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Lorentz Symmetry Violating Extensions of the Nuclear Shell Model

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004887/00001

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Title: Lorentz Symmetry Violating Extensions of the Nuclear Shell Model
Physical Description: Book
Language: English
Creator: Wilcox, Steven
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

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Subjects / Keywords: Lorentz Violations
Nuclear Physics
Shell Model
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The invariance of physical laws with respect to Lorentz symmetry is one of the most well supported notions in all of physics. Nevertheless, as physicists search for clues for understanding the nature of gravity at quantum mechanical scales and for resolving unexplained phenomena such as the observed antisymmetry of matter and antimatter in the universe, the violation of Lorentz covariance has become an increasingly attractive indicator for new discoveries and unconventional physics. The motivation for this search comes from theoretical predictions of leading theories beyond the standard model, such as string theory and loop quantum gravity, as well as from the fact that most other known symmetries besides CPT and Lorentz symmetry are broken in some physical processes. To accommodate this search, a framework called the Standard Model Extension (SME) has been developed and extensively studied Colladay and Kostelecký(1998). The Standard Model Extension includes all first order Lorentz violating couplings of Standard Model fields and has been shown to preserve many of the important features of the Standard Model. In this thesis, a first order non-relativistic expansion of the Lorentz violating perturbations given by the SME are applied to the single particle nuclear shell model, and the implications of these results are discussed.
Statement of Responsibility: by Steven Wilcox
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: McDonald, Patrick; Colladay, Donald

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Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 W66
System ID: NCFE004887:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004887/00001

Material Information

Title: Lorentz Symmetry Violating Extensions of the Nuclear Shell Model
Physical Description: Book
Language: English
Creator: Wilcox, Steven
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

Subjects

Subjects / Keywords: Lorentz Violations
Nuclear Physics
Shell Model
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The invariance of physical laws with respect to Lorentz symmetry is one of the most well supported notions in all of physics. Nevertheless, as physicists search for clues for understanding the nature of gravity at quantum mechanical scales and for resolving unexplained phenomena such as the observed antisymmetry of matter and antimatter in the universe, the violation of Lorentz covariance has become an increasingly attractive indicator for new discoveries and unconventional physics. The motivation for this search comes from theoretical predictions of leading theories beyond the standard model, such as string theory and loop quantum gravity, as well as from the fact that most other known symmetries besides CPT and Lorentz symmetry are broken in some physical processes. To accommodate this search, a framework called the Standard Model Extension (SME) has been developed and extensively studied Colladay and Kostelecký(1998). The Standard Model Extension includes all first order Lorentz violating couplings of Standard Model fields and has been shown to preserve many of the important features of the Standard Model. In this thesis, a first order non-relativistic expansion of the Lorentz violating perturbations given by the SME are applied to the single particle nuclear shell model, and the implications of these results are discussed.
Statement of Responsibility: by Steven Wilcox
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: McDonald, Patrick; Colladay, Donald

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 W66
System ID: NCFE004887:00001


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LorentzSymmetryViolatingExtensionsofthe NuclearShellModel by STEVENWILCOX AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts Undertheco-sponsorshipofPatrickMcDonaldandDonaldColladay Sarasota,FL May,2013

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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

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Acknowledgements IwouldliketoacknowledgemythesissponsorsDr.DonaldColladayandDr.Patrick McDonaldfortheirwillingnesstooverseethisresearchprojectandtoentrustmewith itsexecution.Furthermore,Iwouldliketothankthemfortheirpatienceandguidance throughoutmyundergraduatecareer.TheyweretheprinciplereasonIchoseNewCollege ratherthananotherinstitution.Furthermore,Iwouldliketothankmyparentsandmy sisterfortheirloveandsupport.Itishardtooverestimatethevaluethatyourunconditional kindnesshasaddedtomylife. iii

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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

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BExperimentalDataSupportingtheNuclearShellModel:Figuresand Discussion51 Bibliography58 v

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ListofTables 3.1HarmonicOscillatorSpectrum.........................20 3.2ZerosoftheSphericalBesselFunctions, j l z .................23 3.3Energyspectrumfortherstseveninnitewellstates............24 4.1Clebsch-GordanCoecients..........................29 A.1Clebsch-GordonCoecients..........................50 vi

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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

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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

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hascomeaswellasitspotentialforchange,innovation,adaptation,andunderstanding. Itisthislegacywhichservesasalamppostfortheinquisitivemindsofthisage,bothfor thepurposeofinspirationandforperseverance.Inthisway,thevalueofthishistoryisas muchacatalystforusastheresultsareintheircontinuedapplicationtoscienticresearch. 2

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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

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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

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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

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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

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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

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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

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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

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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
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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

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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

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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

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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
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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

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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
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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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[ 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

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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

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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

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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

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! 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

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! 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

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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

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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

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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

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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

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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

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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

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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

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[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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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,. [Griths]D.Griths, IntroductiontoQuantumMechanics ,2nded.PearsonEducation,Inc,2005. [deShalitandTalmi]A.deShalitandI.Talmi, NuclearShellTheory ,1sted.Dover Publications,Inc,2004. [KosteleckyandLanea]V.A.KosteleckyandC.D.Lane,PhysicalReviewD 60 a. [ElliotandSkyrme]J.ElliotandT.Skyrme,Proc.R.Soc.Lond.A 232 ,561. [Bender etal. Bender,Heenen,andReinhard]M.Bender,P.-H.Heenen,andP.-G. Reinhard,bibeldjournalbibinfojournalRev.Mod.Phys.textbfbibinfovolume 75,bibinfopages121bibinfoyear2003. 58

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[BlattandWeisskopf]J.M.BlattandV.F.Weisskopf, TheoreticalNuclearPhysics 1sted.DoverPublications,Inc,1991. [Mayer50a]M.G.Mayer,PhysicalReview 78 ,16a. [Sakurai]J.Sakurai, ModernQuantumMechanics ,reviseded.Addison-WesleyPublishingCompany,1994. [Speigel etal. Speigel,Lipschutz,andLiu]M.R.Speigel,S.Lipschutz,andJ.Liu, MathematicalHandbookofFormulasandTables ,4thed.McGrawHillCompanies, Inc,2013. [Mayer50b]M.G.Mayer,PhysicalReview 78 ,22b. [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


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