1 / 25

tens of MeV

3D angular momentum and isospin restored calculations w ith the Skyrme EDF. Wojciech Satuła. in collaboration with J. Dobaczewski , W. Nazarewicz & M. Rafalski. Intro :  define fundaments my model is „standing on”

beate
Télécharger la présentation

tens of MeV

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3D angularmomentum and isospinrestoredcalculations with the Skyrme EDF WojciechSatuła incollaborationwith J. Dobaczewski, W. Nazarewicz & M. Rafalski Intro:  definefundaments my model is „standing on” spmean-field (ornuclear DFT)  beyondmean-field (projectionaftervariation) Symmetry (isospin) violation and restoration: • unphysicalsymmetryviolation  isospinprojection • Coulomb rediagonalization(explicitsymmetryviolation) isospinimpuritiesinground-states of e-e nuclei Results structuraleffects SD bandsin56Ni ISB corrections to superallowed beta decay Summary ab initio + NNN + .... tens of MeV

  2. Skyrme-force-inspiredlocal energy densityfunctional (withoutpairing) Y | v(1,2) | Y averageSkyrmeinteraction (in fact a functional!) over the Slater determinant localenergydensityfunctional SV is the onlySkyrmeinteraction Beiner et al. NPA238, 29 (1975) Skyrme (nuclear) interactionconserves:  rotational (spherical) symmetry  isospinsymmetry: Vnn= Vpp= Vnp(in reality approximate) LS LS LS Mean-fieldsolutions (Slaterdeterminants) break (spontaneously) thesesymmetries Symmetry-conserving configurtion Total energy (a.u.) Symmetry-breaking configurations Deformation(q)

  3. Restoration of brokensymmetry Eulerangles in spaceor/and isospace gauge angle Beyond mean-fieldmulti-referencedensityfunctionaltheory rotated Slater determinants are equivalent solutions where

  4. Applytheisospinprojector: in order to creategoodisospin „basis”: AR aC= 1 - |aT=Tz|2 BR aC= 1 - |bT=|Tz||2 Isospinsymmetryrestoration • Therearetwosources of theisospinsymmetrybreaking: • unphysical, causedsolely by the HF approximation • physical, caused mostly by Coulomb interaction • (also, but to much lesserextent, by the strong force isospin non-invariance) Engelbrecht & Lemmer, PRL24, (1970) 607 Findself-consistent HF solution (including Coulomb)  deformed Slater determinant |HF>: See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15 Diagonalizetotal Hamiltonian in „goodisospinbasis” |a,T,Tz>  takesphysicalisospinmixing n=1

  5. eMF = 0 eMF = e Ca isotopes: 0.4 BR SLy4 AR 0.2 0 aC [%] 1 1.0 0.1 0.8 0.01 0.6 60 40 44 48 52 56 0.4 0.2 0 56 40 48 44 52 60 Mass number A Numericalresults: • Isospinimpuritiesingroundstates of e-e nuclei W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009) 012502 Herethe HF issolved without Coulomb |HF;eMF=0>. Herethe HF issolved with Coulomb |HF;eMF=e>. In bothcasesrediagonalization isperformed for thetotal Hamiltonian including Coulomb

  6. DaC ~30% SLy4 BR AR aC [%] 6 N=Z nuclei 5 4 3 1.0 2 0.8 1 E-EHF [MeV] 0.6 0 0.4 0.2 0 20 28 36 44 52 60 68 76 84 92 100 A (II) Isospinmixing & energy inthegroundstates of e-e N=Znuclei: HF tries to reduce the isospin mixing by: AR in order to minimize the total energy BR Projectionincreasesthe ground state energy (the Coulomb and symmetry energiesarerepulsive) BR Rediagonalization (GCM) AR lowerstheground state energy but onlyslightly belowthe HF This is not a single Slater determinat There are no constraints on mixing coefficients

  7. SIII SLy4 SkP 100Sn 35 7 30 aC [%] SkP 6 SLy5 SkP SLy MSk1 SkM* SLy4 5 SIII SkXc SkO’ 25 4 SkO DE ~ 2hw ~ 82/A1/3 MeV 20 y = 24.193 – 0.54926x R= 0.91273 doorway state energy [MeV] 31.5 32.0 32.5 33.0 33.5 34.0 34.5 20 40 60 80 100 Excitationenergyof the T=1 doorwaystate in N=Z nuclei Bohr, Damgard & Mottelson hydrodynamical estimate DE ~ 169/A1/3 MeV mean values E(T=1)-EHF [MeV] Sliv & Khartionov PL16 (1965) 176 Dl=0, Dnr=1  DN=2 based on perturbation theory A

  8. n n p p n n p p n p n p n p T=0 Spontaneousisospinmixing in N=Z nuclei in other but isoscalarconfigs  yetanotherstrongmotivation for isospinprojection four-fold degeneracy of the sp levels Mean-field n p or or n p aligned configuration anti-aligned configuration n p T=1 Isospin projection T=0

  9. Nilsson f5/2 p3/2 [321]1/2 [303]7/2 f7/2 protons neutrons 2 g9/2 pp-h f5/2 p3/2 f7/2 protons neutrons two isospin asymmetric degenerate solutions 1 Isospin symmetry violation in superdeformed bands in 56Ni 4p-4h space-spin symmetric D. Rudolph et al. PRL82, 3763 (1999)

  10. T=1 nph dET centroid pph band 2 dET T=0 Isospin-projection 20 16 8 6 56Ni 4 12 2 Exp. band 1 Exp. band 2 Th. band 1 Th. band 2 8 5 10 15 4 Isospin projection Mean-field aC [%] band 1 Hartree-Fock Excitation energy [MeV] 5 10 15 Angular momentum Angular momentum W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310

  11. -1 ij Isospin-projectionis non-singular: SVD eigenvalues (diagonal matrix) W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310 SVD singularity (ifany) atb=p isinherited by r r =Syi*Oijjj ~ in the worstcase h> 3 1 + |N-Z| - h + |N-Z|+2k kis a multiplicity of zero singularvalues

  12. 40 30 20 10 0 1 3 5 7 42Sc – isospinprojection from [K,-K] configurations with K=1/2,…,7/2 isospin & angularmomentum isospin aC [%] 0.586(2)% 2K Isospin and angular-momentumprojected DFT isill-defined except for the hamiltonian-drivenfunctionals thereis no alternative but Skyrme SV

  13. 1 0.1 0.01 0.001 -1 0.0001 ij 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Why we have to to useSkyrme-V? onlyIP |OVERLAP| IP+AMP p bT [rad] r =Syi*Oijjj T HF sp state space & isospinrotated sp state inverse of the overlap matrix

  14. Primarymotivation of theproject isospincorrections for superallowed beta decay Tz=-/+1 J=0+,T=1 (N-Z=-/+2) t1/2 t+/- Qb J=0+,T=1 (N-Z=0) BR Tz=0 Experiment: Fermi beta decay: 8 8 d5/2 |<t+/->|2=2(1-dC) p1/2 p3/2 2 2 f statisticalratefunctionf (Z,Qb) s1/2 t partialhalf-lifef (t1/2,BR) n n p p GVvector (Fermi) couplingconstant 14N 14O <t+/-> Fermi (vector) matrix element 10 casesmeasuredwithaccuracyft ~0.1% Hartree-Fock 3 casesmeasuredwithaccuracyft ~0.3%

  15. NS-independent g nucleus-independent e ~1.5% 0.3% - 2.0% n n Marciano & Sirlin, PRL96, 032002, (2006) ~2.4% NS-dependent Towner, NPA540, 478 (1992) PLB333, 13 (1994) g e The 13 preciselyknowntransitions, afterincluding theoreticalcorrections, areused to test the CVC hypothesis Towner & Hardy Phys. Rev. C77, 025501 (2008) courtesy of J.Hardy

  16. Withthe CVC beingverified and knowingGm(muondecay) one candetermine mass eigenstates CKM Cabibbo-Kobayashi-Maskawa weakeigenstates |Vud| = 0.97425 + 0.00023  test unitarity of the CKM matrix |Vud|2+|Vus|2+|Vub|2=0.9996(7) 0.9491(4) test of threegenerationquark Standard Model of electroweakinteractions 0.0504(6) <0.0001

  17. „Hidden” model dependence Towner & Hardy Phys. Rev. C77, 025501 (2008) dC=dC2+dC1 Liang & Giai & Meng Phys. Rev. C79, 064316 (2009) shell model mean field spherical RPA Coulomb exchangetreatedinthe Slaterapproxiamtion radialmismatch of thewavefunctions configuration mixing Miller & Schwenk Phys. Rev. C78 (2008) 035501;C80 (2009) 064319

  18. n n n n p p p p n n n p p p n p n p n n p p T=0 Isobaricsymmetryviolation in o-o N=Znuclei Tz=-/+1 J=0+,T=1 (N-Z=-/+2) t1/2 t+/- Qb J=0+,T=1 (N-Z=0) BR Tz=0 MEAN FIELD CORE CORE anti-aligned configurations aligned configurations n p or or ISOSPIN PROJECTION T=1 T=0 Mean-fieldcandifferentiatebetween ground state isbeyondmean-field! and onlythroughtime-oddpolarizations!

  19. Hartree-Fock antialigned state inN=Z (o-o) nucleus ground state inN-Z=+/-2 (e-e) nucleus CPU Project on goodisospin (T=1) and angularmomentum (I=0) (and perform Coulomb rediagonalization) Project on goodisospin (T=1) and angularmomentum (I=0) (and perform Coulomb rediagonalization) ~ few h ~ fewyears t+/- |I=0,T~1,Tz=0> <T~1,Tz=+/-1,I=0| ~ ~ H&TdC=0.330% 14N L&G&MdC=0.181% 14O our:  dC=0.303% (Skyrme-V; N=12)

  20. H&T: 1.4 Ft=3071.4(8)+0.85(85) Tz= 1 Tz=0 1.2 Vud=0.97418(26) 1.0 our (no A=38): 0.8 Ft=3070.4(9) dC[%] 0.6 Vud=0.97447(23) 0.4 0.2 |Vud|2+|Vus|2+|Vub|2= =1.00031(61) 0 A 10 14 18 22 30 34 42 26 38 2.0 Tz=0 Tz=1 1.5 dC [%] 1.0 0.5 W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL106 (2011) 132502 0 A 26 42 50 66 74 34 58

  21. 0.976 0.975 H&T’08 our model 0.974 |Vud| 0.973 Liang et al. n-decay 0.972 superallowedb-decay 0.971 0.970 T=1/2 mirror b-transitions p+-decay

  22. Confidencelevel test based on the CVC hypothesis 2.5 (EXP) Towner & Hardy, PRC82, 065501 (2010) 2.0 Ft dC = 1+dNS - (SV) dC ‚ ft(1+dR) (EXP) dC 1.5 MinimizeRMS deviation between the caluclated and experimentaldC with respect to Ft 1.0 dC [%] 0.5 c2/nd=5.2 for Ft = 3070.0s 75% contribution to the c2comes from A=62 0 Z of daughter 0 5 10 15 20 25 30 35 40

  23. 0 10 20 30 40 50 60 70 80 SUMMARY OF THE CALCULATIONS Ncutoff=12 Ncutoff=10 2.5 Tz Tz+1 A=58 2.0 A=38 1.5 dC [%] 1.0 A=18 0.5 Tz= -1 Tz= 0 0 A

  24. Summary Elementaryexcitationsinbinary systems maydiffer fromsimpleparticle-hole (quasi-particle) exciatations especiallywheninteractionamongparticlesposseses additionalsymmetry (like the isospinsymmetry in nuclei) Projectiontechniquesseem to be necessary to account for thoseexcitations - how to constructnon-singularEDFs? [Isospinprojection, unlike the angular-momentum and particle-number projections, ispractically non-singular !!!] Superallowed 0+0+beta decay:  encomapsextremelyrichphysics: CVC, Vud, unitarity of the CKM matrix, scalarcurrents… connectingnuclear and particlephysics  … thereisstillsomething to do indc business … Pairing & other (shapevibrations) correlationscan be „realtivelyeasily” incorporatedinto the scheme by combiningprojection(s) with GCM

  25. „NEW OPPORTUNITIES” IN STUDIES OF THE SYMMETRY ENERGY: n p a’sym E’sym = a’symT(T+1) In infinitenuclear matter we have: SLy4 SLy4L 1 6 SLy4: SkML* 2 asym=32.0MeV SV 4 SV: a’sym[MeV] asym=32.8MeV SkM*: 2 m asym=30.0MeV m* asym= eF + aint 0 T=1 T=0 SLy4: 14.4MeV SV: 1.4MeV SkM*: 14.4MeV 10 20 30 40 50 A (N=Z)

More Related