Structure of exotic nuclei Takaharu Otsuka University of Tokyo / RIKEN / MSU 7 th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008 A presentation supported by the JSPS Core-to-Core Program “International Research Network for Exotic Femto Systems (EFES)” Day 3

We discussed on how single-particle levels change (shell evolution) as functions of Z and N in exotic nuclei due to various components of the nuclear force. 1.Tensor force 2.Central force Single-particle properties (shell structure) are one of the most dominant elements of nuclear structure. Example : The deformation (of low-lying states) is a Jahn-Teller effect to a good extent. Summary of Day2-1

Summary of Day2 - 2 Some components of NN interactions show characteristic patterns of the shell evolution Tensor force : - variation of spin-orbit splitting - strong  unexpected oblate deformation of a doubly-magic 42 Si Central force : - differentiates different radial nodal structure of single-particle orbits - stronger (< tensor) Tensor + Central combined (typically in the ratio 2:1)  lowering of neutron f5/2 in Ca-Cr-Ni inversion of proton f5/2 and p3/2 in Cu for N~46 54 Ca, 78 Ni, 100 Sn Day-One experiments of RIBF

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

From Day 2 lecture T=1 monopole interaction

T=1 monopole interactions in the pf shell j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

T=1 monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force (  +  exchange) j = j’ Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

Origin and implication of repulsive modification of T=1 monopole components

リチウム１１ nuclei (mass number) stable exotic -- with halo A neutron skin proton halo Proton number  Stable Nuclei Neutron number  Nuclear Chart - Left Lower Part - neutron halo O (Z=8) Drip Line (Existence Limit of Nuclei) F (Z=9) 11 Li Why is the drip line of Oxygen so near ?

This is because the neutron d 3/2 orbit is high for Oxygen. 16 O core 1d 5/2 2s 1/2 1d 3/2 17 O 9 ~ 22 O O 15 ~ 24 O 16 Neutron orbits in Oxygen isotopes 16 O core Neutron orbits in Fluorine isotopes neutron threshold Proton-neutron force, Incl. strong tensor force, due to a proton in d 5/2

16 O core 1d 5/2 2s 1/2 1d 3/2 17 O 9 ~ 22 O O 15 ~ 24 O 16 Neutron orbits in Oxygen isotopes 16 O core Neutron orbits in Fluorine isotopes neutron threshold Why do those neutrons NOT pull down d 3/2 ? Why do those neutrons NOT pull down d 3/2 ?

Effective Single-Particle Energy for Oxygen isotopes Kuo-Brown G-matrix + core-pol. d3/2 d5/2 Neutron number (N) narrowing Wrong drip line

Effective Single-Particle Energy for Oxygen isotopes Less steep USD Kuo-Brown G-matrix + core-pol. d3/2 d5/2 Neutron number (N) narrowing Empirical correction Additional repulsion between d 5/2 and d 3/2 Not enough Neutron number (N) Wrong drip line

Effective Single-Particle Energy for Oxygen isotopes SDPF-M Empirical correction Final correction Less steep USD Kuo-Brown G-matrix + core-pol. d3/2 d5/2 Neutron number (N) narrowing Neutron number (N) Neutron number (N) Finally flat, d3/2 kept high  correct drip line Y. Utsuno, T.O., T. Mizusaki, and M. Honma, Phys. Rev. C 60, (1999).

Question What is the origin of the repulsive modification to T=1 monopole matrix elements ? A solution within bare 2-body interaction is very unlikely (considering efforts made so far)  3-body interaction However 3NF -> attractive effects systematics in results of GFMC, NCSM CC (Hagen et al., Phys. Rev. C76, (2007)

GFMC (Green Function Monte Carlo) by Argonne group 3-body force included 3-body force increases binding energies 3-body force

Nucleons in valence orbits (of low momenta) Nucleons in higher shell (of high momenta) Nucleons in valence orbits (of low momenta) One reason : 3N force with short range produces basically more attraction from the 2 nd order perturbation

The key : Fujita-Miyazawa 3N mechanism  -hole excitation)  particle m=1236 MeV S=3/2, I=3/2  N NN  

Renormalization of NN interaction due to  excitation in the intermediate state  T=1 attraction between NN effectively Modification to bare NN interaction (for NN scattering)

 -hole excitation effect on single-particle energy Renormalization of single particle energy due to  -hole excitation  attractive (more bound)     core valence particle  N N N N N

Pauli blocking effect on the renormalization of single-particle energy Pauli blocking effect on the renormalization of single-particle energy Pauli Forbidden  The effect is suppressed m  m m’ Renormalization of single particle energy due to  -hole excitation  more binding (attractive) m  m m’ single particle states Another valence particle in state m’

Inclusion of Pauli blocking Pauli forbidden (from previous page) This Pauli effect is included automatically by the exchange term.  m m m’  m m

Realization in terms of 3-body interaction + Renormalization of single particle energy m  m m’ same A part of Fujita-Miyazawa 3-body force Pauli blocking m m m’ 

Fujita-Miyazawa 3N mechanism  -hole excitation)  particle m=1236 MeV S=3/2, I=3/2  N NN  

 m m m’ This effect is monopole, because it is about single-particle energies. This effect is repulsive, because it is suppression of more binding. Effective monopole repulsive interaction reproduces the effect. > 0 We look at this effect as an effective interaction between states m and m’. Incorporation of this effect into effective interaction m m m’ v eff

Other diagram included Related effect was discussed by Frisch, Kaiser and Weise for neutron matter See also Nishizaki, Takatsuka and Hiura PTP 92, 93 (1994) T=1 interaction between valence particles  Pauli blocking Particle in the inert core

 -hole excitation may be crucial to neutron matter property Chiral Perturbation incl.  Frisch, Kaiser and Weise

16 O core 1d 5/2 2s 1/2 1d 3/2 17 O 9 ~ 22 O 14 Neutron orbits in Oxygen isotopes neutron threshold Back to the question of high-lying d 3/2 Central : attractive (generally) Tensor : attractive MeV (next page)  -hole induced repulsion ( > tensor ) Next page

Repulsive effective monopole interaction assuming 16 O core  exchange with radial cut-off at 0.7 fm, ΔE =293 MeV f_{πNΔ ｝ /f_{πNN} = \sqrt{9/2} Monopole interaction j j' pion tensor d 5/2 d 3/2 250 keV Tensor  -hole-induced repulsion d 3/2 single-particle energy relative to N=8 neutron number (N) MeV S.P.E. +2 MeV

More binding by 3NF Is this always true ?

2-body empirical fit (~SDPF-M) Pairing cases EFT (V low-k ) result for  ~ conventional  -N-  calculation Contact terms can be evaluated as well Effects may be larger with higher order diagrams.

From EFT (Effective Field Theory) sd shell pf shell

T=1 monopole interactions in the pf shell j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

(Effective) single-particle energies Lowering of f5/2 from Ca to Cr : ~ 1.6 MeV = 1.1 MeV (tensor) MeV (central) KB3G n-n p-n KB interactions : Poves, Sanchez-Solano, Caurier and Nowacki, Nucl. Phys. A694, 157 (01) Rising of f5/2 from 48 Ca to 54 Ca : p3/2-p3/2 attraction p3/2-f5/2 repulsion 3432 new magic numbers ?

KB3 interaction and its family By Strasbourg – Madrid group Started from Kuo-Brown’s G-matrix Monopole part and pairing part are empirically adjusted Recent versions : KBF, KB3G,.... good for lighter pf-shell nuclei

N   If another nucleon (X) is in state m’ and wave functions are coupled antisymmetric, the effect is vanished.  Repulsive force m m’ Density dependent repulsive force - Long-ranged due to  exchange -  

Remark : Multipole interactions … different story  m4m4 m1m1 m2m2 m3m3 Multipole parts  Effective repulsion for monopole m m m’

Contact terms in 3N force Back-ground attraction : still not completely known

Effective monopole repulsive interaction > 0 Effect of higher order diagram Other terms of 3-body force cancel this effect to a certain effect Cancellation strong for T=0 weak T=1 included in EFT calc.  m m m’ Important next order diagram Tensor force Larger correction to T=0

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

Can the shell evolution change the deformation ? If so, how much ?

Nature 435 (2005), Florida/MSU PRL accepted (2007), GANIL Application to controversial 42 Si deformed 44 S -> 42 Si cross section small 42 Si oblate 44 S prolate Cauier et al. Shell Model, Werner et al. Skyrme model, Lalazissis et al. RMF, Peru et al. Gogny model, Rodriguez-Guzman et al. Gogny model

s 1/2 Z=28 gap is reduced also by tensor force proton neutron f 7/2 d 3/2 d 5/2 Potential Energy Surface Si 28 New SM Exp. full Tensor force removed from cross-shell interaction Strong oblate Deformation ? Otsuka, Suzuki and Utsuno, Nucl. Phys. A805, 127c (2008)

Potential Energy Surface of 44 S (triaxial) S isotopes full Tensor removed from cross shell int. Tensor force may not be so crucial for S isotopes.

Effect of tensor force on (spherical) superheavy magic numbers Occupation of neutron 1k17/2 and 2h11/2 1k17/2 2h11/2 Neutron N=184 Woods-Saxon potential Tensor force added Proton single particle levels Otsuka, Suzuki and Utsuno, Nucl. Phys. A805, 127c (2008)

Mean field theories Skyrme, Gogny, RMF, … - Inclusion of tensor Stancu, Brink and Flocard, Phys. Lett. 68B, 108 (1077)  -func. Tensor into Skyrme, no prospect for systematic effect Otsuka, Matsuo and Abe, Phys. Rev. Lett. 97, (2006) Tensor force with Gogny interaction First systematic studies along with the shell evolution idea due to the tensor force, new parameter being searched Brown, Duguet, Otsuka, Abe and Suzuki, Phys. Rev. C 74, , (2006). Re-visit to SBF, systematics, strange T=1 tensor from fit - Inclusion of 3-body force open for Fujita-Miyazawa type + many other works afterwards e.g. Zalewski, Satula and Dobaczewski, Phys. Rev. C (2008)

Does the shell evolution remain in continuum ? Tsukiyama has given a short presentation. The nuclear force shifts resonance-type peaks of neutron spectra. The width changes also. More to be done in a close connection to the nuclear force. (Some works have been done with simpler forces.)

1 + and 2 + states of 24 O to be seen in charge exchange from 24 F Tsukiyama, Otsuka, Fujimoto 2008

Summary of Day-3 Three-body force : - repulsive monopole modification for T=1 channel - suppression of  -hole excitation (monopole part of Fujita-Miyazawa force) - effect seen in single-particle energies (Ca) and drip lines (O) in neutron-rich exotic nuclei with high T (isospin) same physics as neutron matter (through NNN force) - features consistent with modern nuclear forces, e.g. EFT (Effective Field Theory)  oxygen drip line closer to the  stability line new magic N=34 in (or maybe only around) Ca

Summary Nuclear force is rich Although it is known to a good extent, - some properties are still uncertain for instance, 3-body force - many-body effects unknown aspects More is different

Collaborators T. Suzuki Nihon U. R. Fujimoto U. Tokyo (Hitachi) M. Honma U. Aizu Y. Utsuno JAEA Y.Akaishi RIKEN H. Grawe GSI A.Schwenk TRIUMF B.Holt TRIUMF