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

Big computational challenge Shell model side Cluster side Big computational challenge

Cluster model can describe the cluster states with large principal quantum number of the harmonic oscillator Second 0+ state of 12C --------------------------------------------------------------------------- Principal Quantum Number 8 10 12 14 16 18 20 22 Probability 0 11 12 12 10 8 7 6 Y. Suzuki, K. Arai, Y. Ogawa, and K. Varga, Phys. Rev. C 54 2073 (1996)

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. 569 012063 (2014).

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

Superposition of many Slater determinants – mean field model Y. Fukuoka, S. Shinohara, Y. Funaki, T. Nakatsukasa, and K. Yabana, Phys. Rev. C 88 014321

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

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

Shell model side Cluster side Big computational challenge Our strategy

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 71 064307 (2005).

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

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

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

quasi cluster 12C -Y axis Z axis X axis

Transforming all alpha clusters to quasi clusters Tadahiro Suhara, Naoyuki Itagaki, József Cseh, and Marek Płoszajczak Phys. Rev. C 87, 054334 (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)

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

Energy surfaces of 12C R (fm) R (fm) Λ Λ 0+ energy spin-orbit energy Minimum point R = 0.9 fm, Λ = 0.2 - 89.6 MeV (spin-orbit about -20 MeV)

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

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

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

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

Basic knowledge 2 ``Spherical harmonics”    

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

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

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

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

How we can generalize it? Magic numbers 2, 8, 20 corresponding to the closure of 3 dimensional harmonic oscillator can be described by the Brink model, but how about beyond those ones, for instance 28 and 50?

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

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

Optimal Λ for each R 20Ne R (fm)

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)

28Si for d5/2 at R = 0.01 fm j=5/2, l=2, s=1/2, N. Itagaki, H. Matsuno, and T. Suhara, http://arxiv.org/abs/1507.02400 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 - 6 - 3/4) / 2 = 1

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

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

3α state around the 40Ca core?? Tz. Kokalova et al. Eur. Phys. J. A 23 19 (2005). 28Si+24Mg  52Fe

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, http://arxiv.org/abs/1507.02400 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 - 12 - 3/4) / 2 = 3/2

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, http://arxiv.org/abs/1507.02400

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 - 12 - 3/4) / 2 = 3/2

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

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 - 20 - 3/4) / 2 = 2

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