Clustering in light neutron-rich nuclei

Clustering in light neutron-rich nuclei
Y. Kanada-En’yo (Kyoto Univ.) Lesson1: Introduction of cluster phenomena Lesson2: Antisymmetrized molecular dynamics Lesson3: clustering in neutron-rich Be Lesson4: clustering in 12C Lesson5: Monopole & Dipole excitations in light nuclei Today I will talk about cluster phenomena focusing on light neutron-rich nuclei These are my collaborators.

Lesson1.Introduction

Nuclear system atom nucleus Analogy & Differences A finite quantum
many-body system of protons and neutrons atom nucleus electrons proton neutron nuclear force: Attractions Analogy & Differences Electron motion Nucleon motion orbit, shell Self-bound Confined by the external field A nucleus is a finite quantum many-body system consisting of protons and neutrons which are interacting through nuclear force. Of course, nucleons are fermions, and one may find some analogy and differences between nuclear system and atomic system. Most important difference is that nuclear force is attractive and a nucleus is a self-bound system. One of the essential features of nuclear systems is independent-particle behavior in a mean field resulting in shell structures. On the other hand, spatial correlations among nucleons are rather strong due to the attractive nuclear force. A typical example of spatial correlation is "Cluster" that is a sub unit formed by spatially correlating nucleons. That means, in nuclear system, these two kinds of nature, independent-particle feature in a mean field and multi nucleon correlations coexists . Because of this coexistence, rich phenomena appears depending on proton and neutron numbers and excitation energy. 1. Independent-particle feature in self-consistent mean-field 2. Strong nucleon-nucleon correlations cluster

Cluster & Mean field v.s. Shell structure・MF Cluster Cluster:
Mean field, shell structure Independent single-particle Cluster: Many-body correlation Shell structure・MF v.s. Cluster Independent particle in MF Cluster excitation Cluster formation Let me briefly review cluster physics. What is the “clustering” in nuclear systems. In nuclear systems, the mean-field fearture and the cluster feature coexist in low-energy region. The mean-field feature is characterized by independent single-particle motion, and the cluster aspect comes from many-body correlations. The key word is “correlation” The coexistence of two kinds of feature, mean-field and cluster, brings rich phenomena depending on proton and neutron numbers and also depending on the excitation energy and the density. If there is no correlation between nucleons, all nucleons behave as independent particle in a mean field and there exists the shell structure. However, in reality, the nuclear force is attractive and when we switch on the residual interaction, correlation between nucleons occurs to form cluster cores at the nuclear surface. This is nothing but the cluster formation in low-lying states, so-called ground state correlation. there, clusters are largely overlapping with each other and the system is a compact state with the normal density. Once clusters are formed in the system, inter-cluster motion can be activated easily with small amount of energy. Then spacially developed cluster structures appear in excited states because of the excitation of inter-cluster motion, and the system goes to a low-density state. no correlation many-body correlation Developed cluster 12C ground state 12C excited states

Cluster structures 12C r0 r0/3~ r0/5 Nuclear structure + high density
0 MeV Energy 10 MeV 100 MeV Shell & cluster corr. cluster excitation p3/2-closed & 3a 12C + Z=N=6 Nucleon liquid Fermi gas, BCS Cluster gas, crystal Nucleon gas high density low density r0 12C is a typical example of coexisiting cluster and mean-field features. The ground state of 12C is the mean-field state dominated by the sub-shell closed configuration with a mixing of 3-alpha cluster core structure. That is, the alpha clusters are partially formed in the ground state. At approximately 100 MeV, all twelve nucleons in the 12C can dissociate, and the system goes to a free nucleon gas state in a low density limit. At approximately 10 MeV, much lower than the energy of free nucleon gas, three alpha clusters develop spatially in excited states of 12C. The energy of 3 alpha cluster excitation is much smaller than that of the nuclear gas state. This imply that the mean-field and cluster states coexist in the low-energy levels of 12C. %%%% Moreover, with the increase of the excitation energy from the ground to the excited states, the density of the system varies from normal density to low density. As the density changes, we find the remarkable cluster feature at the low density region typically around one third of the normal density we see the cluster enhancement. This structure change is similar to the phase transition in infinite nuclear matter as a function of density. r0/3~ r0/5 Cluster enhancement at low density Nuclear surface Light-mass nuclei Excited states Heavy ion collision

Energy rule for cluster states (Extended) Ikeda diagram
Remarkable cluster states are predicted to appear near the threshold energy Extended Ikeda diagram for n-rich nuclei von Oertzen et al (2006) Ikeda diagram Ikeda et al. PTP464-S (1968) from Phys. Report 432 (2006)

Cluster structures in stable and unstable nuclei
Typical cluster structures known in stable nuclei 7Li 8Be 12C 20Ne 16O* 12C + a a + t a + a 3a 16O + a Heavier nuclei Unstable nuclei Si-Si, Si-C, O-C, O-O a-cluster excitation 3a linear chain a variety of cluster states have been discovered not only in stable nuclei but also in unstable nuclei, in particular, neutron-rich nuclei, in which excess neutrons play an important role in weakening and enhancement of cluster structures. a a-cluster 14C* 36Ar-a, 24Mg-a, 28Si-a Molecular orbital 40Ca*, 28Si*, 32S* Be, C, O, Ne, F

Z-, N-dependence of g.s. structure
7Li 9Li 11Li Z=3 Vanishing of Magic number N=8 8Be 10Be 12Be Z=4 Cluster Cluster 19B 11B 13B 15,17B Z=5 a variety of cluster states have been discovered not only in stable nuclei but also in unstable nuclei, in particular, neutron-rich nuclei, in which excess neutrons play an important role in weakening and enhancement of cluster structures. 10C 12C 14C 16C 18C 20C Neutron Skin Z=6 Shape difference N

Two kinds of remarkable clustering in n-rich Be* and Ne* isotopes
Soic et al., Freer et al., Saito et al., Curtis et al., Milin et al., Bohlen et al., Scholz et al., Rogachev et al., Goldberg et al., Ashwood et al., Yildiz et al., Descouvemont, Kimura, 10Be 22Ne Seya, Von Oerzten, Descouvemont et al., Itagaki et al., Dote et al., K-E et al. Arai et al., M. Ito et al. 18O+a 6He+He Atomic: Cluster resonance 16O Weak coupling a a a Molecular Orbital: s-bond structure 16O Strong coupling a a a Let me here focus on the cluster structures in Be isotopes. In these decades, structures of ground and excited states of neutron-rich Be have been intensively studied theoretically and experimentally. It was found that two kinds of cluster structures exist in Be. In neutron-rich Be, two alpha cluster cores are formed even in the ground state. In the case of 10Be, the second 0+ state has the molecular orbital structure with a large deformaion. Here the valence neutrons occupy the longitudinal molecular orbital called sigma-orbital around the 2 alpha, and two alpha clusters are tightly bonded by the sigma-orbital neutrons. So we call this MO sigma-bond structure. In highly excited states, dinuclear-type cluster resonances of 6He and alpha clusters have been suggested very recently. What is the difference between cluster resonances and the molecular sigma-bond structure. In the MO sigma-bond structure, two valence neutrons are moving around both alpha clusters bonding two alpha clusters, while in the case of cluster resonances, valence neutrons are localized around one of alpha clusters to form a 6He cluster and another cluster is moving around the 6He cluster. Similar cluster phenomena have been also suggested in in neutron-rich sd-shell nuclei such as Ne, F, O isotopes. The MO sigma-bond structure and cluster resonances were suggested. 16O shell model-like a a a Normal states Be isotopes Ne, F, O isotopes

MO bond and Cluster res. in 22Ne
Exp Scholz et al., Rogachev et al., Goldberg et al.,Ashwood et al., Yildiz et al., Theor: Descouvemont, Kimura, AMD study by Kimura, PRC75 (2007) 22Ne 18O+a 16O Cluster res. a 16O MO s-bond a As an example, here I show you the cluster structures in 22Ne Studied by Kimura by means of AMD method. What he found is that the molecular orbital sigma-bond structure is formed in the excited state having a developed O+alpha cluster system. Because of the large deformation, the sigma-bond structure constructs a rotational band. Above the molecular orbital sigma-bond structure, he obtained cluster resonances of 18O+alpha, where valence neutrons are localized around the 16O core to form the 18O cluster. EXP AMD

a-cluster states in n-rich nuclei
Cluster resonances a New states discovered and suggested at Ex = several ~ 10 MeV in a-decay, a-transfer, a-scattering 6,8He+a in Be 10Be+a in 14C 14C+a in 18O 18O+a in 22Ne Exp: Soic et al., Freer et al., Saito et al., Curtis et al., Milin et al., Bohlen et al., Theor: Seya, von Oerzten, Descouvemont et al.,Itagaki et al., K-E et al. Arai et al., M. Ito et al. Exp Soic 04, von Oertzen ‘04, Price 07, Haigh 08, Theor: Suhara ‘10 Exp Scholz et al., Rogachev et al., Goldberg et al.,Ashwood et al., Yildiz et al., Theor: Descouvemont, Kimura, Exp Scholz ‘72, Rogachev ‘01, Goldberg ‘04, Ashwood ‘06, Yildiz et al., Theor: Descouvemont ’88, Kimura ‘07 -> information of nucleus-nucleus potential and valence neutron effects there.

multi-cluster: gas, triangle, linear chain
Bosons in low density a particles occupy a same orbit. 03 + 22 + 31 - b 02 + R0 7.65 MeV BEC in nuclear matter 8Be+a condensation may occur in a dilute nuclear matter Roepke et al., PRL(1998) cluster gas of 3a 01 + +

multi-cluster: gas, triangle, linear chain
16O* Tohsaki et al., Yamada et al., Funaki et al. Wakasa et al., chain？ triangle？ α 12C α α 4a gas α 03 + 22 + 31 - 11C*, 11B* K-E. , Kawabata et 02 + α 7.65 MeV 2a+t gas α 8Be+a t cluster gas of 3a Itagaki, Suhara 14C* 01 + + Triangle, linear a-hain

Rich cluster phenomena in nuclear systems
as functions of proton&neutron numbers and excitation energy Cluster formation/breaking in low-lying states MO s-bond in neutron-rich nuclei Cluster resonances Many clusters : cluster gas, a-chain New types of clusters We will see rich cluster phenomena as functions of proton and neutron numbers and excitation energy. Our aim is to make systematic study of these various cluster phenomena and to search for new modes in neutron-rich nuclei. For this aim we apply a method of AMD, antisymmetrized molecular dynamics. 6,8He+He in Be, 10Be+a in 14C,14C+a in 18O, 18O+a in 22Ne A theoretical method: AMD (antisymmetrized molecular dynamics)

Cluster and mean-field aspects
End of Lesson1 Cluster and mean-field aspects Rich cluster phenomena as functions of proton/neutron numbers and excitation energy

Lesson 2 A theoretical model: AMD
An approach for nuclear structure to study cluster and mean-field aspects Stable and unstable nuclei Ground and excited states What is AMD ? The AMD is an approach for nuclear structure study which can describes cluster and mean-field aspects in ground and excited states of stable and unstable nuclei.

Model setting: AMD wave function
AMD wave fn. Similar to FMD wave fn. Variational parameters: 　　Gauss centers, spin orientations det isospin Gaussian wave packet Intrinsic spins det AMD is a model based on the variational method using effective nuclear forces. An AMD wave function is given by a Slater determinant of single-particle wave functions written by a Gaussian wave packet localized at a certain position Zi. These parameters, centers of Gaussian wave packets for all nucleons, are independently treated as variational parameters. det Cluster and MF formation/breaking A variety of cluster st. Shell-model states 17

Energy variation Variation, Jp -projection Variation after/before
Model wave fn. Phenomenological effective nuclear force For Volkov, MV1, Gogny etc. Randomly chosen Initial states Variation, Jp -projection Energy surface Variation after/before Jp –projection (VAP/VBP) Constraint AMD+GCM With this model wave function, a variety of cluster structure can be explained by the spatial configuration of Gaussian centers. Also, mean field states can be described deu to the antisymmetrization of single-partcle wave function. Then we perform energy variation hin the AMD model space. In this model, we do not assume existence of any clusters but treat all nucleons independently. Nevertheless, if a cluster structure is favored in a system, such the structure is automatically obtained in the energy variation because the model space contains various cluster strucutres. This method is one of the powerful tools to study the formation or breaking of various clusters as well as mean field features in ground and excited states of general nuclei. Model space (Z plane) Energy minimum 18

Cluster limit v.s. Shell-model limit of AMD functions
Identical 2 fermions Anti- Symm. with 2 Gaussians at short distance s-orbit ＋＝ (0s)(0p) config. p-orbit ＋＝ 2a system d→0 limit large d Developed 2a s4p4 (0s)4 (0p)4 Strongly correlated

Cluster: strongly correlated state
Basis expansion from a center ｓmall Large Small Mixing of high angler momentum

8Be (g.s.) large d Developed 2a Strong spatial correlation involves higher-shell components. SVM by Arai. et al. PRC (1996) 8Be (g.s.) Developed cluster states are usually beyond mean-field

End of Lesson 2 A theoretical model: AMD
An approach for nuclear structure to study cluster and mean-field aspects Stable and unstable nuclei Ground and excited states What is AMD ? The AMD is an approach for nuclear structure study which can describes cluster and mean-field aspects in ground and excited states of stable and unstable nuclei.

Lesson 3. cluster phenomena in neutron-rich Be isotopes
Today I would like to discuss some topics of cluster phenomena in light nuclei.

Contents of Lesson 3 Z-,N-dependence of g.s. structure
Molecular orbital structure magic number breaking Cluster resonances The first topic is the cluster structure of neutron-rich Be isotopes.

Z-,N-dependence of g.s. structure Molecular orbital structure
magic number breaking Cluster resonances The first topic is the cluster structure of neutron-rich Be isotopes.

7Li 9Li 11Li Z=3 Vanishing of Magic number N=8 8Be 10Be 12Be Z=4 Cluster Cluster 19B 11B 13B 15,17B Z=5 10C 12C 14C 16C 18C 20C Neutron Skin Z=6 Shape difference N

How to experimentally probe N-dependence of clustering
Clustering in g.s. Deformation 　Q, B(E2) Charge radii a-transfer, a knock-out ? Clustering in excited states a-decay width, a-transfer bp, rp enhanced isotope shift: Be charge changing s : B, C

Minimum proton radii? N=8 Largely deformed: proton radii enhanced by clustering ? 8Be 10Be 12Be Z=4 Cluster 19B 11B 13B 15,17B Z=5 Neutron Skin 10C 12C 14C 16C 18C 20C Z=6 Stable proton radii? N

proton radii along isotope chain
Exp data: Isotope shift W. Nortershauser　et al. PRL102, (2009); A. Krieger, PRL108, (2012). CC :Terashima et al. minimum at N=6 (10Be) new magic? Increases in N>6. bp Another experimental probe useful to pin down the cluster structure change along the isotope chain is charge radius The systematics of the charge radii of Be has been measured by isotope shift. In the theoretical calculation, the charge radius is minimum at 10Be for N=6 and it increases 11Be and 12Be because of the cluster development. The calculated charge radii are consistent with the tendency of the experimental data. The FMD calculation. proton radii(charge radii) reflect N dependence of clustering. Good probe for cluster structure 10Be 12Be

N dependence of rp in Be,B,C AＭＤ & exp. data
exp :determined by isotope shift exp-𝑟𝑐𝑐 :determined by charge changing(cc) cross section eval-𝑟𝑐𝑐 : evaluated by relation 𝜎𝑐𝑐∝𝜋 ( 𝑅 𝑝 𝑃 + 𝑅 𝑚 (𝑇)) 2 using cc cross section

magic number breaking Cluster resonances The first topic is the cluster structure of neutron-rich Be isotopes.

Cluster structures in neutron-rich Be
10Be: energy levels cluster res. 6He+4He AMD calc. Y. K-E, et al. PRC (98) 3,4 + Exp: Milin et al. ’05, Freer et al. ’06 Ito et al. Kobayashi et al. Kuchera et al. 10Be 2 + MO s-bond Let me first explain the cluster structures of 10Be again. The ground state is the normal state having a cluster core structure. Above the ground state, there exists the molecular orbital bond structure in the second 0+ state, in which 2 alpha clusters are bonded by valence neutrons in the longitudinal molecular orbitals. This state has the well developed cluster structure with a large deformation, constructs the rotational band. Experimentally, the band members 2+ and 4+ states of this rotational band starting from the second 0+ have been discovered in these years. The energy slope of the experimental band is consistent with the theoretical prediction supporting the large deformation of this band. Another interesting phenomena is the 6He+alpha cluster resonances, which have been suggested very recently around 10 MeV. 1 + Normal

Molecular orbital(MO) structure in Be
Seya PTP65(81), von Oertzen ZPA354(96) N. Itagaki PRC61(00), Y. K-E.. Ito PLB588(04) 2a-core formation MO formation MO s-bond state + - + α α s-orbital MO formation Gain kinetic energy in developed 2α system Normal state + α α The idea of molecular orbital has been proposed by Seya et al. and von Oertzen et al. In neutron-rich Be, two alpha cluster cores are spontaneously formed because of the many-body correlation in the A-nucleon system. Once two alphas cores are formed, molecular orbitals are constructed by the linear combination of p-orbit around each alpha cluster, and valence neutrons occupy the molecular orbitals. One is the normal pi-type orbital and the other is the higher nodal sigma orbital. If valence neutron occupy the pi orbital, the pi-orbital neutrons pull two alpha clusters to gain the potential energy. On the other hand, when valence neutron occupy the sigma orbital, the sigma orbital neutrons push two alpha clusters outward, because the sigma orbital has two nodes along the alpha-alpha direction and therefore its gain its kinetic energy with the increase of two-alpha cluster distance. In the normal ordering, the higher nodal sigma orbital is higher than the pi orbital. But because of this lowering mechanism the sigma orbital comes down to the lower energy region in the developed cluster system. As a result, the level inversion occurs and the neutron magic number breaks down in neutron-rich Be. Level inversion in 11,12,13Be - p-orbital vanishing of magic number in 11Be, 12Be, 13Be

Cluster formation, MO, and Cluster resonance
6He+4He 6,8He+6,4He 0+3,4? 3 + a a Exp: Kuchera et al. Theor: Ito et al. Kobayashi et al. a a Exp: Freer PRL.82(99)(06) Saito NPA738 (04) Yang PRL112 (14） 0+2 a a MO s-bond 0+2 a a Let me discuss the systematics of cluster structures in Be isotopes. 0+1 0+1 Normal a a a a 10Be 12Be

MO & magic number breaking in low-energy region
cluster reso. 6He+4He 6,8He+6,4He 0+3,4? 3 + a a Exp: Kuchera et al. Theor: Ito et al. Kobayashi et al. a a Exp: Freer PRL.82(99)(06) Saito NPA738 (04) Yang PRL112 (14） 0+2 MO s-bond a a 0+2 a a Normal We first focus on the low-lying state. These low-lying states are understood well by the molecular orbital picture. 0+1 0+1 Normal a a MO s-bond a a 10Be 12Be

Magic number breaking Cluster resonances The first topic is the cluster structure of neutron-rich Be isotopes.

N=8 magic number breaking
Level inversion in developed cluster system N=8 shell vanishes at N=7,8　(11Be, 12Be) Positive- and negative-parity states degenerate New magic N=6 appears? s-orbital neutron enhances clustering 8 6 Let me show you a schematic figure for the level inversion in the developed clustering. In the spherical shell model limit, the N=8 shell gap exists between p-shell and sd-shell. With the enhancement of the cluster structure, the p3/2 and p1/2 goes to the ls-favored and ls-unfavored pi-orbitals, and a d orbit becomes the sigma-orbital. In the well developed 2alpha cluster system, the longitudinal sigma orbital gain its kinetic energy and come down below the ls-unfavored pi orbital. Therefore, in the MO configuration, the N=8 magic number disappears. Here I show you the systematics of the low-energy spectra for 10Be, 11Be, 12Be, 13Be, and 14Be. The experimental levels are shown for 10Be to 12Be, whereas, the theoretical predictions are shown for 13Be and 14Be. The sigma orbital configurations come down to the lowest in 11Be and 12Be indicating the breaking of N=8 magicity. Also for 13Be, the neutron magic number breaking is predicted theoretically. The level ordering looks abnormal in the spherical shell mode configuration, but it is rather natural in the MO configuration. In the MO configuration, the magic number is 6 and the major shell consists of the sigma-orbital and the ls-unfavored pi-orbitals. Then, 10Be is the MO closed nuclei, whereas, 11Be, 12Be, 13Be are regarded as the open-shell nuclei. Indeed, these low-lying states are described by the major-shell configurations. spherical enhanced cluster core

11Be(g.s.) p1/2 p1/2 s2 sd s1/2 6 s1 p3/2 p3/2 ? p3/2 2a s s0 Let me show you a schematic figure for the level inversion in the developed clustering. In the spherical shell model limit, the N=8 shell gap exists between p-shell and sd-shell. With the enhancement of the cluster structure, the p3/2 and p1/2 goes to the ls-favored and ls-unfavored pi-orbitals, and a d orbit becomes the sigma-orbital. In the well developed 2alpha cluster system, the longitudinal sigma orbital gain its kinetic energy and come down below the ls-unfavored pi orbital. Therefore, in the MO configuration, the N=8 magic number disappears. Here I show you the systematics of the low-energy spectra for 10Be, 11Be, 12Be, 13Be, and 14Be. The experimental levels are shown for 10Be to 12Be, whereas, the theoretical predictions are shown for 13Be and 14Be. The sigma orbital configurations come down to the lowest in 11Be and 12Be indicating the breaking of N=8 magicity. Also for 13Be, the neutron magic number breaking is predicted theoretically. The level ordering looks abnormal in the spherical shell mode configuration, but it is rather natural in the MO configuration. In the MO configuration, the magic number is 6 and the major shell consists of the sigma-orbital and the ls-unfavored pi-orbitals. Then, 10Be is the MO closed nuclei, whereas, 11Be, 12Be, 13Be are regarded as the open-shell nuclei. Indeed, these low-lying states are described by the major-shell configurations. 11Be(1/2-) p3s0 11Be(1/2+) p2s1 N=6 N=8 N=10 Parity inversion at N=7

12Be(g.s.) p1/2 p1/2 s2 sd s1/2 6 s1 p3/2 p3/2 ? p3/2 2a s s0 Let me show you a schematic figure for the level inversion in the developed clustering. In the spherical shell model limit, the N=8 shell gap exists between p-shell and sd-shell. With the enhancement of the cluster structure, the p3/2 and p1/2 goes to the ls-favored and ls-unfavored pi-orbitals, and a d orbit becomes the sigma-orbital. In the well developed 2alpha cluster system, the longitudinal sigma orbital gain its kinetic energy and come down below the ls-unfavored pi orbital. Therefore, in the MO configuration, the N=8 magic number disappears. Here I show you the systematics of the low-energy spectra for 10Be, 11Be, 12Be, 13Be, and 14Be. The experimental levels are shown for 10Be to 12Be, whereas, the theoretical predictions are shown for 13Be and 14Be. The sigma orbital configurations come down to the lowest in 11Be and 12Be indicating the breaking of N=8 magicity. Also for 13Be, the neutron magic number breaking is predicted theoretically. The level ordering looks abnormal in the spherical shell mode configuration, but it is rather natural in the MO configuration. In the MO configuration, the magic number is 6 and the major shell consists of the sigma-orbital and the ls-unfavored pi-orbitals. Then, 10Be is the MO closed nuclei, whereas, 11Be, 12Be, 13Be are regarded as the open-shell nuclei. Indeed, these low-lying states are described by the major-shell configurations. 12Be(1-) p3s1 12Be(0+2) p4s0 N=6 N=8 N=10 12Be(0+1) p2s2 Largely deformed ground state at N=8

Experimental evidence of N=8 shell vanishing in 12Be
Y.K-E.PRC (03),(12) , Ito PRL(08) Dufour NPA(10) Fortune PRC(06), Blanchon PRC(10) Energy 12Be deformation in 12Be(gs) 12Be g.s. 0+ Inelastic scat. life time: Iwasaki PLB481(00), Imai PLB673(09) Normal 0hw 2 + p4 intruder config. in 12Be(gs) 1n-knockout reac.: Navin PRL85(00), Pain PRL96(06) s2p2 Intruder 2hw 1 + 12Be(02+) with p-shell config. Shimoura PLB654 (07) B(GT) with charge ex.: Meharchand PRL108 (12)

Inelastic scattering p(12Be,12Be*)
+ 0hw Inelastic scattering 9 + 2 Takashina, Y.K-E., et al. PRC77 (08) Exp: Iwasaki PLB481 (00) + 2 + 1 2 + 1 2 2hw 7.9(2.9) 13 + 1 + 1 s2p2 exp AMD B(E2) e2fm4 B(GT) ratio: B(GT;0+2)/B(GT;0+1)=1.16 (exp) =1.1 (AMD)

How about 13Be (N=9) ? 13Be(g.s.) ? 2a p3s2 s2 6 s1 s0 13Be(1/2+,5/2+)
sd s1/2 6 ? s1 p3/2 3/2 p3/2 2a s s0 Normal N=9 Let me show you a schematic figure for the level inversion in the developed clustering. In the spherical shell model limit, the N=8 shell gap exists between p-shell and sd-shell. With the enhancement of the cluster structure, the p3/2 and p1/2 goes to the ls-favored and ls-unfavored pi-orbitals, and a d orbit becomes the sigma-orbital. In the well developed 2alpha cluster system, the longitudinal sigma orbital gain its kinetic energy and come down below the ls-unfavored pi orbital. Therefore, in the MO configuration, the N=8 magic number disappears. Here I show you the systematics of the low-energy spectra for 10Be, 11Be, 12Be, 13Be, and 14Be. The experimental levels are shown for 10Be to 12Be, whereas, the theoretical predictions are shown for 13Be and 14Be. The sigma orbital configurations come down to the lowest in 11Be and 12Be indicating the breaking of N=8 magicity. Also for 13Be, the neutron magic number breaking is predicted theoretically. The level ordering looks abnormal in the spherical shell mode configuration, but it is rather natural in the MO configuration. In the MO configuration, the magic number is 6 and the major shell consists of the sigma-orbital and the ls-unfavored pi-orbitals. Then, 10Be is the MO closed nuclei, whereas, 11Be, 12Be, 13Be are regarded as the open-shell nuclei. Indeed, these low-lying states are described by the major-shell configurations. 13Be(1/2+,5/2+) or N=6 N=8 N=10 Parity inversion 13Be(1/2-) p3s2

Breaking of magic number in 13Be
13Be: unbound 13Be spectra measured by 1n knock-out reactions at GSI(Simon et al. 2007) and RIKEN(Kondo et al.,2010) AMD calc. Y. K-E. PRC(12) 13Be Er (MeV) EXP. (1/2-) 3.04 (5/2+) (5/2+) 2.39 2 2.0 AMD Kondo10 Simon07 1 12Be+n 1/2- ? (1/2-) d5/2 1/2+ virtual st. 0.51 intruder 1/2- is lowest in 13Be p1/2 s1/2 12Be+n Simon07 Kondo10 p3/2 s1/2

13Be spectra Kondo et al. 2010 They observed resonances in p- and d-waves, and s-wave continuum or virtual state, and concluded that Intruder 1/2- state is the lowest: N=8 shell breaking

Change of cluster structure
s-orbital neutron enhances clustering p-orbital neutron suppresses clustering s2 s1 Let me show you a schematic figure for the level inversion in the developed clustering. In the spherical shell model limit, the N=8 shell gap exists between p-shell and sd-shell. With the enhancement of the cluster structure, the p3/2 and p1/2 goes to the ls-favored and ls-unfavored pi-orbitals, and a d orbit becomes the sigma-orbital. In the well developed 2alpha cluster system, the longitudinal sigma orbital gain its kinetic energy and come down below the ls-unfavored pi orbital. Therefore, in the MO configuration, the N=8 magic number disappears. Here I show you the systematics of the low-energy spectra for 10Be, 11Be, 12Be, 13Be, and 14Be. The experimental levels are shown for 10Be to 12Be, whereas, the theoretical predictions are shown for 13Be and 14Be. The sigma orbital configurations come down to the lowest in 11Be and 12Be indicating the breaking of N=8 magicity. Also for 13Be, the neutron magic number breaking is predicted theoretically. The level ordering looks abnormal in the spherical shell mode configuration, but it is rather natural in the MO configuration. In the MO configuration, the magic number is 6 and the major shell consists of the sigma-orbital and the ls-unfavored pi-orbitals. Then, 10Be is the MO closed nuclei, whereas, 11Be, 12Be, 13Be are regarded as the open-shell nuclei. Indeed, these low-lying states are described by the major-shell configurations. s0 N=6 N=8 N=10

cluster enhancement by s-orbital neutrons
p-orbital neutrons suppress clustering. s-orbital neutrons enhance clustering

cluster enhancement by s-orbital neutrons
deformed ground states p-orbital neutrons suppress clustering. s- orbital neutrons enhance clustering

Magic number breaking Cluster resonances The first topic is the cluster structure of neutron-rich Be isotopes.

Cluster resonances cluster reso. 6He+4He 6,8He+6,4He 0+3,4? 0+2
3 + a a Exp: Kuchera et al. Theor: Ito et al. Kobayashi et al. a a Exp: Freer PRL.82(99)(06) Saito NPA738 (04) Yang PRL112 (14） 0+2 MO bond a a 0+2 a a Normal Let me next discuss cluster resonances in highly excited states. 0+1 0+1 Normal a a MO bond a a 10Be 12Be

Spectra of 10Be cluster res. 10Be MO s-bond Normal 10Be: energy levels
6He+4He AMD calc. Y. K-E, et al. PRC (98) 3 + Exp: Milin et al. ’05, Freer et al. ’06 MeV 4+ Ito et al. Kobayashi et al. Kuchera et al. 4+ 2+ 2 + MO s-bond 0+ exp AMD 1 + 2+ 10Be Normal J(J+1)

Theoretical predictions of cluster resonances in 0+ spectra of 10Be
MO bond In many theoretical calculations, cluster resonances are predicted around 10 to 15 MeV excitation energy. It is a challenging problem to experimentally search for the 6He and 4He resonances. Theor. Y. K-E. et al. PRC60, (1999) T. Suhara and Y. K-E. et al., PTP123, 303 (2010) F. Kobayashi and Y. K-E., Phys.Rev. C 86, (2012) Arai et al., PRC69, (2004). Ito et al., PLB636, 293 (2006). P. Descouvemont et al., NPA 699 (2002) 463 Exp. Kuchera et al.PRC 84, (2011).

Energy spectra of 12Be 12Be 12Be 6He+6He, 8He+4He?
VAP calculation with AMD method 12Be positive parity states withnormal spins 6He+6He, 8He+4He? + 3 12Be Cluster reso. + 2 normal + 1 In 12Be, the MO bond structure comes down to the lowest state. Because of the large deformation, it construct the rotational band. The normal p-shell closed state appears as the second 0+ state. This indicates the vanishing of the N=8 neutron magic number in 12Be, which have been supported by various experiments. In highly excited states, cluster resonances of 6He+6He and 8He+alpha cluster have been theoretically predicted, and indeed observed experimentally in the He+He decays. intruder MO s-bond state Breaking of N=8 magicity Formation of 2a+molecular orbitals Y.K et al., PRC 68, (2003)

Theoretical predictions and observed states in 12Be
AMD by K-E. Kp=0+3 Kp=0- 8He+a 12Be 6He+6He a a a a 8He+a Kp=1- 1p-1h Kp=0+2 a a a a Kp=0+1 a a GCM by Dufour GTCM by Ito Exp. Freer PRL.82(99)(06) Saito NPA738 (04) Yang PRL112 (14） For 12Be, many rotational bands have been predicted by AMD, GCM, and GTCM calculations. It is interesting that two kinds of negative parity bands are predicted. One is the K=1- rotational band arising from the 1p-1h excitation, and the other is the K=0- band caused by the cluster excitation as the parity partner of the 8He+alpha clustering. There are some candidates for the low-lying bands and also He+He cluster resonances. Theor. Y. K-E. et al. PRC68 (03) Ito et al., PRC85 (12) Dufour et al., NPA836 (10) 242

a-cluster states in n-rich nuclei
Cluster resonances a New states discovered and suggested at Ex = several ~ 20 MeV in a-decay, a-transfer, a-scattering 6,8He+a in Be* 10Be+a in 14C* 14C+a in 18O* 18O+a in 22Ne* Exp: Soic et al., Freer et al., Saito et al., Curtis et al., Milin et al., Bohlen et al., Theor: Seya, von Oerzten, Descouvemont et al.,Itagaki et al., K-E et al. Arai et al., M. Ito et al. Exp Soic 04, von Oertzen ‘04, Price 07, Haigh 08, Fritsch ’16, Yamaguchi Theor: Suhara ‘10 Exp Scholz et al., Rogachev et al., Goldberg et al.,Ashwood et al., Yildiz et al., Theor: Descouvemont, Kimura, Exp Scholz ‘72, Rogachev ‘01, Goldberg ‘04, Ashwood ‘06, Yildiz et al., Theor: Descouvemont ’88, Kimura ‘07 Universal phenomena? Further experimental and theoretical studies are requested.

End of Lesson 3 Cluster phenomena in n-rich Be
A variety of cluster phenomena in the ground and excited states. valence neutrons play important roles What is AMD ? The AMD is an approach for nuclear structure study which can describes cluster and mean-field aspects in ground and excited states of stable and unstable nuclei.

Lesson 4. Clustering in 12C

Contents of Lesson 4 Introduction Cluster states in 12C
Rotation of cluster gas a-chain The first topic is the cluster structure of neutron-rich Be isotopes.

Cluster states in low energy
Bending chain? 100 MeV 3a triangle Nucleon gas nucleon breakup 12C 6 protons 6 neutrons 3a 3a Cluster gas low density 12C(0+2) 10 MeV 3a-cluster Hoyle state cluster excitation Impact to nucleosynthesis Saturation density + 0 MeV Liquid drop Shell & cluster formation(correlation)

Cluster states in excited states
Various modes of cluster excitation, and their rotation 12C Geometric ? Bending chain? 31 - 03 + 03,4 + triangles 22 + 31 - FMD, AMD 7.65 MeV 02 + 8Be+a dilute cluster gas bosonic behavior Hoyle st. Uegaki et al. (1977) Funaki et al. (2003) Tohsaki et al.(2001) 01 + Cluster breaking analogy to BEC in nuclear matter 3a Roepke et al., PRL(1998) + 3a p3/2-shell

A long history of the Hoyle state: 12C(0+2)
1954: predicted by Hoyle in astrophysical interest. 3a reaction in nucleosynthesis. 1953, 57:Experimental discovery (Dunbar,Cook) 1956: Possible linear 3a-chain structure (Morinaga) 1972: linear 3a-chain was excluded for the Hoyle state from its large a-decay width (Suzuki) 1970’s~ 3a-cluster model calculations 1977 GCM cal.: dilute gas of bosonic particles (Uegaki) 2003: a condensed state (Tohsaki) 2002-:Ａｌｇｅｂｒａｉｎｃ approaches by Bijker and Iacchello 1998: Model calc. from nucleonic d.o.f., 3a formation (K-E.) 2011: Ab initio (ChEFT) calc. for 12C(0+2) (Epelbaum)

Recent experimental works on 12C
positive parity states Freer PRC83, (11) ~ 10 MeV newly observed 4+ 4+ 13.3 4+(13.3) (2+) M. Ito, NPA738, 268(04) M. Ito PRC84, (11) M. Freer PRC83, (11) S. Hydgaard PRC81, (10) W.R.Zimmerman, PRC84,027394(11) 2+ 11.1 2+(11.1) 0+ (10.6) 10.3 2+ (9.8) (0+) 0+ (9.0) 7.6 7.6 Questions: 0+ 0+ 2+2 :Rotation of 0+2 (Hoyle)? 0+3,4 (~10 MeV) a-chain? Impact to nucleosynthesis possible effect of 3a reaction at T9 >2 to abundance of heavy elements B(E2; 2+2 -> 0+1) 2+ 2+ 0+ 0+ old data new data

Theoretical studies of 12C
E.Uegaki, Y.Abe, S.Okabe, H.Tanaka, PTP 62, 1621 (1979) P.Navratil, J.P.Vary, B.R.Barrett, PRL 84, 5728 (2000) 3a-cluster model no-core shell model 3a-cluster states are missing Shell model expansion is not suitable to developed clustering (strongly correlated states) 3a-RGM: Fukushima,Kamimura(’78,’80) 3a-GCM: Uegaki(‘78),Descouvemont(‘87) 3a-OCM: Horiuchi(‘74),Kurokawa(‘05),Ogata... 3a-potential:Fedrov(‘96),Nguyen(‘12),Ishikawa(‘13),...

AMD Results of 12C 1. Cluster states of 12C 2. Rotation of cluster gas

AMD results of 12C: Energy levels
3a-cluster models

AMD results of 12C: Intrinsic structures
3- 1+ 1- 0+3 α α α 2+2 Bending chain ? 0+2 + + +.... 2+1 Cluster gas α α α α 0+1 α p3/2 + α α Cluster-shell mixing

Band assignment for 2+2 + 03 + 22 + 02 + 21 + 01 12C
Bending chain？ Q1:Is 2+2 rotation of 0+2? What is rotation of cluster gas ? Q2:Possible a-chain state? 03 + 22 + - 02 + 21 + cluster gas of 3a 01 + +

3a-cluster calculations w and w/o p3/2 component
Ground state sub-shell closure: p3/2 s p3/2 p1/2 3 a Energy levels: global energy shift of 0+ relative to 2+ by 3a breaking in g.s.

3a-cluster calculations w and w/o p3/2 component
B(E2): changed by alpha breaking components -> band assignment for 2+2 B(E2) 3a 3a+p3/2 AMD 2+2 -> 0+2 183 76 102 2+2 -> 0+3 64 165 311

Rotational(?) band from cluster gas
Y. K-E., PRL, PTP discussed in D3 session Also discussion for 4a gas Ohkubo et al., PLB684(2010) Funaki et al. PTPS196 (2012) 03 + 4 + 22 + Exp: 22 + 02 + Itoh et al.(2011) Excitation energy (MeV) 4 + Freer et al.(2011) 12C spin J(J+1) Structure change as J goes up 03 + a deformation rotation Bending chain a a a a 02 + a a a a a a spin geometric Spherical gas a

Cluster gas&chain in other nuclei
Tohsaki et al., Yamada et al., Funaki et al. Wakasa et al., K-E. et al., Suhara et al 12C Bending chain？ 16O 11B α α t 03 + - A.Tohsaki et al., (2001) Funaki et al.(2003) 02 + Dilute cluster gas 7.65 MeV Bosonic behavior 8Be+a cluster gas of 3a Q1:Band on cluster gas Can cluster gas rotate? Q2 Possible a-chain state? 01 + +

2a+t cluster in 11B(3/2-3) - - - 12C 11B, 11C triangle？ 2a+t gas
AMD by Y.K-E., Suhara triangle？ 11B, 11C PRC75, (2007) PRC85, (2012) 12C 03 + 2a+t gas 22 + 31 - 2a+t 3/23 - 02 + 7Li+a 7.65 MeV 8.5 3/22 - 8Be+a cluster gas of 3a Strong E0 Weak M1,GT 3/21 - 01 + Kawabata et al. PLB646, 6 (2007) +

Rotational(?) band from cluster gas
Structure change with spin increasing -> change of moment of inertia Y. K-E., PRL, PTP Suhara and Y. K-E., PRC discussed in D3 session 4 + 03 + 9/22 - Freer et al.(2011) 22 + Yamaguchi et al. (2011) Excitation energy (MeV) 02 + Itoh et al.(2011) 3/23 - 12C 11B spin J(J+1) spin J(J+1) 03 + bending chain a deformation rotation a a 02 + a a a a rotation of 3a, 4a gas Ohkubo et al., PLB684(2010) Funaki et al. PTPS196 (2012) Spherical gas a spin geometric a

Linear chain state in 14C*
Neutron-rich AMD　by T.Suhara and Y.K-E, Phys.Rev.C82:044301,2010. 14C 12C 14,16C 3a linear chain 12C* not linear 14C*,16C* proton neutron 03 + Y.K-E.et al., T. Neff et al. Linear chain? von Oertzen ZPA354(96) EPJA21(04),PR 432(06) Itagaki PRC64(01) Suhara PRC82(10) Maruhn NPA833(10) 02 + Tohsaki et al., Funaki et al. Energy (Mev) + 14C 12C g.s. 14C g.s.

Experimental searching for linear chain states
UK group: up reactions Price et al.PRC75(2007), Be+a break up Haigh et al. PRC78 (2008) Be+a break up Freer et al. PRC90(2014) Be+a scattering US group: Fritsch et al., PRCC93 (2016) Be+a scattering. Jpn group: Yamaguchi et al. PLB766,11 (2017). 10Be+a scattering.

Linear chain state in 14C*
AMD　by T.Suhara and Y.K-E, Phys.Rev.C82:044301,2010. Recent experiments of 10Be+a scattering reported candidates for the linear-chain band members. A. Fritsch et al., PRC93, (2016) H. Yamaguchi et al. PLB766,11 (2017) 14C 3a linear chain proton neutron Exp. AMD Energy (Mev) Yamaguchi et al. 14C

Linear chain in n-rich C
Meta stable for bending motion Stretching effect in rotation AMD by Y.K-E, PR432(2006) 15C*(19/2-) b=0.78 MO model for 14,16C Itagaki et al. PRC64(01) HF calc. for 16,20C by Maruhn et al. NPA833(10) suggested to be a yrast state 11Be+a Largely deformed 11Be with MO-bond

a-breaking effect in g.s. and excited states
4. Summary a-breaking effect in g.s. and excited states B(GT) strengths large level spacing B(E2; 2+2 -> 0+2) < B(E2; 2+2 -> 0+3) i.e., Band assignment of the 2+2 state small level spacing Bent chain structure of 0+3 state i.e. stability against the bending angle.

Lesson 5. Monopole and dipole excitations in light nuclei
Today I would like to discuss some topics of cluster phenomena in light nuclei.

Contents of Lesson 5 Introduction shifted basis AMD + cluster-GCM
IVD(E1) & ISD in 9,10Be ISM & ISD excitations in 12C The first topic is the cluster structure of neutron-rich Be isotopes. Y. K-E. PRC89, (2014) Y. K-E. PRC (2016) Y. K-E. PRC (2016) Y. K-E. PRC93, (2016)

decoupling of dof(scale)
IS0,E1,IS1 resonances transition strength Separation of LEDR From GDR GMR, GDR, IS-GDR LER New excitation modes decoupling of dof(scale) Energy T.Yamada, et al. PRC85 (2012). GMR Cluster valence neutrons In the dipole resonances as well as the monopole resonances, giant resonances are known in stable nuclei. GR is the collective vibration mode. In neutron-rich nuclei, the low-energy dipole resonances are discovered below the GDR. What is the origin and the excitation mode of the low-energy dipole resonances. That is the question. The existence of the LEDR separated from the GR may be a signal of the new excitation modes probably decoupling from the GR mode. For example, the valence neutron motion against the core. core Core+Xn GDR

Giant monopole and dipole resonances
IS monopole (IS0): IV dipole (E1): IS dipole (IS1): GMR IVGDR ISGDR Compressive, breathing translational compressive Response to external perturbation GR: broad bump in HE region ISM:10-20 MeV, IVGDR:10-30MeV Collective oscillation of system coherent 1p-1h excitation Physical aspects:matter properties ISGMR: Incomressibility, IVGDR:Symmetry energy strength GR LER Energy

History of LE dipole strengths see review papers
Exp. Eur. Phys. J. A (2015) 51: 99 Gamma decay of pygmy states from inelastic scattering of ions A. Bracco, F.C.L. Crespi, and E.G. Lanza Progress in Particle and Nuclear Physics 70 (2013) 210–245、”Experimental studies of the Pygmy Dipole Resonance” D. Savran a,b,∗, T. Aumannc,d, A. Zilges Theor. Rep. Prog. Phys. 70 (2007) 691–793 Exotic modes of excitation in atomic nuclei far from stability Nils Paar, Dario Vretenar, Elias Khan and Gianluca Col`o “Toroidal resonance: relation to pygmy mode, vortical properties and anomalous deformation splitting”, V.O. Nesterenko, et al., Phys.Atom.Nucl. 79 (2016)

Measurements of LE dipole strengths
E1 and IS1 excitations Stable nuclei：(g,g’),(p,p’),(a,a’)… Unstable nuclei：p, a, 17O, Au, Pb target Physical interests of LE-IVD astrophysical r-process photodisintegration (g,n): n sourse neutron capture rate neutron matter properties neutron-skin mode (Pigmy dipole ) strength GR LER

Long history of LE-E1(IVD)
1960’s～70’s LE-E1 measured in 208Pb etc. 1969 Brzosko named “Pigmy resonance” 1971 neutron skin oscillation against a Z=N core (Mohan with hydro model) 1980’s～1990’s: soft E1 strengths of halo nuclei loosely bound 1n or 2n (halo neutron) motion non-resonant strengths (threshold enhancement) or resonant strengths (Soft E1 “resonances”) 2000’s～ systematic studies of LE-E1 (PDR) in neutron-rich nuclei experimental & theoretical sides

History of LE-ISD Exp: 1980’s IS-LED in 16O , 40Ca ,208Pb (4-5% of EWSR) 1990’s~ detailed observations of IS-GDR, IS-LED Theor: 1980’s Torus, Vortical mode with hydro-dynamical models 2000’s~ microscopic calculations (Semenko, Ravenhall, etc.) Toroidal dipole (TD) LE-IS1 in 𝑁=𝑍 LE-E1 in 𝑁≠𝑍 neutron skin oscillation (Pigmy dipole) in 𝑁≠𝑍 nuclei LE-E1, LE-IS1 Compressive dipole (CD) IS-GDR A. Repko et al., PRC 87, (2013)

Vortical Nature in LED 208Pb 40Ca LE-E1 in 208Pb
LE-IS1 in 16O,40Ca,100Sn LE-E1 in 208Pb N. Ryezayeva et al., Phys. Rev. Lett. 89, (2002). P. Papakonstantinou, et al. EPJ. A 47, 14 (2011). E1 with RPA cal. ISD & E1 with RPA, Gogny D1S 40Ca Toroidal LED Compressive GDR 208Pb PDR GDR Volticity E1

LE-ISM, LE-ISD for cluster states
strength GR LER Energy Isoscalar monopole (IS0): LE GR Inter-cluster motion compressive Isoscalar dipole (ISD=CD): Inter-cluster motion compressive

IS0 and IS1: sensitive probes for cluster excitations
IS0 case: Yamada et al. PRC85, (2012) coherent sum of single-particle strengths GMR Inter-cluster excitation (relative motion) Internal ISM excitations in clusters coherent sum of 1p strengths of four nucleons Also for IS1: Chiba PRC93 (2016) no.3,

LE-IS0 for cluster states in 16O
Cluster mode decouples from collective vibration mode. GR feeds low-energy strengths. Strong ISM for cluster states 4-alpha OCM: Yamada et al. PRC85, (2012) exp:Y.-W. Lui, et al. PRC64, (2001). Low-lying IS0 strengths OCM cal. Cluster mode exp (a,a’) 12C-a radial motion High-lying IS0 (GMR)： Collective vibration coherent 1p-1h motion RPA cal. SRPA calc. Breathing mode D. Gambacurta, et al. PRC81, (2010).

Possible origin of LED excitations
HED Neutron skin only N>Z nuclei IVGDR Toroidal ISGDR Cluster

Problem to be solved Shifted basis AMD + cluster-GCM
Do LE strengths appear separating from the GR ? If so, what is the origin of the LE modes? Which operators (IS0, IS1, E1) are good to probe LE Roles of cluster structure and excess neutrons in LE modes of light neutron-rich nuclei What they tell us about cluster structure? We need a theoretical framework which describes coherent 1p1h excitations in the GRs and the LE (cluster) modes. Shifted basis AMD + cluster-GCM small amp. (1p1h) large amp. cluster mode Y. K-E, Phys.Rev. C89 (2014) , Y. Chiba et al., arXiv: (2015)

Formulatoion of sAMD+GCM
Shifted basis AMD

AMD method for structure study
AMD wave fn. Slater det. Variational parameters: 　　Gauss centers, spin orientations Gaussian spatial det isospin Intrinsic spins Gaussian wave packet Energy Variation Initial states AMD is a model based on the variational method using effective nuclear forces. An AMD wave function is given by a Slater determinant of single-particle wave functions written by a Gaussian wave packet localized at a certain position Zi. These parameters, centers of Gaussian wave packets for all nucleons, are independently treated as variational parameters. Jp-VAP to get the ground state wave function. Energy minimum Model space 93

Shifted basis AMD (sAMD)
Ground st. wave functions small shift of spatial part A shifted basis Small shift for 8 orientations det (8A basis) 8A basis is enough for IS0,E1,IS1 in 12C and Be

sAMD+GCM VAP sAMD GCM sAMD+GCM: all bases are superposed. g.s. cluster
correlation Ground state wave function sAMD 1p1h excitations on g.s. GRs various cluster configurations GCM R1 R2 q 3a-GCM For 12C a-GCM for 10Be and 16O R Large amp. cluster motion in LE sAMD+GCM: all bases are superposed. Jp-projection, cm motion are treated microscopically

Results 1.9,10Be:sAMD+GCM(5,6He+a) for B(E1) & B(ISD)
2.12C: sAMD+ GCM(3a) for B(ISM) & B(ISD) Effective force: Central (MV1 force)+ Spin-orbit+Coulomb finite-range 2-body & zero-range 3-body

Results 1 E1 and ISD in 8Be,9Be, 10Be Method: sAMD+aGCM

E1 excitations in 8Be,9Be,10Be
GDR in 8Be core two peaks in prolate state 8Be total z-dir. Lower peak not affected higher peak broadened 9Be GDR in core total z-dir. LEDR: Coherent two-neutron motion coupling with 6He+a LED B(E1), B(ISD) To investigate the dipole strengths of Be isotopes, we apply a method of extended version of AMD, the shifted basis AMD. Here I show you the E1 strength obtained by the sAMD calculation for 8Be, 9Be, and 10Be. The E1 strength of 8Be shows GDR with the two peak structure. The ground state of 8Be is the 2alpha cluster state and it has prolate intrinsic deformation. The lower peak comes from the longitudinal vibration and the higher peak comes from the transvers vibration. In the E1 strengths of 9Be and 10Be, low-energy resonances appear below the GDR because of the valence neutron around two alpha core. The GDR part comes from the vibration of the 8Be core part. The lower peak of the GDR for the longitudinal mode is almost same as that of 8Be, because the valence neutrons do not disturb the longitudinal vibration of the core. The higher peak for the transverse vibration becomes broader and broader in 9Be and 10Be as the excess neutron increases. It is easily understood because the transverse vibration can be affected by the excess neutrons. Let me come back to the low-energy dipole resonances. The LEDR in 10Be comes from the coherent two neutron oscillation against the 8Be core. Interestingly, this coherent two-neutron motion strongly couples with the cluster mode of 6He+alpha inter-cluster motion. As a result of the coupling with the large amplitude cluster motion, E1 and ISD strengths are enhanced. 10Be total z-dir. GDR in core + LED smearing factor 2MeV

Roles of excess neutrons in deformed neutron-rich nuclei
GDR: two peak structure B(E1) Not affected LEDR? broadening energy LEDR is contributed by the valence neutron motion against the core. The higher peak of the GDR becomes broader in the presence of the valence neutrons. Surface neutron oscillation against core

B(E1) in 9Be compared with experimental data
Ahrens et al.(1975) Goryachev et al. (1992) Utsunomiya et al.,(2015) Photonuclear cross section v.s. sAMD+aGCM calc.

Toroidal, E1 properties of LED in 10Be

What’s “toroidal” mode
ISD operators: Compressive ISD (CD): 𝜌 𝑟 3 𝑌 1𝜇 = 𝛻∙𝒋 𝑟 3 𝑌 1𝜇 Toroidal ISD (TD) 𝛻×𝒋 ∙ 𝑟 3 𝒀 11𝜇 LED? LED? ISGDR skin oscillation Toroidal Compressive A. Repko, et al., PRC87, (2013 Toroidal or Vortical nature of LED has been discussed in MF cal. N. Ryezayeva et al., Phys. Rev. Lett. 89, (2002). P. Papakonstantinou, et al. Eur. Phys. J. A 47, 14 (2011).

E1 & IS1 in 10Be: two LED modes
ISD TD(CD) excites LE(HE) parts 1-1 Ex~8 MeV TD dominance 1-2 Ex~15 MeV E1 dominance E1

Two LED modes in 10Be 1-1 10Be(g.s.) 1-2 1-1 1-2 6He+a (K=1) (K=0)
Toroidal neutron current Enhanced B(TD) Surface neutron oscillation Enhanced B(E1)

Results 2 ISM and ISD in 12C Method: sAMD+3a-GCM

ISM and ISD strengths in 12C: sAMD+GCM(3a)
0+ 0+ John et al.PRC68, (2003) Low-energy strengths for cluster states separately from GRs. Why the LE strengths are fragmented into a few cluster states? 1- 1-

Mode analysis: sAMD+aGCM for 12C
12C(sAMD)+GCM(3a) 12C(g.s.)= + VAP result contains the g.s. cluster correlation 12C(sAMD): 1p1h excitations on g.s. GCM(3a) R1 q R2 R2 R1 (A) Distance mode (B) Distance & rotation mode 12C(0+2) 12C(0+3) Significant to Significant to Details of 3a mode are investigated by Y. Yoshida.

+ B(ISM) of 12C sAMD (A) 3a: distance mode sAMD+GCM(A)
B(IS0) (fm4) sAMD: small amplitudes single-particle motion -> GMR (A) 3a: distance mode R1 R2 Large amplitude cluster mode -> LE-IMS strength sAMD+GCM(A) B(IS0) (fm4) (B) 3a: rotation mode sAMD+GCM(B) B(IS0) (fm4) Coupling of 8Be rotation with distance mode q R2 plit LE-ISM R1

12C B(ISM) of 12C LE-ISM cluster modes door-way E0 g.s. coupling with
rotation LE-ISM cluster modes door-way similar geometry E0 g.s. 12C

sAMD contains g.s. cluster corr. LE cluster mode & GR
B(ISD) of 12C sAMD sAMD contains g.s. cluster corr. LE cluster mode & GR sAMD+GCM(A) (A) 3a: distance mode R2 door-way R1 GR feeds strength of LE cluster mode sAMD+GCM(B) (B) 3a: rotation mode q Split ISD of LE cluster mode R2 R1

IS-LED in 12C, 16O 16O(1-1) TD dominant LED ! 12C 16O 3% of EWSR

Summary sAMD+GCM was applied to investigate IS0, IS1, E1 excitations.
Dipole excitations of Be isotopes were discussed: LE dipole excitations appear below the GDR because of valence neutron motion against the 2a core. LED in 10Be: TD dominant 1-1 and E1 dominant 1-2 states. TD nature of LE-ISDs: rotation of deformed clusters in 10Be, 12C, 16O

General trend of IS0 and ISD
IS0, ISD collective vibration cluster mode energy LER GR LE cluster mode strengths appear separated from GRs. Splitting(fragmentation) of LE strengths because of the cluster rotation (internal degrees of freedom).

Systematic study in a wide region
Excitation energy multidimension * proton number * neutron number * excitation energy * density Low density proton number In such the expanding nuclear chart, we should also stress another axis “excitation energy” on the nuclear chart, because a variety of phenomena appears also in excited states in each of nucleus. Namely, our research target is in this three dimensional space. Neutron number

Temperature/Excitation energy
Rich phenomena in nuclear many-body systems Temperature/Excitation energy a condensation Heavy-ion collision Liquid-gas phase Fusion/fission Nuclear matter Nucleon number Super-heavy element Giant resonance High spin Resonances, cluster decay α-caputure Threshold Resonance, continuum vibration cluster Weak-binding 陽子・中性子の割合いやエネルギーに依存して多様な現象 Shell evolution 2 neutron deformation Particle-hole neutron halo/skin Ground state proton-rich neutron-rich Stable nuclei

Effective interactions
Central force : MV1 parameterization two-range Gaussian 2-body+zero-range 3-body similar to Gogny central force in a sense LS force : two-range Gaussian 2-body from G3RS Coulomb force is also added. Matter properties: r0=0.192 fm-2, E0/A=17.9 MeV, K=245 MeV, m*=0.59m B.E. of nuclei: a 12C 16O 2C+a thres. Cal. (MeV) 27.8 87.6 123.5 8.2 Exp. (MeV) 28.3 92.2 127.6 7.16

Effective nuclear interactions
Central force: MV1 force two-range Gaussian 2-body + zero-range 3-body forces ls force: term of G3RS force two-range Gaussian 2-body (3O) Coulomb force: 7-range Gaussians Matter properties of MV1 force (case-1 with m=0.62, b=h=0) r0=0.192 fm-2, E0/A=17.9 MeV, K=245 MeV, m*=0.59m a 12C 16O 2C+a thres. Cal. (MeV) 27.8 87.6 123.5 8.2 Exp. (MeV) 28.3 92.2 127.6 7.16

Clustering in light neutron-rich nuclei

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