1. A. Gargano Eurisol User Group Wokshop – Firenze 2008 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 Realistic shell model and 132 Sn 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

2. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 3. Two-body matrix elements 1. Model space 2. Single-particle energies 4. Construction and diagonalization of the energy matrices Shell-model calculations 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 Modern codes: Oxbash - widely distributed and used A. Brown … Oslo - m-scheme; dimension ~10 6 T. England Antoine - m-scheme; dimension ~10 9 E. Caurier… Nathan - coupled scheme; ~10 6 Redstick - for 2-and 3- body ME P. Navratil, W.E. Ormand

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

4. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli The role of the NNN potential haa been evidenced by investigation on light nuclei 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) Tjon line Inclusion of three–body forces in the shell model approach not yet attempted NCSM Navratil 2007 Nogga 2004

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

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

7. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Modern (phase-shift equivalent) NN potentials Nijmegen I - (P D = 5.66%) - 41 parameters - 2 /N data = 1.03 Nijmegen II - (P D = 5.64%) - 47 parameters - 2 /N data = 1.03 Argonne V 18 - (P D = 5.76%) - 40 parameters - 2 /N data = 1.09 CD Bonn - (P D = 4.85%) - 43 parameters - 2 /N data = 1.02 (1999 NN Database: 5990 pp and np scattering data) π ρ ω σ1σ2 π ρ ω σ1σ2 based upon the OBE model

8. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 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, N 3 LO potential E. Epelbaum, W. Glöckle, and U.-G. Meissner, 2005: N 3 LO potential N 3 LO potential (Entem & Machleidt): Effective chiral 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 P D = 4.51% - χ 2 /N data = 1.10 (np database below 290 MeV); χ 2 /N data = 1.50 (pp database below 290 MeV) Chiral potentials Important advantage of the chiral perturbation theory: it generates NNN forces (starting from 3rd order)

9. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Matrix elements of V NN in the 1 S 0 channel NN potentials are not completely constrained by low-energy NN data

10. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Renormalization of the NN potential Difficulty in the derivation of V eff from any modern NN potential: existence of a strong repulsive core which prevents its direct use in nuclear structure calculations Traditional approach to this problem: Brueckner G-matrix method New approach: construction of a low-momentum NN potential V low-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). = + +... ~~~ GV NN Infinite sum of ladder diagrams

11. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Derived from the original V NN by integrating out the high- momentum components of the original V NN potential V low-k preserves the physics of the original NN interaction up to the cutoff momentum Λ: the deuteron binding energy scattering phase-shifts Iterative method V low-k : Low-momentum potential confined within a momentum-space cutoff 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). V low-k approach

12. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Matrix elements of V low-k with Λ=2.1 fm -1 in the 1 S 0 channel 1.5 1.8 2.0 2.05 2.2 2.5 187 269 332 350 401 518 k V low-k is free from high momentum modes V low-k is mooth potential suitable to be used directly as input for derive the effective interactionin V low-k gives an approximately unique representation of the NN potential

13. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli G matrix vs V low-k G matrixV low -- k Energy dependent NO energy dependent Model space dependent NO model space dependent No direct connection to the original V NN potential It is a real effective potential - in the k-space it reproduces all the two body problem data

14. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Realistic effective interaction: folded-diagram expansion Calculation of the : collection of irreducible and valenced linked diagrams at any order in V 2-body 2nd order diagrams: V V 2p V 1p1h V 2p2h auxliary 1b potential Sum of the folded-diagram series : (by Kuo-Krengiglowa or Lee-Suzuki iterative technique)

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

16. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 50 82...... 132 Sn i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 h11/2 s1/2 d3/2 d5/2 g7/2 space 126...... 133 Sb 133 Sn V NN : CD-Bonn V low-k with Λ= 2.2 fm -1 V eff : 2 nd order calculation and Lee-Suzuki method intermediate states composed 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 V low-k harmonic-oscillator basis

17. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Across the N=82 shell gap Anomalous behavior of the first 2 + state in even Sn isotopes Anomalous behavior of the first 2 + state in even Sn isotopes " in even Te isotopes " in even Te isotopes Anomalous behavior of the B(E2; 0 + 2 + ) value in even Sn isotopes Anomalous behavior of the B(E2; 0 + 2 + ) value in even Sn isotopes " in even Te isotopes " in even Te isotopes Anomalous behavior of the first 5/2 + in Sb isotopes Anomalous behavior of the first 5/2 + in Sb isotopes Anomalously low position of the 1 - state in 134 Sb Anomalously low position of the 1 - state in 134 Sb Z=52 134 Te 135 Te 136 Te 137 Te 138 Te 139 Te Z=51 133 Sb 134 Sb 135 Sb 136 Sb Z=50 132 Sn 133 Sn 134 Sn N=82N=83N=84N=85N=86N=87 N/Z=1.68 N/Z=1.67

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

19. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 134 Te σ(keV)=115σ(keV)=143σ(keV)=128 Expt Cd-BonnNijmIIArgonne V 18

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

21. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli = 42 keV 134 Sb Calc. Expt. g 7/2 f 7/2 d 5/2 f 7/2

22. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Calc. Expt. 134 Sb Small space: intermediate states composed of particle and hole states restricted to two major shells above and below the FS Large space g 7/2 f 7/2

23. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli V 1p1h V 2p2h V 2p V eff V low-k Diagonal matrix elements of interaction for the g7/ f7/2 configuration J J Large space Small space

24. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 136 Sb Calc. Expt. Yrast states with J π from 0 - to 7 - dominated by the g 7/2 (f 7/2 ) 3 G.S. Simpson et al. 2007, ILL Grenoble

25. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 134 Sb 136 Sb The g 7/2 f 7/2 multiplet in compared to that in 136 Sb 134 Sb

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

27. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli 75% g7/2 ( f7/2) 2 +... 45% d5/2 ( f7/2) 2 + 23% g7/2 ( f7/2) 2 +... J =0+ and J =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 135 Sb: wave functions J=0+ J=2+ J=0+

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

29. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli E2 e eff ( ) = 0.70e e eff ( ) = 1.55e M1 effective operator: 2nd order core & no meson-exchange corrections Comparison of the experimental and theoretical B(E2) [e 2 fm 4 ] and B(M1) [μ 2 N ] new measurements yield values which are higher: 50% HRIBF-ORL 20% ISOLDE-CERN

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

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

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

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

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

35. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Summary The V low-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 with quantitative accuracy the spectroscopic properties of exotic nuclei near closed shells. This gives confidence in their predictive power in these regions. At present no real evidence of shell modifications near 132 Sn. It is of key importance to gain more experimental information. Theoretical open problems: single-particle energies from the theory; role of genuine and effective three-body forces for heavy nuclei.

36. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Low-momentum NN interaction in few-nucleon systems - S. Fujii, E. Epelbaum, H. Kamada, R. Okamoto, K. Suzuki, W. Glöckle, Phys. Rev. C 70, 024003 (2004). - A. Nogga, S. K. Bogner, A. Schwenk, Phys. Rev. C 70, 061002(R) (2004). - L. Coraggio, A. Covello, A. Gargano, N. Itaco, T.T.S. Kuo, Phys. Rev. C 73, 014304 (2006). 1.5 1.8 2.0 2.05 2.2 2.5 187 269 332 350 401 518 Cutoff momentum Λ Choice of Λ

37. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli = 2.1 fm -1

38. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Correlation plots fra gli e.m. di NijmII e Cd-Bonn nella base di HO e le corrispondenti matrici G T per la shell sd

39. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli and the multiplet and the multiplet h 9/2 g 9/2 in 210 Bi? 210 Bi? L S h 9/2 g 9/2

40. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli structure of states Mixed-symmetry state (antisimmetric with respect to interchanges between proton and neutron pairs ) unkown

41. A. Gargano Eurisol User Group Wokshop – Firenze 2008 Napoli Second step - sum of the folded - diagram series (by Kuo-Krengiglowa or Lee-Suzuki iterative technique) H 1 eff contains both 1- and 2-body contributions subtraction procedure to remove from H 1 eff the 1-body terms and sp energies from experiment