クラスターガス状態とモノポール励起 - osaka...

クラスターガス状態とモノポール励起

関東学院大・工山田泰一

1. α-particle condensation in 4n nuclei 2. Hoyle-analog states in A≠4n nuclei 3. IS monopole transitions to search for cluster states “Dual nature of ground states” IS monopole strength function of 16O (typical case) : α-cluster states at low energy mean-field-type states at higher energy 4. Summary

Contents of my talk

Introduction • Cluster picture as well as mean-field picture is important

viewpoint to understand structure of light nuclei. • Structure of light nuclei Cluster states + Shell-model-like states Microscopic cluster models, AMD,.... 8Be=2α, 12C=3α , 16O=12C+α / 4α, … • α-condensate states in 4n nuclei: α-gas-like state described by a product state of α-particles, all with their c.o.m. in (0S) orbit. Typical states: 12C: Hoyle state (2nd 0+)

Cluster gas α cluster

0 03 5ρ ρ～

R ≈ 3.8 fm Tohsaki, Horiuchi, Schuck, Roepke, Phy. Rev. Lett.87 (2001) Funaki et al., PRC (2003) Yamada et al., EPJA (2005), Matsumura et al.,NPA(2004)


0 03 5ρ ρ～

R ≈ 3.8 fm

shell-model –like structure

Condensed into the lowest orbit

3(0 )S α

70%

+2Brink wf exact 0 state

Uegaki et al, PTP57(1977): 3α GCM The 0+

2 state has a distinct clustering and has no definite spatial configuration.

Chernykh, Feldmeir et al., PRL98(2007)

UCOM + FMD, 3α RGM

About 55 components of the Brink-type wave functions are needed to reproduce the full RGM solution for the Hoyle state.

Uegaki et al, PTP57(1977)

Overlap 3α GCM

Tohsaki,Horiuchi,Schuck,Roepke, PRL87(2001)

THSR wave function: α-condensate-type cluster wf 1 base THSR :

2

3 (3 RGM) 99%THSRα αΦ ≈+

2exact 0 state

Kamimura et al., (1977): 3α RGM

Funaki et al., PRC67, (2003)

α-gas-like nature of Hoyle state

α

α α G

X3

X1 X2

3 2

3 21

2( ) exp ( )i ii

B XBα φ α

=

Φ = − ∏

THSR wave function Tohsaki, Horiuchi, Schuck, Roepke, PRL (2001)

3(0 )S α


3(0 )S α

B → ∞3 2

3 21

2( ) exp ( )i ii B

B Xα φ α=

Φ → − ∏

B b→ 4 83 ( ) (0 ) (0 )B s pαΦ →

b: nucleon size parameter

B: parameter

2THSR3 ( ) (3 RGM) 0.9Bα αΦ ≈+

2exact 0 state3α RGM: Kamimura & Fukushima (1978)

Deformation (Bx,By,Bz) 0.999 R ≈ 3.8 fm: alpha-gas-like structure

Funaki et al., PRC (2003)

Occupation probabilities of α-orbits in 12C

12C(g.s) Hoyle state 70%

(λµ)=(04)-nature 0S-orbit

Yamada & Schuck EPJA26 (2005) with 3α OCM wf Matsumura & Suzuki, NPA739 (2004) Funaki et al., PRC (2010) with 3α THSR wf

Single cluster density matrix:

' ( , ') ( ') ( ),ad α α αρ ϕ λ ϕ=∫ r r r r r ( )ϕ r : single-cluster orbital w.f. λ : occupation probability

ρ(r,r')

SU(3) nature: confirmed by no-core shell model , Dytrych et al., PRL98 (2007)

α-density distribution and α-momentum distribution in 12C

Density × r2 Momentum distribution

01+ (g.s)

02+

02+

01+ (g.s)

Compact (01+) vs. Dilute (02

+) 02+ state: δ function-like

Similar to dilute atomic cond. Yamada & Schuck EPJA26 (2005) with 3α OCM wf

Single α-orbits in 12C(0+)

Large oscillation : strong Pauli blocking effect Compact structure SU(3) model: [f](λµ)=[444](04)-like structure

Small oscillation: weak Pauli blocking effect Long tail: dilute structure Radial behavior: Gaussian form with a= 0.04 fm-2

01+ (g.s.)

D-orbit (Nα=35%)

G-orbit (Nα=27%)

S-orbit (Nα=35%)

02+

S-orbit (Nα=70%)

20ref.: ( ) exp( )r N a ar× −


0 03 5ρ ρ～

Condensed into the lowest orbit R ≈ 3.8 fm

3(0 )S α

shell-model –like structure

70%

P. Navr’atil et al., PRC62 (2000) 12C

Family of the Hoyle state No-core shell model calculation

Exp. for 2nd 2+ state

Itoh et al., NPA738, 268 (2004)

Freer et. al. PRC80, (2009)

M. Gai et. al, (2011)

Kurokawa, Kato, PRC71 (2005)

Funaki et al., EPJA24 (2005) Yamada et al., EPJA26 (2005)

Family of the Hoyle state

12C

12C

3−

1−

Precursor of a 3α condensate state: 3-, 1-

(0P)α3 : superfluid?

vortex?


3(0 )S α

70%

3a OCM: Occup. Prob. ~ 40% precursor of 3α condensate

Yamada et al., EJPA26, 185 (2005)

Structure of 16O

1. 4α condensate ? 2. 4α linear chain state ? 3. Monopole excitation (later)

Funaki’s talk

(0Sα)3 : 70%

Cluster-model analyses of 16O

• α+12C OCM

• α+12C GCM

• 4α THSR wf

• 4α OCM

Y. Suzuki, PTP55 (1976), 1751

M. Libert-Heinemann, D. Bay et al., NPA339 (1980)

Tohsaki, Horiuchi, Schuck, Roepke, PRL87 (2001) Funaki, Yamada et al., PRC82(2010)

Funaki, Yamada et al., PRL101 (2008)

Not include α+ 12C configuration.

4α-gas, α+ 12C, shell-model-like configurations

Reproduction of lowest six 0+ states up to 4α threshold (15MeV)

16O=12C+α cluster model Even-parity Odd-parity

EXP CAL EXP CAL

12C+α : molecular states

Y. Suzuki, PTP55 (1976), 1751

12C+α

4α

Funaki et al., PRL101 (2008)

Suzuki, PTP55 (1976)

Experimental data 4α OCM

Ex [MeV]

R [fm]

M(E0) [fm2]

Γ [MeV]

R [fm]

M(E0) [fm2]

Γ [MeV]

0+1 0.00 2.71 2.7

0+2 6.05 3.55 3.0 3.9

0+3 12.1 4.03 3.1 2.4

0+4 13.6 no data 0.6 4.0 2.4 0.60

0+5 14.0 3.3 0.185 3.1 2.6 0.20

0+6 15.1 no data 0.166 5.6 1.0 0.14

20% of total EWSR

over 15% of total EWSR

4α cond. state

12C(01+)+α(S)

12C(21+)+α(D)

12C(01+)+α(S): higher nodal

12C(11-)+α(P)

shell-model-like: (0s)4(0p)12


4α OCM calculation

strong candidate 4(0 )S α 60%

α-particle condensates in heavier 4n nuclei than 16O

1. THSR approach 2. Gross-Pitaevskii approach

THSR approach Energy curve of HW equation

Tohsaki, Horiuchi, Schuck, Roepke, NPA738, 259 (2004)

nα condensate

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

22

2 2 22

1

3

1, 12

1 21

2

n

ii

U rm n

n nU r n d d v

n

v d

α

φ εφα

φ φ φ

φ=

− − ∇ + =

− −′ ′ ′ ′′ ′ ′′ ′ ′′ ′= − +

Φ =

∫ ∫

∏ r r

r r r , r r r r r r , r , r

r

Single-α potential given via Gross-Pitaevskii approach

・Coulomb barrier → quasi-stable states

・Weakly interacting “gas-like” states

・Pure bosons ・nα states are assumed

Yamada & Schuck, PRC 69, 024309 (2004)

α condensates : up to about 10α (40Ca)

Hoyle-analog states in A ≠ 4n nuclei (11B and 13C)

Search for Hoyle-analogue states in A≠4n nuclei

Definition of Hoyle-analogue state: A cluster-gas-like state described mainly by a product state of clusters, all with their c.o.m. in respective 0S orbits

• A=4n nuclei 12C=3α : (0Sα)3 70% 16O=4α : (0Sα)4 60% • A≠4n nuclei, for example, 11B=2α+t : (0Sα)2(0St) 13C=3α+n : (0Sα)3(0Sn)

Condensed into lowest orbit 3(0 )S α

exit or not?

Purposes of present study • Cluster structures in A≠4N nuclei: 11B (13C)

• Hoyle-analogue states? Conditions of appearance? 11B=2α+t : (0Sα)2(0St) 13C=3α+n : (0Sα)3(0Sn) • Which states of 11B (13C) correspond to the Hoyle-analogue? • What happens in 11B (13C) when an extra triton (neutron) is added into 8Be=2α (Hoyle=3α) state?

R.B. Wiringa et al., PRC(2000)

8Be density α

α

11B

8Be(2α) + p-orbit

s-orbit reduced:

not so changed:

2α part in 11B triton Jπ (11B)

1/2+

1/2-,3/2-

Thus, one can expect cluster-gas-like states in 1/2+.

α-t potentials: parity-dependent odd waves : attractive enough to make resonances/bound states even waves: weakly attractive

αα α S-orbit

α

12C 2nd 0+ (Hoyle)

α α t S1/2-orbit

α t

11B 1/2+ : Hoyle-analogue exists or not ?

S-orbit

α-n phase shifts α-triton phase shifts

α-n potential (1) P-wave: attractive

(2) S-wave: weakly attractive

α-triton potential

(1) P-, F-waves: attractive

(2) S-, D-waves: weakly attractive

Parity-dependent potentials

states in 11B 3 ,− +2 1 2

Results of OCM calculations

Yamada and Funaki, PRC82 (2010)

Structure study of 11B

• α+α+t OCM with H.O. basis

Nishioka et al., PTP62 (1979)

• AMD calculation

Enyo-Kanada, PRC75 (2007)

• No-core shell model

Navratil et al., JPG36 (2009)

Not well understood for even-odd states of 11B

No-core shell model Navratil et al., JPG36(2009)

7Li+α

11B energy levels (T=1/2)

Experimentally, three 3/2- states

No-core shell model Navratil et al., JPG36(2009) Not studied well for even-parity states

11B energy levels (T=1/2)

α+α+t

7Li+α

Even-parity states

11 (12,3)

, ,

(23,1) (31,2)

, ,

(23,1) (31,2) (23,1) (31,2)

( , )

( B) ( , ) ( , )

( , ) ( , ) ,

( , ) ( , ) ( , ),

( , ) ( , ) ( , ) , ( , ) ( ) exp

L c cc

c cc

c c c

ij kc ij k mL

A

B

N r

ν µ

ν µ

λ

ν µ ν µ

ν µ ν µ

ν µ ν µ ν µ

ν µ ϕ ν ϕ µ ϕ ν ν ν

+

+

Φ = Φ

+ Φ

Φ = Φ + Φ

Φ = = −

∑

∑

r r r ( )2 ( ),mr Y r

Total wave function: 2α +t OCM

Hamiltonian

( ) ( )

2 12 2 12 2( ) (23),(31)

2(2) (2) 2

12 12

( ) ( ) ( ) ( ) ( ) ,

2( ) exp , ( ) erf

lim ( ) ( ) ( ) (

Coul c LS Coult ij t ij ij t ij t Pauli

ij

Coulx n n x

n

Pauli n n n ij n ij

H T V r V r V r V r V r V V

xeV r V r V r arr

V u r u r u r u r

α α α α α α

α α

λ

β

λ

+ + +=

+ +

→∞

= + + + + ⋅ + + +

= − =

= +

∑

∑

l s

2 4 2 3 (23),(31)

) ,n n ij+ < + < =

∑ ∑ ∑

[ ] 0=Φ−Φ HEδEquation of motion

: Pauli blocking operator

α

t

α α

t

α α

t

α

⊕ ⊕

α+α , α+t phase shifts 7Li:3/2-,, 1/2-, 7/2-, 1/2-

2, ( 7,8)

3( ) : 3( ) :tL Q

V SU L Q SU L Qπ

π πα η λµ λµ

=

= ∑ : effective 2α+t force

OCM+GEM (Gaussian expansion method)

Single-cluster motions in 11B(α+α+t)

' ( , ') ( ') ( ),ad α α αρ ϕ λ ϕ=∫ r r r r r

α

α

triton ( )

1Gr

( )2

Gr

( )3

Gr G

( )ϕ r : single-cluster orbital w.f. λ : occupation probability

1α

λ =∑

11( B)JΦ : α+α+t OCM w.f.

(1 ) ( )1 11

1

21( B) ( ') ( ) ( B)2

( , ') ,G GnJ n J

nα δ δρ

=Φ − − Φ= ∑ r rr rr r

( )11 11( )3 3( B) ( ') ( ) ( B)( , ') ,t J J

G Gδ δρ Φ − − Φ= r rr rr r

alpha:

triton:

' ( , ') ( ') ( ),t t t td ρ ϕ λϕ=∫ r r r r r 1t

λ =∑

11B=α+α+t

Single-cluster density matrix:

Suzuki et al., PRC65 (2002); Matsumura et al., NPA739 (2004) Yamada et al., EPJA 26 (2005); Funaki, Yamada et al., PRL101 (2008) Yamada et al., PRA (2008), PRC(2009)

Energy levels of 11B

3/2- and 1/2+ states Even-parity states

7Li+α

α + α + t OCM α + α + t OCM

B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4

R=2.43 fm R=2.22 fm

R=3.0 fm

E0 E0

vs. Hoyle state: B(E0,IS)=120±9 fm4

11B = α +α + t OCM

T. Kawabata et al., PRC70 (2004)

shell-model-like state

shell-model-like state

cluster state α+7Li(3/2-) with S-wave

But 7Li part has a distorted α+t structure

-3 2

B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4

R=2.43 fm R=2.22 fm

R=3.0 fm

E0 E0


Overlap amplitude with α+ 7Li(g.s) channel

1st 1/2+

3rd 3/2-

1st 3/2-

3/2-3: α+7Li(3/2-) with S-wave

main configuration

But 7Li part has a distorted α+t structure

Yamada & Funaki, PRC82(2010)

11B = α +α + t OCM -3 2


B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4

R=2.43 fm R=2.22 fm

R=3.0 fm

E0 E0

11B: 3rd 3/2-

α triton

No concentration on a single orbit !

NOT cluster condensate


12C: Hoyle state

70%

S-orbit

3(0 )S α

11B = α +α + t OCM Occupation probabilities


B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4

R=2.43 fm R=2.22 fm

R=3.0 fm

E0 E0

11B: 3rd 3/2-

α triton

No concentration on a single orbit !

NOT cluster condensate


12C: Hoyle state

70%

S-orbit

3(0 )S α

11B = α +α + t OCM Occupation probabilities


3 MeV

states in 11B +1 2

Structure of 1/2+ states

α α t

α t

exists or not ?

7Li(3/2-) + α : P-wave Parity doublet partner of 3rd 3/2-

Rrms = 3.14[fm] vs. 3.00 fm for 3rd 3/2-

11B 1/2+ : Hoyle-analogue

S-orbit S1/2-orbit

0

-5

5

2α+t

11 2+

7Li+α (1 2 )+

1 2 ( 1 2)T+ =

EXP

11 2+

Candidate ? CAL

21 2+

E2α+t [MeV] 2α+t OCM

Overlap amplitudes with α+ 7Li(g.s) channel

1st 1/2+

3rd 3/2-

1st 3/2-

Structure of 1/2+ states

α α t

α t

exists or not ? 11B 1/2+ : Hoyle-analogue

S-orbit S1/2-orbit

0

-5

5

2α+t

11 2+

7Li+α (1 2 )+

1 2 ( 1 2)T+ =

EXP

11 2+

Candidate ? CAL

21 2+

E2α+t [MeV] 2α+t OCM

1/2+: Ex=12.6 MeV (E2a+t=1.5 MeV)

Soic et al. NPA742 2004 7Li(9Be,α7Li)5He T=1/2

10Be(p,γ), 9Be(3He,p) T=3/2 Goosman et al. PRC1 1970 Zwiegliski NPA389 1982

Curis et al. PRC72 2006

7Li(7Li,11B*)t T=1/2

⇔Ajenberg-Selove

Rrms = 5.98[fm] : dilute !

7Li(3/2-) + α : P-wave Parity doublet partner of 3rd 3/2-

Rrms = 3.14[fm] vs. 3.00 fm for 3rd 3/2-

11B* → α+7Li

Complex-scaling method for 1/2+ with α+α+t OCM

Bound state

Resonant state

α+α+t 7Li(1/2-)+α

7Li(3/2-)+α

1st 1/2+

2nd 1/2+

2nd 1/2+ exists as a resonant state.

Ex(cal)= 11.9 MeV, Γ=190 keV Ex(exp)=12.56 MeV, Γ=210±20 keV

Occupation probabilities of cluster orbits

1st 1/2+

Concentration on S-wave orbits !

2nd 1/2+

α α triton triton

12C(g.s) Hoyle state 70%

52% 96%

2nd 1/2+: Hoyle-analogue state

T. Yamada et al., EPJA 2005

P-orbit S-orbit S-orbit

structure7 Li + α

S-orbit

Occupation probabilities of cluster orbits

1st 1/2+

Concentration on S-wave orbits !

2nd 1/2+

α α triton triton

12C(g.s)

52% 96%

2nd 1/2+: Hoyle-analogue state

P-orbit S-orbit S-orbit

structure7 Li + α

2nd 1/2+: 65% (0Sα)2(0St)

65% = (2×52%+96%)/3

(0Sα)2(0St): 65%

(0Sα)3 : 70% (0Sα)4 : 60%

Hoyle-analogue states

states in 13C ,− +1 2 1 2

• What happens in 12C when an extra neutron (n) is added into the Hoyle state (3α) ? • Hoyle-analogue states?

• 3α+n OCM with GEM

α α n

α n

αS-orbit S1/2-orbit (0Sα)3(Sn) π +J = 1 / 2

13C 12C(Hoyle) +

p-orbit

s-orbit reduced:

not so changed:

3α part in 13C neutron

11B

8Be(2α) + p-orbit

s-orbit reduced:

not so changed:

2α part in 11B triton

Jπ (13C) 1/2-,3/2-

1/2+

Jπ (11B)

1/2+

1/2-,3/2-

Thus, one can expect cluster-gas-like states in 1/2+.

α-n and α-t potentials: parity-dependent odd waves : attractive enough to make resonances/bound states even waves: weakly attractive

α-n phase shifts α-triton phase shifts

α-n potential (1) P-wave: attractive

(2) S-wave: weakly attractive

α-triton potential

(1) P-, F-waves: attractive

(2) S-, D-waves: weakly attractive

Parity-dependent potentials

3α + n OCM for 13C

J=1/2-, 1/2+ : Yamada et al., Int.J.Mod.Phys. E17 (2008) reproduction of M(E0), Hoyle-analogue state

energy spectra of 13C 12C (Hoyle) + n

9Be(1/2+) + α

9Be(3/2-,1/2-) + α [larger than 12C (Hoyle) + n]

12C (g.s) + n neutron-halo-like

R=2.68fm

1/2+ states Probability

Sα-orbit

20%

28%

55%

R=4.0fm

R=5.4fm

Hoyle-analogue states

(0Sα)3 : 70% (0Sα)4 : 60%

(0Sα)2(0St): 65%

(0Sα)3(Sn): 55%

IS monopole transition in light nuclei

1. Useful to search for cluster states 2. Dual nature of ground states 3. typical case: 16O IS monopole strength function: α-cluster states at low energy mean-field-type states at higher energy

0

10

5

15

30+

20+

10+

30+40+

6.06

12.05 ( 1.5 keV)Γ =

[11.26 ( =2.6 MeV)]Γ

13.5

xE (MeV)

14.4 V4 4 Meα

12 7.16 eC M Vα +

Exp.

Monopole strengths M(E0) in 16O and 12C

12C(01+)+α (S)

12C(21+)+α (D)

2( 0) 3.55 0.21 fm (3%)M E = ±

16O

50+14.0

EWSR values

EWSR values

0

10

5

15

10+

14+

30+

20+

12+4.44

7.66

14.1

10.322+

xE (MeV)

3 7.272 MeVα

12C

2( 0) 5.4 0.2 fm (16%)M E = ±

EWSR value

Single particle M(E0)~5.4 fm2

2( 0) 4.03 0.09 fm (8%)M E = ±

2( 0) 3.3 fm (6%)M E =

60+15.1

12C(1-)+α (P)

Strong candidate (4α condensate)

higher nodal

Isoscalar Monopole Excitations in 16O

α cluster states at low energy and mean-field states at higher energy

• Isoscalar Monopole Excitation ↔ density fluctuation Typical example IS-GMR (heavy, medium-heavy nuclei)

208Pb(α,α’) IS-GQR

IS-GQR IS-GMR

breathing mode Ex=13.7 MeV Γ=3.0 MeV

Harakeh et al., PRL(1977)

exhaust EWSR ~ 100%

IS-GMR is well reproduced by RPA cal.

Blaizot, Gogny, Grammticos, NPA(1976)

208Pb: RPA

Ex=18.0 MeV collective motion with coherent 1p-1h states

N=10 h.o.quanta

N=12

N=14

• Light Nuclei (p-, sd-shell,,,) Isoscalar monopole strengths are fragmented. • For example, 16O, (12C,11B,13C,24Mg,…) IS-monopole response fun. of 16O(α,α’) (i) discrete peaks in Ex ≤ 15 MeV ~20% of EWSR large M(E0) states ⇔ cluster states (ii) three-bump structure : E=18, 23, 30 MeV

IS Monopole Strength Function of 16O Exp. vs Cal.

Exp: histogram Lui et al., PRC 64 (2001) discrete peaks in Ex≤15 MeV three bumps in 18, 23, 30 MeV Cal: real line Relativistic RPA Ma et al., PRC 55 (1997) Multiplied by 0.25 Shifted by 4.2 MeV

16O(α,α’)

Relativistic RPA cal.

Not well reproduced by RRPA cal.

Exp. condition: Ex > 10 MeV

Non-rela. Mean-field calculations of IS-monopole strengths for 16O

(1) RPA calculation: 1p-1h excitations Blaizot et al., NPA265 (1976) (2) Second-order RPA (SRPA) calculations: : 1p1h + 2p2h i) Drozdz et al., PR197(1980): D1-force ii) Papakonstantino et al., PLB671(2009): UCOM iii) Gambacurta et al., PRC81(2010) : full SRPA calculation with Skyrme force

Blaizot et al., NPA(1976)

RPA calculation Isoscalar

16O(α,α’)


Experiment


RPA(1p1h)

SRPA(1p1h+2p2h)

SRPA RPA

Gambacurta et al., PRC(2010)

61 Exp. condition: Ex > 10 MeV

Exp: histogram Lui et al., PRC 64 (2001)


10 20 30 40

10 20 30 40 50 0 10 20 30 40 0

Papakonstantino et al., PLB(2009)

Ex [MeV] Ex [MeV]

Experiment 4α OCM Ex

[MeV] R

[fm] M(E0) [fm2]

Γ [MeV]

R [fm]

M(E0) [fm2]

Γ [MeV]

0+1 0.00 2.71 2.7

0+2 6.05 3.55 3.0 3.9

0+3 12.1 4.03 3.1 2.4

0+4 13.6 no data 0.6 4.0 2.4 0.60

0+5 14.0 3.3 0.185 3.1 2.6 0.20

0+6 15.1 no data 0.166 5.6 1.0 0.14

RPA(1p1h)

SRPA(1p1h+2p2h)

SRPA RPA

Papakonstantino et al., PLB(2009)

Gambacurta et al., PRC(2010)

20+

No sharp peak !

20+

No sharp peak !


Exp: histogram Lui et al., PRC 64 (2001)


10 20 30 40

10 20 30 40 50 0 10 20 30 40 0

Ex [MeV] Ex [MeV]

RPA calculations: (1) Reproduction of 3-bump structure, but the energy positions are by about 3-5 MeV higher than the data. (2) No reproduction of discrete peaks at Ex≤15 MeV

SRPA calculations: (1) Reproduction of 3-bump structure

(2) Discrete peaks at Ex≤15 MeV are not reproduced well.

In particular, M(E0: g.s – 2nd 0+) is not seen in SRPA cal. Ex(2nd 0+) = 6.1 MeV This peak should exit sharply at Ex=6.1 MeV because Γ ≤ 1 eV.

Purposes of my talk

• What kind of states contribute to the discrete peaks? • Recently, 4α OCM succeeded in describing the structure of the lowest six 0+ states up to 4α threshold (Ex≈15 MeV). • The discrete peaks correspond to cluster states ? • We study the strength function of IS M(E0) with the 4α OCM framework.

Funaki’s talk

4α cond. state

12C(01+)+α(S)

12C(21+)+α(D)


12C(11-)+α(P)



4α OCM calculation


16O(α,α’)


Exp: histogram Lui et al., PRC 64 (2001) Cal: real line Ma et al., PRC 55 (1997) Multiplied by 0.25 Shifted by 4.2 MeV

60+

50+40+

30+


10 20 30 40

IS monopole strength function of 16O within 4α OCM

Monopole Strength Function

2

12 2

1( ) Im[ ( )]

/ 21 (0 0 )( ) ( / 2)

nn

n n n

S E R E

ME E

π

π+ +

= −

Γ= −

− + Γ∑

IS-monopole m.e.:

Energy weighted sum rule (EWSR): 22 2

1 12( ) (0 0 ) 16 ,n n

nE E M R

m+ +− − = × ×∑ 16

21 cm 1

1

1 0 ( ) 016 i

iR + +

=

= −∑ r R

+n0 : resonance state with En - iΓn/2

rms radius of 16O

Energy weighted sum rule within 4α OCM (OCM-EWSR)

4α OCM and OCM-EWSR

Total w.f. : internal w.f.s of α clusters × relative motion

[ ]1 1 1 2 2 2 3 3 312( ) ( , ) (( ) , ) ( , )c L J

f SJν

π ν ϕ ν ϕ ν ϕ νΨ × = ∑ ξ ξ ξ

( ) 0F Ju π =Ψ : Orthogonality condition to Paul forbidden states

4 4( ) (Coul)

OCM 2 21

4

3 4

( , ) ( , )

( , , ) (1, 2,3, 4)

Ni cm

i i j

i j k

H T T V i j V i j

V i j k V

α α

α α

= <

< <

= − + +

+ +

∑ ∑

∑

Hamiltonian for ( )J πΨ : 4-body problem with orthogonality condition



IS monopole matrix element

where we used

rms radius of α

α α

α α

16 nucleons

Interesting characters of IS monopole operator

4 42

4(11 1

21)

4

cm4( ( ) ) i k kk ki

kαα+ −= = =

−= − +∑ ∑∑ R Rr R

internal part of each α-cluster relative parts acting on relative motions of 4α’s with respect to c.o.m. of 16O

4 162 2

1 5

212 12C C( ) ( ) 3( )i i

i iαα

= =

− += − −+∑ ∑ R Rr R r R

internal part of α internal part of 12C relative part acting on relative motion of α and 12C

Decomposition into Internal part and relative part plays an important role in understanding monopole excitations of 16O.

α 12C

α α

α α

16 nucleons



EWSR of IS monopole transition in 4α OCM: OCM-EWSR

where 4

2OCM cm

14( ) ,

kk

O α=

= −∑ R R16

21 cm 1

1

1 0 ( ) 016 i

iR + +

=

= −∑ r R

: r.m.s radius of α-particle

OCM-EWSR

: r.m.s radius of 16O


Ratio of OCM-EWSR to total EWSR:

4α OCM shares over 60% of the total EWSR value. ( c.f. α+12C OCM : 31%) This is one of important reasons that 4α OCM works rather well in reproducing IS monopole transitions in low-energy region of 16O.

22 21 1

2( ) (0 0 ) 16n nn

E E M Rm

+ +− − = × ×∑

Yamada et al.

IS monopole strength function S(E) within 4α OCM framework

Monopole Strength Function

2

12 2

1( ) Im[ ( )]

/ 21 (0 0 )( ) ( / 2)

nn

n n n

S E R E

ME E

π

π+ +

= −

Γ= −

− + Γ∑

+n0 : resonance state with En - iΓn/2

: calculated by 4α OCM 2

n2

n= Γ (OCM) + (exp. resolution)Γ

nE : experimental energy of n-th 0+ state

50 keV

Experiment 4α OCM Ex

[MeV] R

[fm] M(E0) [fm2]

Γ [MeV]

R [fm]

M(E0) [fm2]

Γ [MeV]

0+1 0.00 2.71 2.7

0+2 6.05 3.55 3.0 3.9

0+3 12.1 4.03 3.1 2.4

0+4 13.6 no data 0.6 4.0 2.4 0.60

0+5 14.0 3.3 0.185 3.1 2.6 0.20

0+6 15.1 no data 0.166 5.6 1.0 0.14

16O(α,α’)


IS monopole S(E) with 4α OCM

30+

40+

50+

60+

20+

Exp. vs. Cal.


It is likely to exist discrete peaks on a small bump in Ex < 15MeV This small bump may come from the contribution from continuum states of α+12C

10 20 30 40

16O(α,α’)



30+

40+

50+

60+

20+

Exp. vs. Cal.

Excitation to cluster states (α-cluster type)

Monopole excitation of mean-field type (RPA)

Exp. condition: Ex > 10 MeV 10 20 30 40


30+

40+

50+

60+

20+

Exp. vs. Cal.



Two features of monopole excitations They come from a dual nature of the ground state of 16O


30+

40+

50+

60+

20+

Exp. vs. Cal.



Two features of monopole excitations They come from a dual nature of the ground state of 16O

next discuss!

Dual nature of ground state of 16O

mean-field character and α-clustering character

Ground state of 16O

Dominance of doubly-closed-shell structure: (0s)4(0p)12 Cluster-model calculations: 4α OCM, 4α THSR, α+12C OCM,... Mean-field calculations : RPA, QRPA, RRPA,......

Supported by no-core shell model calculations: Dytrych et al., PRL98 (2007)

Bayman-Bohr theorem :

Doubly-closed-shell w.f, (0s)4(0p)12, is mathematically equivalent to a single α-cluster w.f.

This means that the ground state w.f. of 16O originally has an α-clustering degree of freedom together with single-particle degree of free dom.

We call it duality of α-clustering and mean-filed characters in g.s.

= SU(3)(0,0)

SU(3)[f](λµ) = cluster-model wf

[ ]

{ }

{ }

14 12cm cm

120 40 3 0

122 42 23 0

0

1 det (0 ) (0 ) ( )16!

12!4! ( ,3 ) ( C) ( )16!

12!4! ( ,3 ) ( C) ( )16!

J

J

L

L

s p

N A u

N A u

φ

ξ ν φ φ α

ξ ν φ φ α

=

−

=

==

×

=

=

R3/4

2cm cm cm

32( ) exp( 16 )νφ νπ

= −

R R

Bayman-Bohr theorem

c.o.m. w.f. of 16O

G.S. has mean-field-type and α-cluster degrees of freedom.

Nucl. Phys. 9, 596 (1958/1959)

Excitation of mean-field-type degree of freedom in g.s 1p1h states (3-bump structure)

Excitation of α-cluster degree of freedom in g.s α+12C cluster states: 2nd 0+, 3rd 0+

4 162 2

1 5

212 12C C( ) ( ) 3( )i i

i iαα

= =

− += − −+∑ ∑ R Rr R r R

relative part internal parts

4α cond. state

12C(01+)+α(S)

12C(21+)+α(D)


12C(11-)+α(P)



4α OCM calculation


Monopole excitation to 0+2,3 and 0+

6 state

0+2,3 states: 12C(01

+,21+)+α(S) main configuration

Bayman-Bohr theorem: (0s)4(0p)12 has an α + 12C(01

+,21+) degree of freedom

relative motion (α-12C) is excited by IS monopole operator

0+6 state: 4α-gas like structure

Bayman-Bohr theorem: (0s)4(0p)12 has a 4α degree of freedom

relative motion among 4α is excited by IS monopole operator

main configuration

(0s)4(0p)12

16O(g.s.) α

SU(3)(00) wf

= α α (N2α,L2α)=(4,0)

(N2α-α,L2α-α)=(4,0)

0+1

α (N3α-α,L3α-α)=(4,0)

Bayman-Bohr theorem

Bayman-Bohr theorem

G.S. has a 4α-cluster degree of freedom.

(0s)4(0p)12

16O(g.s.) α

SU(3)(00) wf

= α α (N2α,L2α)=(4,0)

(N2α-α,L2α-α)=(4,0)

0+1

α (N3α-α,L3α-α)=(4,0)

α α

α α

4α gas-like or α+12C(Hoyle) ≈ 0+6 state

4 42

4( 1)1 1

42

1cm( ) ( 4 ) i k kk

k i kαα+ −

= = =

−+−= ∑∑ ∑ R Rr RMonopole transition

relative part

coherent excitation

Bayman-Bohr theorem

internal part

4α -gas-like M(E0)=1.0 fm2 by 4α OCM

Dominance of SU(3)(04) : no-core shell model by Dytrych et al., PRL98 (2007)

α

α α

(0s)4(0p)8

α

SU(3)(04) wf

= α α (N44,L44)=(4,0)

(N84,L84)=(4,0)

12C(g.s.)

0+1

3α gas-like = 0+2 (Hoyle) state

3 42

4(

1

1 1

2

1)

3

cm4(

( )

)

i k kk

k

i

kα

α+ −= =

=

−

= −

+

∑

∑

∑

R R

r R

Monopole transition relative part

coherent excitation

internal part 2( 0) 5.4 0.2 fm (16%)M E = ±

portion of EWSR

Yamada et al., PTP120 (2008)

12C

Bayman-Bohr theorem

Bayman-Bohr theorem

12 4 120 internal

0 40 2 40 10

( C) (0 ) (0 ) ;( , ) (0, 4), 0

4!4!4! 8 ( , ) ( , 2 ) ( ) ( ) ( )12! 3

J

J

s p J

N A u u

πφ λ µ

ν ν φ α φ α φ α

+=

=

= = =

= ξ ξ

Dominance of SU(3)(04) : confirmed by no-core shell model Dytrych et al., PRL98 (2007)

This means that the ground state w.f. of 12C originally has an α-clustering degree of freedom together with single-particle degree of free dom.

12C

Monopole excitation to 0+5 and 0+

4 state

0+5 state: 12C(11

-)+α(P) main configuration

Bayman-Bohr theorem: (0s)4(0p)12 has no configuration of 12C(1-)+α

Why this state is excited?

Coupling with 12C(0+,2+)+α and 12C(Hoyle)+α configuration

Coherent contribution from these configurations

0+4 state: 12C(01

+)+α(S): higher nodal

Coherent contributions from 12C(0+,2+)+α and 12C(Hoyle)+α configurations

Components of α+12C(Lπ) channels in 0+ states of 16O

SU(3)-nature (0s)4(0p)12 12C(01

+)+α(S) 12C(21+)+α(D)

12C(01+)+α(S)

higher nodal 12C(11

-)+α(P) 12C(02+)+α(S): 4α gas-like

R=2.7fm

Two features of IS monopole excitations in 16O

IS monopole S(E): 4α OCM

30+

40+

50+

60+

20+

Excitation to cluster states (α-cluster type) Monopole excitation

of mean-field type (RPA)

Fine structure Ex < 16 MeV

3-bump structure 16 < Ex < 40 MeV

4α OCM OK difficult

Mean-field difficult OK

Origin: dual nature of G.S. of 16O (1) α-clustering degree of freedom (2) mean-field-type one (0s)4(0p)12 : SU(3)(00)= 12C+α : Bayman-Bohr theorem

Two features in IS monpole exci.

Huge model space is needed to reproduce S(E) in the whole energy region.

cluster basis + collective basis

Dual nature of the ground states in 12C and 16O : common to all N=Z=even light nuclei

20Ne

Excitation of mean-field degree of freedom

Excitation of α-cluster degree of freedom

α+16O cluster states of Kπ = 0+4, 0

- bands

α+16O comp. = 80% for low spins: Q=10 quanta

Almost pure α+16O structures for low spins: Q=9 quanta

20Ne, 24Mg, 32S, ….., 44Ti,….

M. Kimura, PRC69, 044319 (2004)

AMD+GCM 20Ne

Two features of IS monopole excitation

Cluster states at low energy, Mean-field-type states at higher energy

These features will persist in other light nuclei.

Increasing mass number, effect of spin-orbit forces becomes stronger: some SU(3) symmetries are mixed. Goodness of nuclear SU(3) symmetry: gradually disappearing. Two features of IS monopole excitation will be vanishing. Eventually only 1p1h-type collective motions are strongly excited.

Hope: Systematic experiments!!

Summary • α-condensation in 12C, 16O, heavier 4n nuclei. • Hoyle-analog states in 11B, 13C • IS monopole tran. : useful to search for cluster states • IS monopole excitations have two features: 16O (typical) (i) α-cluster type: discrete peaks in Ex≤15 MeV (ii) mean-field type: 3-bump structure (18,23,30 MeV)

• The origin: dual nature of the ground state of 16O. (0s)4(0p)12 : SU(3)(00) = 12C+α : proved by Bayman-Bohr theorem Dual nature is common to all N=Z=even light nuclei • クラスターガス状態: 新しい物質形態拡がりと深さの探求: 通常核（4n核）, 中性子過剰核

クラスターガス状態と モノポール励起 - osaka...

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クラスターガス状態とモノポール励起 - osaka...