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: Monopole & Dipole excitations in light nuclei

Lesson1.Introduction

Cluster & Mean field

Mean field, shell structure

Independent single-particle

v.s.

Cluster:

Many-body correlation

Independent particle

in MF

Developed cluster

Cluster excitation

Shell structure・MF Cluster

Cluster formation

many-body correlation no correlation

12C excited states 12C ground state

Cluster structures

12C

Nucleon liquid

Fermi gas, BCS

0 MeV 100 MeV

Nucleon gas

Energy 10 MeV

r0

Shell & cluster corr.

low density

cluster excitation

+

r0/3~ r0/5

Cluster gas, crystal

high density

Nuclear

structure

Cluster

enhancement

at low density

Nuclear surface

Light-mass nuclei

Excited states

p3/2-closed & 3a

Z=N=6

Cluster structures

in stable and unstable nuclei Typical cluster structures known in stable nuclei

8Be 12C 20Ne

a + a 3a 16O + a

16O*

12C + a

7Li

a + t

a-cluster

40Ca*, 28Si*, 32S*

36Ar-a, 24Mg-a, 28Si-a

Si-Si, Si-C, O-C, O-O

Molecular

orbital Be, C, O, Ne, F

a-cluster

excitation

14C*

3a linear chain

Unstable nuclei Heavier nuclei

a

Z-, N-dependence of g.s. structure

8Be

Z=4

Z=5

Z=6

10Be 12Be

11B 13B 15,17B 19B

10C 12C 14C 16C 20C 18C

Shape difference N

Cluster

Z=3

7Li

Neutron Skin

N=8

11Li

Vanishing of Magic number

Cluster

9Li

16O

Two kinds of remarkable clustering

in n-rich Be* and Ne* isotopes

Ne, F, O isotopes

a a

Seya, Von Oerzten, Descouvemont et al.,

Itagaki et al., Dote et al., K-E et al.

Arai et al., M. Ito et al.

Be isotopes

a a

a a

Molecular Orbital:

s-bond structure

Atomic:

Cluster resonance

10Be

shell model-like

a

16O a

6He+He 18O+a

Strong

coupling

Weak

coupling

Normal states

Scholz et al., Rogachev et al., Goldberg et al.,

Ashwood et al., Yildiz et al.,

Descouvemont, Kimura,

Soic et al., Freer et al., Saito et al.,

Curtis et al., Milin et al., Bohlen et al.,

16O a

22Ne

MO bond and Cluster res. in 22Ne

16O a

16O a

MO s-bond

Cluster res.

22Ne 18O+a AMD study by Kimura, PRC75 (2007)

Exp Scholz et al., Rogachev et al., Goldberg et al.,Ashwood et al., Yildiz et al.,

Theor: Descouvemont, Kimura,

EXP

AMD

a-cluster states in n-rich nuclei

a

Cluster resonances

6,8He+a in Be

10Be+a in 14C

14C+a in 18O

18O+a in 22Ne

Ex = several ~ 10 MeV

New states discovered and suggested at

-> information of nucleus-nucleus potential and valence neutron

effects there.

Theor: Seya, von Oerzten, Descouvemont et al.,Itagaki et al., K-E et al.

Arai et al., M. Ito et al.

Exp: Soic et al., Freer et al., Saito et al., Curtis et al., Milin et al., Bohlen et al.,

in a-decay, a-transfer, a-scattering

Exp Scholz ‘72, Rogachev ‘01, Goldberg ‘04, Ashwood ‘06, Yildiz et al.,

Theor: Descouvemont ’88, Kimura ‘07

Exp Scholz et al., Rogachev et al., Goldberg et al.,Ashwood et al., Yildiz et al.,

Theor: Descouvemont, Kimura,

Exp Soic 04, von Oertzen ‘04, Price 07, Haigh 08,

Theor: Suhara ‘10

multi-cluster: gas, triangle,

linear chain

01

+

02

+

7.65 MeV

12C

8Be+a

cluster gas of 3a

22 +

03 +

triangle？

31 -

+

11C*, 11B*

14C*

α

α

α

α

Tohsaki et al., Yamada et al.,

Funaki et al. Wakasa et al.,

K-E. , Kawabata et

Itagaki, Suhara

α

α

t

Triangle, linear a-hain

16O*

4a gas

2a+t gas

chain？

Rich cluster phenomena in

nuclear systems

Cluster formation/breaking in low-lying states

MO s-bond in neutron-rich nuclei

Cluster resonances

Multi-cluster systems: cluster gas, a-chain

New types of clusters

as functions of proton&neutron numbers and excitation energy

6,8He+He in Be, 10Be+a in 14C,14C+a in 18O, 18O+a in 22Ne

A theoretical method:

AMD (antisymmetrized molecular dynamics)

An approach for nuclear structure to study

• cluster and mean-field aspects

• Stable and unstable nuclei

• Ground and excited states

Lesson 2

A theoretical model:

AMD

AMD wave fn.

Gauss centers, spin orientations

isospin Intrinsic spins

det

Gaussian wave

packet

Similar to FMD wave fn.

A variety of cluster st. Shell-model states

det det Cluster and MF

formation/breaking

Variational parameters:

Model setting: AMD wave function

Model space (Z plane)

Randomly chosen

Initial states

Energy minimum

Energy surface

Model wave fn. Phenomenological effective nuclear force

Energy variation

For Volkov, MV1, Gogny etc.

Variation, Jp -projection

Variation after/before

Jp –projection (VAP/VBP)

Constraint AMD+GCM

Cluster limit v.s. Shell-model limit of AMD functions

Identical 2 fermions

Anti- Symm.

2 Gaussians at short distance ＋ ＝

＋ ＝

s-orbit

p-orbit (0s)(0p)

config.

large d

(0s)4 (0p)4

d→0 limit

s4p4 Developed 2a

Strongly correlated

with

2a system

Basis expansion from a center Strongly correlated state

Cluster: strongly correlated

state

Mixing of high angler momentum components

Large Small

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14

prob

abili

ty (%

)

Principal quanta Δ <N>

8Be(0+)

SVM by Arai. et al. PRC (1996)

N

8Be (g.s.)

large d

Strong spatial correlation involves higher-shell

components.

Developed 2a

8Be (g.s.)

Developed cluster states are usually beyond mean-field

Lesson 3.

cluster phenomena in neutron-rich Be isotopes

Z-,N-dependence of g.s. structure

Molecular orbital structure

magic number breaking

Cluster resonances

Contents of Lesson 3

Z-,N-dependence of g.s. structure

Molecular orbital structure

magic number breaking

Cluster resonances

Contents of Lesson 3

Z-, N-dependence of g.s. structure

8Be

Z=4

Z=5

Z=6

10Be 12Be

11B 13B 15,17B 19B

10C 12C 14C 16C 20C 18C

Shape difference N

Cluster

Z=3

7Li

Neutron

Skin

N=8

11Li

Vanishing of

Magic number

Cluster

9Li

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

Z-, N-dependence of g.s. structure

8Be

Z=4

Z=5

Z=6

10Be 12Be

11B 13B 15,17B 19B

10C 12C 14C 16C 20C 18C

N

Cluster

Neutron Skin

N=8 Largely deformed:

proton radii enhanced

by clustering ?

Minimum

proton radii?

Stable proton radii?

proton radii along isotope chain

bp minimum at N=6 (10Be)

new magic?

Increasing in N>6.

10Be 12Be

Exp data:

Isotope shift W. Nortershauser et al. PRL102, 062503 (2009); A. Krieger, PRL108, 142501 (2012).

proton radii(charge radii) reflect

N dependence of clustering.

Good probe for cluster structure

CC :Terashima et al.

exp :determined by isotope shift

N dependence of rp in Be,B,C

AＭＤ & exp. data

Z-,N-dependence of g.s. structure

Molecular orbital structure

magic number breaking

Cluster resonances

+

-

+ - + s-orbital

α α

α α

±

p-orbital

Molecular orbital(MO) structure in Be

MO s-bond state

Normal state

2a-core formation

MO formation

Level inversion

in 11,12,13Be

Gain kinetic energy

in developed 2α system

vanishing of magic number in 11Be, 12Be, 13Be

MO formation

Seya PTP65(81), von Oertzen ZPA354(96)

N. Itagaki PRC61(00), Y. K-E.. Ito PLB588(04)

N=8 magic number breaking

spherical enhanced

cluster core

8

6

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

N=8 magic number breaking

s2

s1

s0

N=6 N=8 N=10

?

6

s1/2

p1/2

p3/2

sd

p3/2

p1/2

s 2a

11Be(1/2+) p2s1

11Be(1/2-) p3s0

Parity inversion at N=7

11Be(g.s.)

p3/2

N=8 magic number breaking

s2

s1

s0

N=6 N=8 N=10

?

6

s1/2

p1/2

p3/2

sd

p3/2

p1/2

s 2a

12Be(0+1) p2s2

12Be(0+2) p4s0

Largely deformed ground state at N=8

12Be(g.s.)

12Be(1-) p3s1

p3/2

Energy

0 1

+

0 2

+

12Be

Experimental evidence of

N=8 shell vanishing in 12Be

Normal 0h

deformation in 12Be(gs)

Intruder 2h

Y.K-E.PRC (03),(12) , Ito PRL(08) Dufour NPA(10)

12Be g.s. 0+

s2p2

p4

Inelastic scat. life time:

Iwasaki PLB481(00),

Imai PLB673(09)

B(GT) with charge ex.:

Meharchand PRL108 (12)

intruder config. in 12Be(gs)

12Be(02

+) with p-shell config.

1n-knockout reac.:

Navin PRL85(00),

Pain PRL96(06)

Fortune PRC(06), Blanchon PRC(10)

Shimoura PLB654 (07)

Takashina, Y.K-E., et al. PRC77 (08)

Exp: Iwasaki PLB481 (00)

Inelastic scattering

0 + 2

+ 1 0

+ 1 2

+ 1 0

+ 1 2

0 + 2

2 + 2

exp AMD

13

9

B(E2) e2fm4

Inelastic scattering p(12Be,12Be*) 12Be

7.9(2.9)

B(GT) ratio:

B(GT;0+2)/B(GT;0+

1)=1.16 (exp)

=1.1 (AMD)

0h

2h

s2p2

p4

How about 13Be (N=9)

s2

s1

s0

N=6 N=8 N=10

? 6

3/2

s1/2

p1/2

p3/2

sd

p3/2

p1/2

s 2a

13Be(1/2-) p3s2

13Be(1/2+,5/2+)

Parity inversion

13Be(g.s.) ?

or

Normal N=9

Breaking of magic number in 13Be

12Be+n

Er

(MeV)

0

1

2 2.0

(5/2+)

(1/2-)

3.04

Simon07

1/2+ virtual st.

Kondo10

(1/2-)

0.51

2.39

(5/2+)

1/2- ?

13Be: unbound 13Be spectra measured by 1n knock-out reactions

at GSI(Simon et al. 2007) and RIKEN(Kondo et al.,2010)

13Be

p1/2

s1/2

p3/2

d5/2

s1/2

1/2- is lowest

in 13Be intruder

EXP.

AMD calc. Y. K-E. PRC(12)

Kondo10 Simon07 AMD 12Be+n

Change of cluster structure

s2

s1

s0

N=6 N=8 N=10

s0 s1 s2 s2 s2

s-orbital neutron enhances clustering

clustering, beta, charge radius in g.s.

Z-,N-dependence of g.s. structure

Molecular orbital structure

Magic number breaking

Cluster resonances

0

5

10

0 5 10 15 20

Spectra of 10Be

0 1

+

Exp: Milin et al. ’05, Freer et al. ’06

10Be: energy levels

J(J+1)

0+

2+

4+

2+

4+

10Be

6He+4He

Ito et al.

Kobayashi et al.

Kuchera et al.

MeV

AMD

exp

cluster

res.

MO s-bond

Normal

0 2

+

0 3

+ AMD calc. Y. K-E, et al. PRC (98)

a a

0+2

0+1

a

0+3,4?

6He+4He

Exp: Kuchera et al.

Theor: Ito et al.

Kobayashi et al.

a a

Normal

MO bond

a

a a

a a

0 3

+

6,8He+6,4He

0+2

0+1

Exp:

Freer PRL.82(99)(06)

Saito NPA738 (04)

Yang PRL112 (14）

a a

10Be 12Be

Cluster resonances

MO bond

Normal

cluster reso.

Breaking of N=8 magicity

Y.K et al., PRC 68, 014319 (2003)

positive parity states withnormal spins

Formation of 2a+molecular orbitals

VAP calculation with AMD method

0 + 3

+ 1 0

intruder

12Be 6He+6He,

Energy spectra of 12Be

MO s-bond state

8He+4He?

Cluster reso.

0 + 2 normal

12Be

a-cluster states in n-rich nuclei

a

Cluster resonances

6,8He+a in Be*

10Be+a in 14C*

14C+a in 18O*

18O+a in 22Ne*

Ex = several ~ 20 MeV

New states discovered and suggested at

Universal phenomena?

Further experimental and theoretical studies are requested.

Theor: Seya, von Oerzten, Descouvemont et al.,Itagaki et al., K-E et al.

Arai et al., M. Ito et al.

Exp: Soic et al., Freer et al., Saito et al., Curtis et al., Milin et al., Bohlen et al.,

in a-decay, a-transfer, a-scattering

Exp Scholz ‘72, Rogachev ‘01, Goldberg ‘04, Ashwood ‘06, Yildiz et al.,

Theor: Descouvemont ’88, Kimura ‘07

Exp Scholz et al., Rogachev et al., Goldberg et al.,Ashwood et al., Yildiz et al.,

Theor: Descouvemont, Kimura,

Exp Soic 04, von Oertzen ‘04, Price 07, Haigh 08, Fritsch ’16, Yamaguchi

Theor: Suhara ‘10

Lesson 4.

Monopole and dipole

excitations

in light nuclei

Introduction

shifted basis AMD + cluster-GCM

IVD(E1) & ISD in 9,10Be

ISM & ISD excitations in 12C

Contents of Lesson 4

Y. K-E. PRC89, 024302 (2014)

Y. K-E. PRC93 054307 (2016)

Y. K-E. PRC94 024326 (2016)

Y. K-E. PRC93, 024322 (2016)

IS monopole (IS0): IV dipole (E1): IS dipole (IS1):

GMR

Compressive,

breathing

Giant monopole and dipole resonances

IVGDR

compressive

ISGDR

translational

Energy

strength

GR LER

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

Response to external perturbation

Measurements of

LE dipole strengths

strength

GR

LER

Physical interests of LE-IVD

astrophysical r-process

photodisintegration (g,n): n sourse

neutron capture rate

neutron matter properties

neutron-skin mode (Pigmy dipole )

E1 and IS1 excitations

Stable nuclei：(g,g’),(p,p’),(a,a’)… Unstable nuclei：p, a, 17O, Au, Pb target

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) 842-850

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

Compressive

dipole (CD)

IS-GDR

(Semenko, Ravenhall, etc.)

A. Repko et al.,

PRC 87, 024305 (2013)

LED v.s. GDR

Energy

Low-energy (<15 MeV)

strengths below GRs

observed by (g,g’),(p,p’),(a,a’)…for stable nuclei p, a, 17O, Au, Pb targets for unstable nuclei

stable nuclei: IVD(E1) since 70’s, ISM, ISD since 80’s

neutron-rich nuclei: IVD(E1) since 90’s

LED <15 MeV

Separation of LE strengths from GRs

New excitation modes decoupled from GRs.

strength

GMR, GDR,

IS-GDR

Origins of LE strengths have not clarified yet.

Various origins?

Isoscalar monopole (IS0):

Isoscalar dipole (ISD=CD):

LE GR

compressive

compressive Inter-cluster motion

Inter-cluster motion

LE-ISM, LE-ISD for cluster states

Energy

strength

GR LER Yamada et al. PRC85, 034315 (2012)

Chiba et al. PRC93 (2016) no.3, 034319

GMR coherent sum of

single-particle strengths

Internal ISM excitations in clusters

coherent sum of 1p strengths of

four nucleons

Inter-cluster excitation

(relative motion)

IS0 case: Yamada et al. PRC85, 034315 (2012)

IS0 and IS1: sensitive probes

for cluster excitations

Also for IS1: Chiba PRC93 (2016) no.3, 034319

Dipole excitations

1.Isovector dipole (E1):

(

protoni

iiYrEM r̂);1( 1

2.Isoscalar dipole (ISD=CD):

( i

ii YrISDM r̂);( 1

3

translational

p-n ocsillation

3. IS(IV) Toroidal dipole (TD)

LE? GR in HE

compressive Inter-cluster motion

Neutron skin

oscillation:Core-Xn

• Toroidal

• Compressive

Density conserved

-> LED?

hydrod models , Semenko(1981)

microsciopic MF calc. 2000’s~

N. Ryezayeva et al., PRL89,

272502 (2002).

P. Papakonstantinou,

EPJA 47, 14 (2011).

P. Papakonstantinou, et al.

EPJ. A 47, 14 (2011).

ISD & E1 with RPA, Gogny D1S

40Ca

Toroidal LED Compressive

GDR

LE-IS1 in 16O,40Ca,100Sn

PDR

GDR

Volticity

E1

E1 with RPA cal.

N. Ryezayeva et al., Phys. Rev. Lett. 89,

272502 (2002).

208Pb

LE-E1 in 208Pb

Vortical Nature in LED

What are natures of LED?

Neutron skin

only N>Z nuclei

Toroidal

HED LED

IVGDR

ISGDR

Cluster

IVD

(E1)

ISD

Method: Shifted basis AMD method+cluster-GCM Y. K-E, Phys.Rev. C89 (2014) , Y. Chiba et al., arXiv:1512.08214(2015)

Shifted basis AMD

2.Formulatoion of

sAMD+GCM

Energy Variation

AMD wave fn.

Variational parameters:

Gauss centers, spin orientations

spatial

isospin Intrinsic spins

Slater det.

Gaussian

det

Gaussian

wave packet

Model space

Initial

states

Energy

minimum

Jp-VAP to get the ground

state wave

function.

AMD method for structure study

Shifted basis AMD (sAMD)

Ground st. wave functions

small shift of spatial part

det

A shifted basis

Small shift for

8 orientations

8A basis is enough for IS0,E1,IS1 in 12C and Be

(8A basis)

sAMD+GCM VAP

1p1h excitations on g.s.

various cluster configurations

GRs

Large amp. cluster

motion in LE

g.s. cluster

correlation sAMD

Ground state wave function

GCM

sAMD+GCM: all bases are superposed.

R1

R2 q

3a-GCM For 12C

a-GCM for 10Be and 16O

R

Jp-projection, cm motion are treated microscopically

3.Results

E1 and ISD in 8Be,9Be, 10Be

2a+Xn

GDR in 8Be core two peaks in prolate state

GDR in core

LED

LED LEDR: Coherent two-neutron motion

coupling with 6He+a

smearing factor 2MeV

higher peak

broadened Lower peak

not affected

+

B(E1), B(ISD) total z-dir.

total z-dir.

total z-dir.

9Be

10Be

8Be

GDR in core

E1 excitations in 8Be,9Be,10Be

B(E1) in 9Be compared with

experimental photo nuclear s Ahrens et al.(1975), Goryache et al. (1992), Utsunomiya et al.,(2015)

Decoupling of

LE & HE modes

Toroidal, E1 properties of LED

in 10Be

Two LED modes in 10Be

1-1 E~8 MeV

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

EB

(ISD

)/dE

(fm6 )

Energy (MeV)

CDTDISD

6He Toroidal mode

0

2

4

6

8

10

0 10 20 30 40 50 60

EB(E

1)/d

E (fm

2 )

Energy (MeV)

E1

E1 1-2 E~15 MeV

E1 mode

1-2

1-1

Toroidal Compressive

-4 -3 -2 -1 0 1 2 3 4

-4-3

-2-1

0 1

2 3

4

Two LED modes in 10Be

(K=1)

10Be(g.s.)

6He+a

Toroidal neutron current

Enhanced B(TD)

-4 -3 -2 -1 0 1 2 3 4

-4-3

-2-1

0 1

2 3

4 Surface neutron oscillation

Enhanced B(E1)

1-2 (K=0)

6He

1-1

Cluster interpretation

of two LED in 10Be

10Be(g.s.)

6He+a 1-2

6He 1-1

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

EB(IS

D)/d

E (fm

6 )

Energy (MeV)

CDTD

Inter-cluster mode

E1 & CD

cluster rotation mode

coupling w inter-cluster motion

TD & CD