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Neutron-Rich Nuclei within a realistic shell-model approach Angela Gargano Napoli

Neutron-Rich Nuclei within a realistic shell-model approach Angela Gargano Napoli. L. Coraggio (Napoli) A. Covello (Napoli) N. Itaco (Napoli) T.T.S. Kuo (Stony Brook ).

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Neutron-Rich Nuclei within a realistic shell-model approach Angela Gargano Napoli

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  1. Neutron-Rich Nuclei within a realistic shell-model approach Angela Gargano Napoli L. Coraggio (Napoli) A. Covello (Napoli) N. Itaco (Napoli) T.T.S. Kuo (Stony Brook) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  2. Plan of the talk • Theoretical framework: • Renormalization of the bare NN potential by means of the Vlow-k approach • Derivation of the model space effective interaction by means of the plus folded diagram method • Outline of calculations • Results: • neutron-rich nuclei northeast of 132Sn and comparison with conterpart nuclei in the stable 208Pb region • neutron-rich Ca isotopes • neutron-rich C isotopes • Summary A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  3. Shell-model calculations defined within a reduced model space and acting only between valence nucleons 1. Model space 2. Single-particle energies (takenfrom the experimental spectra of nuclei with one-valence nucleon or treated as free parameters) 3. Two-body matrix elements 4. Construction and diagonalization of the energy matrices Traditional approach: • Two-body matrix elements treated as free parameters • Empirical effective interactions containing adjustable parameters [e.g., s-d shell nuclei,B. A.Brown and B. H. Wildenthal, Ann. Rev. Nucl. Part. Sci.38, 28(1988)] A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  4. Challenge for nuclear physics: understand the properties of nuclei starting from the forces among nucleons Ab-initio calculations: nuclear properties, such as binding and excitation energies, are calculated directly from first principles of quantum mechanics, using an appropriate computational scheme Green’s function Montecarlo method, no-core shell model, coupled cluster method, UCOM (three-nucleon interactions have been also taken into account) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  5. Realistic shell-model calculations: We start from where U is a one-body auxiliary potential introduced to define a convenient single-paticle basis and define the effective shell-model Hamiltonian through themodel-space Schrödinger equation where the E and the corresponding  are required to be a subset of the eigenvalues and eigenvectors of the original Hamiltonian The P operator projects into the chosen model space A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  6. Derivation of Veff ● Choice of the nucleon-nucleon potential CD-Bonn, Argonne V18 , Chiral potentials,… all modern NN potentials fit equally well the deuteron properties and the NN scattering data up the inelastic threshold 2/Ndata~ 1 Notethese potentials cannot be used directly in the derivation of Veff due to their strong short-range repulsion, but a • Renormalization procedure is needed A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  7. Renormalization of the NN interaction Traditional approach: Brueckner G-matrix method Vlow-k approach:construction of a low-momentum NN potential Vlow-k confined within a momentum-space cutoff 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). L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135 A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  8. Vlow-k approach • Derived from the original VNN by integrating out the high-momentum components of the original VNN potential down to the cutoff momentum  Vlow-kdecouples high- and low-energy degrees of freedom preserves the physics of the original NNinteraction • the deuteron binding energy • scattering phase-shifts up to the cutoff momentum Λ How to contruct Vlow-k? Effective interaction technique based on the the Lee-Suzuki similariry transformation (Prog. Theor Phys 64 (1980) 2091)  low-momentum space Q complementary space X similarity transformation Decoupling equation solved by the iterative procedure proposed by Andreozzi(Phys Rev. C 54 (1996) 684) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  9. Features of Vlow-k • real effective potential in the momentumspace (indipendent from the starting energy or from the model space, as instead the case of the G matrix defined in the nuclear medium) • eliminates sources of non-perturbative behavior  can be used directly in nuclear structure calculations • gives an approximately unique representation of the NN potential for   2 fm-1 ELab 350 MeV- Vlow-k’s extracted from various phase-shift equivalent potentials are very similar to each other NoteVlow-k is developed for the two-body system • for A>2 the low-energy observables are not the same of the original NN potential and depend (to a certain extent) on the value of  This may be removed complementing the two-body Vlow-k with three- and higher-body components A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  10. + folded diagram method developed within the framework of the time-dependent perturbative approach by Kuo and co-workers T.T.S. Kuo and E. Osnes, Lecture Notes in Physics, vol 364 (1990) L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135 collection of irreducible valence-linked diagrams with Vlow-k replacing VNN in the interaction vertices Fi i-folded diagrams (expressed in terms of derivatives) Veff ,constructed for two valence particles, is defined -in the nuclear medium -in a subspace of the Hilbert space  accounts perturbatively for • configurations excluded from the chosen model space • excitations of the core nucleons A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  11. Diagramatic expression of the + … + Construction of Veff • Calculation of : • inclusion of diagrams up to a finite order in the interaction • truncation of the intermediate-state summation • Sum of the folded series by the Lee-Suzuki method [Prog. Theor. Phys. 64, 2091 (1980)] Note TB component of the shell-model Hamiltoniam  single-particle energies A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  12. Results A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  13. neutrons protons n-rich nuclei nuclei beyond the N=82 shell magic nature of 132Sn? n-rich Ca isotopes N=34 shell closure? n-rich C isotopes location of the neutron drip line? A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  14. 132Sn region Input of our calculations • 132Sn core Valence neutron levels: 1f7/2, 2p3/2, 0h9/2, 2p1/2, 1f5/2, 0i13/2 Valence proton levels: 0g7/2, 1d5/2, 1d3/2, 0h11/2, 2s1/2 CD-Bonn + Vlow-k : second-order calculation Single-particle energies from the experimental spectra of 133Sb and 133Sn n-rich Ca isotopes • 40Ca core Valence neutron levels: 0f7/2, 0f5/2, 1p3/2, 1p1/2 CD-Bonn + Vlow-k : third-order calculation Single-neutron energies from a fit to exp energies of 47Ca and 49Ca n-rich C isotopes • 14C core Valence neutron levels: 0d5/2, 0d3/2, 1s1/2 N3LOW [chiral potential with a sharp momentum cutoff at 2.1 fm-1] : third-order calculation Theoretical single-neutron energies A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  15. 134Sn Coulex (Oak Ridge) B(E2;0+2+) = 0.029(4) e2b2 Theory B(E2;0+2+) = 0.033 e2b2 134Sn from the f7/2p1/2 configuration their location below the 8+ due to the new position of the p1/2 level measured @ ORNL • p1/2=1.36 MeV (old value: 1.66 MeV) [Nature 465 (2010)] 0.726 Theory Expt. A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  16. 134Sn 132Sn + 2 (f7/2)2 multiplet J 210Pb208Pb + 2 (g9/2)2 multiplet J A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  17. 136Sn 132Sn + 4 212Pb 208Pb + 4 J A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  18. N/Z 1.68 1.70 1.72 1.74 124Sn(stable) with N/Z=1.48 BE/A=8.46 *M. Dworschak et al. Phys. Rev. Lett. 100, (2008) 072501 Old value (Fogelberg et al., 1999): 6.365 MeV neutron shell gap at N= 82 restored A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  19. 134Sn (Theoretical predictions) B(E2;42 ) = 1.64 W.u. B(E2;64) = 0.81 W.u. B(E2;222) = 0.34 W.u. B(E2;224) = 0.22 W.u. Q(2) = -1.3 efm2 µ(2) = -0.56 nm A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  20. 136Sb Theory Expt 136Sb is at present the most exotic open-shell nucleus beyond 132Sn for which information exists on excited states A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  21. 50Ca 52Ca Expt Theory Expt Theory J.J. Valiente Dobón et al. PRL 102, 242502 (2009) L. Coraggio et al. Phys. Rev. C 80, (2009) 044311 A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  22. Ca isotopes - Ground-state energy per valence neutron A A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  23. Ca isotopes Excitation energies of the J =21+ states * no shell gap at N=34 A * M. Honma et al. EPJ A 25, 499 (2005) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  24. Effective single particle energies of the f5/2 and p1/2 levels (relative to the p3/2 level ) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  25. C isotopes from 16C to 24C – ground-state energy (relative to 14C ) • • 22C is the last bound isotope K. Tanaka et al PRL 104, 062701 (2010) S2n(evaluation)=420 keV S2n(calc)=601 keV -427 keV* • • • • *to reproduce the exp g.s. energy of 15C relative to 14C Egs(calc)=-0.79 ; Egs(exp)= -1.22 MeV L. Coraggio et al. PR C 81, 064303 (2010) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  26. C isotopes Excitation energies of the J =21+ states ESPE (MeV) N no subshell closure at N=14 g.s. in 20C: 14% of (d5/2)6 configuration A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

  27. Summary • Reliability of “realistic shell-model calculations” for light  heavy nuclei This outcome gives confidence in its predictive power, and may stimulate and be helpful to future experiments. • Three-body forces seem to contribute mainly to the absolute energy of the single-particle. Role of three-body forces needs futher investigations • It is ofkey importanceto gain more experimental information A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

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