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Cluster shell and DFT Naoyuki Itagaki Yukawa Institute for Theoretical Physics, Kyoto University

Excitation energy decaying threshold to subsystems cluster structure with geometric shapes mean-field, shell structure (single-particle motion)

But the second 0+ state has turned out to be gas like state rather than the state with geometrical configuration

P. Chevallier et al. Phys. Rev 160, 827 (1967)

[ ] J=0 Strong-coupling picture The kinetic energy of 8 Be subsystem increases compared with that of the free 8 Be [[ ] J=0 Weak-coupling picture There is no definite shape

The effect of Pauli principle [ ] J=0 The second and third alpha-clusters are excited to higher-nodal configurations. If linear-chain is stable, there must exist some very strong mechanism in the interaction side.

How can we stabilize geometric cluster shapes like linear chain configurations? • Adding valence neutrons • Rotating the system

How can we stabilize geometric cluster shapes like linear chain configurations? • Adding valence neutrons • Rotating the system

N. Itagaki, S. Okabe, K. Ikeda, and I. Tanihata Phys. Rev. C 64 014301 (2001).

σ-orbit is important for the linear chain, but not the lowest configuration around 3 alpha linear chain N. Itagaki, S. Okabe, K. Ikeda, and I. Tanihata Phys. Rev. C 64 014301 (2001).

Mean field models • Quite general models designed for nuclei of all the mass regions (exotic cluster structure is not assumed a priori). • Appearance of cluster structure as results of studies using such general models give us more confidence for their existence. Many people started analyzing cluster states with mean field models

α-8 Be impact parameter 0 fm E (center of mass) = 2 Me. V A. S. Umar, J. A. Maruhn, N. Itagaki, and V. E. Oberacker Phys. Rev. Lett. 104, 212503 (2010).

α-8 Be impact parameter 0. 2 fm E (center of mass) = 2 Me. V A. S. Umar, J. A. Maruhn, N. Itagaki, and V. E. Oberacker Phys. Rev. Lett. 104, 212503 (2010).

Lifetime of linear chain as a function of impact parameter A. S. Umar, J. A. Maruhn, N. Itagaki, and V. E. Oberacker Phys. Rev. Lett. 104, 212503 (2010).

20 C alpha chain states , Ex ~ 15 Me. V region Skyrme Hartree-Fock calculation Sk. I 3 Sly 6 Sk. I 4 Sk. M* J. A. Maruhn, N. Loebl, N. Itagaki, and M. Kimura, Nucl. Phys. A 833 (2010).

J. A. Maruhn, N. Loebl, N. Itagaki, and M. Kimura, Nucl. Phys. A 833 (2010).

Stability of 3 alpha linear chain with respect to the bending motion Time Dependent Hartree-Fock calculation Geometric shape is stabilized by adding neutrons in (σ)2 16 C (π)4 20 C (π)4(δ)2(σ)2

How can we stabilize geometric cluster shapes like linear chain configurations? • Adding valence neutrons • Rotating the system

4 alpha linear chain in rotating frame Pioneering work, but no spin-orbit, no path to bending motion

Linear chain configuration appears when angular momentum is given, however…. . • Initial state is one-dimensional configuration stability with respect to the bending motion was not discussed • Spin-orbit interaction was not included in the Hamiltonian

Cranked Hartree-Fock calculation

T. Ichikawa, J. A. Maruhn, N. Itagaki, and S. Ohkubo, Phys. Rev. Lett. 107, 112501 (2011).

Ichikawa et al. ＭＯＩ = 0. 06 Me. V Ex(0) = 40 Me. V

Tohsaki interaction

The advantages of Tohsaki interaction • Saturation property is satisfied • Size parameter dependence of 4 He is small and radius and binding energy of 4 He are reasonably reproduced • 4 He- 4 He scattering phase shift is reproduced • It is rather easy to perform angular momentum projection and/or superposition of different states, since the Hamiltonian is in the operator form (density dependence is expressed as finite-range three-body interaction)

α-α scattering phase shift Tohsaki F 1, F 2 BB, SII A. Tohsaki, Phys. Rev. C 49 1814 (1994)

4 alpha chain using Tohsaki interaction MOI = 0. 07 Me. V Ex(0) = 38 Me. V

Y. Iwata, T. Ichikawa, N. Itagaki, J. A. Maruhn, and T. Otsuka, Phys. Rev. C 92 011303(R) (2015)

Y. Iwata, T. Ichikawa, N. Itagaki, J. A. Maruhn, and T. Otsuka, Phys. Rev. C 92 011303(R) (2015)

Y. Iwata, T. Ichikawa, N. Itagaki, J. A. Maruhn, and T. Otsuka, Phys. Rev. C 92 011303(R) (2015)

Y. Iwata, T. Ichikawa, N. Itagaki, J. A. Maruhn, and T. Otsuka, Phys. Rev. C 92 011303(R) (2015)

Y. Iwata, T. Ichikawa, N. Itagaki, J. A. Maruhn, and T. Otsuka, Phys. Rev. C 92 011303(R) (2015)

Y. Iwata, T. Ichikawa, N. Itagaki, J. A. Maruhn, and T. Otsuka, Phys. Rev. C 92 011303(R) (2015)

Coherent effect of adding neutrons and rotating the system P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015). Exotic shape in extreme spin and isospin • Code – TAC 3 D Cartesian harmonic oscillator basis with N=12 major shells • Density functional DD-ME 2 A. Lalazissis, T. Nikšić, D. Vretenar, and P. Ring, Rev. C 71, 024312 (2005). G. Phys.

P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

Rigid rotor J = I ω P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

neutrons P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

valence neutrons P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

protons P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

P. W. Zhao, N. Itagaki, and J. Meng, Phys. Rev. Lett. 115 022501 (2015).

Summary The stability of exotic cluster state can be studied with mean field models as well as cluster models Two mechanisms to stabilize the rod shape, rotation (high spin) and adding neutrons (high Isospin) coherently work in C isotopes Coherent effects: Rotation makes the valence neutron-orbit in the deformation axis (σ-orbit) lower Enhances the prolate deformation of protons (kinetic) Pull down the proton orbits in one dimension (interaction) In 15 C-20 C σ orbit(s) is occupied as a lowest configuration of neutrons around 3 alpha linear chain in the rotating frame