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n-p pairing in N=Z nuclei

the Wigner energy and the generalized. blocking phenomenon. cranking in isospace - response of t=0 pairing. against rotations in isospace. reality or fiction ?. n-p pairing in N=Z nuclei. Motivation & fingerprints ( basic concepts ): . W. Satuła University of Warsaw.

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n-p pairing in N=Z nuclei

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  1. theWigner energy and thegeneralized blockingphenomenon • crankinginisospace - responseof t=0 pairing againstrotationsinisospace reality orfiction? n-ppairinginN=Znuclei Motivation & fingerprints (basicconcepts): W. Satuła University of Warsaw • symmetricnuclearmattercalculations • bindingenergies - mean-fieldcrisisaroundN~Zline • elementaryisobaricexciatationsinN~Znuclei • – a need for isosopinsymmetryrestoration • high-spin signatures of pn-pairing

  2. Neutron - Structure of nucleonic pairs • N=Z  nucleons start to occupy „identical” spatialorbitals • Nuclear interaction favoures L=0 coupling • Pair-structureisgoverned by thePauliprinciple: • Isovector(or1S0) pairs T=1, S=0 - Isoscalar (deuteron-likeor3S1) pairsT=0, S=1 Proton + Tz+1 0 -1 Sz +1 0 -1

  3. 3S1-3D1(coupled)pairing gap insymmetricNuclearMatterfromParis VNN free-spacesp spectrum BHF sp spectrum (in-medium corrections) tensor-force enhancement From M. Baldo et al. Phys. Rev. C52, 975 (1995)

  4. Gapsfromlocaleffectivepairinginteraction DDDI usedinSkyrme-HFBcalculations by Terasaki et al. NPA621 (1997) 706. Isoscalarpairing Tensor force enhancement Cut-off!!! (otherwisedivergent!) ro DDDI: E. Garrido et al. PRC60, 064312 (1999) PRC63, 037304 (2001)

  5. 3S1-3D1(coupled)pairing gap insymmetricNuclearMatterincludingrelativisticcorrections Saturation density Includesrelativisticin-mediumcorrections (splevelsfromDirac-Brueckner-HF) O. Elgaroy, L.Engvik, M.Hjorth-Jensen, E.Osnes, Phys.Rev.C. 57 (1998) R1069

  6. Empirical NN interactioninN~Z • T=1 channel: • J=0coupling dominates • T=0 channel: • J=1 and J=2j aresimilar • T=0 is, on theaverage, strongerthan T=1 by a factor of ~1.3 N.Anantaraman & J.P. Schiffer PL37B (1971) 229 Dufour & Zucker, Phys. Rev. C54, 1641 (1996)

  7. The model: deformedmean-field plus pairing: 0 0 Pairs p-n and p-ñ Pairs: ~ Pairs ñ-n and p-p « usual » ; T=1 ~ Pairs: p-ñ + n-p; T=1 ~ Pairs: p-ñ – n-p ; T=0 Hamiltonian BCS: N.Anantaraman and J.P. Schiffer PL37B (1971) 229

  8. Comparisonwithdelta-forcetowards a localtheory M.Moinester, J.P. Schiffer, W.P. Alford, PR179 (1969) 984

  9. A.L.Goodman Nucl. Phys. A186 (1972) 475 BCStransformation BCS transformationtakesthefollowing form : real complex wherethevariationalparametersare: i 2 Densitymatrix(occupation) and thepairing tensor Generalization:BCSHFB UiU &ViV matrices of dimension 4N

  10. BCS Solution Energy(Routhian) Variational equationinN=Z system (without Coulomb) Occupationprobabilities; quasiparticleenergies: Pairgaps: Gap T=0 aã Gap T=0 aa n-ñ, p-p T=1 n-p + p-ñ ~ ~

  11. X X T=0/T=1 (no)mixing X= / 48Ca • Incompletemixing? • T=1, Tz=+/-1 andTz=0 • T=1, Tz=+/-1andT=0 W.S. &R.Wyss PLB 393 (1997) 1

  12. Energy gain as a functionof T=0/T=1 pairing’smixing „x” Energy gain: DMass =E(T=0+1)- E(T=1) Thomas-Fermi X=1.1 X=1.2 X=1.3 X=1.4 X= / generalizedblockingeffect n-excess blocks pn-pairs scattering Wignerterm from Myers & Swiatecki neutrons protons Satuła & Wyss PLB393 (1997) 1

  13. Wignereffectfromself-consistentSkyrme-HF N=Z Exp. HFBCS T=1 Sph. HFBCS T=1 Def. (SIII) • Defficiency of conventionalself-consistentmodels: HF or HFBincluding standard T=1, |Tz|=1 ~ p-p & n-npairs: • (N-Z)2 ~ T2 termisOK! • no (orveryweak) |N-Z| ~ term |N-Z|=2,4 (black) A.S. Jensen, P.G.Hansen, B.Jonson, Nucl.Phys. A431(1984) 393 o-o e-e

  14. TheWignereffect total 1 2 w / w DE= asymT(T+x) 25 A=48 20 B (MeV) 15 48Cr 10 5 1.0 w 0 0.8 N-Z -4 0 4 0? 1?? 1.25??? exp. inN~Z 4 ???? Wigner SU(4) 0.6 0.4 24Mg 0.2 X= 0.0 0 1 2 3 4 5 6 7 Jmax

  15. Isobaricexcitations inN~Z nuclei 2.0 1.5 1.0 0.5 47/A [MeV] W(A) [MeV] • Thelowest: • T=0, T=1 & T=2 in e-e nuclei • T=0 & T=1 statesin o-o nuclei GT=0 1.4 GT=1 • The model needs to be extended to • includeisospinprojection isospincranking 0.6 strong T=0 pairing limit! A 30 40 J.Janecke,Nucl. Phys. A73 (1965) 73 A. Macchiavelli et al.Phys. Rev.C61(2000) 041303(R) P.Vogel,Nucl. Phys. A662 (2000) 148

  16. Theextremes.p.model: 4-fold degenerated equidistant s.p. spectrum  Energy:  Eigen-states (routhians) are 2-fold (Kramers) degene- rated „stright lines”: Crossings form simple arithmetic serie: „inertia” defined through mean level spacing !!!

  17. T=2 states in e-e nuclei 20 14 28 1 2 DE=deT2 20 15 10 5 0 20 30 40 50 DET=2[MeV] hWS+HT=1 +HT=0-wtx T=2 hWS+HT=1 -wtx iso-cranking A Iso-cranking gives excitation energy which goes like: + Epair vacuum mean level spaceing at the Fermi energy

  18. (iso)Coriolis antipairingeffect iso-MoI Tx 1.5 1.0 48Cr 0.5 0 0 1 2 3 D/e = 0.001 3 6 e=1 iso-moment of inertia 2 1 D/e = 0.5;1.0;1.5 0 3 0.7 DT=0 2 D [MeV] Tz 0.6 DT=1 0 iso-moment of inertia 1 1 0.5 2 3 0 0.4 4 0 1 2 3 hw [MeV] 0.3 hw

  19. T=1 statesine-e N=Znuclei • T=1 states: 2qp+ isocranking

  20. odd-Tsequence Isocranking N=Z odd-oddnuclei T de 5 de de 2de 6de 4de 4 hw even-Tsequence 3 de 2 de 1 de 0 hw de 3de 5de iso-signature selection rule Eeven-T = 1/2deTx2 Eodd-T = 1/2deTx2 - 1/2de

  21. T=0 vs T=1 statesino-o N=Znuclei T=0 T=1 1.0 2qp cranking vacuum 0.5 DET=1 - DET=0 [MeV] 0.0 -0.5 exp th 20 30 40 50 60 70 A

  22. Neutron-proton pairing collectivity (a fit plus three easy steps) (III) ET=1 - ET=0 (even-even) ET=1 - ET=0 (odd-odd) (II) • Wigner energy linked to the n-p pairing collectivity • T=2 states in even-even nuclei obtained from isocranking • T=1 states in even-even nuclei obtained as 2qp excitations • T=1 states in odd-odd nuclei obtained from isocranking • T=0 states in odd-odd nuclei obtained as 2qp excitations Fit of GT=0 /GT=1 ET=2 - ET=0 (even-even) (I) W. Satuła & R. Wyss Phys. Rev. Lett., 86, 4488(2001); Phys. Rev. Lett., 87, 052504(2001)

  23. Schematic isospin-isospin interaction: extreme sp model even-even vacuum de H=hsp- wT+ kTT l 2 de de de de+k 3de 3(de+k) hw + kT E= (de+k)T2 1 1 1 E= (de+k)T2 2 2 2 seee.g. Bohr & Mottelson „NuclearStructure” vol. I Neergard PLB572 (2003) 159 1 mean - • field • (Hartree) HMF=hsp- (w - k T )T iso-cranking with isospin-dependent frequency!!! Hartree Hartree- -Fock

  24. Pairinginfastrotatingnuclei Muller et al., Nucl. Phys. A383 (1982) 233 Resistance of nucleonic paires against fastrotation:

  25. 48Cr ; HFB calculationsincluding T=0 & T=1 pairing -1 d3/2 g9/2 4 4 [nf7/2pf7/2] 16+ J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998) • Skyrmeinteractioninp-h • DDDI inp-p channel • fullyself-consistenttheory • no sphericalsymmetry • two-classes of solutions: - T=0 dominatedat I=0 - T=1 dominatedat I=0 isoscalar pairing Non-collective (oblate) rotation no T=0 at low spins Collective (prolate) rotation T=1 collapses (termination) exp

  26. (1) 73Kr – manifestationof (dynamical) T=0 pairing? 3qp 2.5 30 2.0 3qp 1.5 25 1.0 20 0.5 15 0.0 -0.5 10 5 g 40 0.5 0.5 0.5 1.0 1.0 1.0 1.5 1.5 1.5 fp R.Wyss, P.J. Davis, WS, R. Wadsworth Conventional TRS calculations involving only T=1 pairing: negative parity negative parity positive parity Ix (-,-) (+,+) 73Kr 73Kr (-,-) Ew [MeV] 5qp 1qp 1qp 73Kr: Kelsall et al., Phys. Rev. C65 044331 (2005) hw[MeV] hw[MeV] |1qp> = a+n(fp)|0> |3qp> = a+ng a+pg a+p(fp)|0> <1qp|E2|3qp> ~ 0 (one-body operator)

  27. (2) 73Kr – manifestationof (dynamical) T=0 pairing? 1.0 n(fp) p(fp) p(fp) 0.5 pg9/2 pg9/2 ng9/2 0 n(fp) p(fp) p(fp) 30 ng9/2 pg9/2 pg9/2 25 20 15 10 5 0 1.4 0.4 0.8 1.0 1.2 1.6 0.2 0.6 What makes the 1qp and 3qp configurations alike? Scattering of a T=0 np pair TRS involving T=0 and T=1 pairing in 73Kr Dn Dp D [MeV] DT=0 73Kr n(fp)(-) vacuum 1qp configuration Ix n(fp) theory ng9/2 exp n(fp) ng9/2 ng9/2(+) pg9/2 p(fp)(-) 3qp configuration hw [MeV]

  28. 2.0 30 1.5 25 1.0 20 0.5 15 0.0 10 -0.5 5 0.5 1.0 1.5 0.5 0.5 1.0 1.0 1.5 1.5 (3) 73Kr – manifestationof (dynamical) T=0 pairing? Conventional TRS calculations involving only T=1 pairing in neighbouring nuclei: negative parity positive parity all bands Ix (-,+) 75Rb 3qp 75Rb (+,+) Ew [MeV] 1qp 3qp 1qp hw[MeV] hw[MeV] Excellent agreement was obtained in: Tz=1 : 74Kr,76Rb, D. Rudolph et al. Phys. Rev. C56, 98 (1997) Tz=1/2: 75Rb, C. Gross et al. Phys. Rev. C56, R591 (1997) Tz=1/2: 79Y, S.D. Paul et al. Phys. Rev. C58, R3037 (1998)

  29. SUMMARY Part of T=0 correlationsinN~Znucleiisdefinitely beyond standard formulation of mean-field (Wigner energy) Adding T=0 pairinghelps but cannotsolvethe problem of theWigner energy (symmetry energy) inN~Znuclei whichseems to be beyondmean-field Thereis no convincingarguments for coherency of the T=0 phase Theoreticaltreatment of T=1 statesin e-e nuclei and T=0 states o-o nucleirequiresangularmomentum and isospinprojections

  30. (**) (*) sn-1 aw 2as sn-1 x a 4asT(T+x); x=aw/2as 0.153 1.33 0.239 6 1/2 8 0.125 1.27 0.213 11 2/3 14 0.107 1.24 0.196 38 1 47 0.106 1.26 0.196 31 0.95 39 Independent least-square fits of: aw|N-Z|/Aa the Wigner energy strength: as(N-Z)2/Aa the symmetry energy strength: Głowacz, Satuła, Wyss, J. Phys. A19, 33 (2004) very consistent with: Janecke, Nucl. Phys. (1965) 97 Fit includes N~Z nuclei with: Z>10; 1<Tz<3 excluding odd-odd Tz=1 nuclei - - - (*) See: Satuła et al. Phys. Lett. B407 (1997) 103 (**) Based on double-difference formula: J.-Y Zhang et al. Phys. Lett. B227 (1989) 1

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