Structure of Neutron-rich Isotopes and

Roles of Three-body Forces

Toshio Suzuki Nihon University

Trento, July 13, 2011

○ Shell-model interactions important roles of tensor force need more repulsion in T=1 monopoles need more attraction in T=0 monopoles

1. Repusive Corrections in T=1 Monopoles and Structure of C isotopes with the use of a ‘phenomenological’ interaction

Three-body forces → repulsion

2. ・ Structure of O and Ca isotopes and three-body forces 　 ‘G + FM-3N (Δ excitaions by 2π exchanges)’ ・ He, Sn isotopes and remaining problems

1. Repusive Corrections in T=1 Monopoles and Structure of C isotopes

・ Important roles of tensor forces e.g. a new p-shell Hamiltonian: SFO ・ Need for repulsion in T=1 monopoles G-matrix vs. phenomenological interactions・ Monopole-based-universal interaction (VMU)・ Phenomenological shell model interaction for neutron-rich carbon isotopes: SFO-tls・ Structure of C isotopes

New shell model Hamiltonians→ success in better description of spin modes in nuclei

● Important roles of 　 tensor force → SFO (p, p-sd)(Suzuki-Fujimoto-Otsuka)

・ Shell evolutions ・ GT transitions and 　 magnetic moments

● Monopole-based universal interaction (VMU)　

Monopole terms in Vnn

tensor force

1 2 1 2T JM 1 2

J

(2J 1) j j ;JT | V | j j ;JTV ( j j )

(2J 1)

SFO p-sd shell C T LSV V (central V (tensor V (LS ） ） ）

Tensor components

Suzuki, Fujimoto, Otsuka, PR C67 (2003)

Shell evolution inN=8 isotone

πp3/2

N=6

N=8

B(GT) for 12C →12N

present = SFO Suzuki, Fujimoto, Otsuka, PR C67 (2003)

Magnetic moments of p-shell nuclei

SFO*: gAeff/gA=0.95

B(GT: 12C)_cal = experiment

SFO

Space: up to 2-3 hw

PR C55, 2078 (1997)

Suzuki, Chiba, Yoshida,Kajino, Otsuka, PR C74, 034307, (2006).

more repulsion than G in T=1more attraction than G in T=0

● Tensor force + repulsive corrections in T=1 monopoles → SFO-tls

・ Structure of neutron-rich C isotopes ・ Exotic M1 transitions in 17C

● 3 body forces induced by Δ excitations → repulsion in T=1 monopoles

VMU= Monopole based Universal Interaction

Otsuka, Suzuki, Honma,Utsuno, Tsunoda, Tsukiyama, Hjorth-Jensen PRL 104 (2010) 012501

Tensor: bare≈renormalized16

20

• Modification of SFO Full inclusion of tensor force ・ p-sd: tensor-> LS ->

・ sd: Kuo G-matrix T=1 monopole terms more repulsive → SFO-tls

C T LS

T

LS

V V V V

V V V

V V

3=0d3/25=0d5/21=1s1/2

neutron

N dependent en 0.33 0.27 0.22

ESP

• M1 transitions in 17C Anomalous suppression of B(M1) strength

D. Suzuki et al., PL B666 (2008)

2 21 10( ) ( )NB M

Suzuki, Otsuka,PR C78 (2008) 061301(R)

2. Structure of O and Ca isotopes and three-body forces

Shell model G-matrix vs. G-matrix + three-body force

G = BonnC, CD-Bonn for Ca; 3rd-order Q-box G = Kuo, BonnC, CD-Bonn for O Hjorth-Jensen, Kuo, Osnes Phys. Rep. 261 (1995) 125.

FM (Fujita-Miyazawa) three-body force Δ-excitation by two-pion exchange ・ Effective neutron single-particle energies ・ Ground state energies ・ Ex (2+) ・ M1 transition in 48Ca

core-polarization effects

+3rd-order

etc.Hjorth-Jensen et al., Phys. Rep. 261, 125 (1995) T. T. S. Kuo, Nucl. Phys. A103, 71 (1967)

Kuo (HJ): 2nd-order, up to 2hwBonnC: 3rd-order, up to 2-4 hwCD-BonnC: 3rd-order, up to 18hw

－ = －

X X

Monopole terms from 3-body force induced by Δ excitations and short-range terms

j j’ 　 j’ j j’ j

j j’ j j’ j j’

21| ' | | | 0pj V hj

E

repulsive

－ ．．．

Monopole terms from 3-body force induced by Δ excitations（ A ）

j j’ j’ j j j’ j j’

－ －

＋ ＋

j’ j j’ j j’ j j’ j

2| | | ' ' |pj V hj j h

（ B ） j j’

－ ．．．

….

j j’

2| | | ' ' |ph V jj j j

（ C ）　 j’ j j j’

j j’ 　 j j’

| | ' | | ' ' |pj V jh hj pj V j h hj

Monopoles for sd-shell: T=1

● Oxygen isotopes

ESPE of Oxygen Isotopes

3N →repulsion

E(2+)

Multipoles vs. monopoles

Energies of O isotopes

3-body force → drip line at 24OOtsuka, Suzuki, Holt, Schwenk, Akaishi,

PRL 105, 032501 (2010)

Effects of breaking of 16O core

0hw 2hw 16O 83% 17%

20O 91%24O 97%28O 99%

p-sdp, p-sd: SFOsd: G

How double magic is 24O?Cal: closed (p-d5/2-s1/2 ） core 87%

Monopoles

3-body force →repulsion

● Ca isotopes

Energies of Ca isotopes

E(2+)

48

3N → Shell closure at 48Ca

Multipoles vs. monopoles

EXP. ： Steffen et al.NP A404, 413 (1983)

B(M1)

+3N (multipole) → concentration of M1 strength

(A/42)-0.35

Energy levels of odd Ca isotopes

Important roles of multipole components

SPE=PKUO p1/2: 3.8282 MeV p3/2: 1.744 MeV

(spe) : p3/2: +0.6MeV

● He isotopes

Erosion of N=64 magic New magic at N=76?

Remaining Problems

• T=0 monopoles Need attractive correction• Microscopic derivation of single-particle energies

(J. D. Holt)• Extension of the configuration space sd -> sd+f7/2,p3/2 (J. D. Holt) fp -> fp+g9/2 (J. D. Holt) G-matrix for non-degenerate orbits (Tsunoda) p-sd, sd-pf, pf-g9/2

－ －

Higher order terms

Monopoles for π(AV8’) Core=4He

1 2 1 2

T=1 T=0 1 : 3x(-3)=-9

Monopoles in T=0

Summary• Three-body force can describe well

the g.s. energies of O and Ca (and He) isotopes, drip-line at 24O, shell closure at 48Ca, as well as M1 transition strength in 48Ca.

• Structure of C isotopes can be well described by an improved Hamiltonian with proper tensor forces and repulsive corrections in T=1 monopoles.

Collaborators T. Otsuka Univ. of Tokyo J. D. Holt ORNL A. Schwenk Darmstadt

殻模型

H = T + U(r) + Σi>jVij = H0 + V　　　一体場　 + 　残留相互作用 U(r) = Uc(r) +ULS(r)L ・ S

殻模型相互作用 ・ Microscopic interaction derived from NN interaction 1. Renormalization of repulsive core part of NN interaction G-matrix:

　　　

V_{low-k} integrating out high momentum components of two-nucleon interaction

2p

0

2pa a a

0

a b a b

a b

QG(E) V V G(E)

E H

QG

E H

| G | | V |

cf . | V |

sum of ladders

Good energy levels except for a few cases: e.g. closed-shell struture of 48Ca can not be obtained Problems in saturation (binding energies)

・ Phenomenological interaction single particle energies + fitted two-body matrix elements e.g. p-shell: Cohen-Kurath p-sd: Millener-Kurath sd: USD

core-polarization effects

+3rd-order etc.

Hjorth-Jensen et al., Phys. Rep. 261, 125 (1995)

Monopoles for sd-shell: T=1

● Oxygen isotopes

Monopoles for sd-shell: T=1

● Oxygen isotopes

ESPE of Oxygen Isotopes

3N →repulsion

ESPE of Oxygen Isotopes

3N →repulsion

E(2+)

Energies of O isotopes

3-body force → drip line at 24OOtsuka, Suzuki, Holt, Schwenk, Akaishi,

PRL 105, 032501 (2010)

Effects of breaking of 16O core

0hw 2hw 16O 83% 17%18O 87%20O 91%22O 95%24O 97%26O 98%28O 99%

p-sdp, p-sd: SFOsd: G

Energies of Ca isotopes

E(2+)

48

3N → Shell closure at 48Ca

B(GT) for 14N -> 14C

SFO

Negret et al., PRL 97 (2006)

KVI

RCNP

14C → 14N

SFO

14C -> 14Ng.s. Bonn-B 0hw

SFO- ｔｌｓ

*TV f V

－ －

Higher order terms

Monopoles for π(AV8’) Core=4He

1 2 1 2

T=1 T=0 1 : 3x(-3)=-9

Monopoles in T=0

Erosion of N=64 magic New magic at N=76