1. 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

2. Big computational challenge
Shell model side
Cluster side
Big computational challenge

3. Cluster model can describe the cluster states
with large principal quantum number
of the harmonic oscillator
Second 0+ state of 12C
Principal Quantum Number
Probability
Y. Suzuki, K. Arai, Y. Ogawa, and K. Varga, Phys. Rev. C (1996)

4. Recent Monte Carlo shell model calculation for Be isotopes
0+1
10Be
0+2
T. Yoshida, N. Shimizu, T. Abe, and T. Otsuka,
J. Phys.: Conf. Ser (2014).

6. microscopic 3α models in ‘70s
Supplement of Prog. Theor. Phys. 82 (1980)
12C
3 alpha threshold
Successful but the moment of inertia of the ground band was not

7. Superposition of many Slater determinants – mean field model
Y. Fukuoka, S. Shinohara, Y. Funaki, T. Nakatsukasa, and K. Yabana,
Phys. Rev. C

8. α-cluster model Each 4He:
(0s)4 configuration at some localized position
Non-central interactions between nucleons
(spin-orbit, tensor) do not contribute
S=0
S=0
S=0

9. Cluster model partially covers the model space of the shell modelx y
Elliott SU(3) limit x
This is (s)4(px)4(py)4, but not (s1/2)4(p3/2)8

11. Shell model side
Cluster side
Big computational challenge
Our strategy

12. quasi cluster model exp[-ν( r – Ri )2]
How we can include the spin-orbit contribution?
spatial part of the single particle wave function
exp[-ν( r – Ri )2]
In the Brink’s model,
4 nucleons share the same Ri value in each α 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)
quasi cluster model
N. Itagaki, H. Masui, M. Ito, and S. Aoyama, Phys. Rev. C (2005).

13. quasi cluster
8Be
-Y axis
Z axis
quasi cluster
X axis

14. 8Be R = 3 fm Λ = 0 R = 0.01 fm Λ = 0 R = 0.01 fm Λ = 0.5

15. jj-coupling shell state cluster state Large R Zero Λ R  0 Λ 1
SU(3) limit
(3 dim.
Harmonic
Oscillator)
cluster state
Large R
Zero　Λ
zero Λ
zero R
increasing Λ
decreasing R
Spin-orbit contribution
can be taken into account

16. quasi cluster
12C
-Y axis
Z axis
X axis

17. Transforming all alpha clusters to quasi clusters
Tadahiro Suhara, Naoyuki Itagaki, József Cseh, and Marek Płoszajczak
Phys. Rev. C 87, (2013)
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)

18. R = 3 fm
Λ = 0
R = 0.01 fm
Λ = 0
12C
R = 0.01 fm
Λ = 1

19. Energy surfaces of 12C R (fm) R (fm) Λ Λ 0+ energy spin-orbit energy
Minimum point
R = 0.9 fm, Λ = 0.2
MeV (spin-orbit about -20 MeV)

20. the breaking of threefold symmetry
Comparison with AMD+GCM (superposition of β-γ constrained AMD wave functions)
overlap
1% is coming from
the breaking of threefold symmetry

21. 12C 0.06 0.60 0.56 1.28 Σ li ∙ si Various configurations
Λ ≠ 0
Various configurations
of 3α’s with Λ=0

22. Mathematical explanation for the quasi cluster model
This part is just for better understanding of the model, and not needed in practical calculations

23. Basic knowledge 1 ``spin-orbit contribution”
j = l + 1/2  j-upper, spin-orbit attractive
j = l - 1/2  j-lower, spin-orbit repulsive

24. Basic knowledge 2 ``Spherical harmonics”

27. Basic knowledge 3 ``stretched configuration”
Yll ↑
Yl-l ↓

28. Spatial part of the single particle wave function
for the four nucleons in the quasi cluster
exp[-ν( r – R )2]
R = R (ex＋i Λ ey )
R = R (ex - i Λ ey )
for the spin-up nucleons
for the spin-down nucleons
+  spin-up
-  spin-down

29. at Λ = 1 Spatial part of the single particle wave function
for the four nucleons in the quasi cluster (spin-up)
for spin-up
at Λ = 1

30. at Λ = 1 (s1/21/2+p3/23/2+d5/25/2+f7/27/2+.…) exp[ -𝜈r2 ]
Spatial part of the single particle wave function
for the four nucleons in the quasi cluster (spin-up)
for
spin-up
at Λ = 1
(s1/21/2+p3/23/2+d5/25/2+f7/27/2+.…) exp[ -𝜈r2 ]
Single particle wave function of a nucleon in the quasi cluster
 coherent state of j-upper orbits

31. How we can generalize it?
Magic numbers 2, 8, 20 corresponding to the closure of dimensional harmonic oscillator can be described by the Brink model, but how about beyond those ones, for instance 28 and 50?

32. Starting point　– 16O
Four alpha clusters forming a tetrahedron configuration
correspond to the closed shell configuration of the p-shell
in the limit of small relative distance

33. 20Ne case Cluster model – 16O+alpha model
Present model – 16O+quasi cluster

34. Optimal Λ for each R
20Ne
R (fm)

35. Generality of the model - 28Si
Both representations become identical
(jj subclosure of d5/2) at R  0 and Λ  1
With finite R distances, there appears difference
(0+1 oblate 0+2 prolate)

36. 28Si for d5/2 at R = 0.01 fm j=5/2, l=2, s=1/2,
N. Itagaki, H. Matsuno, and T. Suhara,
28Si
at R = 0.01 fm
for d5/2
j=5/2, l=2, s=1/2,
l*s= ( j(j+1) - l(l+1) - s(s+1) )/2 = (35/ /4) / 2 = 1

37. Important message
We can prepare the jj coupling shell model wave functions of 12C and 16O, the jj coupling wave function of 28Si can be generated at the zero distance limit of these two
There exists a path going from 28Si prolate state to 12C(jj-coupling) +16O cluster configuration
The oblate state (pentagon shape) is connected to which cluster configuration?  future problem
Discovering such paths from jj-coupling shell model states to cluster different configurations would be an interesting subject

38. 52Fe  40Ca + 3 quasi clusters (12C)
For heavier nuclei,
we can just replace
the 16O core with 40Ca
44Ti  40Ca + 1 quasi cluster
52Fe  40Ca + 3 quasi clusters (12C)
56Ni  40Ca + 4 quasi clusters

39. 3α state around the 40Ca core??
Tz. Kokalova et al. Eur. Phys. J. A (2005).
28Si+24Mg
 52Fe

40. 52Fe 44Ti for f7/2 at R = 0.01 fm j=7/2, l=3, s=1/2,
N. Itagaki, H. Matsuno, and T. Suhara,
52Fe
at R = 0.01 fm
44Ti
for f7/2
j=7/2, l=3, s=1/2,
l*s= ( j(j+1) - l(l+1) - s(s+1) )/2 = (63/ /4) / 2 = 3/2

41. 44Ti projected to 0+ 40Ca+α threshold dotted line Λ=0 solid line
optimal Λ
for each R
N. Itagaki, H. Matsuno, and T. Suhara,

42. Λ 56Ni for f7/2 at R = 0.01 fm j=7/2, l=3, s=1/2,
l*s= ( j(j+1) - l(l+1) - s(s+1) )/2 = (63/ /4) / 2 = 3/2

43. For 100Sn we add 5 quasi cluster around the 80Zr core
For even heavier nuclei,
we can just replace
the 40Ca core with 80Zr
For 100Sn we add 5 quasi cluster around the 80Zr core ~α
80Zr ~α ~α ~α ~α

44. Λ 100Sn for g9/2 at R = 0.01 fm for j=9/2, l=4, s=1/2,
( j(j+1) - l(l+1) - s(s+1) )/2 = (99/ /4) / 2 = 2

45. Conclusion for this part
We found a way to transform α cluster model wave function to jj-coupling shell model within a single Slater determinant, and spin-orbit interaction can be taken into account.
The optimal state of 12C is just in between the 3 alpha state and subclosure configuration of p3/2 shell. Although 3 α’s are changed into 3 quasi clusters, the threefold symmetry is still there.
This approach can be applied to various cases, and discovering a paths from jj-coupling states to (jj-coupling) subsystems will be an important future subject