Shell-model studies on exotic nuvlei around 132Sn

<strong>Shell</strong>-<strong>model</strong> <strong>studies</strong>on exotic nuclei around 132 SnLuigi CoraggioIstituto Nazionale di Fisica Nucleare - Sezione di NapoliINPC 2007 - Tokyo, June 3-8 2007 – p.1

The 132 Sn region• Neutron-rich nuclei: an interesting laboratory to test the roleof the residual NN interactionINPC 2007 - Tokyo, June 3-8 2007 – p.2

The 132 Sn region• Neutron-rich nuclei: an interesting laboratory to test the roleof the residual NN interaction• Approaching the neutron drip-line: is there some newphysics coming out?Radioactive ion beams facilities: more data in the pipelineINPC 2007 - Tokyo, June 3-8 2007 – p.2

The 132 Sn region• Neutron-rich nuclei: an interesting laboratory to test the roleof the residual NN interaction• Approaching the neutron drip-line: is there some newphysics coming out?Radioactive ion beams facilities: more data in the pipeline• Astrophysical interest: decay <strong>studies</strong> of the N = 82r-processINPC 2007 - Tokyo, June 3-8 2007 – p.2

The 132 Sn regionINPC 2007 - Tokyo, June 3-8 2007 – p.3

The 132 Sn regionThe doubly-magic 132 Sn is a robust coreINPC 2007 - Tokyo, June 3-8 2007 – p.4

The 132 Sn regionThe doubly-magic 132 Sn is a robust coreNuclear shell <strong>model</strong> is the best tool to investigate nuclei with⇓valence particle/hole out/inside 132 SnINPC 2007 - Tokyo, June 3-8 2007 – p.4

The 132 Sn regionThe doubly-magic 132 Sn is a robust coreNuclear shell <strong>model</strong> is the best tool to investigate nuclei with⇓valence particle/hole out/inside 132 SnIf shell <strong>model</strong> works, this region is particurarly suited forcalculation with realistic NN potentialsINPC 2007 - Tokyo, June 3-8 2007 – p.4

The 132 Sn regionRealistic shell-<strong>model</strong> calculations have been widely performedsince mid 1990sF. Andreozzi, L. Coraggio, A. Covello, A. Gargano, A. Porrino, and T. T. S. Kuo,Phys. Rev. C 56 (1997)A. Holt, T. Engeland, E. Osnes, M. Hjorth-Jensen, and J. Suhonen, Nucl. Phys.A 618 (1997)INPC 2007 - Tokyo, June 3-8 2007 – p.5

The 132 Sn regionRealistic shell-<strong>model</strong> calculations have been widely performedsince mid 1990sF. Andreozzi, L. Coraggio, A. Covello, A. Gargano, A. Porrino, and T. T. S. Kuo,Phys. Rev. C 56 (1997)A. Holt, T. Engeland, E. Osnes, M. Hjorth-Jensen, and J. Suhonen, Nucl. Phys.A 618 (1997)At present, the derivation of realistic shell-<strong>model</strong> interactions hasreached a good quality level, and calculations are able todescribe a large set of experimental dataINPC 2007 - Tokyo, June 3-8 2007 – p.5

The shell-<strong>model</strong> effectiveinteractionSchrödinger equation for A nucleons in the whole Hilbert space:HΦ i (1,...,A) = E i Φ i (1,...,A)INPC 2007 - Tokyo, June 3-8 2007 – p.6

The shell-<strong>model</strong> effectiveinteractionSchrödinger equation for A nucleons in the whole Hilbert space:HΦ i (1,...,A) = E i Φ i (1,...,A)Nuclear shell-<strong>model</strong> scheme: 2 nucleons interacting in the <strong>model</strong>space PPH eff PΨ α = ẼαPΨ α{Ẽα}∈ {E j }INPC 2007 - Tokyo, June 3-8 2007 – p.6

The shell-<strong>model</strong> effectiveinteractionA very useful way to derive V eff is the time-dependentperturbative approach as developed by Kuo and is co-workers inthe 1970s(see T. T. S. Kuo and E. Osnes, Lecture Notes in Physics vol. 364(1990))In this approach the effective hamiltonian H eff is expressed asH eff = ˆQ − ˆQ ′ ∫ˆQ + ˆQ ′ ∫∫ˆQˆQ − ˆQ ′ ∫∫ˆQ∫ˆQˆQ · · · ,INPC 2007 - Tokyo, June 3-8 2007 – p.7

The ˆQ-boxThe ˆQ-box diagrams are divided into a one-body component,called Ŝ-box: INPC 2007 - Tokyo, June 3-8 2007 – p.8

... and a two-body component:The ˆQ-boxM. Hjorth-Jensen, E. Osnes, and T. T. S. Kuo, Phys. Rep. 261 (1995)INPC 2007 - Tokyo, June 3-8 2007 – p.9

The shell-<strong>model</strong> effectiveinteractionKuo and Krenciglowa have proved that∫ˆQˆQ = − d ˆQ(ω)dωˆQ(ω)INPC 2007 - Tokyo, June 3-8 2007 – p.10

The shell-<strong>model</strong> effectiveinteractionKuo and Krenciglowa have proved that∫ˆQˆQ = − d ˆQ(ω)dωˆQ(ω)The above relation is the at basis of two iterative techniques tosum up the folded-diagram perturbative series:INPC 2007 - Tokyo, June 3-8 2007 – p.10

The shell-<strong>model</strong> effectiveinteractionKuo and Krenciglowa have proved that∫ˆQˆQ = − d ˆQ(ω)dωˆQ(ω)The above relation is the at basis of two iterative techniques tosum up the folded-diagram perturbative series:• The Krenciglowa-Kuo partial summationINPC 2007 - Tokyo, June 3-8 2007 – p.10

The shell-<strong>model</strong> effectiveinteractionKuo and Krenciglowa have proved that∫ˆQˆQ = − d ˆQ(ω)dωˆQ(ω)The above relation is the at basis of two iterative techniques tosum up the folded-diagram perturbative series:• The Krenciglowa-Kuo partial summation• The Lee-Suzuki partial summationINPC 2007 - Tokyo, June 3-8 2007 – p.10

Realistic NN potentialsFundamental ingredients for a realistic shell-<strong>model</strong> calculation:INPC 2007 - Tokyo, June 3-8 2007 – p.11

Realistic NN potentialsFundamental ingredients for a realistic shell-<strong>model</strong> calculation:• a realistic nucleon-nucleon potential V ij• CD-Bonn• Argonne V18• chiral potentialsINPC 2007 - Tokyo, June 3-8 2007 – p.11

Realistic NN potentialsFundamental ingredients for a realistic shell-<strong>model</strong> calculation:• a realistic nucleon-nucleon potential V ij• CD-Bonn• Argonne V18• chiral potentials• possibly, a three-body force V ijk• Urbana IX• Illinois 1-5• Chiral many-body potentialsINPC 2007 - Tokyo, June 3-8 2007 – p.11

Realistic NN potentialsFundamental ingredients for a realistic shell-<strong>model</strong> calculation:• a realistic nucleon-nucleon potential V ij• CD-Bonn• Argonne V18• chiral potentials• possibly, a three-body force V ijk• Urbana IX• Illinois 1-5• Chiral many-body potentialsAll the two-body potentials show a “hard core” repulsion in theshort-range component: we need to renormalize itINPC 2007 - Tokyo, June 3-8 2007 – p.11

The G-matrixThe standard way to renormalize the short-range repulsion is toresort to the theory of the Brueckner reaction matrix GINPC 2007 - Tokyo, June 3-8 2007 – p.12

The G-matrixThe standard way to renormalize the short-range repulsion is toresort to the theory of the Brueckner reaction matrix GINPC 2007 - Tokyo, June 3-8 2007 – p.12

The G-matrixThe standard way to renormalize the short-range repulsion is toresort to the theory of the Brueckner reaction matrix GIt can be written by way of an integral equation:G(ab,cd) = V NN (ab,cd) + 1 2or in operatorial form:∑αβV NN (ab,αβ)G(αβ,cd)ɛ c + ɛ d − ɛ α − ɛ β,G(ω) = V NN + V NNQ 2pω − H 0G(ω)INPC 2007 - Tokyo, June 3-8 2007 – p.12

The V low−kInspiration to renormalize V NN :INPC 2007 - Tokyo, June 3-8 2007 – p.13

The V low−kInspiration to renormalize V NN :• Effective field theory (EFT)• Renormalization group (RG)INPC 2007 - Tokyo, June 3-8 2007 – p.13

The V low−kInspiration to renormalize V NN :• Effective field theory (EFT)• Renormalization group (RG)from EFT: we restrict the configurations of V NN (k,k ′ ) to thosewith k,k ′ < k cutoff = ΛS. Bogner, T. T. S. Kuo, L. Coraggio, A. Covello, and N. Itaco, Phys. Rev. C 65,051301(R) (2002)INPC 2007 - Tokyo, June 3-8 2007 – p.13

The deuteron hamiltonian:In the full space:The V low−k〈k|(T + V NN )|k ′ 〉〈k ′ |Ψ n 〉 = E n 〈k|Ψ n 〉0 < k,k ′ < ∞In a reduced <strong>model</strong> space P = ∑ k

The deuteron hamiltonian:In the full space:The V low−k〈k|(T + V NN )|k ′ 〉〈k ′ |Ψ n 〉 = E n 〈k|Ψ n 〉0 < k,k ′ < ∞In a reduced <strong>model</strong> space P = ∑ k

The deuteron hamiltonian:In the full space:The V low−k〈k|(T + V NN )|k ′ 〉〈k ′ |Ψ n 〉 = E n 〈k|Ψ n 〉0 < k,k ′ < ∞In a reduced <strong>model</strong> space P = ∑ k

The V low−kNumerical test: deuteron binding energy with the CD-BonnpotentialΛ (in fm −1 ) PV eff P (in MeV) V NN (in MeV)1.6 -2.225 -2.2251.8 -2.2252.0 -2.2252.2 -2.225INPC 2007 - Tokyo, June 3-8 2007 – p.15

The V low−kNumerical test: phase shifts in the 1 S 0 channel (in degrees)E lab (MeV) CD-Bonn V low−k Expt.1 62.1 62.1 62.110 60.0 60.0 60.025 50.9 50.9 50.950 40.5 40.5 40.5100 26.4 26.4 26.8150 16.3 16.3 16.9200 8.3 8.3 8.9250 1.6 1.6 2.0300 -4.3 -4.3 -4.5INPC 2007 - Tokyo, June 3-8 2007 – p.16

Cut-off momentum ΛGeneral criterion• small enough so to give a reasonably smooth potentialsuitable to be used directly in a perturbative scheme• large enough so that V low−k reproduces the samephase-shifts of the original V NN up to the anelasticthreshold (E lab ≃ 350 MeV)E lab ≤ 2 2 Λ 2 /M → Λ ≃ 2.1fm −1 INPC 2007 - Tokyo, June 3-8 2007 – p.17

<strong>Shell</strong>-<strong>model</strong> calculationsINPC 2007 - Tokyo, June 3-8 2007 – p.18

<strong>Shell</strong>-<strong>model</strong> calculationsModel space outside 132 Sn coreproton0g7/2 0.000 (-9.663)1d5/2 0.9621d3/2 2.4402s1/2 2.8000h11/2 2.793neutron0h9/2 1.5611f7/2 0.000 (-2.461)1f5/2 2.0012p3/2 0.8642p1/2 1.656 (1.5)0i13/2 2.700 - unbound?INPC 2007 - Tokyo, June 3-8 2007 – p.19

<strong>Shell</strong>-<strong>model</strong> calculations• Experimental single-particle energiesINPC 2007 - Tokyo, June 3-8 2007 – p.20

<strong>Shell</strong>-<strong>model</strong> calculations• Experimental single-particle energies• Input realistic potential: V low−k derived from the CD-Bonnpotential and a cutoff Λ = 2.2 fm −1 INPC 2007 - Tokyo, June 3-8 2007 – p.20

<strong>Shell</strong>-<strong>model</strong> calculations• Experimental single-particle energies• Input realistic potential: V low−k derived from the CD-Bonnpotential and a cutoff Λ = 2.2 fm −1• Effective interaction derived at 2nd order of the theoryINPC 2007 - Tokyo, June 3-8 2007 – p.20

<strong>Shell</strong>-<strong>model</strong> calculations• Experimental single-particle energies• Input realistic potential: V low−k derived from the CD-Bonnpotential and a cutoff Λ = 2.2 fm −1• Effective interaction derived at 2nd order of the theory• Number of intermediate states large enough to obtain stabletwo-nucleon matrix elements:up to 5 h.o. shells above the Fermi surfaceINPC 2007 - Tokyo, June 3-8 2007 – p.20

<strong>Shell</strong>-<strong>model</strong> calculationsFundamental test for V eff : spectroscopy of the two-valencesystemINPC 2007 - Tokyo, June 3-8 2007 – p.21

<strong>Shell</strong>-<strong>model</strong> calculationsFundamental test for V eff : spectroscopy of the two-valencesystemINPC 2007 - Tokyo, June 3-8 2007 – p.21

<strong>Shell</strong>-<strong>model</strong> calculationsFundamental test for V eff : spectroscopy of the two-valencesystemINPC 2007 - Tokyo, June 3-8 2007 – p.21

<strong>Shell</strong>-<strong>model</strong> calculationsFundamental test for V eff : spectroscopy of the two-valencesystem1.51.0π d 5/2ν f 7/2E(MeV)0.50.0π g 7/2ν f 7/20 1 2 3 4 5 6 7J134 Sb proton-neutron multipletsINPC 2007 - Tokyo, June 3-8 2007 – p.21

<strong>Shell</strong>-<strong>model</strong> calculationsTest for the quality of the predicted wavefunctions: calculation ofthe electromagnetic transition rates (e effp= 1.55, e effn = 0.7)INPC 2007 - Tokyo, June 3-8 2007 – p.22

<strong>Shell</strong>-<strong>model</strong> calculationsTest for the quality of the predicted wavefunctions: calculation ofthe electromagnetic transition rates (e effp= 1.55, e effn = 0.7)134 Te Calc. Expt.B(E2;0 + 1 → 2+ 1B(E2;4 + 1 → 2+ 1) (in W.u.) 20 24(3)) (in W.u.) 4.3 4.3(0.3)B(E2;6 + 1 → 4+ 1 ) (in W.u.) 1.9 2.05(0.03) INPC 2007 - Tokyo, June 3-8 2007 – p.22

<strong>Shell</strong>-<strong>model</strong> calculationsTest for the quality of the predicted wavefunctions: calculation ofthe electromagnetic transition rates (e effp= 1.55, e effn = 0.7)134 Te Calc. Expt.B(E2;0 + 1 → 2+ 1B(E2;4 + 1 → 2+ 1B(E2;6 + 1 → 4+ 1) (in W.u.) 20 24(3)) (in W.u.) 4.3 4.3(0.3)) (in W.u.) 1.9 2.05(0.03)134 SnB(E2;0 + 1 → 2+ 1 ) (in e2 b 2 ) 0.033 0.029(4)INPC 2007 - Tokyo, June 3-8 2007 – p.22

<strong>Shell</strong>-<strong>model</strong> calculationsTest for the quality of the predicted wavefunctions: calculation ofthe electromagnetic transition rates (e effp= 1.55, e effn = 0.7)134 Te Calc. Expt.B(E2;0 + 1 → 2+ 1B(E2;4 + 1 → 2+ 1B(E2;6 + 1 → 4+ 1) (in W.u.) 20 24(3)) (in W.u.) 4.3 4.3(0.3)) (in W.u.) 1.9 2.05(0.03)134 SnB(E2;0 + 1 → 2+ 1 ) (in e2 b 2 ) 0.033 0.029(4)134 SbB(E2;2 − 1 → 0− 1 ) (in e2 fm 4 ) 123 429(238)B(E2;3 − 1 → 1− 1 ) (in e2 fm 4 ) 115 118(26)B(M1;3 − 1 → 2− 1 ) (in nm2 ) 1.4 2.0(4)INPC 2007 - Tokyo, June 3-8 2007 – p.22

<strong>Shell</strong>-<strong>model</strong> calculationsCalculations with effective interactions derived starting from aCD-Bonn G-matrixINPC 2007 - Tokyo, June 3-8 2007 – p.23

<strong>Shell</strong>-<strong>model</strong> calculationsCalculations with effective interactions derived starting from aCD-Bonn G-matrixB. A. Brown, N. J. Stone, I. S. Towner, and M. Hjorth-Jensen, Phys. Rev. C 71, 044317(2005)INPC 2007 - Tokyo, June 3-8 2007 – p.23

<strong>Shell</strong>-<strong>model</strong> calculationsCalculations with effective interactions derived starting from aCD-Bonn G-matrixM. P. Kartamyshev, T. Engeland, M. Hjorth-Jensen, and E. Osnes,ArXiv:nucl-th/0610017v1 (2006)INPC 2007 - Tokyo, June 3-8 2007 – p.23

<strong>Shell</strong>-<strong>model</strong> calculationsCalculations with effective interactions derived starting from aCD-Bonn G-matrixJ.Shergur et al., Phys. Rev. C 71, 064321 (2005)INPC 2007 - Tokyo, June 3-8 2007 – p.23

<strong>Shell</strong>-<strong>model</strong> calculationsG-matrixINPC 2007 - Tokyo, June 3-8 2007 – p.24

<strong>Shell</strong>-<strong>model</strong> calculationsV low−kINPC 2007 - Tokyo, June 3-8 2007 – p.24

<strong>Shell</strong>-<strong>model</strong> calculationsMain difference the two calculations: with the V low−k a largernumber of intermediate states can be includedINPC 2007 - Tokyo, June 3-8 2007 – p.25

<strong>Shell</strong>-<strong>model</strong> calculationsMain difference the two calculations: with the V low−k a largernumber of intermediate states can be includedEmploying the same reduced Q-space of intermediate states inthe calculation of the ˆQ-box diagrams, the quality of the results issimilarINPC 2007 - Tokyo, June 3-8 2007 – p.25

<strong>Shell</strong>-<strong>model</strong> calculations134 Sb proton-neutron multipletsINPC 2007 - Tokyo, June 3-8 2007 – p.26

<strong>Shell</strong>-<strong>model</strong> calculations134 Sb proton-neutron multiplets1.51.0π d 5/2ν f 7/2INPC 2007 - Tokyo, June 3-8 2007 – p.26E(MeV)0.50.0π g 7/2ν f 7/20 1 2 3 4 5 6 7J

<strong>Shell</strong>-<strong>model</strong> calculationsINPC 2007 - Tokyo, June 3-8 2007 – p.27

<strong>Shell</strong>-<strong>model</strong> calculationsCalc.Expt.B(E2;0 + 1 → 2+ 1 ) (in e2 b 2 ) 0.18 0.103(15)INPC 2007 - Tokyo, June 3-8 2007 – p.27

<strong>Shell</strong>-<strong>model</strong> calculationsCalc.Expt.B(E2;0 + 1 → 2+ 1 ) (in e2 b 2 ) 0.18 0.15(1) new!INPC 2007 - Tokyo, June 3-8 2007 – p.27

<strong>Shell</strong>-<strong>model</strong> calculations136 Te interesting feature: low-lying mixed symmetry states| 136 Te; 2 + 1 〉 = +0.82|D ν × S π 〉 + 0.45|S ν × D π 〉| 136 Te; 2 + 2 〉 = +0.38|D ν × S π 〉 − 0.76|S ν × D π 〉N. Shimizu, T. Otsuka, T. Misuzaki, and M. Honma, Phys. Rev. C 70, 054313(2004)INPC 2007 - Tokyo, June 3-8 2007 – p.28

<strong>Shell</strong>-<strong>model</strong> calculations136 Te interesting feature: low-lying mixed symmetry states| 136 Te; 2 + 1 〉 = +0.82|D ν × S π 〉 + 0.45|S ν × D π 〉| 136 Te; 2 + 2 〉 = +0.38|D ν × S π 〉 − 0.76|S ν × D π 〉N. Shimizu, T. Otsuka, T. Misuzaki, and M. Honma, Phys. Rev. C 70, 054313(2004)Using V low−k :| 136 Te; 2 + 1 〉 = +0.73|134 Te; g.s.〉| 134 Sn; 2 + 1 〉 + 0.36|134 Sn; g.s.〉| 134 Te; 2 + 1 〉| 136 Te; 2 + 3 〉 = −0.31|134 Te; g.s.〉| 134 Sn; 2 + 1 〉 + 0.78|134 Sn; g.s.〉| 134 Te; 2 + 1 〉INPC 2007 - Tokyo, June 3-8 2007 – p.28

<strong>Shell</strong>-<strong>model</strong> calculationsCalculations for nuclei with many valence-nucleons: the N = 82isotonic chainINPC 2007 - Tokyo, June 3-8 2007 – p.29

<strong>Shell</strong>-<strong>model</strong> calculationsCalculations for nuclei with many valence-nucleons: the N = 82isotonic chainINPC 2007 - Tokyo, June 3-8 2007 – p.29

INPC 2007 - Tokyo, June 3-8 2007 – p.30

<strong>Shell</strong>-<strong>model</strong> calculationsThe monopole component of the effective interaction: groundstate energies as a function of the valence protonsINPC 2007 - Tokyo, June 3-8 2007 – p.31

<strong>Shell</strong>-<strong>model</strong> calculationsThe monopole component of the effective interaction: groundstate energies as a function of the valence protonsINPC 2007 - Tokyo, June 3-8 2007 – p.31

<strong>Shell</strong>-<strong>model</strong> calculationsThe monopole component of the effective interaction: groundstate energies as a function of the valence protonsINPC 2007 - Tokyo, June 3-8 2007 – p.31

Single-particle propertiesHot topic: the role of tensor-force contribution to determine thesingle-particle spacings, and consequently the evolution of shellstructure of nucleiB. A. Brown, T. Duguet, T. Otsuka, D. Abe, and T. Suzuki, Phys. Rev. C 74,061303(R) (2006)INPC 2007 - Tokyo, June 3-8 2007 – p.32

Single-particle propertiesHot topic: the role of tensor-force contribution to determine thesingle-particle spacings, and consequently the evolution of shellstructure of nucleiB. A. Brown, T. Duguet, T. Otsuka, D. Abe, and T. Suzuki, Phys. Rev. C 74,061303(R) (2006)Realistic potentials include one-pion exchange and more: areeffective realistic interactions able to reproduce single-particlespacings?INPC 2007 - Tokyo, June 3-8 2007 – p.32

Preliminary resultsComparison between theoretical and experimental133 Sb single-particle relative spectraINPC 2007 - Tokyo, June 3-8 2007 – p.33

Preliminary resultsComparison between theoretical and experimental133 Sb single-particle relative spectraINPC 2007 - Tokyo, June 3-8 2007 – p.33

Preliminary resultsComparison between theoretical and experimental133 Sb single-particle relative spectraINPC 2007 - Tokyo, June 3-8 2007 – p.33

• Effective interactions derived from modern NN potentialsare able to describe with quantitative accuracy thespectroscopy of exotic nuclei near closed shells. This givesconfidence in their predictive power in these regions• At present, no real evidence of shell <strong>model</strong> modificationsnear 132 Sn• It is of key importance to gain more experimentalinformation for nuclei around 100 Sn and 132 Sn. This is anexciting physics to be done with RIB’sINPC 2007 - Tokyo, June 3-8 2007 – p.34

CollaboratorsUniversità di Napoli “Federico II” and INFNL. C.A. CovelloA. GarganoN. ItacoSUNY - Stony Brook:T. T. S. KuoUniversity of Idaho - Moscow:R. MachleidtUniversidad de SalamancaD. R. EntemINPC 2007 - Tokyo, June 3-8 2007 – p.35