N. Itagaki Yukawa Institute for Theoretical Physics, Kyoto University

1. Exotic cluster structure in light nuclei N. Itagaki Yukawa Institute for Theoretical Physics, Kyoto University

2. weakly interacting states of strongly bound subsystems Excitation energy decay threshold to clusters single-particle motion of of protons and neutrons Nuclear structure

3. Synthesis of 12C from three alpha particles 0+2 Ex =7.65MeV Γα 3αthreshold Ex = 7.4 MeV Γγ 2+ Γγ 0+ The necessity of dilute 3alpha-cluster state has been pointed out from astrophysical side, and experimentally confirmed afterwards

5. “Lifetime” of linear chain as a function of impact parameter

6. How can we stabilize geometric shapes like linear chain configurations? • Adding valence neutrons

7. 10Be πσ （σ）2 （π）２ N. Itagaki and S. Okabe, Phys. Rev. C 61 044306 (2000)

8. 1/2+ 3/2- S. Okabe and Y. Abe Prog. Theor. Phys. (1979)

9. N. Itagaki, S. Okabe, K. Ikeda, and I. Tanihata PRC64 (2001), 014301

10. Linear-chain structure of three clusters in 16C and 20C J.A. Maruhn, N. Loebl, N. Itagaki, and M. Kimura, Nucl. Phys. A 833 1-17 (2010) Cluster study with mean-field models

11. Stability of 3 alpha linear chain with respect to the bending motion Solid -- 20C Dotted -- 16C Geometric shape is stabilized by adding neutrons

12. weakly interacting state ofclusters Excitation energy decay threshold to clusters cluster structure with geometric shapes mean-field, shell structure (single-particle motion)

13. How can we stabilize geometric shapes like linear chain configurations? • Adding valence neutrons • Orthogonalizing to other low-lying states (14C could be possible  by Suhara) • Rotating the system

17. weakly interacting state ofclusters Excitation energy decay threshold to clusters cluster structure with geometric shapes single-particle motion of protons and neutrons

18. THSR wave function

19. Discussion for the gas-like state of alpha’s moves on to the next step – to heavier regions Gas-like state of three alpha’s around 40Ca? Tz. Kokalova et al. Eur. Phys. J A23 (2005) 28Si+24Mg  52Fe Hoyle state around the 40Ca core?

20. Virtual THSR wave functionN.Itagaki, M. Kimura, M. Ito, C. Kurokawa, and W. von Oertzen, Phys. Rev. C 75 037303 (2007) Gaussian center parameters are randomly generated by the weight function of

21. 3α Solid, dotted, dashed, dash-dotted  σ = 2,3,4,5 fm r.m.s. radius of 12C (fm)

22. Two advantages of this treatment • Coupling with normal cluster states can be easily calculated

23. 0+ states of 5α system 16O-α + 5αgas 16O-αmodel N. Itagaki, Tz. Kokalova, M. Ito, M. Kimura, and W. von Oertzen, Phys. Rev. C 77 037301 1-4 (2008).

24. Two advantages of this treatment • Coupling with normal cluster states can be easily calculated • Adding core nucleus is easily done

25. T. Ichikawa, N. Itagaki, T. Kawabata, Tz. Kokalova, and W von Oertzen Phys. Rev. C 83, 061301(R) (2011). 24Mg = 16O+2alpha’s 0+ Energy E0 7th state, candidate for the resonance state Large E0 transition strength

26. Squared overlap with 16O+2alpha’s (THSR)

27. 28Si = 16O+3alpha’s T. Ichikawa, N. Itagaki, Y. Kanada-En'yo, Tz. Kokalova, and W. von Oertzen, Phys. Rev. C 82 031303(R) (2012)

28. How about Fermion case? • Calculation for a 3t state in 9Li, where the coupling effect with the alpha+t+n+n configuration, is performed • Not gas-like and more compact?

29. Alpha+t+n+n t+t+t

34. Threshold rule: gas like structure clusters Excitation energy cluster-threshold cluster structure with geometric shape Competition between the cluster and shell structures mean-field, shell structure

35. α-cluster model • 4He is strongly bound (B.E. 28.3 MeV) Close shell configuration of the lowest shell  This can be a subunit of the nuclear system We assume each 4He as (0s)4 spatially localized at some position Non-central interactions do not contribute

36. 12C 0+ energy convergence N. Itagaki, S. Aoyama, S. Okabe, and K. Ikeda, PRC70 (2004)

37. How we can express the cluster-shell competition in a simple way? We introduce a general and simple model to describe this transition The spin-orbit interaction is the driving force to break the clusters

38. 12C case 3alpha model Λ = 0 2alpha+quasi cluster Λ = finite

39. Slater determinant exp[-ν( r – Ri )2] spatial part of the single particle wave function In the cluster model, 4nucleons share the same Ri value in each alpha cluster The spin-orbit interaction: (r x p) • s r  Gaussian center parameter Ri p  imaginary part of Ri (r x p) • s = (s x r) • p For the nucleons in the quasi cluster: Ri Ri＋i Λ (e_spin x Ri)

40. -Y axis Z axis X axis

41. Single particle wave function of nucleons in quasi cluster (spin-up): Quasi cluster is along x Spin direction is along z Momentum is along y the cross term can be Taylor expanded as:

42. the single particle wave function in the quasi cluster becomes for the spin-up nucleon (complex conjugate for spin-down)

43. 12C Various configurations of 3α’s with Λ=0

44. 12C Λ ≠ 0 Various configurations of 3α’s with Λ=0

45. H. Masui and N.Itagaki, Phys. Rev. C 75 054309 (2007). 3α cluster state is important in the excited states Λ = 0.8 and 0.0 Λ = 0.8 0+ states of 16C

46. We need to introduce an operator and calculate the expectation valueof α breaking What is the operator related to the α breaking? one-body spin-orbit operator for the proton part

47. 12C 0.03 0.30 0.28 0.64 one body ls Λ ≠ 0 Various configurations of 3α’s with Λ=0

48. 16C One-body LS 0.44 0.51 1.45 1.39

49. 18C One-body LS 0.66 0.64 1.16 1.15 1.09

50. Breaking all the clusters Introducing one quasi cluster Rotating both the spin and spatial parts of the quasi cluster by 120 degree (rotation does not change the j value) Rotating both the spin and spatial parts of the quasi cluster by 240 degree (rotation does not change the j value)