1. Exotic clustering in neutron-rich nuclei and connection to the shell structure
Naoyuki Itagaki
Yukawa Institute for Theoretical Physics
Kyoto University, Japan
CNS Summer School 2015

2. Content of the talk Exotic clustering in neutron-rich nuclei
Brief overview for the history of the cluster study
Cluster model v.s. Density functional theory for the study of the stability of the cluster states
Connection between cluster structure and shell structure

3. Content of the talk Exotic clustering in neutron-rich nuclei
Brief overview for the history of the cluster study
Cluster model v.s. Density functional theory for the study of the stability of the cluster states
Connection between cluster structure and shell structure

4. Nuclear structure weak binding state of strongly bound subsystems
Excitation energy
decaying threshold　　　 to subsystems
single-particle motion of
protons and neutrons
Nuclear structure

5. α-cluster structure 4He is strongly bound (B.E. 28.3 MeV)
　　Closed shell configuration of the lowest s shell
 This can be a subunit of the nuclear system

6. 8Be
rigid rotor of alpha-alpha

7. 2+ 0+ Synthesis of 12C from three alpha particles 0+2 Ex =7.65 MeV Γα
3α threshold
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

8. Let us move on to the neutron-rich side
9Be
Sn= MeV
Qα= MeV
No bound state for the two-body subsystems
-- Boromian system

9. 1/2+ 3/2- around the threshold ground state
1/ /2-
around the threshold ground state
S. Okabe and Y. Abe, Prog. Theor. Phys (1979).

10. 中性子過剰核で p1/2とs1/2が逆転している？

11. Nilsson diagram Bohr-Nottelson Nuclear Structure

12. 1/2[220] means J=1/2 N=2 Nz=2 Lz=0
Neutron stays on the deformation axis, where the curvature of the potential and corresponding ħω value for that axis is smaller than those of other axes
E ~ ħ ( nxωx + nyωy + nzωz )

13. lowering
ħω = 20 MeV
ħω = 10 MeV

14. In the case of Be isotopes
The optimal distance of alpha-alpha core just corresponds to the super deformation region
The alpha-alpha clustering of the core could be a clue for the lowering of ½+ orbit (another effect is ``halo” nature of the s-wave)

16. 10Be Introducing molecular orbits for valence neutrons
around two α clusters
10Be

17. ``New” physics in 10Be
The α-α clustering of the 8Be core is even enhanced compared with free 8Be, when the two valence neutrons occupy (σ)2 configuration
This enhanced cluster configuration forms a rotational band structure starting with the second 0+ state of 10Be (the level spacing is much smaller than 8Be)

18. α+α+n+n model for 10Be large (σ)2 contribution  thick line
N. Itagaki and S. Okabe, Phys. Rev. C (2000).

19. 10Be
（σ）2 πσ
（π）２
N. Itagaki and S. Okabe, Phys. Rev. C (2000)

20. A.A. Korsheninnikov et al., Phys. Lett. B 343 53 (1995).

21. M. Freer et al., Phys. Rev. Lett. 82 1383 (1999).

22. Breaking of N=8 magic number in 12Be
N. Itagaki, S. Okabe, K. Ikeda, Physical Review C62 (2000)
α＋α＋4N model for 12Be
Breaking of N=8 magic number in 12Be

23. Disappearance of N=8 magic number in 12Be
Observation of a low-lying 1- state at 2.68(3) MeV at RIKEN
H. Iwasaki et al., Phys. Lett. B491 (2000) 8.

24. Observation of the 0+2 state of 12Be at Ex = 2.24 MeV

25. 12Be M. Ito, N. Itagaki, H. Sakurai, and K. Ikeda
Phys. Rev. Lett. 100, (2008).

26. Content of the talk Exotic clustering in neutron-rich nuclei
Brief overview for the history of the cluster study
Cluster model v.s. Density functional theory for the study of the stability of the cluster states
Connection between cluster structure and shell structure

27. The α particle model
The emission of α particles has been known already in the end of 19th century (even before the discovery of atomic nuclei), and it was quite natural to consider that α particles are basic constituents of nuclei
However independent particle motion became the standard picture of nuclear structure in 1950s, since the shell structure is established
The cluster studies restarted in 1960s

28. Mysterious 0+ problem in 16O
The first excited state of 16O is 0+2??

29. Inversion doublet structure (12C+α model)
Y. Suzuki
Prog. Theor. P hys (1976)

30. Large scale shell-model calculation (6hω) (diagonalization of 86,000 states)
Four single particle energies
are adjusted to fit
the calculated energy levels
W.C. Haxton and Calvin Johnson,
Phys. Rev. Lett (1990).

31. 12C+12C
molecular states

32. Ikeda daiagram Alpha gas states around the thresholds
Inversion doublet (Mysterious 0+)
Molecular resonances

33. Examples of the cluster study until 1980s (mainly α nuclei)
8Be 　 α+α
9Be　　 α+α+n　
12C　　　３α
16O　　　α＋12C
20Ne　 　α＋16O
24Mg C+12C, α＋α＋16O
44Ti α＋40C

34. Various 3α models 3α calculations by RGM, GCM, OCM
Supplement of Prog. Theor. Phys. 82 (1980).

35. If you expand cluster wave function using Harmonic oscillator basis….
Y. Suzuki, K. Arai, Y. Ogawa, and K. Varga, Phys. Rev. C 54 (1996) 2073.

36. Resonating Group Method (RGM) J.A. Wheeler, Phys. Rev. 52 (1937)
GCM and RGM Ｒ α α
Resonating Group Method (RGM) J.A. Wheeler, Phys. Rev. 52 (1937)
Ψ = A [Φ(r’1, r’2, r’3)Φ(r’4 , r’5, r’6)] χ(r)
Internal wave function
of each α-cluster
Relative wave function
between α-clusters
Generator Coordinate Method (GCM) J.A. Wheeler, Phys. Rev. 52 (1937), H. Margenau, Phys. Rev. 59 (1941)
Ψ = ∑A [Φ(r1-R/2)Φ(r2-R/2)Φ(r3-R/2)Φ(r4-R/2)
Φ(r5+R/2)Φ(r6+R/2)Φ(r7+R/2)Φ(r8+R/2)]

37. Resonating Group Method (RGM) Jacobi coordinate is introduced
GCM and RGM Ｒ α α
Resonating Group Method (RGM) Jacobi coordinate is introduced
Generator Coordinate Method (GCM) No Jacobi coordinate and center of mass
wave function is mixed
Ψ = ∑A [Φ(r1-R/2)Φ(r2-R/2)Φ(r3-R/2)Φ(r4-R/2)
Φ(r5+R/2)Φ(r6+R/2)Φ(r7+R/2)Φ(r8+R/2)]

38. Brink’s wave function (1965) = GCM Ψ= P[A(Φ1(r1)Φ2(r2)・・・・)]
P: Angular momentum and parity projection operator A: Antisymmetrizer
Φ1(r) = exp[-ν(r - R1)2]χ1
Gaussian-center parameter
spin-isospin wave function
αcluster is expressed as four nucleons (p,p,n,n) sharing the same R
exp[-(x-X)2] = ∑ Xn Hn(x) exp(-x2) / n!
Local Gaussian corresponds to the coherent state of many excited orbits of the shell-model

39. Two nucleon’s case (with the same spin and isospin (χ))
Similarity between shell model wave functions and cluster wave functions
Two nucleon’s case (with the same spin and isospin (χ))
Φ1 = exp[ -ν(r – X)2 ] χ
Φ2 = exp[ -ν(r + X)2 ] χ
A [Φ1Φ2] ∝ A [(Φ1+Φ2)(Φ1 - Φ2) ]
At X  0
(Φ1+Φ2)  exp[ -νr2]
(Φ1 - Φ2) / |X|  r exp[ -νr2]
Cluster (local Gaussian) wave function coincides
with the lowest shell-model wave function at X  0

40. Restoration of the symmetry 1 -- Parity projection
Ψ+ = (Ψ{R} + Ψ{-R}) / 2
Ψ- = (Ψ{R} - Ψ{-R}) / 2

41. Restoration of the symmetry 2 -- Angular momentum projection
ΨI = PIMK Ψ

42. Summary for this part
Cluster structure is important also in the neutron-rich systems
Clustering is enhances depending on the neutron-orbit, and this is related to the disappearance of magic number N=8
Brief history and basic frameworks of cluster studies were introduced