1. クラスター・シェル競合の新展開 板垣 直之 ( 京都大学基礎物理学研究所 )

2. Shell structure; single-particle motion of protons and neutrons decay threshold to clusters Excitation energy Nuclear structure Cluster-shell competition

3. 3 alpha model for 12 C T. Yoshida, N. Itagaki, and T. Otsuka, Phys. Rev. C 79 034308 (2009).

4. α-cluster model We assume each 4 He as (0s) 4 configuration with respect to some localized position Non-central interactions do not contribute

5. In the low-lying state….. Spin-orbit interaction is the driving force

6. x This is (s) 4 (p x ) 4 (p y ) 4, but not (s 1/2 ) 4 (p 3/2 ) 8 y x y Cluster model covers the model space of the shell model?

8. How we can include jj-coupling shell model-like correlations in the cluster model? If we treat all the Gaussian centers parameters of the nucleons independently and superpose different Slater determinants, this is possible as in anti-symmetrized molecular dynamics (AMD) and Fermionic molecular dynamics (FMD)

10. How we can describe cluster-shell competition in a simple way? Is there a simple transformation from Brink’s cluster model wave function to the lowest configuration of jj-coupling shell model within a single Slater determinant? We introduce a general and simple model to describe this transition

11. The spin-orbit interaction: (r x p) s = Gaussian center parameter Ri = imaginary part of Ri For the nucleons in the quasi cluster: Ri  Ri ＋ i Λ (e_spin x Ri) exp[-ν( r – Ri ) 2 ] In the cluster model, 4 nucleons share the same Ri value in each alpha cluster How we can include the spin-orbit contribution In the cluster model?? spatial part of the single particle wave function Simplified method to include the spin-orbit contribution (SMSO) N. Itagaki, H. Masui, M. Ito, and S. Aoyama, Phys. Rev. C 71 064307 (2005).

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

13. X axis Z axis -Y axis

14. 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:

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

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

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

18. We must measure the breaking of alpha cluster in each state What is the operator related to the α breaking? one-body spin-orbit operator for the proton part

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

20. For C isotopes The R and Λ values for the 12 C core are randomly generated The positions of Gaussian center parameters for the valence neutrons are randomly generated, and we superpose many different configurations for the valence neutrons

21. 16 C One-body LS 0.44 0.51 1.45 1.39

23. 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)

24. Single particle energy of spin-up protons R = 0.01 [fm] p 3/2 s 1/2 Energy [MeV] Λ

25. Energy sufaces 0+ energy Minimum point R = 0.9 fm, Λ = 0.2 - 89.6 MeV LS force Tadahiro Suhara, Naoyuki Itagaki, Jozsef Cseh, and Marek Ploszajczak arXiv nucl-th 1302.5833

26. A single state has different features

27. Comparison with β-γ constraint AMD overlap SMSOAMD energy- 89.6 [MeV]- 90.1 [MeV] # of freedom 2 （ R, Λ ） 6A （ 3×2×A ） xyz complex

28. Tensor contribution in C isotopes using SMSO (MeV) 11 C 12 C 13 C 14 C 16 C With tensor -64.8 -89.3 -93.8 -106.2 -109.2 Without T -65.6 -91.4 -95.5 -106.3 -111.6 Δ +0.9 +2.1 +1.7 +0.1 +2.4 We can take into account first order-like tensor contribution, if we break alpha’s to take into account the spin-orbit effects

29. How we can incorporate the 2p2h nature of the tensor contribution in the cluster model in a simple way? “Simplified method to include the tensor contribution in alpha-cluster model ” N. Itagaki, H. Masui, M. Ito, S. Aoyama, and K. Ikeda Phys. Rev. C 73 034310 1-7 (2006).

30. We would like to break the α cluster in a simplified way p↑ n↑ n↓ Tensor interaction works between proton spin-up and neutron spin-up, but this is cancelled by the spin-sown neutron located at the same position Necessary to break αas 2+1+1 system (2p2p) However, the tensor interaction does not work even if we break αto 3 nucleons + 1 nucleon system

31. p↑ n↑n↑ n↓n↓ p↓ Model 2 Spin (z) direction -33.83 MeV compared to -27.57 MeV of the (0s) 4 model Simplified method to include Tensor contribution (SMT)

32. 12 C

33. Summary Cluster-shell competition and role of spin-orbit interaction in light nuclei can be studied in a simplified way We can transform Brink’s wave function to jj-coupling shell model (jj subclosure states) by introducing two parameters The model is rather general and we can apply to heavier nuclei Description of particle-hole states is going on First-order-like effect of the tensor interaction can be taken into account if we break the alpha clusters to taken into account the spin-orbit contribution To take into account the 2p2h nature of the tensor interaction, we need another model, which is under development

34. For the study of heavier systems, we must simplify the THSR wave function THSR wave function

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

36. r.m.s. radius of 12 C (fm) 3α3α Solid, dotted, dashed, dash-dotted  σ = 2,3,4,5 fm

37. σ = 3 fm σ = 3.5 fm σ = 4 fm