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Structure of heavy neutron-rich nuclei Angela Gargano Napoli

Structure of heavy neutron-rich nuclei Angela Gargano Napoli. Realistic shell-model calculations: where do we stand?. Realistic effective interaction: Renormalization of the bare NN potential - the V low-k approach; Many body theory to construct V eff – folding expansion.

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Structure of heavy neutron-rich nuclei Angela Gargano Napoli

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  1. Structure of heavy neutron-rich nuclei Angela Gargano Napoli Realistic shell-model calculations: where do we stand? Realistic effective interaction: • Renormalization of the bare NN potential - the Vlow-k approach; • Many body theory to construct Veff – folding expansion Realistic shell model and 132Sn neighbors: Results for nuclei beyond the N=82 shell closure Realistic shell model and nuclei with several valence nucleons: Results for the N=82 isotonic chain Summary A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  2. Shell-model calculations 3. Two-body matrix elements 1. Model space 2. Single-particle energies 4. Construction and diagonalization of the energy matrices Modern codes: • Oxbash - widely distributed and used A. Brown … • Oslo - m-scheme; dimension ~106T. England • Antoine - m-scheme; dimension ~109E. Caurier… • Nathan - coupled scheme; ~106 • Redstick - for 2-and 3- body ME P. Navratil, W.E. Ormand Shell-model TB effective interaction: • TBME based on “simple potentials” • TBME treated as parameters Cohen-Kurath 1965 - p shell nuclei A. Brown et al. 2006 - sd shell nuclei • Realistic TBME derived from the bare potential among nucleons Kuo-Brown 1966 • Semi-realistic TBME realistic with some adjustements A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  3. Realistic shell-model calculations Veff from the bare potential A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  4. The role of the NNN potential haa been evidenced by investigation on light nuclei NCSM Navratil 2007 Tjon line Nogga 2004 Inclusion of three–body forces in the “shell model” approach not yet attempted • effective interaction derived from the NN potential without any adjustement (to assess the quality and reliability of realistic effective interactions and the possible need for improvement) • effective interaction with some modified ME (for instance monopole changes; Caurier et al. 2005) A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  5. Realistic shell-model calculations Veff from the bare potential Nuclear many-body Schroedinger equation Model-space Schroedinger equation Model space defined by the operator A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  6. Realistic shell-model calculations Basic ingredients : Nucleon-nucleon potential Many-body theory to derive the effective interaction A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  7. πρωσ1σ2 Modern (phase-shift equivalent) NN potentials Nijmegen I - (PD = 5.66%) - 41 parameters - 2/Ndata = 1.03 Nijmegen II - (PD = 5.64%) - 47 parameters - 2/Ndata =1.03 Argonne V18 - (PD = 5.76%) - 40 parameters - 2/Ndata =1.09 CD Bonn - (PD = 4.85%) - 43 parameters - 2/Ndata =1.02 based upon the OBE model (1999 NN Database: 5990 pp and np scattering data) A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  8. Chiral potentials NN potential derived from chiral effective field theory • S. Weinberg (1990)“Nuclear forces from chiral lagrangians” D. R. Entem and R. Machleidt (2001- 2003):Idaho potential, N3LO potential E. Epelbaum, W. Glöckle, and U.-G. Meissner, 2005: N3LO potential N3LO potential (Entem & Machleidt): • Effectivechiral πN Lagrangian • One and two-pion exchange contributions. TPE contributions up to fourth order of chiral perturbation theory • Short-range force parametrized in terms of 24 contact terms • Total number of parameters: 29 • PD = 4.51% -χ2/Ndata= 1.10 (np database below 290 MeV); χ2/Ndata = 1.50 (pp database below 290 MeV) • Important advantage of the chiral perturbation theory: it generates NNN forces (starting from 3rd order) A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  9. Matrix elements of VNN in the 1S0 channel NN potentials are not completely constrained by low-energy NN data A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  10. New approach:construction of a low-momentum NN potential Vlow-k S. Bogner,T.T.S. Kuo,L. Coraggio,A. Covello,N. Itaco, Phys. Rev C 65, 051301(R) (2002). S. Bogner, T.T.S. Kuo, A. Schwenk, Phys. Rep. 386, 1 (2003). Renormalization of the NN potential Difficultyin the derivation of Veff from any modern NN potential: existence of a strong repulsive core which prevents its direct use in nuclear structure calculations Traditionalapproach to this problem: Brueckner G-matrix method Infinite sum of ladder diagrams VNN G VNN ~~~ + = + ... VNN A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  11. Vlow-k approach Vlow-k:Low-momentum potentialconfined within a momentum-space cutoff ● Derived from the original VNN by integrating out the high- momentum components of the original VNN potential ● Vlow-k preserves the physics of the original NNinteraction up to the cutoff momentum Λ: the deuteron binding energy scattering phase-shifts ●Iterative method S. K. Bogner, T.T.S. Kuo, L. Coraggio, Nucl. Phys. A684, 432c (2001). S.K. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello, N. Itaco, Phys. Rev. C 65, 051301(R) (2002). A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  12. Vlow-k is free from high momentum modes 1.5 1.8 2.0 2.05 2.2 2.5 187 269 332 350 401 518 • Vlow-k is mooth potential suitable to be used directly as input for derive the effective interactionin • Vlow-k gives an approximately unique representation of the NN potential Matrix elements of Vlow-k with Λ=2.1 fm-1 in the 1S0 channel k A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  13. G matrix vs Vlow-k A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  14. Realistic effective interaction: “folded-diagram expansion” auxliary 1b potential ● Calculation of the : collection of irreducible and valenced linked diagrams at any order in V 2-body 2nd order diagrams: VV2p V1p1h V2p2h ● Sum of the folded-diagram series : (by Kuo-Krengiglowa or Lee-Suzuki iterative technique) A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  15. Results 132Sn neighbors beyond N=82 N=82 isotonic chain A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  16. 126 harmonic-oscillator basis i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 . . . 133Sn 82 h11/2 s1/2 d3/2 d5/2 g7/2  space 133Sb . . . 50 132Sn  space VNN : CD-Bonn Vlow-k with Λ= 2.2 fm-1 Veff : ● 2nd order calculation and Lee-Suzuki method ● intermediate statescomposed of all possible hole states and particle states restricted to 5 shells above the Fermi surface (which guarantees the stability of results when increasing the number of intermediate states) Coulomb force for protons added to Vlow-k A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  17. N/Z=1.67 N/Z=1.68 Across the N=82 shell gap • Anomalous behavior of the first 2+ state in even Sn isotopes " in even Te isotopes • Anomalous behavior of the B(E2; 0+2+) value in even Sn isotopes " in even Te isotopes • Anomalous behavior of the first 5/2+ in Sb isotopes • Anomalously low position of the 1- state in 134Sb A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  18. 2+ in Te isotopes 2+ in Sn isotopes odd Sb isotopes Downshift of the d5/2 proton level relative to the g7/2 one? Onset of a modification in the shell structure? 7/2+ 5/2+ N A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  19. 134Te Expt Cd-Bonn NijmII ArgonneV18 σ(keV)=115 σ(keV)=143 σ(keV)=128 A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  20. 134Sn 86% (f7/2)2 81% (f7/2)2 Expt Calc  = 70 keV A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  21. 134Sb • Calc. ▲Expt. d5/2f7/2 g7/2f7/2  = 42 keV A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  22. 134Sb • Calc. ▲Expt. g7/2f7/2 Large space g7/2f7/2 Small space: intermediate states composed of particle and hole states restricted to two major shells above and below the FS A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  23. Diagonal matrix elements of interaction for the g7/f7/2 configuration Large space J Small space J A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  24. Yrast states with Jπ from 0- to 7- dominated by the g7/2(f7/2)3 136Sb • Calc. ▲Expt. G.S. Simpson et al. 2007 , ILL Grenoble A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  25. The “g7/2f7/2 multiplet” in compared to that in 136Sb 134Sb 134Sb 136Sb A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  26. 135Sb 0.282 anomalously low position Calc Expt  = 72 keV A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  27. 135Sb: wave functions J=0+ 75% g7/2(f7/2)2 +... 45% d5/2(f7/2)2 + 23% g7/2(f7/2)2 +... J=2+ J=0+ • J=0+andJ=2+ ME in the (f7/2)2 configuration differ only by 300 keV • J= 1- ME: (g7/2f7/2)  -600 keV (d5/2f7/2)  -500 keV A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  28. 136Te Expt Calc A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  29. Comparison of the experimental and theoretical B(E2) [e2fm4] and B(M1) [μ2N] E2 eeff() = 0.70e eeff() = 1.55e M1 effective operator: 2nd order core & no meson-exchange corrections new measurements yield values which are higher: ● 50% HRIBF-ORL ● 20% ISOLDE-CERN A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  30. N=82 isotones 134Te 136Xe 138Ba 140Cs 142Nd 144Sm 146Gd 148Dy 150Er 152Yb 154Hf Zval from 4 to 22 Zval=2testof Veff Role of three-body forces both genuine and effective A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  31. J=2+ A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  32. J=4+ A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  33. J=6+ A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  34. Ground-state binding energy per valence proton Zval The Expt and Theor behaviors of the g.s. binding energy per valence proton diverge  contribution of many-body effective forces A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

  35. Summary The Vlow-k approach to the renormalization of the bare NN potential is a valuable tool for nuclear structure calculations. This potential may be used directly in shell-model calculations without the need of first calculating the Brueckner G-matrix. Effective interactions derived from modern NN potentials are able to describe withquantitative accuracythe spectroscopic properties of exotic nuclei near closed shells. This gives confidence in their predictive power in these regions. At presentno real evidenceof shell modifications near 132Sn.It is ofkey importanceto gain more experimental information. Theoretical open problems: ▪ single-particle energies from the theory; ▪ role of genuine and effective three-body forces for heavy nuclei. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli

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