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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 Napoli Eurisol User Group Wokshop – Firenze 2008 Shell-model calculations 1. Model space 2. Single-particle energies 3. Two-body matrix elements 4. Construction and diagonalization of the energy matrices Modern codes: •Oxbash - widely distributed and used •Oslo - m-scheme; dimension ~106 •Antoine - m-scheme; dimension ~109 •Nathan - coupled scheme; ~106 •Redstick - for 2-and 3- body ME A. Brown … T. England E. Caurier… 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 Napoli Eurisol User Group Wokshop – Firenze 2008 Realistic shell-model calculations Veff from the bare potential H T VNN VNNN A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 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 Napoli Eurisol User Group Wokshop – Firenze 2008 Realistic shell-model calculations Veff from the bare potential H T VNN VNNN Nuclear many-body Schroedinger equation Hi (1,, A) (T VNN ) Ei i (1,, A) Model-space Schroedinger equation Model space defined by the operator d P i i 1 the complement being Q 1 P 1 d 1 i i PHeff Pi Ec P( H 0Veff ) Pi Ei Pi A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 Realistic shell-model calculations Basic ingredients : Nucleon-nucleon potential Many-body theory to derive the effective interaction A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 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 π ρ ω σ1σ2 (1999 NN Database: 5990 pp and np scattering data) A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 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): • 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 • 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 Napoli Eurisol User Group Wokshop – Firenze 2008 Matrix elements of VNN in the 1S0 channel NN potentials are not completely constrained by low-energy NN data A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 Renormalization of the NN potential Difficulty in the derivation of Veff 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 Infinite sum of ladder diagrams G( ) V NN VNN Q 1 Q G( ) Q TQ G ~~~ = VNN VNN + + ... VNN 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). A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 Vlow-k approach Vlow-k: Low-momentum potential confined within a momentum-space cutoff kΛ ● Derived from the original VNN by integrating out the highmomentum components of the original VNN potential ● Vlow-k preserves the physics of the original NN interaction 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 Napoli Eurisol User Group Wokshop – Firenze 2008 Vlow-k is free from high momentum modes •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 2 ELab M 0.012 ELab 2 2 ( fm1 ) 1.5 1.8 2.0 2.05 2.2 ELab ( MeV ) 187 269 332 350 401 518 A. Gargano Napoli 2.5 2.1 fm-1 Eurisol User Group Wokshop – Firenze 2008 G matrix vs Vlow-k G matrix Vlow -- k ● Energy dependent ● NO energy dependent ● Model space dependent ● NO model space dependent ● No direct connection to the original VNN potential ● It is a real effective potential in the k-space it reproduces all the two body problem data A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 Realistic effective interaction: “folded-diagram expansion” H T V (T U ) (V U ) H0' H1 d Model space: P i i with Q 1 P 1 d 1 1 auxliary 1b potential i i ● Calculation of the Qˆ box : collection of irreducible and valenced linked diagrams at any order in V 2-body 2nd order diagrams: V V2p V1p1h ● V2p2h Sum of the folded-diagram series : (by Kuo-Krengiglowa or Lee-Suzuki iterative technique) A. Gargano Napoli V eff Qˆ Fi (Qˆ ) i 1 Eurisol User Group Wokshop – Firenze 2008 Results 132Sn neighbors beyond N=82 N=82 isotonic chain A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 126 133Sn i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 harmonic-oscillator basis 7.88 MeV . . . 82 h11/2 s1/2 d3/2 d5/2 g7/2 space . . . 50 space 133Sb 132Sn VNN : CD-Bonn Vlow-k with Λ= 2.2 fm-1 Veff : ● 2nd 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 Vlow-k A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 Z=52 134Te 135Te 136Te 137Te Z=51 133Sb 134Sb 135Sb 136Sb Z=50 132Sn 133Sn 134Sn N=82 N=83 N=84 N=85 138Te 139Te N=86 N=87 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 A. Gargano Napoli 134Sb Eurisol User Group Wokshop – Firenze 2008 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 Napoli Eurisol User Group Wokshop – Firenze 2008 134Te Expt Cd-Bonn σ(keV)=115 A. Gargano Napoli NijmII σ(keV)=143 Argonne V18 σ(keV)=128 Eurisol User Group Wokshop – Firenze 2008 134Sn 86% (f7/2)2 81% (f7/2)2 Expt Calc = 70 keV A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 134Sb • Calc. ▲ Expt. d5/2f7/2 g7/2f7/2 = 42 keV A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 134Sb • Calc. ▲ Expt. g7/2f7/2 Large space g7/2f7/2 Small space: intermediate states composed of particle and hole states restricted to two major shells above and below the FS A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 Diagonal matrix elements of interaction for the g7/f7/2 configuration Large space J V1p1h V2p2h Small space V2p Veff Vlow-k A. Gargano Napoli J Eurisol User Group Wokshop – Firenze 2008 Yrast states with Jπ from 0- to 7dominated by the g7/2(f7/2 )3 136Sb • Calc. ▲ Expt. G.S. Simpson et al. 2007 , ILL Grenoble A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 The “g7/2f7/2 multiplet” in 136Sb compared to that in 134Sb 134Sb 136Sb A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 135Sb anomalously low position 0.282 Expt Calc = 72 keV A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 135Sb: wave functions J=0+ 7 2 75% g7/2 (f7/2)2 +... 5 2 45% d5/2 (f7/2)2 + 23% g7/2 (f7/2)2 + ... J=0+ J=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 A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 136Te Expt A. Gargano Napoli Calc Eurisol User Group Wokshop – Firenze 2008 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 Napoli Eurisol User Group Wokshop – Firenze 2008 N=82 isotones 134Te Zval=2 test of 136Xe 138Ba 140Cs 142Nd 144Sm 154Hf Veff 146Gd 148Dy 150Er 152Yb Zval from 4 to 22 Role of three-body forces both genuine and effective A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 J=2+ A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 J=4+ A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 J=6+ A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008 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 Napoli Eurisol User Group Wokshop – Firenze 2008 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 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 importance to gain more experimental information. 132Sn. It is of key Theoretical open problems: ▪ single-particle energies from the theory; ▪ role of genuine and effective three-body forces for heavy nuclei. A. Gargano Napoli Eurisol User Group Wokshop – Firenze 2008