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Nuclear Superfluidity PairinginFiniteSystems NuclearSuperfluidityisthefirstmoderntextdevotedexclusivelytopair-correlationsinnuclei. It begins by exploring pair-correlations in a variety of systems including superconductivity in metalsatlowtemperaturesandsuperfluidityinliquid3Heandinneutronstars. Thebookgoes ontointroducebasictheoreticalmethods,symmetrybreakingandsymmetryrestorationinfinite many-bodysystems. Thelastfourchaptersaredevotedtointroducingnewresultsontheroleof polarizationeffectsinthestructureofbothnormalandexoticnuclei. Centraltothisdiscussion isthefactthatwhilebarenucleon–nucleoninteractionsareessentialfortheproductionofpair- correlationsinnuclei,thecouplingofpairsofnucleonstolow-energynuclearcollectiveexcitations alsoplaysanimportantrole. This book will be essential reading for researchers and students in both experimental and theoretical nuclear physics, and related research fields such as metal clusters, fullerenes and quantumdots. DAVID M. BRINKobtainedhisfirstdegreeattheUniversityofTasmaniain1951andhisD.Phil. atOxfordUniversityin1955.Between1958and1993heheldacademicpositionsintheUniversity ofOxford,includingaFellowshipatBalliolCollegeandtheMoseleyReadershipinTheoretical Physics,andtaughtmanybranchesofphysicsatgraduateandundergraduatelevel. From1993 to1998hewasProfessoroftheHistoryofPhysicsattheUniversityofTrentoinItaly. Professor BrinkisaFellowoftheRoyalSocietyandin1982wasarecipientoftheRutherfordMedalof the Institute of Physics. He has published several books including Semi-classical Methods in Nucleus–NucleusScattering(CambridgeUniversityPress,1985). RICARDO A. BROGLIA earned his Ph.D. at the University of Cuyo, Argentina, in 1965. Following positions at the University of Buenos Aires, the Niels Bohr Institute and the Uni- versity of Minnesota, he joined the staff of the Niels Bohr Institute in 1970, where he is now adjunctProfessor. HehasheldvisitingpositionsattheStateUniversityofNewYorkatStony Brook,BrookhavenNationalLaboratory,LosAlamosScientificLaboratoryandOakRidgeNa- tionalLaboratory. In1985hewascalledtooccupythechairofNuclearStructureattheUniversity ofMilan. ProfessorBroglia’sresearchinterestsincludenuclearstructureandnuclearreactions, thephysicsofmetalclustersandfullerenes,andthefoldingandaggregationofproteins. Hehas publishedseveralbooksonthesesubjectsincludingFiniteQuantumSystemsjointlywithGeorge Bertsch(CambridgeUniversityPress,1994). CAMBRIDGEMONOGRAPHSON PARTICLEPHYSICS NUCLEARPHYSICSANDCOSMOLOGY 24 GeneralEditors:T.Ericson,P.V.Landshoff 1. K.Winter(ed.):NeutrinoPhysics 2. J.F.Donoghue,E.GolowichandB.R.Holstein:DynamicsoftheStandardModel 3. E.LeaderandE.Predazzi:AnIntroductiontoGaugeTheoriesandModernParticlePhysics,Volume 1:ElectroweakInteractions,the‘NewParticles’andthePartonModel 4. E.LeaderandE.Predazzi:AnIntroductiontoGaugeTheoriesandModernParticlePhysics,Volume 2:CP-Violation,QCDandHardProcesses 5. C.Grupen:ParticleDetectors 6. H.GrosseandA.Martin:ParticlePhysicsandtheSchro¨dingerEquation 7. B.Andersson:TheLundModel 8. R.K.Ellis,W.J.StirlingandB.R.Webber:QCDandColliderPhysics 9. I.I.BigiandA.I.Sanda:CPViolation 10. A.V.ManoharandM.B.Wise:HeavyQuarkPhysics 11. R.K.Bock,H.Grote,R.Fru¨hwirthandM.Regler:DataAnalysisTechniquesforHigh-Energy Physics,Secondedition 12. D.Green:ThePhysicsofParticleDetectors 13. V.N.GribovandJ.Nyiri:QuantumElectrodynamics 14. K.Winter(ed.):NeutrinoPhysics,Secondedition 15. E.Leader:SpininParticlePhysics 16. J.D.Walecka:ElectronScatteringforNuclearandNucleonStructure 17. S.Narison:QCDasaTheoryofHadrons 18. J.F.LetessierandJ.Rafelski:HadronsandQuark–GluonPlasma 19. A.Donnachie,H.G.Dosch,P.V.LandshoffandO.Nachtmann:PomeronPhysicsandQCD 20. A.Hofmann:ThePhysicsofSynchrotronRadiation 21. J.B.KogutandM.A.Stephanov:ThePhasesofQuantumChromodynamics 22. D.Green:HighPt PhysicsatHadronColliders 23. T.Hatsuda,K.YagiandY.Miake:Quark–GluonPlasma 24. D.M.BrinkandR.A.Broglia:NuclearSuperfluidity Nuclear Superfluidity Pairing in Finite Systems DAVIDM.BRINK and RICARDOA.BROGLIA cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521395403 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 isbn-13 978-0-511-11365-9 eBook (NetLibrary) isbn-10 0-511-11365-x eBook (NetLibrary) isbn-13 978-0-521-39540-3 hardback isbn-10 0-521-39540-2 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Formyfamily–DMB ForAngela,Donatella,GianandreaandBettina–RAB Contents Preface pagexi 1 Introduction 1 1.1 Pairinginnuclei,superconductors,liquid3Heandneutronsstars 1 1.2 Macroscopicwavefunctionandphaserigidity 3 1.3 Brokensymmetryandcollectivemodes 6 1.4 Superfluid4He(HeII) 8 1.5 Criticalvelocityforsuperconductors 13 1.6 Pairinginnuclei 14 1.7 Superconductivity 19 1.8 Superfluidityofliquid3He 25 1.9 Comparisonofpairinginnucleiwithsuperconductivity 26 1.10 Neutronstars 30 2 Thepairingforceandseniority 33 2.1 Evidenceforpairingcorrelations 33 2.2 Thepairinginteraction 36 2.3 Theδ-functionnucleon–nucleonpotential 39 2.4 Thedegeneratemodelandquasi-spin 42 2.5 Pairingbindingenergyformula 44 2.6 Quasi-spinwavefunctions 45 2.7 Pairingrotations 47 2.8 ExactsolutionofthepairingHamiltonian 48 3 TheBCStheory 52 3.1 TheBCSwavefunction 52 3.2 Theenergy 55 3.3 Excitedstatesandquasiparticles 57 3.4 Themean-fieldHamiltonian 60 vii viii Contents 3.5 Thecorrelationenergy 61 3.6 Pairingcorrelationsinthewavefunction 64 3.7 ThedegeneratemodelintheBCSapproximation 65 3.8 Gaugeinvariance 66 3.9 Matrix elements of one-body operators 67 3.10 Pairingandisospin 69 4 Spontaneoussymmetrybreaking 72 4.1 General background 72 4.2 Pairing in atomic nuclei (0D systems; ξ(cid:2)R)75 4.3 Infinite 3D neutral superconductors (ξ(cid:3)L)88 5 Pairingvibrations 92 5.1 The two-level model 92 5.2 Applications 102 5.3 Multipolepairingvibrations 108 6 Phasetransitions 117 6.1 Theexperimentalsituation 119 6.2 Staticpairingcorrelations: theBCStheoryofpairingphasetransitions instronglyrotatingnuclei 122 6.3 Pairingfluctuations 138 6.4 Momentsofinertia 141 6.5 Condensation-inducedtunnelling 144 6.6 ResponsefunctiontechniquetocalculateRPAfluctuations 145 7 Plasticbehaviourofnucleiandotherfinitesystems 154 7.1 Exoticdecay 155 7.2 Avarietyofapplications 163 7.3 Low-lyingsurfacevibrations 165 7.4 Fissioninmetalclusters 168 8 Sourcesofpairinginnuclei 170 8.1 Thebarenucleon–nucleonpotentialandthepairinginteraction 171 8.2 Mean-fieldtheory 177 8.3 Randomphaseapproximation 184 8.4 Correlationenergycontributiontonuclearmasses 199 9 Beyondmeanfield 204 9.1 Doorwaystates 204 9.2 Effectivemass(ω-mass) 211 9.3 Theω-massandtheinducedinteraction 215 10 Inducedinteraction 219 10.1 Simpleestimates 219 10.2 Microscopiccalculations 223 Contents ix 10.3 Slabmodel 231 10.4 Inducedpairinginteraction,effectivemassandvertexcorrection processes 239 10.5 Superfluidityintheinnercrustofneutronstars 244 11 Pairinginexoticnuclei 257 11.1 Thehalonucleus11Li 258 11.2 Thehalonucleus12Be 275 AppendixA Abriefre´sume´ ofsecondquantization 280 A.1 Fermions 280 A.2 Particlesandholes 286 A.3 Bosons 288 A.4 Quasi-bosons 290 AppendixB Singleparticleinanon-localpotential 292 B.1 Singleparticleinanon-local,ω-dependentpotential 294 AppendixC Usefulrelationsinthetreatmentofcollectivemodes 297 C.1 Limitonthemultipolarityofcollectivesurfacevibrations 297 C.2 Therelationbetween Fˆ andαˆ 297 AppendixD Particle-vibrationcoupling 299 D.1 Estimateof(cid:4)lj||Y ||lj(cid:5) 301 L D.2 Asimpleestimateof(cid:4)R ∂U(cid:5) 303 0∂r AppendixE Modelofthesingle-particlestrengthfunction 305 AppendixF SimplemodelofPauliprinciplecorrections 308 AppendixG Pairingmean-fieldsolution 310 G.1 SolutionofthepairingHamiltonian 310 G.2 Two-quasiparticleexcitations 315 G.3 Minimization 317 G.4 BCSwavefunction 318 AppendixH Pairinginasinglej-shell 320 H.1 BCSsolution 320 H.2 Crankingmomentofinertia 322 H.3 Two-particletransfer 323 H.4 Polarizationeffects 324 AppendixI Fluctuationsandsymmetryrestoration 327 I.1 Conjugatevariables 327 I.2 Rotationaboutanaxis 328 I.3 Rotationsingaugespace 329 I.4 Symmetryrestoringfluctuationsandpairingrotations 330