1. Structure of Neutron-rich Isotopes and Roles of Three-body Forces Toshio Suzuki Nihon University Trento, July 13, 2011

2. ○ 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

3. 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

4. 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 V nn tensor force

5. SFO p-sd shell Tensor components Suzuki, Fujimoto, Otsuka, PR C67 (2003) Shell evolution in N=8 isotone πp3/2 N=6 N=8

6. B(GT) for 12 C → 12 N present = SFO Suzuki, Fujimoto, Otsuka, PR C67 (2003) Magnetic moments of p-shell nuclei SFO*: g A eff /g A =0.95 B(GT: 12 C)_cal = experiment SFO Space: up to 2-3 hw PR C55, 2078 (1997) Suzuki, Chiba, Yoshida,Kajino, Otsuka, PR C74, 034307, (2006).

7. more repulsion than G in T=1 more 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 17 C ● 3 body forces induced by Δ excitations → repulsion in T=1 monopoles

8. VMU= Monopole based Universal Interaction Otsuka, Suzuki, Honma, Utsuno, Tsunoda, Tsukiyama, Hjorth-Jensen PRL 104 (2010) 012501 Tensor: bare≈renormalized 16 20

9. Modification of SFO Full inclusion of tensor force ・ p-sd: tensor->  LS ->  ・ sd: Kuo G-matrix T=1 monopole terms more repulsive → SFO-tls 3=0d3/2 5=0d5/2 1=1s1/2

10. neutron N dependent e n 0.33 0.27 0.22 ESP

11. M1 transitions in 17 C Anomalous suppression of B(M1) strength D. Suzuki et al., PL B666 (2008) Suzuki, Otsuka, PR C78 (2008) 061301(R)

12. 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; 3 rd -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 ・ E x (2 + ) ・ M1 transition in 48 Ca

13. core-polarization effects +3 rd -order etc. Hjorth-Jensen et al., Phys. Rep. 261, 125 (1995) T. T. S. Kuo, Nucl. Phys. A103, 71 (1967) Kuo (HJ): 2 nd -order, up to 2hw BonnC: 3 rd -order, up to 2-4 hw CD-BonnC: 3 rd -order, up to 18hw

14. 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 ’ repulsive

15. 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 （ B ） j j ’ （ C ） j ’ j j j ’ j j ’ j j ’

16. Monopoles for sd-shell: T=1 ● Oxygen isotopes

17. ESPE of Oxygen Isotopes 3N →repulsion

18. E(2 + )

19. Multipoles vs. monopoles

20. Energies of O isotopes 3-body force → drip line at 24 O Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL 105, 032501 (2010)

21. Effects of breaking of 16 O core 0hw 2hw 16 O 83% 17% 20 O 91% 24 O 97% 28 O 99% p-sd p, p-sd: SFO sd: G How double magic is 24O? Cal: closed (p-d5/2-s1/2 ） core 87%

22. Monopoles 3-body force →repulsion ● Ca isotopes

23. Energies of Ca isotopes

24. E(2 + ) 48 3N → Shell closure at 48 Ca

25. Multipoles vs. monopoles

26. EXP. ： Steffen et al. NP A404, 413 (1983) B(M1) +3N (multipole) → concentration of M1 strength

27. (A/42 )-0.35

28. Energy levels of odd Ca isotopes Important roles of multipole components

29. SPE=PKUO p1/2: 3.8282 MeV p3/2: 1.744 MeV (spe) : p3/2: +0.6MeV ● He isotopes

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

31. 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

32. －－ Higher order terms Monopoles for π(AV8 ’ ) Core= 4 He T=1 T=0 1 : 3x(-3)=-9 Monopoles in T=0

33. Summary Three-body force can describe well the g.s. energies of O and Ca (and He) isotopes, drip-line at 24 O, shell closure at 48 Ca, as well as M1 transition strength in 48 Ca. Structure of C isotopes can be well described by an improved Hamiltonian with proper tensor forces and repulsive corrections in T=1 monopoles.

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

36. 殻模型 H = T + U(r) + Σ i>j V ij = H 0 + V 一体場 + 残留相互作用 U(r) = U c (r) +U LS (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 sum of ladders

37. 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 +3 rd -order etc. Hjorth-Jensen et al., Phys. Rep. 261, 125 (1995)

38. Monopoles for sd-shell: T=1 ● Oxygen isotopes

40. Monopoles for sd-shell: T=1 ● Oxygen isotopes

41. ESPE of Oxygen Isotopes 3N →repulsion

42. ESPE of Oxygen Isotopes 3N →repulsion

43. E(2 + )

47. Energies of O isotopes 3-body force → drip line at 24 O Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL 105, 032501 (2010)

48. Effects of breaking of 16 O core 0hw 2hw 16 O 83% 17% 18 O 87% 20 O 91% 22 O 95% 24 O 97% 26 O 98% 28 O 99% p-sd p, p-sd: SFO sd: G

50. Energies of Ca isotopes

51. E(2 + ) 48 3N → Shell closure at 48 Ca

52. B(GT) for 14 N -> 14 C SFO Negret et al., PRL 97 (2006) KVI RCNP 14 C → 14 N SFO

53. 14 C -> 14 N g.s. Bonn-B 0hw

54. SFO- ｔ ｌｓ

55. －－ Higher order terms Monopoles for π(AV8 ’ ) Core= 4 He T=1 T=0 1 : 3x(-3)=-9 Monopoles in T=0

56. Erosion of N=64 magicNew magic at N=76