16O 　における 4α 凝縮状態

Yasuro Funaki (RIKEN)

Taiichi Yamada (Kanto Gakuin)

Peter Schuck (IPN, Orsay, Paris-Sud Univ.)

Hisashi Horiuchi (RCNP)

Akihiro Tohsaki (RCNP)

Gerd Röpke (Rostock Univ.)

原子核・ハドロン物理：横断研究会 ,KEK, 19 - 21 November, 2007

Appearing of cluster gas state and ``BEC’’ state in finite nuclei

Condensed into the lowest orbit

12C

Lowest energy state

0 MeV

～ 100 MeV

Energy

～ 10 MeV

E/A ～ 8 MeV

luquid

Appearing near the 3αthreshold

Lowest configuration of gas phase

0

Cluster gas

αcluster

0 03 5 ～

12 nucleons break-up threshold

E/A ～ 1 MeV

3 α break-up threshold

A. Tohsaki et al., PRL 87, 192501 (2001).

B

,n b B=

bb b

0s

†1 1 4 4, , VAC

n

n nri r i Cr r

L

2 2 2( 2 )B b

2† 3 †

2exp RC d R B R

r r

Brink’s wave function

22

1

2, expn

n i ii

b XB

r

=A

†1 1 4 4( ) , , ( ) VACri r i B R

rr rL B 4(0 )s

（ 0s ） 4 　

configuration around

b

Rr

Calculation of matrix elementsis owing to Tohsaki’s technique

,n nf b

= , , 0n nb H E b f

=

nαcondensate wave function (THSR-w.f.)

Hill-Wheeler equation (β: generator coordinate, b: fixed)

First example of α condensate state in finite nuclei Hoyle state (02

+ state in 12C (excitation energy : 7.65

MeV) )

3αbreak-up threshold : 7.27 MeV

Indicating 3α condensate character Occupation probability of α-particle orbit

Momentum distribution of α-particle

Hoyle state : red line

Ground state : black lineSemi-microscopic approach (3αOCM)

・ Occupation probability of α-particle orbitHuge 0S occupancy ( ＞ 70 %)

・ Momentum distribution

Delta-function-like behavior

T. Yamada et al., EPJA, 26, 185 (2005).

Microscopic approach (3αcond. model w.f.)

3 3, 0H E A s rRGMThe Solution of 3αRGM eq. of motion,

3

22

1

2, exp ( )i Gi

X XB

s r

is almost equivalent to the 3αcond. w.f.

srA

A 0S

M. Kamimura, NPA 351, 456 (1981).

: c.o.m of -particleiX

3αcond.

Y. F et al., PRC 67, 051306(R) (2003).

First attempt to explore 4α condensate state in 16O

Nucleon density distribution

22

1

2expn

n i ii B

A X

A. Tohsaki, H. Horiuchi, P. Schuck and G. Röpke, PRL 87, 192501 (2001).

nα condensate model w.f. (THSR-w.f.)

n=4 case

Very dilute density.4 αcondensate state

+3 THSR

0 :0

10

5

15

20

10

30

6.06

12.1+(11.3, 0 )

13.6

xE (MeV)

14.4 V4 4 Me

12 7.2 eVC M

14.04050

0.08

10.34

14.94

Low lying 0+ levels of 16O

Exp.

+1 THSR

0

+2 THSR

0

+3 THSR

0

４ αcond. w.f.

15.1 60

First attempt to explore 4α condensate state in 16O

Nucleon density distribution

22

1

2expn

n i ii B

A X

A. Tohsaki, H. Horiuchi, P. Schuck and G. Röpke, PRL 87, 192501 (2001).

nα condensate model w.f. (THSR-w.f.)

n=4 case

Very much dilute density.4 αcondensate state

+4 THSR

0 :0

10

5

15

20

10

30

6.06

12.1+(11.3, 0 )

13.6

xE (MeV)

14.4 V4 4 Me

12 7.2 eVC M

14.04050

0.08

10.34

14.94

Low lying 0+ levels of 16O

Exp.

+1 THSR

0

+2 THSR

0

+3 THSR

0

４ αcond. w.f.

15.1 6017.3 +

4 THSR0

13.6 MeV state, T. Wakasa, E. Ihara, M. Takashina and Y. F. et al, PLB 653, 173 (2007).

22

1

2, expn

n i ii

b XB

r

=A

n α cond. w. f. (microscopic)

12 +20 ( C(0 ))

12 +40 ( C(2 )) 3(0 ) ?

△・

・・ ×

4 α cond. w.f. hardly describes these states

Need to solve full 4 α problem,4 α OCM (semi microscopic)

・ Need to check the existence of 4 α cond. state in larger (4α)model space.・ 4αcond. w. f. may have a difficulty to represent the 12C+α structure.

Motivation for 4α OCM (Orthogonality Condition Model)

0

10

5

15

20

10

30

6.06

12.1+(11.3, 0 )

13.6

xE (MeV)

14.4 V4 4 Me

12 7.2 eVC M

14.04050

0.08

10.34

14.94

Low lying 0+ levels of 16O

Exp.

+1 THSR

0

+2 THSR

0

+3 THSR

0

４ αcond. w.f.

15.1 6017.3 +

4 THSR0

12C(0+)+α

12C(2+)+α

Fully solving 4 α-particles relative motions (4αOCM)

Gaussian basis

2( , ) ( ) exp ( )m mN r r Y r r

Present: Larger model space

3 3 3( , ) r

1 1 1( , ) r

2 2 2( , ) r

A

Approximately taken into account

121 2 3K-type and H-type, [[ , ] , ] :l Ll l lAdopted angular momentum channels, totally 10 channels

0 0[[0,0] ,0] 2 0[[2,0] , 2] 0 0[[2,2] ,0]2 0[[0,2] , 2]

Hamiltonian

2 2 3 4( ) ( )Coulij ij Pauli

i j

H T V r V r V V V

Cal. (MeV) Exp. (MeV)

12C(g.s.)

－ 7.32 － 7.27

16O(g.s.)

－ 14.2 － 14.44

4α ＯＣＭ（ H. O. basis ）

12C+α coupled channel ＯＣＭ （ H. O. basis ）

Y. Suzuki, PTP 55, 1751 (1976); 56, 111 (1976).

K. Fukatsu and K. Kato, PTP 87, 151 (1992).

12C+α: succeededdilute 4α: not reproduced

1 0[[0,1] ,1] 1 0[[2,1] ,1] Added in only K-type

3 3 3( , ) r

1 1 1( , ) r

2 2 2( , ) r

or

(K-type) (H-type)

2 2

3 4

( ) : 2-body force (folding MHN), : Coulomb force

& : phenomenological 3-body and 4-body forces (repulsive)

: pauli operator (to eliminate 0s, 1s, 0d w.r.t. rel. motion)

Coul

pauli

V r V

V V

V

Energies from 4αthreshold

Correct 12C(g.s.)+αand 4α threshold energies

Energy levels, rms radii, monopole matrix elements and density distribution.

0

10

5

15

20

10

30

6.06

12.1+(11.3, 0 )

13.6

xE (MeV)

4 threshold

12C threshold

14.04050

Low lying 0+ levels of 16O

Exp.

+1 OCM

0

+2 OCM

0

+4 OCM

0

４ αOCM

+3 OCM

0

+5 OCM

0

Rrms (fm) M(E0)(fm2)

M(E0)(fm2) Exp.

2.7

3.1 4.2 02+: 3.55

2.9 4.1 03+: 4.03

3.9 2.4 04+: no data

3.1 2.0 05+: 3.3

5.4 1.4

+2 OCM

0

+3 OCM

0

+4 OCM

0

+5 OCM

0

+1 OCM

0

Density distribution

2 1r r dr

(MeV)60

15.2 +

6 OCM0

+6 OCM

0

Large monopole matrix element can be the evidence of cluster states (Yamada, Y.F. et el., nucl-th/0703045)

3 OCM(0 )

2 OCM(0 )

Reduced width amplitude for the lower states12 16

1, 0 00( ) ( ) ( C) ( O)

k k kL LL J J

r r Y

rr Y

12C(01+) +α component is dominant

in surface region ( ～ 5 fm).

12C(21+) +α component is dominant

in surface region ( ～ 5 fm).

α ＋１２Ｃ ( ０ 1+)

structureα ＋１２Ｃ (21

+) structure

Consistent with the previous Suzuki and Kato results !

5 OCM(0 )

Reduced width amplitudes of 04+ and 05

+ states obtained via 4αOCM(partial waves between α and 12C contained in 16O w.f.)Defined as

12 16

, 0 00( ) ( ) ( C) ( O)

k k kL LL J J

r r Y

rr Y

・ Large amplitude in surface region( ～5 fm)　 ・ developed αcluster structure ・ mixing of 12C(1-)+α, 12C(3-)+αand 12C(02

+)+α structures

4 OCM(0 )

4αcond. state obtained with THSR-w.f.

3 THSR(0 )

+4 OCM(0 )

Very well developed α cluster structure Furthermore, 　 corresponds to 4 αcondensate state previously obtained usingTHSR-w.f.

6 OCM(0 )4 OCM(0 )

Reduced width amplitudes of 04+ and 05

+ states obtained via 4αOCM(partial waves between α and 12C contained in 16O w.f.)Defined as

12 16

, 0 00( ) ( ) ( C) ( O)

k k kL LL J J

r r Y

rr Y

4αcond. state obtained with THSR-w.f.

3 THSR(0 )4 THSR(0 )

Single particle occupancy and single particle orbit for the 04

+ and 06+ states obtained with 4αOCM (Only

the S orbit (L=0))04

+ state : the largest (n=1) 0.87/4 =22 %

06+ state :

the largest (n=1) 2.33/4 =58 %

Large 0S occupancy !

Not 0S orbit

Strongly suggesting the 06+

state is 4α condensate

4* ' ' ' ' '

1 4 1 4 1 1 1 41

' ( , ' ) ( ') ( )

( , ') ( , , ) ( , , )

= ( , ), : single particle occupancy, ( ) : single particle orbit

i i i ii

dr r r f r f r

r r r r r r r r dr dr dr dr

n L f r

the second largest (n=2) 0.41/4 =10 % 0S orbit

Possible assignment of the two calculations andobservations

0

10

5

15

20

10

30

6.06

12.1+(11.3, 0 )

13.6

xE (MeV)

4 threshold

12C threshold

14.0

4050

Low lying 0+ levels of 16O

Exp.

+1 OCM

0

+2 OCM

0

+4 OCM

0

４ αOCM

+3 OCM

0

+5 OCM

0

0.08

10.34

14.94

THSR- w.f.

Γ(04+)theor ～ 0.2 MeV

Γ(05+)theor < 0.05 MeV

(calculated based on R-matrix theory)

Γ(04+ at 13.6 MeV)= 0.6

MeVΓ(05

+ at14.0 MeV)= 0.17 MeV

4α cond. state

12C(0+)+α

12C(2+)+α

All 0+ states up to 4 α thresholdare simultaneously reproduced !

6015.2 +

6 OCM0

17.3

4α cond. state 12C(1-)+α

12C(0+)+α(higher nodal)

Summary ・ 4αcondensate w. f. (4αTHSR-w.f.) predicts the existence of 4α

condensate state. (not as the third 0+ state but as the fourth 0+ state)

・ Two new resonance states are obtained near the ４ α threshold. One has a developed αcluster structure (Rrms ～ 3.0 fm) in which 12C(1-)+α, 12C(3-)+α and 12C(02

+)+α components are mixed. The other has a very well developed αcluster structure (Rrms ～ 4.0 fm). 12C(0+)+α(higher nodal) ⇒ corresponding to the observed 04

+and 05+ states,

respectively

Analysis by using ４ αOCM （ orthogonality condition model ） in order to describe both 12C+α, 4αgas states and others, if any, in larger model space.

・ Successfully reproducing the well known 12C(g.s.)+α(6.05 MeV) and 12C(2+)+α(12.05 MeV) structures,

For future work, ・ Analyses of condensate fraction and 4αCSM are necessary for more reliable conclusion.　・ 4αlinear chain structure (the band head is estimated at 16.7 MeV) We are able to discuss it simultaneously with the lower structures !

4α condensate state and other cluster states are simultaneously obtained.Large 0S-occupancey, 60 %

Assignment of the 4αcondensate state to one of the observed levels

０ 2＋ state ( 6.05 MeV ) ：　 α+12C( ０ 1

＋ ) structure０ 4

＋ state (12.05 MeV) ：　 α+12C(21＋ ) structure

Well investigated

Large monopole matrix element can be the evidence of cluster states (Yamada et el., nucl-th/0703045)

Exp. (fm2)

M(E0: 01+ － 02

+ (6.06 MeV)) 3.55±0.21

M(E0: 01+ － 04

+ (12.05 MeV) )

4.03±0.09

M(E0: 01+ － 05

+ (13.05 MeV) )

Missing !

M(E0: 01+ － 06

+ (14.0 MeV) )

3.3±0.7

０ 5＋ state ( 13.5 MeV, Γ=0.6 MeV ) ：　 recently observed

０ 6＋ state ( 14.0 MeV, Γ=0.17 MeV) ：　 large M(E0)

On the other hand,

+2 THSR

0

+3 THSR

0

(α,α’) cross section

M(E0: (01+)theor.- (03

+)theor=2.5 fm2) )

Puzzling situation !

Both can be the candidate !

Γ(03+)theor=1.5 MeV

(calculated based on R-matrix theory)

T. Wakasa, E. Ihara, M. Takashina and Y. F. et al, nucl-ex/0611021

Asymmetric shape

　　・・・　 13.6 MeV state might be hidden 　

13.6 MeV state must have a largeMonopole matrix element.Similar magnitude of cross section

M(E0: 01+ － 04

+ (12.05 MeV) )

= 4.03±0.09

Phys. Lett. B, 29 (1969), 306.

THSR-w.f. predicts the existence of only one state around the 4α threshold. . . .

Hamiltonian of 4α ＯＣＭ

Pauli blocking operator on α-α motions

2 4

lim ( ) ( )Pauli n ij n ijn ij

V u r u r

Pauli forbidden state: h.o,w.f.

0s0p

8 Be( ) (Q=0) + (Q=0)Q 4

relQ 4

α-α motion with Qrel < 4 should be removed.

16 16( O) ( O) 0L LE H

Equation of motion

Cal. (MeV) Exp. (MeV)

12C(g.s.)

－ 7.34 － 7.28

12C(21+) － 4.88 － 2.84

12C(41+) 2.06 　 6.43

12C(02+) 0.698 　 0.38

16O(g.s.)

－ 14.4 － 14.44

Qrel : H.O. quantum number regarding α-αmotion

2 2 3 4( ) ( )Coulij ij Pauli

i j

H T V r V r V V V

Phenomenological 3-body force (repulsive)

(2) (2) 22 ( ) expn n

n

V r V r

2

2

4( ) erfCoul e

V r arr

Phenomenological 4-body force (repulsive)

Coulomb force

2-body force (folding MHN force)

Energies from 4αthreshold

(3) 2 2 23

(3) -2

exp ( )

31.7 MeV, =0.15 fm

ij jk kii j k

V V r r r

V

(4) 2 2 2 2 2 24 12 13 14 23 24 34

(4) -2

exp ( )

12000 MeV, =0.15 fm

V V r r r r r r

V

0.50 0.498590 0.326865 0.164184 0.062899 1.00 0.908864 0.815898 0.572459 0.287296 1.50 0.896297 0.336547 0.322929 0.151992 2.00 0.572353 0.881969 0.703933 0.396726 2.50 0.307156 0.978650 0.000038 0.065416 3.00 0.155720 0.866247 0.651207 0.405040 3.50 0.078273 0.676844 0.899942 0.704774 4.00 0.039853 0.489846 0.980462 0.658007 4.50 0.020752 0.338195 0.968052 0.111590 5.00 0.011094 0.227118 0.897224 0.542085 5.50 0.006096 0.150310 0.793885 0.775540 6.00 0.003442 0.098902 0.677975 0.901844 6.50 0.001995 0.065086 0.563380 0.955530 7.00 0.001185 0.043008 0.458466 0.952726 7.50 0.000721 0.028610 0.367255 0.906961

0R

2 22 2

0 4 1 0 1 4 0 1 0 4 4 0 1 0 0

4 4 01

1/ 2

0 4 0 4 0

ˆ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) ( ) ( )

ˆ ˆ1 (0 ) (0 ) , 1

ˆ( ) ( ) ( )

k k k k k k k k

k

k l ll

k k

R C R P R C R R C R R

P P

C R R P R

1 0( )R 3 0( )R2 0( )R 4 0( )R