeloftheatominwhichelectronswerethoughtto existinacloudsurroundingthenucleus.Nevertheless,classicalinterpretationsoftheplanetaryanalogywhichreplacethegravitationalforcewiththeCoulombinteractionpresent seriousproblems.Inparticular,asimpleclassicalcalculationutilizingLarmor'sformula 3 PAGE 11 showsthatclassicalorbitsofelectronsinthehydrogenatomareunstable.Inparticular, theformulaindicatesthattheboundelectronradiatesenergyataratewhichcausesit tospiralintothenucleusinatimeontheorderof10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(11 seconds.In1913,NielsBohr proposedamodeloftheatomwhichincorporatedRutherford'sdiscoveryandxedthese inconsistencies.TheelectronsinBohr'smodelmovedaroundapositivelychargednucleus inquantizedcircularorbitals.Whilethismodelyieldedunprecedentedapproximations fortheenergyspectrumofhydrogenandotherelements,thescopeofitsapplicationwas limited. Inspiteofthisshortcoming,thefundamentalconceptofelectronsexistinginquantized energystatesperseveredthroughLouisdeBroglie'sworkonmatterwavesin1924and twoyearslaterwiththeadventofthenon-relativisticSchrodingerequation.Furthermore, notonlydidSchrodinger'sequationreformulateBohr'sideasofatomicshellstructure,but, wheninterpretedstatistically,itsetforthatheoryunderauniedmathematicalframework which,atthetime,providedthemostaccuratepredictionspertainingtoquantumscale phenomena.Onlywiththistheoryinplacedidtheunderstandingofnuclearstructure becometractable. Despitethevastprogressthatwasmadeinthenearlythirtyyearswhichseparated Thompson'splumpuddingmodelandtheadventoftheSchrodingerequation,thedevelopmentofinsightfultheoriesofthenucleusremainedelusive.Manysimplefactsregarding nuclearcomposition,suchaswhetherornot particlesexistedwithinthenucleus,remained amystery.Needlesstosay,therewerenomodelswhichcouldexplainthepropertiesoflarge categoriesofnucleiorevenattempttoclassifythemaccordingtoanorganizationscheme. Thischangedin1930,whenWaltherBotheandHerbertBeckerconductedaseriesofexperimentswhichallowedJamesChadwicktodeterminetheexistenceandmassoftheneutron in1932.Withthisnewsubatomicparticlecamethecontemporarynotionthatthenucleus isadensecollectionofneutronsandprotons.Moreover,itputnuclearphysicistsinapositiontoaskquestionsabouttheinteractionsbetweennucleonsandtothinkofhowanswers 4 PAGE 12 tothesequestionscouldprecipitatemorecomprehensivemodels. Asresearchcontinuedintothe1940's,progresswasmadetowardmoregeneralunderstandingofnucleiandanincreasinglylargeamountofexperimentaldataregardingnuclear propertieswascollected.Inparticular,theliquiddropmodelandsemi-empiricalmassformulawereconstructedandenjoyedlimitedsuccessalongwithothertheories.Finally,by 1949,MariaGoeppert-MayeralongwithJ.HansD.JensonandEugeneWignerassembled thenecessaryconceptstocreateatheorywhichexplainedthestabilityaswellasother propertiesofalargenumberofnuclei.Importantly,thegeneralityofthetheoryallowed nucleitobeorganizedinachartinasimilarmannertothewayinwhichtheelementsare organizedintheperiodictable,withthestabilityofthenucleibeingafunctionoftheir protonandneutroncomposition.Forthisreason,thetheoryiscalledthenuclearshell model. Theprimarymotivationforthenuclearshellmodelwastheexistenceofwhatarecalled MagicNumbers. 2,8,20,28,50,82,126 ThesewererstcommentedonbyWalterElsasserinhis1933publications[Elsasser1933] andweresaidtoincreasethestabilityofnucleiwheneitherprotonorneutronnumbers weremagic.WhenworkingonatheoryoftheoriginoftheatomicelementswithEdward Teller,MariaGoeppert-Mayernoticedthattherewereafewnucleiwhichhadagreater isotopicaswellascosmicabundancethanhertheoryoranyotherreasonablecontinuum theorycouldexplain[GoeppertMayer]."Shefurthernoticedthatthosenuclei havesomethingincommon:theyeitherhad82neutrons,whatevertheassociatedproton number,or50neutrons[GoeppertMayer]."Sinceeighty-twoandftyaremagic numbers,sheconcludedthatthestabilityofnucleimusthaveplayedaroleincreatingthe elements.Furthermore,thefactthatthenumberofstableisotonesorisotopestendstobe largerwhentheneutronorprotonnumbercorrespondswithamagicnumberwasalsoan 5 PAGE 13 indicatorofthestabilitypropertieswhichresultfromshellstructureAppendixB. Additionally,Mayerconcludedthatthesignicanceofthemagicnumberswasfurther conrmedbythedatathathadbecomeavailablesinceElsasser's1933publications,indicatingthattheyplayafactorindiversenuclearphenomena.Onepossibleapproachfor explainingthesignicanceofthemagicnumbersistocomparethetrendgivenbythe bindingenergyofthe`lastproton'or`lastneutron'plottedwithrespecttotheprotonor neutronnumberwiththetrendofionizationpotentialplottedagainsttheatomicnumber forelementsintheperiodictable.Intheatomiccase,theshellstructureisevenmore pronounced,andinparticular,thehighionizationpotentialsofthenoblegasesandthe lowionizationpotentialsofthealkalimetalsserveasananalogueforthechangeinbinding energyamongnucleonsbetweennucleiwithmagicprotonandneutronnumberafullshell tothosewithanadditionalnucleonamostlyemptyshell.Inthenuclearcase,theshell structurecanbeseenby,amongotherthings,consideringalphaandbetadecayenergies plottedaboutregionswithneutronnumber N =20and N =50andintheenergyofalpha particlesfromalphadecayexperimentsintheregionof N =126[GoeppertMayer]. AmorecompletediscussionoftheexperimentaldataandshellmodelpredictionsispresentedinAppendixB. Ifonewishestoapplytheanalogybetweennuclearandatomicshellstructureinmore detail,onecantrytocomputethemagicnumbersdirectlybyfollowingthesamemethodologyusedtopredicttheshellstructureinthehydrogenatom.Inthismodel,however, thecenterofthesystemisthenuclearcenterofmass,which,initsgroundstate,canbe consideredtobestationary.Moreover,theCoulombpotentialisreplacedbyaradially symmetricnuclearpotentialwhichapproximatesthemutualinteractionsbetweenpairsof nucleons.Anothernecessarydierenceisthatprotonsandneutronsaretreatedidentically butseparately,inthateachtypeofparticlellsitsownsetofshells.Thissimplemodel iscalledthesingleparticleshellmodelsincethesystemasawholebehaves,intherst approximation,asasystemofindependentfermions.Withthisinmind,theSchrodinger 6 PAGE 14 equationcanbesolvedforeachparticleinsphericalcoordinatesviaseparationoftheangularandradialvariables.Thisprocesswillyieldtwosolutionsets.Therstisthesetof sphericalharmonics,andthesecondisasetofradialfunctionswhichdierbasedonthe formoftheapproximatednuclearpotential. Experimentally,itisobservedthatthenuclearforceexertsitsinuenceoveranite distance.Specically,itisknownthatatseparationsofabout1.5-2femtometersthenuclearforceisattractiveaccordingtotheone-pionexchangepotentialwhichhasananalytic dependenceof V r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(exp )]TJ/F24 7.9701 Tf 6.587 0 Td [(r r 1+ 3 r + 3 r 2 : .1 Forsmallerdistances, r< 0 : 5femptometers,thepotentialbecomesrepulsive,morepions areneededtomediatetheinteraction,andnon-relativisticcalculationsarenolongerjustied.Thisiscalledthehardcorepotential;however,thisregimedoesnotapplytoshell modelcalculations[Heyde]. Forexplicitcalculations,thesimpleharmonicoscillatorandinnitewellpotentials providelowerandupperlimits,respectively,forthestrengthofthispotential.Sinceanalytic solutionstotheSchrodingerequationcanbeobtainedforbothofthesepotentials,they areusefulfordevelopingsimpleattemptstoexplainnuclearshellstructure.Inthecaseof thesimpleharmonicoscillator,thesolutionstotheradialequationgiveenergyeigenvalues withadegeneracywhichcoincideswiththerstthreemagicnumbers.Afterthis,however, theoscillatorspectrumpredictsmagicnumbersof40,70,112,and168,noneofwhich areconrmedbyexperiment.Furthermore,theshellspredictedforaninnitespherical wellpotentialalsofailtopredictthemagicnumberscorrectly.Furthermore,itisnoted byMayerinherNobelPrizelecturethatnoperturbationofthesepotentialscouldcause thecalculationtoaccountfortherestofthemagicnumbers.Asshestated,itwasa jigsawpuzzlewithmanypiecesinplace,andonefeltthatifonehadjustonemorepiece 7 PAGE 15 everythingwouldt[GoeppertMayer]."ForMayer,thatpiecewasdiscoveredina conversationthatshehadwithEnricoFermi. Fermi,whoworkedwithMayer,hadbecomeinterestedinthemagicnumbers,andhe askedheronenight,ashewasleavingheroce,iftherewasanyevidenceofspin-orbit coupling.Mayer,whowasverywellversedinthedataresponded,Yes,ofcourse,and thatwillexplaineverything[GoeppertMayer]."Fermiwasnotpersuaded,however, untilaboutaweeklaterwhenMayerexplainedthemagicnumbersandworkedoutthe remainingimplicationsofthetheory.Atthatpoint,hebecamesoconvincedthathebegan teachingtheshellmodelinhisclassonnuclearphysics[GoeppertMayer]. 8 PAGE 16 Chapter3 TheSingleParticleShellModel Inthischapter,thesingleparticleshellmodelSPSMisdiscussedinmoredetail,and examplesarecomputedusingharmonicoscillatorandinnitesphericalwellpotentials. Thesepotentialsareusefulinthattheyprovideextremecaseswhichconstraintheanalytical formofthepotentialthatisnecessarytoreproducethemagicnumbers.Furthermore, bothadmitanalyticalsolutions.Forcompleteness,someoftheideasfromChapter2are repeated.Inparticular,thediscussionbeginswiththeimplicationsofthesolutionofthe timeindependentSchrodingerequationforthehydrogenatomandhowthisappliestothe conceptofa`shellmodel.' Recallthatthedynamicsofanon-relativisticquantummechanicalcollectionof A particlesisgivenbythestatisticalinterpretationofsolutionstotheSchrodingerequation i ~ @ ~r 1 ;~r 2 ;:::;~r A ;t @t = H ~r 1 ;~r 2 ;:::;~r A ;t ; .1 wheretheHamiltonian, H ,is H = ~ 2 2 m A X i =1 ~ r 2 i + V ~r 1 ;~r 2 ;:::;~r A ;t : .2 Ifweassumethatthepotentialistimeindependent,then.1separatesas ~r 1 ;~r 2 ;:::;~r A ;t = ~r 1 ;~r 2 ;:::;~r A t ; .3 9 PAGE 17 yieldingtheordinarydierentialequations i ~ @' n @t = E n n ; .4 and H n = E n n : .5 Onceboundaryconditionsareimposed,thespinoftheparticlecanbeincludedaspartof thesolutionssothatthegeneralsolutionto.1becomesasuperpositionofeigenfunctions: ~r 1 ;~r 2 ;:::;~r A ;t = X m s 1 X n =1 c n n ~r 1 ;~r 2 ;:::;~r A e )]TJ/F25 5.9776 Tf 7.782 3.527 Td [(iE n t ~ 1 2 ;m s : .6 Ingeneral,the c n arexedbyaninitialcondition,and m s = 1 2 forfermions. Inordertosolve.1forthehydrogenatom,itisassumedthatthenucleusisxed atthecenterofastationarysphericalcoordinatesystemsothat.1reducestoasingle particleproblem.Furthermore,theCoulombpotential, V r = )]TJ/F24 7.9701 Tf 16.961 4.707 Td [(e 2 4 0 r ,isused.Thus, solvingthetimeindependentequationinsphericalcoordinates,thesolutionisgivenby .6,and n ~r separatesintheangularandradialcoordinatestogive n;l;m ~r = Y m l ; R n;l r ; .7 wherethe Y m l ; arethesphericalharmonics,andthe R n;l r aresolutionstotheradial equation.Moreover,theenergyspectrumisgivenby E n = )]TJ/F29 11.9552 Tf 11.291 20.444 Td [(" m 2 ~ 2 e 2 4 0 2 # 1 n 2 ;n =1 ; 2 ; 3 ;::: .8 See[Griths]fordetails. Finally,itisimportanttonotethatthesphericalharmonicsintroducedegeneracyin theenergyspectrumabove.Namely,foreachprimaryquantumnumber n ,wehave l PAGE 18 procedureforhelium.Aswithhydrogen,wewillassumethattheheliumnucleusisxed toastationarycoordinatesystemwithsphericalcoordinatesbeingthemostnaturalchoice forthecomputation. TheHamiltonianforheliumis H = )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m ~ r 2 1 + ~ r 2 2 )]TJ/F23 11.9552 Tf 19.263 8.088 Td [(e 2 4 0 2 r 1 + 2 r 2 )]TJ/F15 11.9552 Tf 30.858 8.088 Td [(1 j ~r 1 )]TJ/F23 11.9552 Tf 11.559 0 Td [(~r 2 j ; .9 and ~r 1 and ~r 2 aregivenbyFigure.Sincethemutualinteractionissucientlysmall, Figure3.1:Heliumatomwithanuclearpointchargeattheorigin e 2 4 0 1 j ~r 1 )]TJ/F24 7.9701 Tf 6.355 0 Td [(~r 2 j canbetreatedasaperturbation,and H 0 = )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m ~ r 2 1 + ~ r 2 2 )]TJ/F23 11.9552 Tf 19.263 8.088 Td [(e 2 4 0 2 r 1 + 2 r 2 ; .10 separatesintotwosingleparticleHamiltoniansgivenby H 1 )]TJ/F35 11.9552 Tf 25.564 8.088 Td [(~ 2 2 m ~ r 2 1 )]TJ/F23 11.9552 Tf 19.262 8.088 Td [(e 2 4 0 2 r 1 and H 2 )]TJ/F35 11.9552 Tf 25.564 8.088 Td [(~ 2 2 m ~ r 2 2 )]TJ/F23 11.9552 Tf 19.262 8.088 Td [(e 2 4 0 2 r 2 : .11 Thisproduces,twosingleparticleSchrodingerequationsgivenby H 1 1 = E 1 1 ; .12 and H 2 2 = E 2 2 : .13 Thus, E n in.8takestheform E ~ n = E n 1 + E n 2 ,where E n 1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 m 2 ~ 2 e 2 4 0 2 # 1 n 2 1 and E n 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 m 2 ~ 2 e 2 4 0 2 # 1 n 2 2 : .14 11 PAGE 19 Thus,withoutfurtheranalysis,thisrstapproximationrepresentstheheliumatomas twoindependentparticleswithenergyspectrumspossessingthesamedegeneracystructure asthehydrogenatom.Namely,thereareatotalof2 n 2 statesforeachprimaryquantum number n .Furthermore,whenheliumisinitsgroundstatethismeansthatbothelectrons willbeinan n =1, l =0state.Onewillhavespin 1 2 andtheotherwillhavespin )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 2 Continuingwiththismethodology,onecouldanalyselargerandlargeratomsandcontinuetoneglectthemutualinteractionsoftheirelectrons.Asbefore,theproblemwould separateintoaseriesofsingleparticleequationsandthegroundstateoftheatomwould begivenbyparticlesllingdegeneratehydrogenlikestatesfromthelowesttothehighest energyeigenvaluesaccordingtothePauliexclusionprinciple.Nevertheless,inspiteofthe promiseofthisconcept,withoutappropriatelyaccountingfortheerrorintroducedbythe mutualinteractionsthismethodwillnotproduceacceptableresultsfortheenergyspectrum ofhelium,muchlessforlargeratomswheretheeectsoftheseinteractionsbecomemore pronounced.Inlightofthis,werecallthattheexperimentalobservationsgoverningthe trendsoftheionizationpotentialsofatomsintheperiodictableshowthatenergyofstates whicharedistributedamongasingleprimaryquantumnumber, n ,aremuchclosertoeach otherthanstateswithdierentprimaryquantumnumbers.Thisistheshellstructure,and itiswhatallowsustotreatthemutualinteractionsperturbatively. Aswasalreadymentioned,hydrogen-likespectrumsaredegenerateasaconsequence ofthedegreesoffreedomassociatedwiththesphericalharmonics.Thus,itismoreenergeticallyfavourableforelectronstooccupyallpossiblestatesofagivenenergyeigenvalue beforeproceedingtothenexthighestone.Inatoms,oncethemutualinteractionsare accountedfor,theperturbationtheorybreaksthisdegeneracy.Nevertheless,bythenature of`perturbation,'thesplittingofthelevelsissmallwithrespecttotheseparationbetween levelsbelongingtodierentprinciplequantumnumbers,thebasicspectralstructureis preserved,andwesaythatthequantumnumber n denesashellpossessingorbitscorrespondingthequantumnumber l .Inordertocompletelyspecifyaparticle'sstate,however, 12 PAGE 20 allfourquantumnumbers n;l;m l ;m s arenecessary. Asitturnsout,thisnotionofasingleparticleshellmodelviaperturbationcansuccessfullybeusedtopredictanumberofnuclearpropertiesaslongastherequiredmodicationsaremade.First,itisimportanttorealizethat,unliketheatomiccase,thenucleus ofanatomhasnonaturalpointofreferenceotherthanitscenterofmass.Forthisreason,elementaryshellmodelcalculationsonlypredictpropertiesofnucleiintheirground statesinceexcitedstatestendtocorrespondtomotionofthecenterofmass[deShalit andTalmi004].Furthermore,theradiallysymmetricpotential,towhichallnucleons aresubjected,canonlybeanapproximationoftheaveragemutualnuclearinteractions. Specically,theseinteractionsarecurrentlybestdescribedbyquantumchromodynamics, whichdoesnotadmitananalyticalrepresentationofthestronginteraction.Thus,very simplepotentialsmodellingqualitativefeaturesofthenuclearforceorwhichtempirical data,suchastheWoods-Saxonpotential,areused.Furthermore,thereisalsotheelectromagneticforcetoconsider,althoughthemodicationsitintroducesareeithernegligibleor oftentreatedasaperturbationformoredetailedcalculations.Inadditiontothesedierences,itisimportanttonotethattherearetwotypesofnucleons,protonsandneutrons, whichinteractidenticallywithrespecttothestrongforce.Thisisfundamentallydierent fromtheatomiccase,inthatsetsofprotonsandneutronsneedonlyobeythePauli exclusionprinciplewithrespecttoparticlesofthesametype.Forthisreason,protonand neutronshellsaretreatedindependentlyinthenuclearshellmodel.Lastly,forthismodel, empiricaldatahasveriedthatthereisastrongspin-orbitcouplingbetweentheangular momentum l andthespin S ofthesingleparticlewavefunctions.Thisperturbationis necessaryandisafundamentalassumptionofthemodel. Withthisinmind,wecandeveloptheseideasmorequantitativelybyconsideringa nucleuswithatomicnumber A .Let N bethenumberofneutronsand Z bethenumber ofprotons.Thenthecompletesolutiontothe A = N + Z particletimeindependent Schrodingerequationisgivenby 13 PAGE 21 H ~r 1 ;~r 2 ;:::;~r A ;t = E ~r 1 ;~r 2 ;:::;~r A ;t .15 whereisospinisincorporatedsimilarlytospin.Hereeachindex i 2f 1 ; 2 ;:::;A g represents anucleonwithcoordinatesgivenbythepositionvector ~r i ,aspincoordinate ~s i ,andan isospincoordinate ~ t i .Thewavefunctionmustalsobetotallyantisymmetricbythesymmetrizationrequirement.Withthisinmind,wecanattempttowrite H asacomposition ofsingleparticleandtwoparticleinteractions,givenby i 2f 1 ; 2 ;:::N g forneutronsor i 2f 1 ; 2 ;:::Z g forprotons.Thus,weexpectthatthegeneralformof H tobegivenby H = A X i =1 )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m i + A X i PAGE 22 with H A X i =1 )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m i + U i A X i =1 h i : .18 AswasdoneoriginallybyMariaGoeppert-Mayer,thesimplesttreatmentofthesingle particleshellmodelaccountsforthisperturbationviathethespin-orbitcouplingterm r L S ,andtheresultingcalculationscoincidewellwiththeexperimentaldata.Furthermore,thedataalsoindicatesthat,iftheshellmodelcanbesuccessfullyapplied,the actualpotentialshouldbebetweenasphericalharmonicoscillatorpotentialandaninnite squarewellpotential.Bothoftheseextremecasesaretreatedinthefollowingsections. 3.1SimpleHarmonicOscillator Inthissection,thesingleparticletimeindependentSchrodingerequationissolvedforthe simpleharmonicoscillatorpotential, V r = 1 2 kr 2 : .19 Thispotentialisconvenientbecauseitadmitsaclosedformsolutiontothesingleparticle timeindependentSchrodingerequation,andshellmodelcalculationswiththispotential havebeenshowntocoincidewiththeexperimentaldatawhichpredictsthemagicnumbers. Furthermore,itisalsospecialbecauseitprovidesamethodbywhichonemightdemand thatthecenterofmassofanucleusbeconnedinspace.Specically,aslongasweareonly interestedinpredictingintrinsicnuclearproperties,apotentialtermcanbeaddedtothe Hamiltonianforthenucleuswhichactsonthecenterofmass,conningittoarestricted regionofspace.Ifthispotentialhastheformof.19,thenbytemporarilytaking r tobe thecenterofmasscoordinatewehave 1 2 kr 2 = 1 2 k 0 B B @ A P i =1 m i r i A P i =1 m i 1 C C A 2 = 1 2 A 2 A X i =1 r 2 i : .20 15 PAGE 23 Forthe r 2 term,wehavetheidentity 1 A 2 r 2 = 2 A 2 A X i =1 r 2 i )]TJ/F15 11.9552 Tf 16.978 8.088 Td [(1 A 2 X i PAGE 24 Forthesakeofconvenience,onecanconverttodimensionlessvariablesusingthefollowing substitutions: u = r = r ~ m! : Thisimplies d dr = du dr d du = 1 d du .25 d 2 dr 2 = du dr 2 d 2 du 2 = 1 2 d 2 du 2 : .26 TheODEbecomes 1 2 d 2 R du 2 + 1 2 dR du )]TJ/F15 11.9552 Tf 15.609 8.088 Td [(1 2 u 2 R )]TJ/F15 11.9552 Tf 15.609 8.088 Td [(1 2 l l +1 u 2 R + 2 E 2 ~ R =0 ; .27 whichsimpliesto d 2 R du 2 + 2 u dR du )]TJ/F23 11.9552 Tf 11.955 0 Td [(u 2 R )]TJ/F23 11.9552 Tf 13.151 8.087 Td [(l l +1 u 2 R + 2 E ~ R =0 : .28 Now,weproceedbyanalysingtheasymptoticbehaviorofthisresultas u !1 .As u !1 weneglecttermsdecreasingin u andnotethatif d 2 R du 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(u 2 R =0 ; .29 then R e )]TJ/F25 5.9776 Tf 7.782 3.258 Td [(u 2 2 : .30 Ontheotherhand,as u 0,wehave d 2 R du 2 + 2 u dR du )]TJ/F23 11.9552 Tf 13.151 8.088 Td [(l l +1 u 2 R =0 : .31 ThisisanEulerequationandadmitssolutions.Assuming R = u s forsome s 2 N givesthe characteristicequation s s )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 u s )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 +2 su s )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 = l l +1 u s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 : .32 17 PAGE 25 Fromthis,weconclude s s +1= l l +1 : .33 Thus,for u 0,weexpect R u l : .34 Takingintoaccounttheasymptoticbehavior,wetryapowerseriessolutionoftheform R = u l 1 X k =0 a k u k e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(u 2 2 = 1 X k =0 a k u l + k e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(u 2 2 : .35 Wecancomputethederivativesseparatelyas dR du = 1 X k =0 a k l + k u l + k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F23 11.9552 Tf 11.956 0 Td [(u l + k +1 e )]TJ/F25 5.9776 Tf 7.782 3.258 Td [(u 2 2 .36 and d 2 R du 2 = 1 X k =0 a k l + k l + k )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 u l + k )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 k +1 u l + k + u l + k +2 e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(u 2 2 : .37 Substitutinginto.28,wehave 1 X k =0 a k l + k l + k )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 u l + k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 k +1 u l + k + u l + k +2 +2 l + k u l + k )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 u l + k )]TJ/F23 11.9552 Tf 11.955 0 Td [(u l + k +2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(l l +1 u l + k )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 + 2 E ~ u l + k e )]TJ/F25 5.9776 Tf 7.782 3.258 Td [(u 2 2 =0 : .38 Collectingliketermsandcancellingtheexponentialfactorgives 1 X k =0 a k k l + k +1] u l + k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 + 2 E ~ )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 k +3 u l + k =0 : Wecaneliminatethe u l + k termbyshiftingthesumindexingsuchthat l + k +2becomes l + k .Observe, 1 X k =0 a k +2 k +2 l + k +3 u l + k + a k 2 E ~ )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 k +3 u l + k =0 : Thisyieldstherecursionrelation a k +2 k +2 l + k +3+ a k 2 E ~ )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 k +3 =0 ; .39 18 PAGE 26 whichgives a k +2 = )]TJ/F21 7.9701 Tf 12.75 13.492 Td [(2 E ~ )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 k +3 k +2 l + k +3 a k : .40 Noticethatas k becomeslarge a k +2 2 k a k .Thisnotonlyviolatesthenormalization condition,itcausestheseriesrepresentationtodiverge.Thus,theonlywaytosalvagethis solutionisfortheseriestoterminate.Thatis,ifforsome k = n r ,wehave 2 E ~ )]TJ/F15 11.9552 Tf 11.955 0 Td [( l +2 n r +3=0 ; orequivalently, E = n r + l + 3 2 ~ !: .41 Replacingfor E intermsof n r and l wehave a k +2 = 2 k )]TJ/F23 11.9552 Tf 11.955 0 Td [(n r k +2 l + k +3 a k : Finally,welet u u 2 n r 2 n r ,and k 2 k .Thisallowsustowrite a k +1 intermsof a k .Thus,weobtain a k +1 = k )]TJ/F23 11.9552 Tf 11.955 0 Td [(n r k +1 l + k + 3 2 a k .42 R n r l = 1 X k =0 a k u l +2 k e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(u 2 2 .43 E = 2 n r + l + 3 2 ~ !: .44 Asasidenote,weobservethatthespectrumagreeswiththesolutiontothisproblemwhen itisposedinCartesiancoordinates.Here n =2 n r + l ,andthespectrumis E n = n + 3 2 ~ !; .45 where n = n x + n y + n z ,where n x n y ,and n z arequantumnumbersforthe x y ,and z oscillatorsrespectively. 19 PAGE 27 ByanalysingthedegeneracyoftheenergyspectrainbothsphericalandCartesian coordinates,weobservethatthepredictedshellshaveoccupanciesof2,8,20,40,and70. Thissolutioncorrespondswiththerstthreemagicnumbers,butfailsbeyondthat.The solutionistoperturbtheHamiltonianwithaspin-orbitcouplingterm, r l S .Further detailsandimplicationsofthiscalculationarediscussedinAppendixB.Thecalculation oftheunperturbedenergyspectrumissummarizedinTable.1. Table3.1:HarmonicOscillatorSpectrum En r ln x ;n y ;n z N Spherical N Cartesian ShellsTotalNo:ofParticles ~ 000001122 ~ 010013368 ~ 10,02002,011661220 ~ 11,03003,210,11110102040 ~ 20,12,04004,310,220,21115153070 20 PAGE 28 3.2InniteSphericalWell Inthissection,thesingleparticletimeindependentSchrodingerequationissolvedforan innitesphericalwellpotential. V r = 8 > < > : 0: r 2 [0 ;a ] 1 : r= 2 [0 ;a ] ; .46 Liketheharmonicoscillatorpotential,itisconvenientbecauseitadmitsananalytical solutionandprovidesanextremecasewithwhichtotestthenuclearshellmodel.Furthermore,calculationsperturbedbyaspin-orbitcouplingtermhavebeenshowntosuccessfully predictthemagicnumberscorrectlyaswellasothernuclearpropertiesAppendixB. Asinthelastsection,theproblemofsolvingthetimeindependentSchrodingerequation reducestosolvingtheradialequationgivenby d 2 R n;l dr 2 + 2 r dR n;l dr + k 2 )]TJ/F23 11.9552 Tf 13.151 8.088 Td [(l l +1 r 2 R n;l =0 ; .47 where0 r a ,and k 2 = 2 mE ~ 2 .Ifwechangevariablesto z = kr ,weobtainspherical Besselfunctionsasthewellknownsolutionstothisequation.UsingRayleigh'sformulas, thesphericalBesselfunctionshavetherepresentations j l z = z l )]TJ/F15 11.9552 Tf 10.553 8.088 Td [(1 z d dz l sinz z .48 y l z = )]TJ/F23 11.9552 Tf 9.299 0 Td [(z l )]TJ/F15 11.9552 Tf 10.552 8.088 Td [(1 z d dz l cosz z : .49 Fromtheseformulas,wecanexplicitlywritetheBesselfunctionsfor0 l 1as 21 PAGE 29 j 0 z = sinz z ; j 1 z = sinz z 2 )]TJ/F23 11.9552 Tf 13.151 8.088 Td [(cosz z ; y 0 z = )]TJ/F23 11.9552 Tf 10.494 8.087 Td [(cosz z ; y 1 z = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(cosz z 2 )]TJ/F23 11.9552 Tf 13.151 8.088 Td [(sinz z : ByplottingthesewithrespecttozandconsideringtheRayleighformulasfor y l we canseethat,forany m 2 N ,the y l solutionsareunboundedon z 2 [0 ; 1 m ].Thus,the y l 'scannotbesolutionstotheradialequation.Thus,thenormalizedsolutionsto.47 Figure3.2:SphericalBesselFunctions j 0 red, j 1 magenta, y 0 green, y 1 blue. mustbeproportionaltoalinearcombinationofthe j l solutions.Sincethefunctions j l are relatedtotheBesselFunctionsoftherstkindby j n r = r 2 r J n + 1 2 r : .50 22 PAGE 30 theysatisfytheorthogonalityproperty, Z a 0 j l z n;l r a j l z n 0 ;l r a r 2 dr =0and 2 Z 1 0 j 2 n )]TJ/F22 5.9776 Tf 7.782 3.259 Td [(1 2 r r 2 dr = a j n + 1 2 a ; .51 wherethenotation z n;l representsthen th zeroofthefunction j l r .Furthermore,theyform acompletesetofsolutionsto.47.Itthereforefollowsthateachsolution, R r ,to.47 mustbeproportionaltoasphericalBesselfunction, j l ,forsome l .Moreover,byimposing theboundaryconditionsthat R =0and R a =0,wendthat z = ka mustbethe n th zeroof j l ,forsome n l 2 N .Inlightofthis,wecanlabelthenormalizedsolutionsto.47 as R n;l whichwillhaveanassociatedenergyeigenvalueof E n;l = z 2 n;l ~ 2 2 ma 2 : .52 Inthetablebelow,therstfewzerosaredisplayedfor0 l 2and0 n 2.Now,as Table3.2:ZerosoftheSphericalBesselFunctions, j l z n =1 n =2 n =3 n =4 l =03.1426.2839.42512.566 l =14.4937.72510.90414.066 l =25.7639.09512.32315.515 l =36.98810.41713.69816.924 l =48.18311.70515.04018.301 anillustration,itisreasonabletoset a =10fmforthewellradius,andlet m =1 : 67 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(27 kgthemassofaneutron.ByorderingthesevensmallestzerosgiveninTable.2,we cancomputetherstsevenenergyeigenvaluesofthespectrumusingequation3.52. FromTable.2wecanseethat,exceptfortheveryrstpointat n;l = ; 0,which 23 PAGE 31 Table3.3:Energyspectrumfortherstseveninnitewellstates n;l z n;l E n;l ; 03.1422 : 05 10 6 eV ; 14.4934 : 19 10 6 eV ; 25.7636 : 90 10 6 eV ; 06.2838 : 20 10 6 eV ; 36.9881 : 01 10 7 eV ; 17.7251 : 24 10 7 eV ; 48.1831 : 39 10 7 eV isseparatedappreciablyfromtheothereigenvalues,thereisnotaclearshellstructure whichcorrelateswellwiththemagicnumbers.Therefore,inthecaseoftheinnitewell potential,itisagainevidentthatperturbationofthespectrumisessentialtothesuccess ofthenuclearshellmodel. 24 PAGE 32 Chapter4 LorentzSymmetryViolating PerturbationsoftheSPSM 4.1TheStandardModelExtension Asphysicsevolvedthroughtheturnofthe20 th century,researchersbeganprobingphenomenawhichprogressivelyescapedourcommonplaceexperiences.Althoughcosmological observationshavebeencarriedoutforhundredsofyears,theadventofquantummechanics revealedaregimeofphysicswhichwascompletelyunobservabletothenakedeyeandcorrelatedwellwithdynamicaltheoriesthatradicallydivergedfromtheclassicalintuitionthat pervadedatthetime.Furthermore,theintroductionoftheSpecialandGeneralTheories ofRelativitybyAlbertEinsteinestablishedadditionalphenomenologicaldomainsinwhich classicalexplanationswhereproventofail.Asquantummechanicaltheoriesdeveloped throughtheearlyandmiddlepartsofthetwentiethcentury,attemptsweremadetounify thesevariousregimeswithinthecontextofasingletheory.Duringthisperiod,quantum eldtheoriesweredevelopedtoextendquantummechanicsintoarelativisticcontext,and theoriginoftheelectromagnetic,strong,andweakforceswereexplained.Moreover,an explanationfortheoriginofmasswasgivenbymeansoftheHiggsmechanism.Bythe 25 PAGE 33 mid1970's,thesetriumphswereformulatedintoanewfundamentaltheory,knownasthe StandardModel. Inspiteofthisachievement,therearestillmanyobservationswhichthistheorycannot explain.Mostprominently,perhaps,isthattheStandardModeldoesnotincludeatheoryof quantumgravity.Forexperimentswhichobserveeventsattheenergiespresentinterrestrial phenomena,thegravitationalforceisnegligiblysmallinparticlephysics.Nevertheless,at higherenergiesthereisreasontobelievethatthisisnotthecaseandthatthefourobserved forcesrevealthemselvesasmanifestationsofasingleentity.Furthermore,asanaidfor understandingthenatureoftheobservedmatter-antimatterasymmetryintheuniverse anddarkmatter,theStandardModelfallsshort. Inlightofthis,muchresearchhasbeendonetondmethodswhichcouldprovidevericationsforphysicaltheoriesbeyondtheStandardModel.Thedicultyisthatmany experimentaltestsrequireenergieswhicharenottechnologicallyfeasible.Thismeans thattheorieswhichmakepredictionsthataretestablewithincurrenttechnologicallimitationsmayprovetohaverareimport.Oneparticularlyinterestingexampleofsuchwork waspublishedbyDonaldColladayandAlanKosteleckyin1998.Thistheoryisknown astheStandardModelExtensionSME,anditprovidesaunique,comprehensiveframeworkwhichdescribeshowviolationstoLorentzsymmetryperturbtheStandardModel Lagrangianwhilepreservingmanyofitsimportantproperties.Theseperturbations,althoughminuteinmagnitude,couldinducevariationsinobservablesthatarewithinthe measuringcapabilitiesofmodernhighprecisionexperiments.Inthisway,theStandard ModelExtensioncouldprovidemuchneededtestsofleadingtheoriesofquantumgravity. Inaddition,ithasbeenproventhataviolationofCPTsymmetrynecessitatesaviolationofLorentzsymmetry.ThisimpliesthatanyphysicalphenomenawhichviolatesCPT symmetryisexpectedtobedescribedbytheStandardModelExtension. Inwhatfollows,therstorderperturbationspredictedbytheStandardModelExtension inanon-relativisticcontextareappliedtosingleparticleshellmodelHamiltonians.The 26 PAGE 34 resultsofthesetermsarecomputedandtheimplicationsoftheperturbationsarediscussed. Therstordernon-relativisticperturbationsforspin 1 2 particlespredictedbytheStandardModelExtensionaregivenby h = m + a 0 )]TJ/F23 11.9552 Tf 13.15 8.088 Td [(mc 00 )]TJ/F23 11.9552 Tf 11.955 0 Td [(me 0 +[ a j )]TJ/F23 11.9552 Tf 11.955 0 Td [(m c 0 j + c j 0 )]TJ/F23 11.9552 Tf 9.298 0 Td [(me j ] p j m )]TJ/F23 11.9552 Tf 11.955 0 Td [(m c jk )]TJ/F23 11.9552 Tf 11.956 0 Td [( jk c 00 p j p k m 2 + )]TJ/F15 11.9552 Tf 10.929 8.088 Td [(1 b j + md j 0 + 1 2 kl j H kl )]TJ/F15 11.9552 Tf 16.512 8.088 Td [(1 2 m kl j g klo + jk b 0 + m d jk )]TJ/F23 11.9552 Tf 11.955 0 Td [( jk d 00 + l kj H 0 l )]TJ/F23 11.9552 Tf 11.955 0 Td [(m lm j 1 2 g lmk )]TJ/F23 11.9552 Tf 11.956 0 Td [( km g l 00 p k m + )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m 2 p 2 b k + md k 0 + 1 2 mn k H mn + 1 2 m mn k g mn 0 jl )]TJ/F23 11.9552 Tf 11.955 0 Td [(m d 0 k + d k 0 jl + m m lj g m 0 k + g mk 0 ] p k p l m 2 + )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m 2 p 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(m d kl )]TJ/F23 11.9552 Tf 11.956 0 Td [( kl d 00 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 m nq l g nqk jm p k p l p m m 3 j : .1 Herethetermsinthevariable p arecomponentsofmomentum,andthe termsarethe Paulimatricesforspin 1 2 particles.Therestofthetermsrepresentcouplingparameters whichcarryoverfromtheStandardModelExtensionLagrangianintothenon-relativistic formulation.Themagnitudeoftheseparametersareboundedbyhighprecisionexperiments lookingfordeviationsfromLorentzsymmetry.Themomentumtermsareproductsof componentsofmomentumoperators,andthe termsarePaulimatricesforspin 1 2 particles. Moreover,byformallysettingvariousparameterstozero,wecanexploretheeectsof variouscombinationsoftheseperturbationswhichrepresenttheeectofaconstantbackgroundeld.Inwhatfollows,weformallycomputetheLorentz-violatingperturbationsthat areduetoaconstantbackgroundeldcouplingtothespinofanucleonviathe sigma term. Wethenseparatelycomputeperturbationsduetotermsquadraticinmomentum.When rstorderperturbationsareperformed,theexpectationsoftheproductsofmomentum termswiththe sigma termsaregivenbytheproductoftheexpectationsofeachseparately.Thisimpliesthatthecombinationsofmomentumandspinperturbationsfrom.1 27 PAGE 35 canbeachievedbycombiningtheresultsinthefollowingtwosections.Formoredetailed explanationoftheStandardModelExtensionsee[ColladayandKostelecky].Foran explanationregardingsomeearlyexperimentalboundsplacedoncouplingparametersin theStandardModelExtension,see[KosteleckyandLanea]. 4.2PerturbationsbyaStaticBackgroundField Inthissection,weanalysetheaectsofperturbationsduetoaconstantbackgroundeld, ,whichcouplestothespinangularmomentumofspin 1 2 particles.Inparticular,we designatetheunperturbedsingleparticleHamiltonianas H 0 andtheperturbationas H 0 ThisallowsustowritetheHamiltonianas H = H 0 + H 0 ,where H 0 = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 2 m + 1 2 m! 2 r 2 .2 and H 0 = r L S + S : .3 Sincethetotalperturbationisthesumoftwooperators r L S and S .Itis instructivetocompute[ L S ; S ].Observethat [ L S ; S ]=[ L S ; x S x ]+[ L S ; y S y ]+[ L S ; z S z ] ; .4 and,forconvenience,choosecoordinatessuchthat = ; 0 ; ,where = p 2 x + 2 y + 2 z Thisimplies 28 PAGE 36 [ L S ; S ]=[ L S ;S z ] = L x S x S z )]TJ/F23 11.9552 Tf 11.955 0 Td [(S z L x S x + L y S y S z )]TJ/F23 11.9552 Tf 11.955 0 Td [(S z L y S y + L z S z S z )]TJ/F23 11.9552 Tf 11.955 0 Td [(S z L z S z = L y [ S y ;S z ]+ L x [ S x ;S z ] = i ~ L y S x )]TJ/F23 11.9552 Tf 11.955 0 Td [(L y S y 6 =0 : Since[ L S ; S ] 6 =0,wecannotndabasisforthesolutionspacetotheunperturbed timeindependentSchrodingerequationsuchthatbothoperatorsaresimultaneouslydiagonal.Sincethe S termcommuteswiththeunperturbedHamiltonianwewilluse orthonormaleigenfunctionsof H 0 forthisperturbationandthenusetheClebsh-Gordon coecientstowritetheseeigenvectorsintermsofanorthonormalbasiswhichdiagonalizes thespin-orbitterm. SincethesolutionspacetotheSchrodingerequationisnotnitedimensional,wesettle foratruncatedHamiltonian.Inpractice,thisdoesnotimposeanyadditionallimitations onthesolutions,sinceparticlesintheground-stateofastablenucleuswillultimately beconnedtosomeshellofniteenergy.Inthisexampleweillustratethisprocessby computingtheseresultsforenergyeigenvalueswhicharerestrictedto n 2. First,recallthattheClebsch-GordancoecientsaregivenasinTable.1. Sincewehaverestrictedtheanalysistoeigenfunctionswith n 2,wecanwrite Table4.1:Clebsch-GordanCoecients m s 1 2 )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 2 j l + 1 2 q l + 1 2 + m 2 l +1 q l + 1 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(m 2 l +1 l )]TJ/F21 7.9701 Tf 13.15 4.707 Td [(1 2 q l + 1 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(m 2 l +1 )]TJ/F29 11.9552 Tf 9.298 15.315 Td [(q l + 1 2 + m 2 l +1 29 PAGE 37 0 2 = N j =1 a j j lm l ij sm s i ; .5 where isaproductoftheeigenfunctionsoftheoperators J 2 l 2 S 2 ,and j z TheClebsch-GordanCoecientsgive j jm j i = a 1 2 1 2 l m l )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 + b 1 2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 2 l m l + 1 2 ; .6 where a = q l m l + 1 2 2 l +1 b = q l m l + 1 2 2 l +1 ,andwenotethatthepositive b coecientcorrespondstothe j = l + 1 2 eigenvalueandthenegative b coecientcorrespondstothe j = l )]TJ/F21 7.9701 Tf 12.499 4.707 Td [(1 2 eigenvalue.Recall,thatwhen n =2and l =0or l =1,weusethistodistinguish theeigenvectorswrittenas jm j .Observe, l =0 8 > > > < > > > : 1 1 2 1 2 = j 00 i 1 2 1 2 2 1 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 = j 00 i 1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 .7 l =1 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : 3 3 2 3 2 = j 11 i 1 2 1 2 4 3 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 2 = j 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 i 1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 5 3 2 1 2 = q 2 3 j 10 i 1 2 1 2 + q 1 3 j 11 i 1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 6 1 2 1 2 = )]TJ/F29 11.9552 Tf 9.298 13.68 Td [(q 1 3 j 10 i 1 2 1 2 + q 2 3 j 11 i 1 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 7 3 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 = q 1 3 j 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 i 1 2 1 2 + q 2 3 j 10 i 1 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 8 1 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 = )]TJ/F29 11.9552 Tf 9.298 13.679 Td [(q 2 3 j 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 i 1 2 1 2 + q 1 3 j 10 i 1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 : .8 Steppingbackforamoment,werecallourgoalwastocalculatethematrix W ij h i j H 0 j j i for H 0 = r L S + S .Again,choosingthe z -directionsuchthat = ; 0 ; ,wewish 30 PAGE 38 tocalculate W ij = h i j r L S j j i + h i j S j j i .9 = h i j r L S j j i + h i j S z j j i : .10 Wedene 1 ij h i j L S j j i and 2 ij h i j S z j j i .Since h i j r L S j j i = Z R 3 R r r R r dr h i j L S j j i ; .11 wecanlet R R 3 R r r R r dr .Thus, W ij = 1 ij + 2 ij : .12 Thismeansthatourtaskbecomestodeterminethematrices 1 ij and 2 ij .Observefrom .7and.8that 1 ij = h i j L S j j i6 =0ifandonlyif i = j .Thus,for1 k 8,we have 1 kk = h k j L S j k i = h k j 1 2 J 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(L 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(S 2 j k i = 1 2 ~ 2 j j +1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(l l +1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(s s +1 h k j k i = 1 2 ~ 2 j j +1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(l l +1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(s s +1 : .13 Inparticular, 1 11 = 1 2 ~ 2 )]TJ/F21 7.9701 Tf 6.675 -4.976 Td [(1 2 1 2 +1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0+1 )]TJ/F21 7.9701 Tf 13.151 4.708 Td [(1 2 1 2 +1 =0.Similarly,wendthat 1 22 =0,and 1 33 = 1 44 = 1 55 = 1 77 = ~ 2 2 ,but 1 66 = 1 88 = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 .Therefore, 1 ij = ~ 2 2 0 B B B B B B B B B B B B B B B B B B B B B B B @ 00000000 00000000 00100000 00010000 00001000 00000 )]TJ/F15 11.9552 Tf 9.298 0 Td [(200 00000010 0000000 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 1 C C C C C C C C C C C C C C C C C C C C C C C A : .14 31 PAGE 39 Now,wecalculate 2 ij usingtherepresentationgivenbythePaulimatrices, S z = ~ 2 z = ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 1 C A : .15 Observe, 2 11 = h 00 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 00 i = ~ 2 ; .16 2 12 = h 00 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 00 i =0 : .17 Furthermore, 2 13 = 2 14 = 2 15 = 2 16 = 2 17 = 2 18 =0.Similarly,weconcludethat 2 2 k =0 for k 6 =2, 2 3 k =0for k 6 =3,and 2 4 k =0for k 6 =4.Since, 5 through 8 aretwoterm linearcombinationsofthe l m l s m s basisvectors,wenotethatdirectcalculationshows thatrows5through8of 2 ij exhibitablockdiagonalstructure.Morespecically, 2 5 k =0 for k 6 =5 ; 6, 2 6 k =0for k 6 =5 ; 6, 2 7 k =0for k 6 =7 ; 8,and 2 8 k =0for k 6 =7 ; 8.Calculating theremainingnonzeromatrixelementswend 2 55 = r 2 3 2 h 10 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 10 i + r 1 3 2 h 11 j 01 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 11 i = 1 3 ~ 2 ; .18 32 PAGE 40 ! 2 56 = )]TJ/F29 11.9552 Tf 9.298 18.786 Td [(r 2 3 r 1 3 h 10 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 10 i + r 2 3 r 1 3 h 11 j 01 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 11 i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p 2 3 ~ 2 ; .19 2 65 = 2 56 ; .20 2 66 = )]TJ/F29 11.9552 Tf 9.299 18.785 Td [(r 1 3 2 h 10 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 10 i + r 2 3 2 h 11 j 01 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 11 i = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 ~ 2 ; .21 2 77 = r 1 3 2 h 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 i + r 2 3 2 h 10 j 01 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 10 i = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 ~ 2 ; .22 33 PAGE 41 ! 2 78 = )]TJ/F29 11.9552 Tf 9.298 18.786 Td [(r 1 3 r 2 3 h 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 i + r 2 3 r 1 3 h 10 j 01 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 10 i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p 2 3 ~ 2 ; .23 2 87 = 2 78 ; .24 2 88 = )]TJ/F29 11.9552 Tf 9.299 18.785 Td [(r 2 3 2 h 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 j 10 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 1 C A 0 B @ 1 0 1 C A j 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 i + r 1 3 2 h 10 j 01 ~ 2 0 B @ 10 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 1 C A 0 B @ 0 1 1 C A j 10 i = 1 3 ~ 2 : .25 Thus, 2 i;j = ~ 2 0 B B B B B B B B B B B B B B B B B B B B B B B @ 10000000 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1000000 00100000 000 )]TJ/F15 11.9552 Tf 9.299 0 Td [(10000 0000 1 3 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 p 2 3 00 0000 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 p 2 3 )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 3 00 000000 )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 p 2 3 000000 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 p 2 3 1 3 1 C C C C C C C C C C C C C C C C C C C C C C C A : .26 34 PAGE 42 Thismeansthat W ij isgivenasfollows: W ij = 1 i;j + 2 i;j = ~ 2 2 0 B B B B B B B B B B B B B B B B B B B B B B B B @ 00000000 00000000 00100000 00010000 00001000 00000 )]TJ/F59 10.9091 Tf 8.485 0 Td [(200 00000010 0000000 )]TJ/F59 10.9091 Tf 8.485 0 Td [(2 1 C C C C C C C C C C C C C C C C C C C C C C C C A + ~ 2 0 B B B B B B B B B B B B B B B B B B B B B B B B @ 10000000 0 )]TJ/F59 10.9091 Tf 8.485 0 Td [(1000000 00100000 000 )]TJ/F59 10.9091 Tf 8.485 0 Td [(10000 0000 1 3 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 p 2 3 00 0000 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 p 2 3 )]TJ/F21 7.9701 Tf 9.68 4.295 Td [(1 3 00 000000 )]TJ/F21 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 p 2 3 000000 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 p 2 3 1 3 1 C C C C C C C C C C C C C C C C C C C C C C C C A = 0 B B B B B B B B B B B B B B B B B B B B B B B B @ ~ 2 0000000 0 )]TJ/F60 10.9091 Tf 8.485 0 Td [( ~ 2 000000 00 ~ 2 2 + ~ 2 00000 000 ~ 2 2 )]TJ/F60 10.9091 Tf 10.909 0 Td [( ~ 2 0000 0000 ~ 2 2 + ~ 6 )]TJ/F35 7.9701 Tf 6.587 0 Td [(~ p 2 3 00 0000 )]TJ/F35 7.9701 Tf 6.587 0 Td [(~ p 2 3 ~ 2 2 )]TJ/F60 10.9091 Tf 10.909 0 Td [( ~ 6 00 000000 ~ 2 )]TJ/F60 10.9091 Tf 10.909 0 Td [( ~ 6 )]TJ/F35 7.9701 Tf 6.586 0 Td [(~ p 2 3 000000 )]TJ/F35 7.9701 Tf 6.587 0 Td [(~ p 2 3 ~ 2 + ~ 6 1 C C C C C C C C C C C C C C C C C C C C C C C C A : .27 35 PAGE 43 4.3MomentumPerturbations Nowthatwehavecalculatedtherstordercorrectionsduethe sigma termperturbationsof theHamiltonian,wewishtocalculatetheresultsoftheperturbationsduetothemomentum termsin.1.Forsimplicity,wewillonlyperturbwavefunctionsinthe l =0state.This casecanbeappliedtonucleiwithanevennumberofprotonsorneutronsandanodd numberofneutronsorprotons,wherethelastoddnucleonresidesinanSorbit.Notice thatanyperturbationofan l =0stateduetoasinglecomponentofmomentumiszero sinceiteithercontainsanangularderivativeoranangulartermwhichaveragestozero. Asimilarargumentcanbemadeforthetermswhichareproductsofthreemomentum coordinates. Withthisinmind,wewillfocusonthesituationinwhichtheperturbationsaredueto productsoftwomomentumcoordinates,andwewillleaveoutsuchtermswhichobviously have`zeroexpectation.'Since[ p i p j ;p j p i ]=0,weonlyneedtocomputeatotalofsix operators.Forradiallysymmetricpotentials,itismostnaturaltorepresentthesingle particlewavefunctionsasproductsofthesphericalharmonicsandsolutionstotheradial equation.Thus,itisparticularlyconvenienttoobtaintheCartesiancomponentsofthe momentumoperator, p = )]TJ/F23 11.9552 Tf 9.298 0 Td [(i ~ ~ r ; .28 insphericalcoordinates. Giventhatthegradientinsphericalcoordinatesis ~ r = @ @r ^ r + 1 r @ @ ^ + 1 r sin @ @ ^ ; .29 wewishndits x y ,and z components,whichwedesignateas p x p y ,and p z .Wecando thisbysubstitutingtheCartesianprojectionsof^ r ^ ,and ^ into.29andcollectinglike 36 PAGE 44 terms.Thetransformationfortheunitvectorsisgivenby ^ r =sin cos ^ x +sin sin ^ y +cos ^ z .30 ^ =cos cos ^ x +cos sin ^ y )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin ^ z .31 ^ = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin ^ x +cos ^ y: .32 Thus,theCartesiancomponentsofthemomentumbecome p x = )]TJ/F23 11.9552 Tf 9.298 0 Td [(i ~ sin cos @ @r + 1 r cos sin @ @ )]TJ/F15 11.9552 Tf 16.373 8.088 Td [(sin r sin @ @ .33 p y = )]TJ/F23 11.9552 Tf 9.298 0 Td [(i ~ sin sin @ @r + 1 r cos sin @ @ + cos r sin @ @ .34 p z = )]TJ/F23 11.9552 Tf 9.298 0 Td [(i ~ cos @ @r )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(sin r @ @ : .35 Sincewehavealreadynotedthattheexpectationsofsinglecomponentsofmomentum andproductsofthreecomponentsofmomentumhavevanishingexpectationvaluesforthe casethat l =0,wecancomputeproductsoftwomomentumcomponents,leavingoutterms whichwillhaveanexpectationofzeroduetoaderivativeinanangularvariable.Observe, p x p y = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 sin cos @ @r + 1 r cos cos @ @ )]TJ/F15 11.9552 Tf 16.373 8.088 Td [(sin r sin @ @ sin sin @ @r + 1 r cos sin @ @ + cos r sin @ @ .36 = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 1 2 sin 2 sin2 @ 2 @r 2 + 1 2 r cos 2 sin2 @ @r )]TJ/F15 11.9552 Tf 15.951 8.088 Td [(1 2 r sin2 @ @r +`termsofzeroexpectation,' p x p z = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 sin cos @ @r + 1 r cos cos @ @ )]TJ/F15 11.9552 Tf 16.373 8.087 Td [(sin r sin @ @ cos @ @r )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 r sin @ @ .37 = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 1 2 sin2 cos @ 2 @r 2 + 1 2 r sin2 cos @ @r )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(sin2 2 r @ @r +`termsofzeroexpectation,' 37 PAGE 45 p y p z = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 sin sin @ @r + 1 r cos sin @ @ + cos r sin @ @ cos @ @r )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 r sin @ @ .38 = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 1 2 sin2 sin @ 2 @r 2 )]TJ/F15 11.9552 Tf 15.951 8.088 Td [(1 2 r sin2 sin @ @r )]TJ/F15 11.9552 Tf 15.951 8.088 Td [(1 2 r sin2 sin @ @r +`termsofzeroexpectation,' p x p x = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 sin cos @ @r + 1 r cos cos @ @ )]TJ/F15 11.9552 Tf 16.373 8.088 Td [(sin r sin @ @ sin cos @ @r + 1 r cos cos @ @ )]TJ/F15 11.9552 Tf 16.373 8.088 Td [(sin r sin @ @ .39 = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 sin 2 cos 2 @ 2 @r 2 + 1 r cos 2 cos 2 @ @r + sin 2 r @ @r +`termsofzeroexpectation,' p y p y = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 sin sin @ @r + 1 r cos sin @ @ + cos r sin @ @ sin sin @ @r + 1 r cos sin @ @ + cos r sin @ @ .40 = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 sin 2 sin 2 @ 2 @r 2 + 1 r cos 2 @ @r + 1 r cos 2 sin 2 @ @r +`termsofzeroexpectation,' and p z p z = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 cos @ @r )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 r sin @ @ cos @ @r )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 r sin @ @ .41 = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 cos @ 2 @r 2 +sin 2 @ @r +`termsofzeroexpectation.' Now,wewishtocomputetheinnerproductofthesetermsgivenby h f j Q j f i = R R 3 f ~r Q f ~r dr 3 ,where dr 3 isthestandardEuclideanmeasureon R 3 and Q isaHermitianoperator.Weobservethatwhenthemeasureisrepresentedinsphericalcoordinates 38 PAGE 46 theinnerproductsoftheoperatorsoftheform p i p j with i 6 = j allcontainasinetermwith adoubleangle,either2 or2 ,whichaveragestozerosinceawavefunctionwith l =0 onlycontributesaconstanttotheangularintegrals.Thus, h j p x p y j i = h j p x p z j i = h j p y p z j i =0. Recall,thatthesphericalharmonicscanbegivenintermsoftheassociatedLegendre polynomials P m l s = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 m )]TJ/F23 11.9552 Tf 11.955 0 Td [(s 2 m 2 d m ds m P l s ; .42 where P l s = 1 2 l l d l ds l [ s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 l ].43 areLegendrePolynomialsinaparameter s .Thisrepresentationyieldstheformula, Y m l ; = s l +1 l )]TJ/F23 11.9552 Tf 11.956 0 Td [(m 4 l + m P m l cos e im : .44 Thus,forthe l =0and l =1states,thesphericalharmonicsare Y 0 0 = 1 2 r 1 Y )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 r 3 2 sin exp )]TJ/F24 7.9701 Tf 6.587 0 Td [(i Y 0 1 = 1 2 r 3 2 cos Y 1 1 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 r 3 2 sin exp i 39 PAGE 47 Usingthelinearityoftheinnerproduct,wecanexpresstheexpectation h j p x p x j i asasumoftheexpectationsofthenonzerotermsin.39.Thisfollowssimilarlyfor h j p y p y j i and h j p z p z j i .Observe, h j p x p x j i [1] Z R 3 p x p x dr 3 = Z R 3 Y m l R n;l p x p x Y m l R n;l dr 3 = Z 0 Z 2 0 Z 1 0 Y m l R n;l sin 2 cos 2 @ 2 @r 2 Y m l R n;l r 2 sin drdd = Z 0 1 4 sin 3 d Z 2 0 cos 2 d Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr = 1 4 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr : [2] Z R 3 Y m l R n;l cos 2 cos 2 1 r @ @r Y m l R n;l dr 3 = Z 0 Z 2 0 Z 1 0 Y m l R n;l cos 2 cos 2 1 r @ @r Y m l R n;l r 2 sin drdd = Z 0 1 4 cos 2 sin d Z 2 0 cos 2 d Z 1 0 R n;l 1 r @ @r R n;l r 2 dr = 1 4 2 3 Z 1 0 R n;l 1 r @ @r R n;l r 2 dr : [3] Z 0 Z 2 0 Z 1 0 Y m l R n;l sin 2 1 r @ @r Y m l R n;l r 2 sin drdd = Z 0 1 4 sin d Z 2 0 sin 2 d Z 1 0 R n;l 1 r @ @r R n;l r 2 dr = 1 4 Z 1 0 R n;l 1 r @ @r R n;l r 2 dr : 40 PAGE 48 Thesearethetermsfor h j p y p y j i [1] Z 0 Z 2 0 Z 1 0 Y m l R n;l sin 2 sin 2 @ 2 @r 2 Y m l R n;l r 2 sin drdd = Z 0 1 4 sin 3 d Z 2 0 sin 2 d Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr = 1 4 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr : [2] Z R 3 Y m l R n;l cos 2 1 r @ @r Y m l R n;l dr 3 = Z 0 Z 2 0 Z 1 0 Y m l R n;l cos 2 1 r @ @r Y m l R n;l r 2 sin drdd = Z 0 1 4 sin d Z 2 0 cos 2 d Z 1 0 R n;l 1 r @ @r R n;l r 2 dr = 1 4 Z 1 0 R n;l 1 r @ @r R n;l r 2 dr : [3] Z R 3 Y m l R n;l cos 2 sin 2 1 r @ @r Y m l R n;l dr 3 = Z 0 Z 2 0 Z 1 0 Y m l R n;l cos 2 sin 2 1 r @ @r Y m l R n;l r 2 sin drdd = Z 0 1 4 cos 2 sin d Z 2 0 sin 2 d Z 1 0 R n;l 1 r @ @r R n;l r 2 dr = 1 4 2 3 Z 1 0 R n;l 1 r @ @r R n;l r 2 dr : Finally,for h j p z p z j i ,wehavethefollowing. 41 PAGE 49 [1] Z 0 Z 2 0 Z 1 0 Y m l R n;l cos @ 2 @r 2 Y m l R n;l r 2 sin drdd = Z 0 1 4 cos sin d Z 2 0 d Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr = 1 4 2 3 Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr : [2] Z 0 Z 2 0 Z 1 0 Y m l R n;l sin 2 1 r @ @r Y m l R n;l r 2 sin drdd = Z 0 1 4 sin 3 d Z 2 0 d Z 1 0 R n;l 1 r @ @r R n;l r 2 dr = 1 4 2 )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(2 3 Z 1 0 R n;l 1 r @ @r R n;l r 2 dr : Therefore,weneedtocalculatethecommonradialintegrals I 1 Z 1 0 R n;l 1 r @ @r R n;l r 2 dr .45 and I 2 Z 1 0 R n;l @ 2 @r 2 R n;l r 2 dr: .46 Intermsoftheseintegrals,themomentumperturbationsaregivenby h j p x p x j i = h j p y p y j i = h j p z p z j i = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 2 3 I 1 + 1 3 I 2 : Furthermore,asacheckoftheseresults,wenotethat h j )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 2 m )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(p 2 x + p 2 y + p 2 z j i = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 2 m I 1 + I 2 .47 shouldreproducetheinnitewellenergieswhentheHamiltonianisgivenasanInnite SphericalWell,sincethepotentialtermiszeroontheinterval ;a 42 PAGE 50 Recallthat,fortheinnitesphericalwellpotential,the l =0solutionstotheradial equationare j l z = sin z z ,where z = kr and k = k n;l isascalardependingon n and l such that R k n;l a =0inaccordancewiththenecessaryboundaryconditions.Withthisinmind, wecandesignatethen th zeroofthesphericalBesselfunction j l as z n;l k n;l a .Furthermore, wenormalize j 0 z withaconstant A 2 R toobtain R r = A sin z z bydemandingthat Z a 0 AR r AR r dr =1 ; .48 whichimplies A = k 3 n;l R z n;l 0 sin 2 z z 2 z 2 dz 1 2 = 2 k 3 n;l z n;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos z n;l sin z n;l 1 2 : .49 Wecannowcalculate I 1 and I 2 explicitlybymakingthechangeofvariables r z .For .45,weget I 1 = A 2 k n;l Z z n;l 0 sin z z 1 z @ @z sin z z z 2 dz = A 2 k n;l Z z n;l 0 sin z z z cos z )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin z z 2 zdz = A 2 k n;l sin 2 z z )]TJ/F23 11.9552 Tf 13.151 8.088 Td [(Si z 2 z n;l 0 = A 2 k n;l sin 2 z n;l z n;l )]TJ/F23 11.9552 Tf 13.151 8.087 Td [(Si z n;l 2 ; .50 wherewehaveusedthenotationforthesineintegral, Si z = R z 0 sin x x dx .Likewise,for .46,weobtain 43 PAGE 51 I 2 = A 2 k n;l Z z n;l 0 sin z z @ 2 @z 2 sin z z z 2 dz = A 2 k n;l Z z n;l 0 sin z z @ @z z cos z )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin z z 2 z 2 dz = A 2 k n;l Z z n;l 0 z sin z )]TJ/F23 11.9552 Tf 9.298 0 Td [(z 3 sin z )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 z 2 cos z +2 z sin z z 4 dz = A 2 k n;l Z z n;l 0 )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 2 z )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(sin2 z z +2 sin 2 z z 2 dz = A 2 k n;l )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 z 2 + z sin2 z +4cos2 z )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 4 z + Si z z n;l 0 = A 2 k n;l )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 z 2 n;l + z n;l sin2 z n;l +4cos2 z n;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 4 z n;l + Si z n;l : .51 Furthermore,wecancarrythiscomputationfurtherbyverifyingtheinnitewellenergiesifweassumethatradiusofthewellis a =10fm.Forthe n =1and n =2zerosof j 0 z weobtaintheapproximations z 1 ; 0 =3 : 142and z 2 ; 0 =6 : 283,respectively.Substituting thesevaluesinto.45and.46,yields I 1 z 1 ; 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 : 4564 10 28 I 2 z 1 ; 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(9 : 5935 10 27 .52 and I 1 z 2 ; 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(9 : 3750 10 28 I 2 z 2 ; 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 0726 10 29 : .53 Computingtheeigenvalues,weget E 1 ; 0 = h 1 ; 0 j H 0 j 1 ; 0 i = h 1 ; 0 j )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 2 m p x p x + p y p y + p z p z j i = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 2 m I 1 z 1 ; 0 + I 2 z 1 ; 0 =2 : 05 10 6 eV; .54 E 2 ; 0 = h 2 ; 0 j H 0 j 2 ; 0 i = )]TJ/F35 11.9552 Tf 9.298 0 Td [(~ 2 2 m I 1 z 2 ; 0 + I 2 z 2 ; 0 =8 : 20 10 6 eV: .55 44 PAGE 52 Thisservesasacheckforthemomentumperturbationcalculation,sincethesevaluescorrespondwiththespectrumobtainedinTable.29usingtheformula E n;l = z n;l ~ 2 2 ma 2 : .56 4.4ConclusionsandFurtherWork Intheprevioussectionsofthischapter,formalcalculationswereperformedtocompute, ingeneral,allrst-ordernon-relativisticLorentzviolatingperturbationstosingleparticle shellmodelwavefunctionsinthecasethat l =0.Thesecalculationscouldbeparticularly applicabletonucleiwhichhaveanevennumberofonetypeofnucleonandhaveanodd numberofanothertypewiththelastoddnucleonresidinginanS-orbit.Inthiscase,the singleparticleshellmodelpredictsthattheangularmomentumofthenucleuswillbegiven bythatofthelastoddnucleon,whichiszero.Thus,inthiscase,allcombinationsofthe termsin.1canbeanalysed.Additionally,thecalculationinSection4.2alsoappliesin thecasethat l 6 =0aslongasthecoecientsofthemomentumtermsin.1havebeen tunedtozero.Forfurtherresearch,perturbationsforangularmomentum l 6 =0canbe calculated. Moreover,itisstillnecessarytoknowtowhatdegreeofaccuracythetermsof.1have beenboundedinthecontextofphenomenawhichcouldpertaintothisthesis.Experimental methodsusedtosearchforLorentzviolatingphenomena,have,inthepast,involvedPenningtrapmethodsaswellasanumberofothertechniqueswhichcouldserveascandidates toinvestigatetheconsequencesoftheseshellmodelcalculations.Nevertheless,beforean experimentcanbeproposedtheaccuracyofexperimentsdesignedtotestthepredictions oftheshellmodelmustbeevaluated.Ifanappropriatemethodisdiscovered,itcanthen bedeterminediffurthercalculationsareneededtotakeintoaccountthephysicsspecic totheexperimentalsetup.Insuchcases,itmaybethattheapproximationofnuclear centerofmassresidinginitsgroundstatemaynolongerbevalid.Ifthisisthecase,simple 45 PAGE 53 shellmodelcalculationssuchastheonesdiscussedinChapter3maynolongerapply.In thiscase,itwouldbenecessarytoincludethecenterofmassmotioninthepreliminary analysis.MethodsfortreatingthissituationhavebeentreatedinotherworksSee[Elliot andSkyrme]. Finally,muchworkhasbeendonetorenethenuclearshellmodelsinceMayer's resultswerepublishedin1949.Thesemethodsnotonlyincludeperturbationsdueto theelectromagneticforce,buttheyalsomakeimprovedpredictionsforcasesinwhich morethanoneparticleinteractoutsideofaclosedshell.Inparticular,[deShalitand Talmi],[Heyde],and[Bender etal. Bender,Heenen,andReinhard]provideamorecomprehensivediscussion. 46 PAGE 54 AppendixA Clebsch-GordonCoecients Inthisappendix,wederiveexplicitformulasfortheClebsch-Gordoncoecientsforspin 1 2 particleswhichallowonetochangebasisbetweenarepresentationofastationarystate intermsofeigenfunctionsoftheoperators S 2 ; l 2 ; j 2 ; and j z toarepresentationinterms ofabasisoftheeigenfunctionsoftheoperators S 2 ;S z ; l 2 ; and l z .Moreprecisely,let beastationarystateofaparticlewithspin S ,orbitalangularmomentum l ,andtotal angularmomentum j l + S .Thenitiseasytoshowthattheoperators S 2 ;S z ; l 2 ;l z and S 2 ; l 2 ; j 2 ;j z commutewitheachother.Furthermore,givenanysingleparticleHamiltonian, H = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 2 m ~ r 2 + V ~r ; theelementsof f S 2 ;S z ; l 2 ;l z g and f S 2 ; l 2 ; j 2 ;j z g commutewith H .Thisindicatesthat elementsof f S 2 ;S z ; l 2 ;l z g[f H g and f S 2 ; l 2 ; j 2 ;j z g[f H g aresimultaneouslydiagonalizable.SincealloftheseoperatorsareHermitian,thespectraltheoremtellsusthatwecan ndasimultaneoussetofeigenfunctionsforeitherofthesetslistedabovewhichforma completeorthonormalset.Wewouldliketondawaytochangebetweentheserepresentations.Thisisusefulforcomputingexpectationvaluesoftwoormoreoperatorswhichdo notcommute.Thistransformationlawcanbederivedbyconsideringastationarystate writtenintermsofeigenfunction, sljm .Now,weindicatethecompletesetoforthonormal 47 PAGE 55 eigenfunctionsoftheset f S 2 ; l 2 ; j 2 ;j z g[f H g by slm s m j ,and,bycompletenesswecan write sljm = X m s m l sm s lm l j sljm j slm s m j ; A.1 where sm s lm l j sljm j 2 R arecalledtheClebsch-Gordoncoecients. Recall,that j 2 sljm j = j j +1 slm s m j and j z sljm = m j sljm Thus,wecanactonA.1toget j z sljm = l z + s z X m s ;ml sm s lm l j sljm j slm s m j = X m s ;ml l z + s z sm s lm l j sljm j slm s m j = X m s ;ml m s + m l sm s lm l j sljm j slm s m j = m j sljm Bytheorthogonalityofthe s,thatis X spincoordinates Z sm s lm l sm 0 s lm 0 l d = m s ;m 0 s m l ;m 0 l ; A.2 wecantakeaninnerproductwithbothsidestocomeupwithanidentityforthecoecients. From, sm s lm l j sljm j 6 =0ifandonlyif m l = m j )]TJ/F23 11.9552 Tf 11.955 0 Td [(m s : Forspin 1 2 particles, m s = 1 2 .Thus,onlytwocoecients,whichwedesignateas a and b arenonzero.Furthermore,welet a = 1 2 ; 1 2 ;l;m l )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 2 j 1 2 ljm j and b = 1 2 ; )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 ;l;m l + 1 2 j 1 2 ljm j 48 PAGE 56 Thus,wecanwriteforthetwonon-zerofunctions sljm j = a + + b )]TJ/F23 11.9552 Tf 7.085 1.793 Td [(; where + = 1 2 l 1 2 m + 1 2 and + = 1 2 l )]TJ/F22 5.9776 Tf 7.782 3.258 Td [(1 2 m )]TJ/F22 5.9776 Tf 7.782 3.258 Td [(1 2 .Byactingon sljm j with j 2 = l 2 + S 2 + l + S )]TJ/F15 11.9552 Tf 9.025 1.794 Td [(+ l )]TJ/F23 11.9552 Tf 7.085 1.793 Td [(S + +2 l z S z andbyusingorthogonalityweobtainasetofequationsfor a and b .Observe, j 2 sljm j = j j +1 a + + b )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( =[ l l +1+ 3 4 ] a + + b )]TJ/F15 11.9552 Tf 7.084 1.793 Td [(+ r l l +1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( m )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 m + 1 2 a )]TJ/F15 11.9552 Tf -320.979 -31.363 Td [(+ r l l +1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( m + 1 2 m )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 b + + m )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 a + )]TJ/F15 11.9552 Tf 11.955 0 Td [( m + 1 2 b )]TJ/F23 11.9552 Tf 7.085 1.793 Td [(: Takingtheinnerproductrstwith + andthenwith )]TJ/F15 11.9552 Tf 7.085 1.793 Td [(,weobtain a j j +1 )]TJ/F29 11.9552 Tf 11.955 16.857 Td [( l + 1 2 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(m # )]TJ/F23 11.9552 Tf 11.956 0 Td [(b s l + 1 2 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(m 2 =0 )]TJ/F23 11.9552 Tf 9.299 0 Td [(a s l + 1 2 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(m 2 + b j j +1 )]TJ/F29 11.9552 Tf 11.955 16.857 Td [( l + 1 2 2 + m # =0 : Fortheseequationstobeconsistentweneed j = l 1 2 .Withoutlossofgenerality,we usetherstequationforthecasewhen j = l )]TJ/F21 7.9701 Tf 13.15 4.707 Td [(1 2 a s l + 1 2 + m + b s l + 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(m =0 : Thisgives a s l + 1 2 + m = N and b s l + 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(m = N: Imposingthenormalizationcondition, a 2 + b 2 =1on a and b weobtainanotherindependent equation,whichgives a = s l + 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(m 2 l +1 and b = s l + 1 2 + m 2 l +1 Thederivationforthecasewhere j = l + 1 2 followssimilarly.Theresultsforbothcases aresummarizedinTableA.1below. 49 PAGE 57 TableA.1:Clebsch-GordonCoecients m s 1 2 )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 2 j l + 1 2 q l + 1 2 + m 2 l +1 q l + 1 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(m 2 l +1 l )]TJ/F21 7.9701 Tf 13.15 4.707 Td [(1 2 q l + 1 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(m 2 l +1 )]TJ/F29 11.9552 Tf 9.298 15.314 Td [(q l + 1 2 + m 2 l +1 50 PAGE 58 AppendixB ExperimentalDataSupportingthe NuclearShellModel:Figuresand Discussion InChapter1,abriefhistoryofthenuclearshellmodelwasgiven.Inthisappendix,selected datasupportingthemodelisexplained,andthepredictions,successes,andlimitationsof themodelaretreatedinfurtherdetail. Theprimarymotivationforthenuclearshellmodelwasthesignicanceofthemagic numbers, 2,8,20,28,50,82,and126. Inparticular,alargeamountofexperimentaldataproducedinbetweenthe1930'sand 1940'sindicatedthatnucleithatpossessedaprotonorneutronnumbercoincidingwithone ofthese`magicvalues'wasparticularlystable.Although,muchwasunknownregarding thestrongnuclearforce,thisseemedtoindicatetheexistenceofanuclearshellstructure. Asimilarstructurewasalreadyknowntobeaprimaryfactorindeterminingtheproperties ofelectronsinatoms,andsupportforthisisexempliedbrilliantlybyconsideringhowthe ionizationpotentialofvariouselementsvarieswithrespecttoatomicnumbersFigureB.1. 51 PAGE 59 FigureB.1:IonizationEnergyofNeutralAtomsasafunctionofAtomicNumber [Heyde] Furthermore,anobservationmadebyMariaGoeppertMayerwasthatthenumberof stableisotopesandisotonesseemedtoincreasewhentheprotonorneutronnumberofa nucleiwasmagic.ThisdataispresentedinFigureB.2forthemagicnumbers,20,28,50, and82;anditprovidesevidencefortheincreasedstabilityofmagicnuclei. Fromthiswealsonoticethattheevidenceforneutronmagicnumbersismuchbetter thanfortheprotonnumbers[BlattandWeisskopf]."Additionally,itisobserved thatthistablealsosupportswhatisknownasHarkin'srule:theclaimthatnucleiwith anevennumberofprotonsorneutronsaremorestablethanthosewithoddnumbers [BlattandWeisskopf].Nevertheless,theredoesnotappeartobeanyevidencefor increasedstabilityfortheprotonnumberof Z =82.Forthis,wecanturntoalpha-particle experiments,whichprovideanothersourceofsupportfortheshellmodel.Infact,support forthemagicprotonnumber Z =82andforthemagicneutronnumberof N =126 isgiven.FigureB.3providessupportforthestabilityassociatedwiththemagicproton 52 PAGE 60 FigureB.2:Numberofstableisotopesandisotonesforvariousprotonandneutronnumbers [Heyde] number Z =82andforthemagicneutronnumberof N =126. Wheninterpretingthisdata,itisimportanttorecognizethatnucleiwiththesame protonnumberareconnectedwithlines,and,fornucleiwith A> 213theenergyofthe emittedalphaparticlestrictlyincreaseswithrespecttoadecreaseinneutronnumber. Thisbecomesmostpronouncedfor At 213 and Po 212 .Inbothofthesedecays,theneutron numberofthedaughternucleiis N =126,whichismagic.Furthermore,immediatelyafter thesetwocases N< 126,thetrendissimilar,butthemaximumalpha-particleenergyis considerablylowerforeachsetofisotopes.Thus,itwouldappearthatnucleiwith N =126 haveespeciallylowenergyandare,therefore,morestable.Thisconclusionissupported bythewellknowfactsthat Po 212 At 213 ,and Bi 209 arestableagainstalpha-decay,and `neutronrich'isotopes, Bi 210 and Bi 211 ,arealpha-radioactive[BlattandWeisskopf]. Furthermore,itispossibletoconcludethat Z =82mustbeparticularlystable,sincethe comparisonbetweenisotopesof At Z =85, Po Z =84,and Bi Z =83showsanet declineintheenergyplotsforthealphaproducts.Thisconclusionisreinforcedbythe 53 PAGE 61 FigureB.3:Energyof -EmissionVersusNeutronNumber[Heyde94] factthat Pb Z =82isespeciallystable.Infact,itisnotevenradioactive[Blattand Weisskopf].Furthermore,itisnoteworthytomentionthatasimilaranalysiscanbe appliedtothedataavailablefrombetaemissions[GoeppertMayer],anditshould alsobestatedthatdataforradiativecapturecross-sectionsprovideadditionalsourcesof supportforthecorrelationbetweenthemagicnumbersandnuclearstability. Inordertofurtherdiscussthepredictionsmadebythesingleparticlenuclearshell model,itshouldbenotedthatthemodelemploysfourprimaryassumptionsuponwhich itsresultsarebased.Thesearenowstatedintheforminwhichtheyoriginallyappeared inMayer'srstarticlepublishedinthe PhysicalReview onthesignicanceofspin-orbit coupling.TherstmajorassumptionisthatThesuccessionofenergiesofsingleparticleorbitsisthatofasquarewellwithstrongspinorbitcouplinggivingrisetoinverted 54 PAGE 62 doublets[Mayera]."Thisassumptionisusuallycategorizedalongwithtworelated assumptions.Therstisthat,Foragiven l ,thelevel j = l + 1 2 hasinvariablylower energyandwillbelledbeforethatfor j = l )]TJ/F21 7.9701 Tf 12.328 4.707 Td [(1 2 [Mayera].Thisassumptionisgiven atheoreticalpremiseinMayer'ssecond PhysicalReview articleinwhichthetwoparticle interactionismodelledbyaDiracdeltafunction.Thesecondstatesthatpairsofspin levelswithinoneshell,whicharisefromadjacentorbitallevelsinthesquarewellinsucha waythatspin-obitcouplingtendstobringtheirenergyclosertogethercan,andveryoften will,cross[Mayera]."ThenextprimaryassumptionisthatAnevennumberofnucleonsinanyorbitwithtotalangularmomentumquantumnumber j willalwayscoupleto givespinzeroandnocontributiontothemagneticmoment[Mayera]."Theoretical supportforthishypothesiscanbefoundin[deShalitandTalmi2004].Thethirdrequirementisthatanoddnumberofidenticalnucleonsinastate j willoccupytogiveatotal spin j andamagneticmomentequaltothatofasingleparticleinthatstate.Finally,the lastassumptionstatesthatforanygivennucleus,thepairingenergy"ofthenucleonsin thesameorbitisgreaterfororbitswithlarger j [Mayera]. Giventheseassumptions,themagicnumberscanbeexplainedwithinthecontextof thesingleparticleshellmodel.AsdiscussedintheChapter3,thepredictedshellstructureintheunperturbedharmonicoscillatorandinnitewellpotentialswasdevelopedby computingdegeneraciesfortheenergyspectrumoftheharmonicoscillatorHamiltonian and`approximate'degeneraciesfortherstfewinnitewellstatesfortheinnitewell Hamiltonian.Fromthis,itwasobservedthatthesepredictionsonlyagreedwiththemagic numbersforlowlyingenergies.Whilewedonotintendtoexplicitlyreproducethecalculationswhichresultfromtheperturbationofthesepotentialsbyastrongspin-orbitcoupling term, r l S ,itisworthwhiletonotethattheresultingtruncatedHamiltonianisproducedinasimilarfashionastheonedevelopedinChapter4fortheperturbationoftheshell modelbyaconstantbackgroundeld.Theprimarydierencebeingthatonewouldneed tosetthebackgroundeldtozero,andtheHamiltonianwouldneedtobetruncatedatthe 55 PAGE 63 principlequantumnumberof n =6inordertoproducethehigherordermagicnumbers. Finally,dierentformsofthecouplingcoecient r maybeusedtopredictthemagic numbers.Specically,Mayernotesin[Mayer1950a]thatacalculationbasedontheMeson theoryofGausindicatesthatthesplittingbetweenthelevelswith j = l )]TJ/F21 7.9701 Tf 12.39 4.707 Td [(1 2 and j = l + 1 2 isproportionalto l +1 A )]TJ/F22 5.9776 Tf 7.783 3.259 Td [(2 3 [Mayera]."'Thisindicatesthatsplittingresponsible formagicnumbersshouldnotbeverydierentfordierentshells[Mayera].' Havingsummarizedthesebasictenets,wepresenttheresultsoftheperturbedsingle particleshellmodelspectrumfortheharmonicoscillatorinFigureB.5.Inaddition,the unperturbedspectrumisincludedforcomparisonFigureB.4. FigureB.4:SingleParticleHarmonicOscillatorSpectrumwithSpin-OrbitCoupling [Heyde] Inadditiontothemagicnumbers,theshellmodelalsoadmitspredictionsofvarious othernuclearproperties.Forinstance,thespinsofmostodd-evennuclei,theparityof nuclei,aswellaspropertiesofnuclearmagneticmomentscanallbeexplainedwithin thissimplesingleparticleframework.Furthermore,theexistence,andregioninthe periodictable,ofnuclearisomerismisalsopredicted[Mayer0a].'Nevertheless,there areseverelimitsregardingtheaccuracyofsimpleshellmodelpredictions.Inparticular, 56 PAGE 64 FigureB.5:UnperturbedSingleParticleHarmonicOscillatorSpectrum[Heyde] eventhoughthemodelpredictsthemagicnumbers,thissimplesingleparticletreatment doesnotaccountfortheorderingoftheenergylevelswithinagivenshell.Furthermore, thenuclearshellmodelisespeciallypooratcalculatingobservableswhichcantakenonintegervalues.Forinstance,inmoredetailedcalculationsofnuclearmagneticmoments, itisobservedthattheagreementisnotevenwithin10percentofthemeasuredvalues. Inthisinstance,themutualnuclearinteractionmustbehandledinmoredetail.This hasbeentreatedinamuchmorethoroughmannerbyTalmi[deShalitandTalmi004]. Furthermore,inthecaseoflightnuclei,theassumptionofstrongspin-orbitcouplingis notevenjustied.Infact,thepredictionsmadebyWigner,whichassumeveryweakspinobiteects,producemoreaccuratepredictionsthantheshellmodel.Inadditiontothis, modelswhichincorporateanintermediatespin-obitcouplinghavealsobeenperformed. Theseproducethebestresultsofthethreeandreinforcethepointthatthesingleparticle shellmodelshouldonlybeseenasapracticalrststeptowardsmoreexactmethods. Additionalinformationonmoresophisticatedmethodscanbefoundinthereferences. 57 PAGE 65 Bibliography [ColladayandKostelecky]D.ColladayandV.A.Kostelecky,PhysicalReviewD. 58 [Heyde]K.L.Heyde, TheNuclearShellModel ,2nded.Springer-Verlag,1994. [Elsasser]W.Elsasser,J.Phys.Radium 4 ,549. [GoeppertMayer]M.GoeppertMayer,enquotebibinfotitleNobellecture:The shellmodel,. 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[Mayer48]M.G.Mayer,bibeldjournalbibinfojournalPhys.Rev.textbfbibinfo volume74,bibinfopages235bibinfoyear1948. [KosteleckyandLaneb]V.A.KosteleckyandC.D.Lane,JournalofMathematical Physics 40 b. [PospelovandRomalis]M.PospelovandM.Romalis,PhysicsToday 57 ,40. 59 |