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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli Plan of the talk Theoretical framework : Renormalization of the bare NN potential by means of the V low-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 132 Sn and comparison with conterpart nuclei in the stable 208 Pb region neutron-rich Ca isotopes neutron-rich C isotopes Summary

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli Shell-model calculations 3. Two-body matrix elements 1. Model space 2. Single-particle energies (taken from the experimental spectra of nuclei with one-valence nucleon or treated as free parameters) 4. Construction and diagonalization of the energy matrices Traditional approach: Two-body matrix elements treated as free parameters Two-body matrix elements treated as free parameters Empirical effective interactions containing adjustable 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)] defined within a reduced model space and acting only between valence nucleons

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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli 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 the model-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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli all modern NN potentials fit equally well the deuteron properties and the NN scattering data up the inelastic threshold 2 /N data ~ 1 Choice of the nucleon-nucleon potential Derivation of V eff Note these potentials cannot be used directly in the derivation of V eff due to their strong short-range repulsion, but a due to their strong short-range repulsion, but a Renormalization procedure is needed Renormalization procedure is needed CD-Bonn, Argonne V 18, Chiral potentials,…

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli Renormalization of the NN interaction Traditional approach: Brueckner G-matrix method V low-k approach: construction of a low-momentum NN potential V low-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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli V low-k approach Derived from the original V NN by integrating out the high-momentum components of the original V NN potential down to the cutoff momentum V low-k decouples high- and low-energy degrees of freedom preserves the physics of the original NN interaction the deuteron binding energy scattering phase-shifts up to the cutoff momentum Λ How to contruct V low-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)

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli Features of V low-k real effective potential in the momentum space (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 E Lab 350 MeV - V low-k s extracted from various phase-shift equivalent potentials are very similar to each other Note V low-k is developed for the two-body system èfor A>2 the low-energy observables are not the same of the original NN potential of the original NN potential and depend (to a certain extent) on the value of and depend (to a certain extent) on the value of This may be removed complementing the two-body V low-k with three- and higher-body components

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli + folded diagram method collection of irreducible valence-linked diagrams with V low-k replacing V NN in the interaction vertices FiFi i-folded diagrams (expressed in terms ofderivatives) 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 V eff,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 developed within the framework of the time-dependent perturbative approach by Kuo and co-workers

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli 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)] + + … Construction of V eff NoteNote Diagramatic expression of the single-particle energies TB component of the shell-model Hamiltoniam

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

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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli 132 Sn region CD-Bonn + V low-k : second-order calculation Single-particle energies from the experimental spectra of 133 Sb and 133 Sn 132 Sn 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 n-rich Ca isotopes CD-Bonn + V low-k : third-order calculation Single-neutron energies from a fit to exp energies of 47 Ca and 49 Ca 40 Ca core Valence neutron levels: 0f7/2, 0f5/2, 1p3/2, 1p1/2 n-rich C isotopes N 3 LOW [ chiral potential with a sharp momentum cutoff at 2.1 fm -1 ] : third-order calculation Theoretical single-neutron energies 14 C core Valence neutron levels: 0d5/2, 0d3/2, 1s1/2 Input of our calculations

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli Theory B(E2;0 + 2 + ) = 0.033 e 2 b 2 134 Sn Coulex (Oak Ridge) B(E2;0 + 2 + ) = 0.029(4) e 2 b 2 Theory 0.726 134 S n Expt. from the f 7/2 p 1/2 configuration their location below the 8 + due to the new position of the p 1/2 level measured @ ORNL p1/2 =1.36 MeV (old value: 1.66 MeV) [Nature 465 (2010)]

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli 134 Sn 132 Sn + 2 ( f 7/2 ) 2 multiplet 210 Pb 208 Pb + 2 ( g 9/2 ) 2 multiplet J J

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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli A 134 Sn 135 Sn 136 Sn 137 Sn BE Calc relative to 132 Sn 5.918.3011.8614.18 BE Expt relative to 132 Sn 5.92* BE/A Calc8.278.238.208.15 N/Z 1.70 1.72 1.74 1.68 124 Sn(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

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

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

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

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

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli A * M. Honma et al. EPJ A 25, 499 (2005 ) * no shell gap at N=34 Ca isotopes Excitation energies of the J =2 1 + states

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Effective single particle energies of the f 5/2 and p 1/2 levels (relative to the p 3/2 level ) A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli

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-427 keV * A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli C isotopes from 16 C to 24 C – ground-state energy ( relative to 14 C ) *to reproduce the exp g.s. energy of 15 C relative to 14 C E gs (calc)=-0.79 ; E gs (exp)= -1.22 MeV 22 C is the last bound isotope K. Tanaka et al PRL 104, 062701 (2010) S 2n (evaluation)=420 keV S 2n (calc)=601 keV L. Coraggio et al. PR C 81, 064303 (2010)

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli C isotopes Excitation energies of the J =2 1 + states ESPE (MeV) N no subshell closure at N=14 g.s. in 20 C: 14% of ( d 5/2 ) 6 configuration

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A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010 Napoli 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 of key importance to gain more experimental information Summary

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