13Cにおける単極遷移強度と

クラスター構造

山田泰一、船木靖郎

Contents

• Monopole strength and cluster structureTypical case : 16O (12C)

• Alpha clustering and condensation in 13C

Monopole strengths in 1/2- states

Structures of 13C: 1/2-, （1/2+）

E0-strength of 16O: Exp. vs Cal.16O(α,α’) 16O(α,α’)

Lui et al., PRC64(2001)

RelativisticRPA cal.

Exp: Lui et al., PRC 64 (2001)Cal: Ma et al., PRC 55 (1997)

Multiplied by 0.25Shifted by 4.2 MeV

TEWSR=48±10%

0

10

5

15

20+

10+

30+40+

6.06

12.05 ( 1.5 keV)Γ =13.5

xE (MeV)

4α

12C α +

Exp.

Monopole matrix elements M(E0) in 16O and 12C

12C(01+)+α(S)

12C(21+)+α(D)

2( 0) 3.55 0.21 fm (3%)M E = ±

16O50+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 M eVα

12C

2( 0) 5.4 0.2 fm (16%)M E = ±

EWSR value

2( 0) 4.03 0.09 fm (8%)M E = ±

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

60+15.1

Monopole Strengths

2.6 (3.50*)4.03±0.090+1－ 0+

3

Cal. [fm2]Exp. [fm2]

5.4±0.2

3.3±0.7

3.55±0.21

0+1－ 0+

2

0+1－ 0+

5

0+1－ 0+

2

6.712C

3.0 (－*）

4.1 (3.98*)

16O

16O: 12C+α OCM*4α OCM

Suzuki, PTP56, 111 (1976)Funaki, Yamada et al., PRL 101 (2008)

12C: 3α RGM Kamimura, Nucl. Phys. A 351, 456 (1981)

No effective charge!

12Be: 0+1 – 0+

2 (shell-model structure), <r2>=0.83 fm2, S. Shimoura et al., PLB560 (2002)

Exotic characters of 3/2-3 of 11B

0.483

0.745

0.588Shell ModelExperiment

－

0.6160.782

< 0.0038.104 (3/2−)

0.487±0.0295.020 (3/2−)

0.398±0.0318.420 (5/2−)

0.453±0.0294.445 (5/2−)0.401±0.0322.125 (1/2−)

B(GT)Ex (MeV)

0.345±0.0080.000 (3/2−)

3/2−3 state has exotic characters.

・ Suppressed GT strength・ Large monopole strength・ Not predicted by the shell-model calculation・ 100-keV below the α-decay threshold.

0

2

4

6

8

10

12

1/2-1

3/2-1

3/2-2

3/2-3

3/2-4

5/2-1

5/2-2

5/2-3

5/2-4

7/2-1

Experiment

Exc

itatio

n E

nerg

y (M

eV)

1/2-1

1/2-2

3/2-1

3/2-2

3/2-3

3/2-4

5/2-1

5/2-2

5/2-3

7/2-1

Shell-Model

Exc

itatio

n E

nerg

y (M

eV)

T. Kawabata et al., Phys. Lett. B 646, 6 (2007).

α+α+t

7Li+α

AMD, 2α+t OCM: 3/2-3 state has α+α+t cluster structure

4( 0, ) 96 16 fm (13%)B E IS = ±

Enyo-Kanada Yamada, Funaki

Monopole strengths to cluster states: ～20% of EWSRHere, we have an interesting question.

Why cluster states are populated from the ground states with mean-field structures by the monopole transitions ?

Non trivial problem !

T. Yamada, Y. Funaki, H. Horiuchi,K. Ikeda, A. Tohsaki, to be published in Prog. Theor. Phys.

Monopole operator

16O: ground statemean-field structure

2nd ,3rd 0+ states: 12C+α structure

Monopole transition between 0+1 and 0+

2 states

0+2

Relative motion is excited.

12C（0+）+ α cluster structure

( ) ( ) ( )12 C

162 22 2

1

1 1 1 1 12 42 2 2 2 16i G

ii i

iC

i αα

∈∈=

×− = − + − +∑ ∑ ∑r r r r r r r

Monopole operator rel.

12C（0+）α

Monopole

(0s)4(0p)12

16O (g.s.)

=0+1

12C（0+）

α

40 ( ,3 )NR r ν

Bayman-Bohr theorem

Q=12

G.S. correlation Q>12: α clustering

activating the seed of α

10% comp.

having an seed of αin G.S.

SU(3)(00)90% comp.

Doubly closed shell-model w.f. 16O

(0s)4(0p)12

16O (g.s.)

=12C（0+）

α

40 ( ,3 )NR r ν

=12C（2+） α

42 ( ,3 )NR r ν

12C(0+,2+,4+) wf.: SU(3)(04) wfα cluster: intrinsic (0s)4

R4L(r,3vN): h.o.w.f. with Q=4

Bayman-Bohr theorem

16O g.s. can be excited throughcluster degree of freedom, namely, 12C+α relative motion, from R4L tohigher nodal states.

SU(3)(00)

[ ]4 1 12(01. . det ( )1

)6

(0!

) G Gg s s p φ −= × r

{ }120 0 40 0

12!4! ( C) ( ) ( )16!g L J

A R rN φ φ α= =⎡ ⎤= ⎣ ⎦

{ }122 2 42 0

12!4! ( C) ( ) ( )16!g L J

A R rN φ φ α= =⎡ ⎤= ⎣ ⎦

howf

So far, our discussion was qualitatively.Next, we study the monopole strengths in 16O quantitatively with use of the 12C+α OCM.

16O=12C+α OCM

12Cα

{ },

12

12!4! ' ( ,3 )16!

( C) ( ) ( )

J NJ NJi i

i N

NJ Ji i NL

Ji I L J

c

A h R r

h Y r

ν

φ φ α

Φ = Φ

Φ =

⎡ ⎤= ⎣ ⎦

∑

12( ) : SU(3)(0,4) w.f. ( 0, 2,4)( , ) : h.o. basis ( 2 4)

L

NL

C LR r N n Lφ

β=

= + ≥

Y. Suzuki, PTP55(1976)

r

SU(3) basis : (0,4)×(N,0) = (N-4,0) + (N-2,2) + (N,4)12C 12C－α

Relative motionexpanded by h.o. basis

N=4 : (0s)4(0p)12 closed shellLarger N : 12C-α clustering

N

1616O=O=1212C+C+α α cluster modelcluster model

Even-parity Odd-parity

EXP CAL EXP CAL

12C+α : molecular states

Y. Suzuki, PTP55 (1976), 1751

12C+α

4α

Am

plitu

de

(0,0)

(2,0)

(4,0)

(4,2)

(6,2)

(8,4)

Full model space

G.S. w.f. : SU(3) expansion

122 *

111 1

1(0 0 ) ( ) ( ) (0 0 0 )2

0k kk i Gi

M r r+ +

=

+ + + +− = − ∝ Φ Φ∑Product of amplitudes

Ground state correlation in 16O (12C+α OCM)

s4p12

G.S. correlation

12C+α OCM

11 %N>4

89 %N=4

comp.

16O g.s wf

Deviation from doubly closed shell w.f.

2

10( / 3 ) ( ) 0 : OCMNP βνβ +

+= Φ

Squared overlap on β/3νΝ

0.9580.847･･･

0.890･･･1/ 3 Nβ ν

P最大値になる

α clustering is activated !

Modified doubly closed shell w.f.

{ }120 400 0

12!4!( ) ( ) ( ) ( , ) ( )16! LN A C R rβ β βφ φ α+ =⎡ ⎤Φ = ⎣ ⎦

/ (3 ) 1Nβ ν = の時、doubly closed shell w.f: (0s)4(0p)12

/ (3 ) 1Nβ ν < α clustering is activated.

Nν : nucleon size parameter

12C40 ( , )R r β

α

Monopole Strengths & G.S. correlation

1 : G.S. wf within quanta model space ( =4,6,0 ; ,30)N N N+

122

1 1*

11

1(0 0 ) (0 ; 0 0) ( );2

0) (k i Gi

N k kN r NM r++ +

=

+ + +− = − ∝ Φ Φ∑

Study effect of the ground-state correlation

(12C-α clustering in g.s.)

1 1

(1 ) 0 ,

0 0 0

0

; 0 ;

Nkk k

k k k

N P

N N N

+

+ + + +

+ = −

⎡ ⎤= −⎣ ⎦

10 ; 0 0kN+ + =k=2,3

2 30 , 0 : obtained with full model space ( =30)N+ +

12C+α structures

12C+α OCM

Dep. of MN(0+1-0+

2,3) on model space of G.S. in 16O

01+ - 03

+

01+ - 02

+

Monopole strengths

Importance of G.S. correlation (12C-α clustering in g.s.)

（model space of ground state)

3.98 fm2 vs. 3.55±0.22 fm2

EXPCAL

3.50 fm2 vs. 4.03±0.09 fm2

Shell-model limit: G.S.=(0s)4(0p)12 T. Yamada et al.,to be published in Prog. Thor. Phys.

16O

12C+α OCM

Quanta N

24 ( 0) ( 0) 0 ( 0)

. : 44 6

G S NN

M E O EO E N

+=

=∝

=

=

Summary (I)Mechanism of M(E0) in light nuclei

(1) Structure of ground statedual aspects in g.s. : mean-field + cluster originally having a seed of α clustering (Bayman-Bohr theorem）

g.s. correlation enhanced α clustering or activating the seed

(2) Monopole operator：exciting relative motions between clusters by 2hw

(3) Cluster states are populated by E0 (about 20% of EWSR).

Monopole strengths are a good tool to explore cluster states.

Cluster structure and α condensation in 13C

1. Structures of 1/2- states and monopole strengths2. Structures of 1/2+ states

T. Yamada and Y. Funaki

Motivations

• 12C, 2nd 0+ (Hoyle); 3α condensate16O, 6th 0+; 4α condensate

• Addition of an extra neutron to Hoyle state (3α cond.)What happens ? Which state has the 3α+n gas-like (condensate) structure ?

: gateway to explore gas-like states composed of bosons and fermions

• 1/2- states excited by monopole transitions in 13C(α,α’): What kinds of structures they have ?

Monopole excitations are a good tool to explore cluster structures.

Sasamoto and Kawabata et al.Mod. Phys. Lett. 21 2393 (2006)Reanalayses: Mar. 2008

13C: monopole strengths

（5%)

（3%)

（4%)

EWSR

Shell modelM(E0) < 0.2 fm2

Monopole excitations in 13C

{ }

{ }

{ }

{ }

1/ 2

3 / 2

00

02

4, 23/ 2

4, 01

13 12

1/ 2

12

1/ 2

9

1/ 2

9

1// 2 2

12!( C) ( C) ( )13!

12! ( C) ( )13!9!4! ( Be) ( ) ( )13!9!4! ( Be) ( ) ( )13!

p

p

N L

N L

GS J

J

J

J

A u

A u

A u

A u

φ

φ

φ φ α

φ φ α

+ −

−

−

−

−+

−

= =

=

=

=

=

= =

Φ =

=

=

=

R

R

R

R

Bayman-Bohr theorem: GS of 13C SU(3)(31)×1/2

12C(3α)+n

9Be+α

13C

12C n

Monopole operator: O(E0:IS)

12C n13Cα

αα12C

n

9Beα

9Be α

9Be α

O(E0)

Monopole-excited cluster states

NeutronHalo (?)

12C(Hoyle)+n

9Be+α cluster

gs

gs

equivalent

13C=3α +n OCM with Gaussian basis

Approximately taken into account: Fully Solving 4-body problem

12C=3α OCM, 9Be=2α+n OCM : successful reproduction of 12C and 9Be

(OCM=Orthogonality Condition Model)

α3 3 3( , )ϕ νr

1 1 1( , )ϕ νr

2 2 2( , )ϕ νr

An

ααΨJ(13C) =

3 3

2 2 3 21 1

( ) ( ) ( )Coulij ij n in n Pauli

i j i

H T V r V r V r V V Vα α α α α +< = =

⎡ ⎤= + + + + + +⎣ ⎦∑ ∑

( )2( , ) ( ) exp ( )m mN r r Yϕ ν ν ν= −r rGaussian bais:

Angular momentum channels:

[ ] 144 84 2nI L Jl l l⎡ ⎤⎡ ⎤⎣ ⎦⎣ ⎦ 30 channels

Vαα: reproduction of α-α phase shiftsVαn: reproduction of α-n phase shifts (Kanada-Kaneko pot.)

K-H coordinates

Overlap amplitudes (reduced width amp.)

12C(gs)+n12C(2+)+n

12C(3-)+n12C(1-)+n9Be(gs)+α

12C(Hoyle)+n 12C(2+)+n9Be+α

13C(1/2-, gs)13C(1/2-, 2nd): M(E0,IS)=4.2 fm2

13C(1/2-, 3rd): M(E0,IS)=5.6 fm2 13C(1/2-, 4-th): M(E0,IS)=8.2 fm2

SU(3) character

12C (Hoyle) + n9Be + α

12C (2+) + n

9Be + α12C (3−) + n

12C (gs, 2+) + n

8.25.64.2

(fm2)M(E0;IS)

8.793.361/2−2

6.374.33

2.82(fm)R3α-n

2.971/2−3

3.191/2−4

2.391/2−1

(fm)RrmsJπ

3α+n OCM

M(E0:IS)

Rms=4.3 fm for Hoyle state

Cluster structure

Fujimura and Fujiwara et al., PRC69, 064327 (2004)

13C(3He,t)13N12C+p decay

1/22- 1/23

-

12C(g.s)+p decay

12C(2+)+p decay

12C(Hoyle)+p decay

13N

12C (Hoyle) + n9Be + α

12C (2+) + n

9Be + α12C (3−) + n

12C (gs, 2+) + nLarge decay branching to the Hoyle and 2+ states.

Large neutron transition strength

8.25.64.2

(fm2)M(E0;IS)

8.793.361/2−2

6.374.33

2.82(fm)R3α-n

2.971/2−3

3.191/2−4

2.391/2−1

(fm)RrmsJπ

3α+n OCM

M(E0:IS)

Rms=4.3 fm for Hoyle state

Summary (2)• Mechanism of M(E0) transition in light nuclei

good tool to explore cluster structures

• 13C（1/2-）, structure and M(E0)： 3α+n OCM2nd 1/2- ; 12C(0+,2+)+n 3rd 1/2- ; 9Be+α, 12C(3-,1-)+n4-th 1/2- ; 12C(Hoyle)+n, 9Be+α, 12C(2+)+n

Nuclear radii =3.0~3.4 fm : cluster structure

• 13C(1/2+)3nd 1/2+ ; strong candidate of dilute α condensation

R~5 fm, Occu. Prob. ~60%• Future : 3/2-, other states, LS-splitting.