N. Itagaki Yukawa Institute for Theoretical Physics, Kyoto University

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

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

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

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

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

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

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

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

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

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

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

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

THSR wave function

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

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

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

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

0 + states of 5α system 16 O-α model 16 O-α + 5α gas N. Itagaki, Tz. Kokalova, M. Ito, M. Kimura, and W. von Oertzen, Phys. Rev. C (2008).

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

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

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

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

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

Alpha+t+n+n t+t+t

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

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

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

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

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

The spin-orbit interaction: (r x p) s r  Gaussian center parameter Ri p  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 (r x p) s = (s x r) p Slater determinant spatial part of the single particle wave function

X axis Z axis -Y axis

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:

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

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

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

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

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

Various configurations of 3α’s with Λ=0 Λ ≠ 0 12 C one body ls

16 C One-body LS

18 C One-body LS

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)

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

One-body spin-orbit operator (p and n)

Comparison with β-γ constraint AMD overlap SMSOAMD energy [MeV] [MeV] # of freedom 2 （ R, Λ ） 6A （ 3×2×A ） xyz 複素数 粒子数

13 C ½- states Λ=0 Λ > 0.1 One-body LS (p)

Summary Nuclear structure changes as a function of excitation energy Geometric configurations are stabilized by adding neutrons or giving large angular momentum Studies of gas-like structure of alpha-clusters are extended to heavier nuclei Cluster-shell competition and role of non-central interactions in neutron-rich nuclei can be studied. We can transform Brink’s wave function to jj-coupling shell model by introducing two parameters