Cluster-shell Competition in Light Nuclei

N. Itagaki, University of Tokyo

S. Aoyama, Kitami Institute of Technology

K. Ikeda, RIKEN

S. Okabe, Hokkaido University

Purpose of the present study

• Construction of an unified model which can express both shell and cluster structures

• To show that the cluster structure becomes more important in neutron-rich nuclei （ weakly bound systems ）

The first step of the construction of the Unified model is

to establish the description of excess nucleons around clusters

Molecular-Orbit is a bridge from Cluster model to Unified model

8Be （ α-α) - core caseα α

We can make a connection from the cluster physics to the physics of Neutron-rich nuclei

The MO approach in nuclear physics“Molecular Viewpoints in Nuclear Structure”J.A. Wheeler, Physical Review 52 (1937)

Applied to 9Be in 1973 Y. Abe, J. Hiura, and H. Tanaka, P.T.P. 49(application of LCAO-SCF method to nuclear systems)

Systematic Analyses of the Be isotopesM. Seya, M. Kohno, and S. Nagata, P.T.P. 65 (1981) π- and σ-orbits are described as linear combination of p-orbits (Gaussian around each α-cluster with zero distance)

2-body effective interaction

Hamiltonian

α ＋ α ＋ 4N model for 12Be

N. Itagaki, S. Okabe, K. Ikeda, PRC62 (2000) 034301

The significance of present analysis on Be isotopes

• The application of MO to nuclear systems has been done

on a large-scale (the idea itself has existed for a long time).• It is shown once again that cluster physics is closely related

to the physics of unstable nuclei

（ importance of cluster structure in neutron-rich nuclei ）• Now the research is extended to more exotic clusters with

excess neutrons

Significance of the present analysis on C isotopes

• We have re-realized the difference between atomic physics and nuclear physics.

In general, it is difficult to stabilize the linear-chain configuration of 3α

　 But there is possibilities when excess neutrons rotate around the core

• Experiments have been done, and some candidates are already observed.

Equilateral triangular shape of 3α

D3h symmetryKπ=0+, 3- rotational bandsare possible

12C 3- at 9.6 MeV4-, 5- have not been observed（ above theα-threshold ）

How about in neutron-rich nuclei?

Energy level of 14C

Alpha threshold

N. Itagaki, T. Otsuka, K. Ikeda, and S. Okabe, Physical Review Letters, in Press

Energy level of 14C

Alpha threshold

N. Itagaki, T. Otsuka, K. Ikeda, and S. Okabe, Physical Review Letters, in Press

Ikeda’s diagram

Cluster structure should appeararound the corresponding threshold energyof the cluster emissionThreshold rule

Appearance of the cluster structure

Excitation energy

Mean-field-like

Cluster structurewith definite shape?

Cluster structurewith αcondensation

αthreshold

MO is a good tool to describe the wave nature of valence neutrons

around the cluster core

But… Number of Slater determinant is huge

Ex. 84 = 4096

Even if the core-degree of freedom is fixed

Further extension of the MO approach

Antisymmetrized Molecular Dynamics (AMD)(H. Horiuchi et al.)

　 Single-particle wave function

Slater determinant 　　　　　　　　　 Φ 　＝　 A [(ψ1χ1)(ψ2χ2)(ψ3χ3) ・・・ ]

　 Projection of parity and angular momentum 　　　　　　　　 　　　　　　 Ψ 　＝　 PJ 　 Pπ 　 Φ　　　　　　　　　　

AMD – Superposition of Selected Snapshots (AMD triple S)

• We superpose many AMD wave functions• They are randomly generated, and cooling equatio

n is applied only for the imaginary part (spin-orbit contribution is incorporated)

• We use the idea of Stochastic Variation Method （ Suzuki, Varga … ） for the selection of the basis

N. Itagaki and S. Aoyama, PRC68 (2003) 054302

AMD cooling equation

AMD triple-S cooling equation

AMD triple-S for 12Be α-α 3 fm

• 0+1 -62.14 MeV, 0+

2 -58.33 MeV

• 1- -57.43 MeV 0- -54.24 MeV

• <N> value for the 4n is 5.43

higher shell mixing

• 0- is much higher than second 0+ and 1-, and it is considered that the observed state at 2.24 MeV must be the second 0+ state.

Results of linear-chain states using AMD SSS (α-α 3 fm)

14C 0+ (MeV) <N> ( valence neutrons) -84.1 2.7 -70.8 4.1 -68.4 5.4

16C 0+ (MeV) <N> ( valence neutrons) -86.5 5.2 -77.9 6.8 -76.2 7.7

Energy of the α-α system

Volkov No.2 M=0.6

It is difficult to shrink α-α systemIf α-cluster structure disappears dissolution of α

N. Itagaki and S. Okabe, Physical Review C61 (2000) 013456

・・・ have dominantly (σ)2 component

α+α+N+N model

Cluster-shell competition

RANK I Spin-orbit interaction

We prepare α-α(3.5 fm) and add α+p+p+n+n model space

The squared overlapbetween the first state(cluster state) and the final state is 0.91

Cluster structure survives

8Be 0+ state

10Be case

　 α+α+n+n

　 α+4n+2p is addedα+α+2n model

Volkov No.2 M=0.62

without spin-orbit

with spin-orbit

12C 0+ state by single AMD wave function

12C case (α+α+4N and α+8N)

α+α+2p+2n(α-α 3.5 fm)

α+4p+4nmodel spaceis added

12C case (3α and α+8N model space)

Nuclear chart of cluster-shell competition

8BeΔ=2.7 MeV

10BeΔ=2.9 MeV

12C Δ=3.8 MeV (Δ3=8.8 MeV)

N

P

Δ= energy of the final solution - energy of the α+α+nucleon model space

10BΔ=3.6 MeV

Cluster-shell competition

RANK II Tensor

Tensor interaction is renormalized

in the central part and the spin orbit part

This should be explicitly treated

in weakly bound systems

Already started by A. Dote, T. Myo, S. Sugimoto….

Present status of the Tensor• Furutani force is included in our AMD SSS approach (V2+

G3RS+Furutani)

VT = Σ (Wn - MnPm) Vn r2 exp[–μr2] Sij

Sij =3(σi ・ r)(σ ｊ・ r) / r2 – (σi ・ σj)

μ(fm-2) Vn (MeV) W M 3E (MeV) 3O (MeV) 0.53 -15.0 0.3277 0.6723 -15.0 5.17

1.92 -369.5 0.4102 0.5898 -369.5 66.36 8.95 1688.0 0.5000 05000 1688.0 0.0

Furutani force is included in our AMD SSS approach (V2+G3RS+Furutani)

For α, we generate 400 basis states and Sz = 0, 1 for the neutrons and Sz = 0, 1 for the protons are prepared. (V

irtual Jacobi coordinate is introduced) Total (0s)4

Total (MeV) -37.7 -27.6 T (MeV) 53.5 43.8 V (MeV) -74.6 -71.3 LS (MeV) 0.4 0.0Tenser (MeV) -17.0 0.0N 0.35 0.0

Simple description of α

• We keep the spatial part as (s1/2)4

-35.57 MeV by 30 Slater determinants

• We use this simple description of α-cluster

to 4N nuclei

α-α model

double(triple)projectionis necessary?

12C α-α 2.5 fm

• Without Tensor -85.7 MeV

• With Tensor -90.6 MeV

Double (triple) projection is necessary?

Conclusion

• Molecular-orbit is a good tool to describe the cluster structure of neutron-rich nuclei.

• To investigate heavier nuclei, we have introduced new AMD method (AMD triple-S), and the previous MO results are justified using large model space.

• The spin-orbit interaction plays a crucial role for the cluster-shell competiton.

• Tensor term is incorporated but we really need to fix the interaction.