Exotic clustering in neutron-rich nuclei and connection to the shell structure Naoyuki Itagaki Yukawa Institute for Theoretical Physics Kyoto University, Japan CNS Summer School 2015

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

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

Nuclear structure weak binding state of strongly bound subsystems Excitation energy decaying threshold　　　 to subsystems single-particle motion of protons and neutrons Nuclear structure

α-cluster structure 4He is strongly bound (B.E. 28.3 MeV) 　　Closed shell configuration of the lowest s shell  This can be a subunit of the nuclear system

8Be rigid rotor of alpha-alpha

2+ 0+ Synthesis of 12C from three alpha particles 0+2 Ex =7.65 MeV Γα 3α threshold Ex = 7.4 MeV Γγ 2+ Γγ 0+ The necessity of dilute 3alpha-cluster state has been pointed out from astrophysical side, and experimentally confirmed afterwards

Let us move on to the neutron-rich side 9Be Sn= 1.66454 MeV Qα= 2.308 MeV No bound state for the two-body subsystems -- Boromian system

1/2+ 3/2- around the threshold ground state 1/2+ 3/2- around the threshold ground state S. Okabe and Y. Abe, Prog. Theor. Phys. 61 1049 (1979).

中性子過剰核で p1/2とs1/2が逆転している？

Nilsson diagram Bohr-Nottelson Nuclear Structure

1/2[220] means J=1/2 N=2 Nz=2 Lz=0 Neutron stays on the deformation axis, where the curvature of the potential and corresponding ħω value for that axis is smaller than those of other axes E ~ ħ ( nxωx + nyωy + nzωz )

lowering ħω = 20 MeV ħω = 10 MeV

In the case of Be isotopes The optimal distance of alpha-alpha core just corresponds to the super deformation region The alpha-alpha clustering of the core could be a clue for the lowering of ½+ orbit (another effect is ``halo” nature of the s-wave)

10Be Introducing molecular orbits for valence neutrons around two α clusters 10Be

``New” physics in 10Be The α-α clustering of the 8Be core is even enhanced compared with free 8Be, when the two valence neutrons occupy (σ)2 configuration This enhanced cluster configuration forms a rotational band structure starting with the second 0+ state of 10Be (the level spacing is much smaller than 8Be)

α+α+n+n model for 10Be large (σ)2 contribution  thick line N. Itagaki and S. Okabe, Phys. Rev. C 61 013456 (2000).

10Be （σ）2 πσ （π）２ N. Itagaki and S. Okabe, Phys. Rev. C 61 044306 (2000)

A.A. Korsheninnikov et al., Phys. Lett. B 343 53 (1995).

M. Freer et al., Phys. Rev. Lett. 82 1383 (1999).

Breaking of N=8 magic number in 12Be N. Itagaki, S. Okabe, K. Ikeda, Physical Review C62 (2000) 034301 α＋α＋4N model for 12Be Breaking of N=8 magic number in 12Be

Disappearance of N=8 magic number in 12Be Observation of a low-lying 1- state at 2.68(3) MeV at RIKEN H. Iwasaki et al., Phys. Lett. B491 (2000) 8.

Observation of the 0+2 state of 12Be at Ex = 2.24 MeV

12Be M. Ito, N. Itagaki, H. Sakurai, and K. Ikeda Phys. Rev. Lett. 100, 182502 (2008).

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

The α particle model The emission of α particles has been known already in the end of 19th century (even before the discovery of atomic nuclei), and it was quite natural to consider that α particles are basic constituents of nuclei However independent particle motion became the standard picture of nuclear structure in 1950s, since the shell structure is established The cluster studies restarted in 1960s

Mysterious 0+ problem in 16O The first excited state of 16O is 0+2??

Inversion doublet structure (12C+α model) Y. Suzuki Prog. Theor. P hys. 56 111 (1976)

Large scale shell-model calculation (6hω) (diagonalization of 86,000 states) Four single particle energies are adjusted to fit the calculated energy levels W.C. Haxton and Calvin Johnson, Phys. Rev. Lett. 65 1325 (1990).

12C+12C molecular states

Ikeda daiagram Alpha gas states around the thresholds Inversion doublet (Mysterious 0+) Molecular resonances

Examples of the cluster study until 1980s (mainly α nuclei) 8Be 　 α+α 9Be　　 α+α+n　 12C　　　３α 16O　　　α＋12C 20Ne　 　α＋16O 24Mg 12C+12C, α＋α＋16O 44Ti α＋40C

Various 3α models 3α calculations by RGM, GCM, OCM Supplement of Prog. Theor. Phys. 82 (1980).

If you expand cluster wave function using Harmonic oscillator basis…. Y. Suzuki, K. Arai, Y. Ogawa, and K. Varga, Phys. Rev. C 54 (1996) 2073.

Resonating Group Method (RGM) J.A. Wheeler, Phys. Rev. 52 (1937) GCM and RGM Ｒ α α Resonating Group Method (RGM) J.A. Wheeler, Phys. Rev. 52 (1937) Ψ = A [Φ(r’1, r’2, r’3)Φ(r’4 , r’5, r’6)] χ(r) Internal wave function of each α-cluster Relative wave function between α-clusters Generator Coordinate Method (GCM) J.A. Wheeler, Phys. Rev. 52 (1937), H. Margenau, Phys. Rev. 59 (1941) Ψ = ∑A [Φ(r1-R/2)Φ(r2-R/2)Φ(r3-R/2)Φ(r4-R/2) Φ(r5+R/2)Φ(r6+R/2)Φ(r7+R/2)Φ(r8+R/2)]

Resonating Group Method (RGM) Jacobi coordinate is introduced GCM and RGM Ｒ α α Resonating Group Method (RGM) Jacobi coordinate is introduced Generator Coordinate Method (GCM) No Jacobi coordinate and center of mass wave function is mixed Ψ = ∑A [Φ(r1-R/2)Φ(r2-R/2)Φ(r3-R/2)Φ(r4-R/2) Φ(r5+R/2)Φ(r6+R/2)Φ(r7+R/2)Φ(r8+R/2)]

Brink’s wave function (1965) = GCM Ψ= P[A(Φ1(r1)Φ2(r2)・・・・)] P: Angular momentum and parity projection operator A: Antisymmetrizer Φ1(r) = exp[-ν(r - R1)2]χ1 Gaussian-center parameter spin-isospin wave function αcluster is expressed as four nucleons (p,p,n,n) sharing the same R exp[-(x-X)2] = ∑ Xn Hn(x) exp(-x2) / n! Local Gaussian corresponds to the coherent state of many excited orbits of the shell-model

Two nucleon’s case (with the same spin and isospin (χ)) Similarity between shell model wave functions and cluster wave functions Two nucleon’s case (with the same spin and isospin (χ)) Φ1 = exp[ -ν(r – X)2 ] χ Φ2 = exp[ -ν(r + X)2 ] χ A [Φ1Φ2] ∝ A [(Φ1+Φ2)(Φ1 - Φ2) ] At X  0 (Φ1+Φ2)  exp[ -νr2] (Φ1 - Φ2) / |X|  r exp[ -νr2] Cluster (local Gaussian) wave function coincides with the lowest shell-model wave function at X  0

Restoration of the symmetry 1 -- Parity projection Ψ+ = (Ψ{R} + Ψ{-R}) / 2 Ψ- = (Ψ{R} - Ψ{-R}) / 2

Restoration of the symmetry 2 -- Angular momentum projection ΨI = PIMK Ψ

Summary for this part Cluster structure is important also in the neutron-rich systems Clustering is enhances depending on the neutron-orbit, and this is related to the disappearance of magic number N=8 Brief history and basic frameworks of cluster studies were introduced