Beyondmean-fieldstudyoncollectivevibrationsandbeta-decay

Yifei Niu���

Collaborators:C.L.Bai,G.Colo,J.Meng,Z.M.Niu,N.Paar,H.Sagawa,E.Vigezzi,D.Vretenar

“Advancedmany-bodyandstatisticalmethodsinmesoscopic systemsIII”September4th – 8th ,2017,Busteni,Romania

SingleParticleExcitation

2

• EmissionSpectraofatomHandFe

• SingleNeutronSpectrumin209PbMuehllehner,etal.,Phys.Rev.159,1039(1967)

p

p

n n

NuclearCollectiveVibrations-- AnExample

3

R.Bramblett,Phys.Rev.133,B869(1964)

• GiantDipole Resonance (GDR)in16O

• Characteristics• Broadresonancewidth~5MeV• Largertransitionprobabilitiesthans.p.• Excitationenergyvariesslowlyandsmoothlywithmassnumber

γ

ImportantsubjectinRA-4• gammaabovetheneutronthresholdexperimentsatELI-NP• (γ,n)experiments

4

Characteristics• Excitationenergy≈neutronthresholdenergy<10MeV• Appearsinneutron-richnuclei

Mysterious•Whatisitsphysicalpicture?Oscillationofexcessofneutronagainstcore?FormationofclusterwithdifferentN/Z?Single-particle shelleffect?•Isitrelatedtothesymmetryenergy,liketheGDR?•Whatistheisospin property?

Fig.D.Vretenar

Importance• nuclearsymmetryenergyneutronskinthickness• neutroncaptureinr-process

• PygmyDipoleResonance(PDR)

Neutronthresholdenergy

PygmyDipoleResonance

PygmyDipoleResonance

5

Characteristics• Excitationenergy≈neutronthresholdenergy<10MeV• Appearsinneutron-richnuclei

Fig.D.Vretenar

Importance• nuclearsymmetryenergyneutronskinthickness• neutroncaptureinr-process

• PygmyDipoleResonance(PDR)

ImportantsubjectinRA-2• nuclearresonancefluorescenceexperimentsatELI-NP

Neutronthresholdenergy

• Degreesoffreedomü Spin

ü Isospin - Uniquedegreeoffreedominnucleus!

Richnessinnuclearsystem

6

• InteractionsinvolvedStrongElectromagneticWeak

Fermion

s =1/2sz =+1/2 sz =-1/2

nucleon

nucleon

protonneutron

t=1/2tz =+1/2 tz =-1/2

Similarmassnearlythesameinteraction

VariousModesofCollectiveVibrations

7

BellHammers(iron,wooden)SoundsoftheBell

reactions:(α,α’ γ),(γ,γ’),(γ,n),(e,e’)

reactions:(p,n)T-,(n,p)T+

NucleusOperators/Probes(e,γ,α,p,n)Modesofcollectivevibrations

• Non-charge-exchangeexcitations

• Charge-exchangeexcitations

StrongEM

Strong

VariousModesofCollectiveVibrations

8

BellHammers(iron,wooden)SoundsoftheBell

reactions:(α,α’ γ),(γ,γ’),(γ,n),(e,e’)

reactions:(p,n)T-,(n,p)T+

NucleusOperators/Probes(e,γ,α,p,n)Modesofcollectivevibrations

• Non-charge-exchangeexcitations

• Charge-exchangeexcitations

StrongEM

Strong

9

Gamow-Tellertransition

operator

Transitionprobability

TransitionstrengthS=smoothedB

ΔS=1ΔL=0ΔT=1

• Gamow-TellerResonance

• β-decaydominatedbylow-energyGTtransition

Gamow-TellerResonanceandβ decay

Wakasa,etal.,PRC85,064606(2012)

Strong

weak

Howweretheheavyelementsmade?

10

Question3Howweretheheavyelementsfromirontouraniummade?rapidneutroncaptureprocess

(r-process)

nuclearphysicsinputs:mass,β-decayhalf-lives,…

settingthetime-scaleofr-process

The11greatestunansweredquestionsofphysics

ImportantsubjectinELI-NP• laserdrivennuclearphysicsatELI-NP

ExperimentalInvestigations

11

82

28

50

5028Neutron Number

Prot

on N

umbe

r

1E-025E-021E-015E-011E+005E+001E+015E+011E+02

r-path

•Developmentofradioactiveion-beamfacilities=>importantadvances• 78NiandaroundHosmer,etal.,PRL94,112501,2005;Xu,etal.,PRL113,032505,2014• veryneutronrichKrtoTcisotopesNishimura,etal.,PRL106,052502,2011• ZnGaisotopesMadurga etal.,PRL109,112501,2012• 110neutron-richnucleiacrossN=82shellgapLorusso,etal.,PRL114,192501,2015• 94neutron-richnucleifromZ=55-67Wu,etal.,PRL118,072701,2017• N=126r-processnuclei?Notreachedyet.ImportantgoalatELI-NPlaserdrivennuclearphysics

• provideagoodtestgroundfortheoreticalmodels

Johnson, Koonin, et al., 1992; Koonin et al., 1997) allowscalculation of nuclear properties as thermal averages,employing the Hubbard-Stratonovich transformation torewrite the two-body parts of the residual interaction byintegrals over fluctuating auxiliary fields. The integra-tions are performed by Monte Carlo techniques, makingthe SMMC method available for basically unrestrictedmodel spaces. While the strength of the SMMC methodis the study of nuclear properties at finite temperature, itdoes not allow for detailed nuclear spectroscopy.

The evaluation of nuclear matrix elements for theFermi operator is straightforward. The Gamow-Telleroperator connects Slater determinants within a modelspace spanned by a single harmonic-oscillator shell (0!"space). The shell model is then the method of choice forcalculating the nuclear states involved in weak-interaction processes dominated by allowed transitions,as complete or sufficiently converged truncated calcula-tions are now possible for such 0!" model spaces. Thepractical calculation of the Gamow-Teller distribution isachieved by adopting the Lanczos method (Wilkinson,1965), as proposed by Whitehead (1980; see also Langa-nke and Poves, 2000; Poves and Nowacki, 2001).

The calculation of forbidden transitions, however, in-volves nuclear transitions between different harmonic-oscillator shells and thus requires multi-!" modelspaces. These are currently feasible only for light nucleiwhere ab initio shell-model calculations are possible(Navratil et al., 2000; Caurier et al., 2001). Such multi-!"calculations have been used for the calculation of neu-trino scattering from 12C (Hayes and Towner, 2000;Volpe et al., 2000). However, for heavier nuclei one hasto rely on more strongly truncated nuclear models. Asthe kinematics of stellar weak-interaction processes areoften such that forbidden transitions are dominated bythe collective response of the nucleus, the random-phaseapproximation (RPA; Rowe, 1968) is usually the method

of choice (Fig. 2). Another advantage of this method isthat, in contrast to the shell model, it allows for globalcalculations of these processes for the many nuclei ofteninvolved in nuclear networks. An illustrative example isthe evaluation of nuclear half-lives based on the calcu-lation of the Gamow-Teller strength function within thequasiparticle RPA model (Krumlinde and Moller, 1984;Moller and Randrup, 1990). The RPA method considersthe residual correlations among nucleons via one-particle/one-hole (1p-1h) excitations in large multi-!"model spaces. The neglect of higher-order correlationsrenders the RPA method inferior to the shell model, formatrix elements between individual, noncollectivestates. A prominent example is the Gamow-Teller tran-sition from the 12C ground state to the T!1 triad in theA!12 nuclei (see, for example, Engel et al., 1996).While the shell model is able to reproduce the Gamow-Teller matrix element between these states (Cohen andKurath, 1965; Warburton and Brown, 1992), RPA calcu-lations miss an important part of the nucleon correla-tions and overestimate these matrix elements by about afactor of 2 (Kolbe et al., 1994; Engel et al., 1996). Recentdevelopments have extended the RPA method to in-clude the complete set of 2p-2h excitations in a givenmodel space (Drozdz et al., 1990). Such 2p-2h RPAmodels have, however, not yet been applied to semilep-tonic weak processes in stars. Moreover, the RPA allowsfor the proper treatment of the momentum dependencein the different multipole operators, as it can be impor-tant in certain stellar neutrino-nucleus processes (see be-low), and for the inclusion of the continuum (Buballaet al., 1991). Detailed studies indicate that standard andcontinuum RPA calculations yield nearly the same re-sults for total semileptonic cross sections (Kolbe et al.,2000). This is related to the fact that both RPA versionsobey the same sum rules. The RPA has also been ex-tended to deal with partial occupation of the orbits sothat configuration mixing in the same shell is includedschematically (Rowe, 1968; Kolbe, Langanke, and Vo-gel, 1999).

III. HYDROGEN BURNING AND SOLAR NEUTRINOS

The tale of the solar neutrinos and their ‘‘famous’’problem took an exciting turn from its original goal ofmeasuring the central temperature of the sun to provid-ing convincing evidence for neutrino oscillations, thusopening the door to physics beyond the standard modelof the weak interaction. In 1946, Pontecorvo suggested(Pontecorvo, 1946, 1991; later independently proposedby Alvarez, 1949) that chlorine would be a good detec-tor material for neutrinos. Subsequently, in the 1950s,Davis built a radiochemical neutrino detector which ob-served reactor neutrinos via the 37Cl(#e ,e")37Ar reac-tion (Davis, 1955). After the 3He($ ,%)7Be cross sectionat low energies had been found to be significantly largerthan expected (Holmgren and Johnston, 1958) and,slightly later, the 7Be(p ,%)8B cross section at low ener-gies had been measured (Kavanagh, 1960), it becameclear that the Sun should also operate by what are now

FIG. 2. (Color in online edition) The most commonly usednuclear models for the calculation of weak processes in starsare the random-phase aproximation (RPA) and the shellmodel (SM). In the RPA, the basis states are characterized byparticle-hole excitations around a given configuration (typi-cally a closed-shell nucleus). In the shell model, all the possibletwo-body correlations in a given valence space are considered.Excitations from the core or outside the model space are ne-glected, but this effect can be included perturbatively usingeffective interactions and operators.

823K. Langanke and G. Martınez-Pinedo: Nuclear weak-interaction processes in stars

Rev. Mod. Phys., Vol. 75, No. 3, July 2003

TheoreticalInvestigations

12

• abinitioapproach:forverylightnuclei Barrett,etal.,PPNP69,1312013• shellmodel:uptoA=60 oraroundmagicregions

Langanke etal.,ADNDT79,12001;Suzuki,etal.,PRC85,015802,2012;LiandRen,JPG41,105102,2014;Zhi,etal.,PRC87,025803(2013)

• RandomPhaseApproximation(RPA)ü Spherical

o Non-self-consistent• FRDM+Quasiparticle RPA(QRPA)

Moller,etal.,ADNDT66,1311997o Self-consistent

• Skyrme QRPAEngel,etal.,PRC60,014302,1999

• RelativisticQRPAMarketin,etal.,PRC93,025805,2016;Niu,etal.,PLB723,172,2013.

ü Quadrupole deformation• deformedQRPAwithrealisticforceNiandRen,PRC89,064320,2014• Skyrme deformedQRPA Sarriguren,etal.,PRC81,064314(2013);Yoshida,JPS

CP,6,020017,2015

DensityFunctionalTheory

13

many-body problem

one-body problem

[ ] EE ==ΨΨ= effHH Φ Φ ρ Φ Slaterdeterminant ⇔ 1-bodydensitymatrixρ

•Nucleus:quantummany-bodysystem• Density Functional Theory (DFT)

Skyrme DensityFunctional

14

Heff = T + Veff• EffectiveHamiltonian

• Skyrme EffectiveInteraction

• Around11parameterstofitthenuclearobservables: ti,xi,W0, α(SkM*,SLy5,…)• Successfultodescribealmostthewholenuclearchart:g.s.andexcitedstates

RandomPhaseApproximation(RPA)

15

keepthelinearterm

• RPA:magicnuclei• QuasiparticleRPA(QRPA):superfluidnuclei

ü Solutiononbasis

• RPA:widelyusedmodelforthedescriptionofcollectivevibrationü Smalloscillation:linearlimitoftimedependentDFTtheory

Harmonicoscillation

TheRPAexcitedstate(collectivevibrationstate)isgeneratedby

β-DecayHalf-Life

16

LifetimeIsaHardProblem…

17

• SkyrmeHFB+QRPA• RHB+QRPA

•Half-livesareoverestimated• Duetothenuclearstructurepart– Gamow-Tellertransition

Engel,etal.,PRC60,014302(1999)

78Ni

.

Niksic,etal.,PRC71,014308(2005)

78Ni

Limitsof(Q)RPADescription

18

RPA

• (Q)RPAcannotdescribethespreadingwidth

• SpreadingWidth(DampingWidth)

• CorrelationsbeyondRPA

energyandangularmomentumofcoherentvibrations→morecomplicatedstatesof2p-2h,3p-3h,…character

E

E

B

B

1p1h

1p1h2p2h

Solution:RPA+PVC

19

Low-lyingvibrationphonons |N>

RPA

• SecondRPAdrozdz etal.,PR197,1(1990)Gambacurta etal.,PRC81,054312(2010)

• RPA+PVC(particlevibrationcoupling)

• RPA+PVCmodelbasedonSkyrme DFTColo etal.,PRC50,1496(1994);Niu etal.,PRC85,034314(2012)

• RPA+PVCmodelbasedonrelativisticDFTLitvinova etal.,PRC75,064308(2007)

RPA+PVC:Gamow-TellerResonance

20

ü Developaspreadingwidthü Reproduceresonancelineshape

Y.F.Niu, G.Colo,andE.Vigezzi,PRC90,054328(2014)

• ImproveddescriptionofGTresonancein208Pb

RPA+PVC:β-DecayHalf-Lives

21

ü Reproduceβ-decayhalf-livesü Reducehalf-livessystematically

Y.F.Niu,Z.M.Niu,G.Colo,andE.Vigezzi,PRL114,142501(2015).

• Improveddescriptionofβ-decayhalf-lives

HowPVCreduceshalf-lives?

22

Exp.:Xu,etal.,PRL113,032505,2014

• Half-life

• PhaseVolume

TakenfromNatureNews&Views4652010

Quasiparticle RPA+quasiparticle vibrationcoupling(QRPA)+(QPVC)ü forthestudyofGamow-Tellerresonanceinsuperfluidnucleiü forthestudyofβ-decayhalf-livesofthewholeisotopicchain

ØToincludepairingcorrelationsforsuperfluid nuclei

23

RPA+PVC:onlyformagicnuclei…

QRPA+QPVCmodel

24

TheQRPA+QPVCequationreads

Step1:HFB+QRPAcalculation(charge-exchange)=>Gamow-Tellerresponse inQRPAlevelHFB+QRPAcalculation(non-charge-exchange)=>vibrationalphonons 1-,2+,3-,4+,5-

Step2:QPVCcalculation=>Gamow-TellerresponseinQRPA+QPVClevel

TheQRPAequationwillgivetheenergy,andwavefunction

where,andthematricescontainthespreadingcontributions, e.g.,

Particle-particleinteraction

TheGTstrengthfunction

SpreadingTerms

25

Thematrixelementsof thespreading terminthequasiparticle basis

whererepresentsadoorwaystate,anda,barequasi-particlestates.

Thevertexreducedmatrixelement:

Isoscalar Pairing

26

f =0:without isoscalar pairingf =1:withisoscalar pairing

Effectofisoscalar pairing

• QRPAlevel• Increasethelow-lyingstrength• Decreasethesplittingbetweentwo

high-lyingstates

• QRPA+QPVClevel• Increasethelow-lyingstrength• Similarprofile

Particle-particleinteraction

Niu,Colo,Vigezzi,Bai,Sagawa,PRC94,064328(2016)

GTStrengthDistribution

27

• (p,n) data:normalizedbyunitcrosssectionSasano,etal.,PRC79,024602(2009)

ü QRPA+QPVC• Developawidthof5.3MeV(6.4MeVfromexp.),reproduceexp.profile inGTR• Overestimatethelow-lyingstrength

CumulativeSum

28

• f=0->f=1ü low-lyingstrengthisincreasedforbothQRPAandQRPA+QPVC

• QRPA->QRPA+QPVC:ü betterreproduces theexp.profileü cumulativestrengthisquenchedby10%atE=25MeVü QRPA+QPVCstrength� 0.75=exp.strength ((p,n) data)atE=25MeV

Cumulativesum Cumulativesumwithnormalizeddata

Niu,Colo,Vigezzi,Bai,Sagawa,PRC 94,064328(2016)

TakenfromNatureNews&Views4652010

Quasiparticle RPA+quasiparticle vibrationcoupling(QRPA)+(QPVC)ü forthestudyofGamow-Tellerresonanceinsuperfluidnucleiü forthestudyofβ-decayhalf-livesofthewholeisotopicchain

ØToincludepairingcorrelationsforsuperfluid nuclei

29

RPA+PVC:onlyformagicnuclei…

β-DecayHalf-LivesinNiisotopes

30

• Isoscalar Pairing:notsoeffectiveforNiisotopes(nucleibeforeN=50closedshell)

• QPVC:reducethehalf-lives

31

β-DecayHalf-LivesinSnisotopes

• Isoscalar Pairing:effectiveforSnisotopes(nucleiaboveN=82closedshell)

• Isoscalar Pairing+QPVC:reducethehalf-lives

SummaryandPerspectives

32

SummaryImportantsubjectinELI-NP(laserdrivennuclearphysicsatELI-NP:r-processnucleonsynthesis)• Gamow-Tellertransitionisanimportantspin-isospinmodeofnucleus;β-

decayismainlydeterminedbylow-energyGTtransition• GoingbeyondRPA:RPA+PVC

ü Spreadingwidthü Reducethehalf-lives

• GoingbeyondQRPA:QRPA+QPVCü EffectofQPVCü Effectofisoscalar pairing

Perspectives• QRPA+QPVC:systematicstudyofisotopicchain• QRPA+QPVC:otherweakinteractionprocesses• QRPA+QPVC:GDR,PDR

gammaabovetheneutronthresholdexperimentsatELI-NP;nuclearresonancefluorescenceexperimentsatELI-NP

Thank you!

33

CumulativeSum

34

• f=0->f=1ü low-lyingstrengthisincreasedforbothQRPAandQRPA+QPVC

• QRPA->QRPA+QPVC:ü betterreproduces theexp.profileü cumulativestrengthisquenchedby10%atE=25MeVü QRPA+QPVCstrength� 0.75=exp.strength ((p,n) data)atE=25MeV

Cumulativesum Cumulativesumwithnormalizeddata

Niu,Colo,Vigezzi,Bai,Sagawa,PRC94,064328(2016)

PairinggapinNiandSnisotopes

35

• SurfacepairingEcut =100MeV• Niisotopes:V0=-457MeVfm3 • Snisotopes:V0=-520MeVfm3

ExperimentalInvestigations

36

TheoreticalInvestigations

37

EffectofIVpairingonhalf-lives

38

Explainthebehaviorofhalf-lives

39

EffectofISpairingonhalf-lives

40

Isospin propertiesofPDR

• E=8.4MeV:IS≈80%, isoscalar character,surface peakedtransition• E>10MeV:isovector character,collectivemotiondominates

• Low-lyingisoscalar stateswithsurfacepeakedtransitiondensity↔(α,α’γ)• Higherisovector states↔ (γ,γ’)

41

N.Paar,Y.F.Niu,D.Vretenar,andJ.Meng,Phys.Rev.Lett. 103,032502(2009).

•Model:RelativisticHartree Bogoliubov +Quasi-particlerandomphaseapproximationRHB+QRPA

PhysicalPictureofPDR

42

• E=8.4MeVInterior:protonandneutron inphaseSurface:onlycontributions fromneutrons

• PhysicalPictureNeutronskinoscillatesagainsttheproton-neutron isospinsaturatedcore

• E=15.1MeVWholenucleus:protonandneutronopposite inphase

• PhysicalPictureProtonsandneutronsoscillateagainsteachother

N.Paar,Y.F.Niu,D.Vretenar,andJ.Meng,Phys.Rev.Lett. 103,032502(2009).

•Model:RelativisticHartree Bogoliubov +Quasi-particlerandomphaseapproximationRHB+QRPA

HowPVCreduceshalf-lives?

43