多体共鳴状態の境界条件によって解析した 3α 共鳴状態の構造

KEK 原子核研究会「現代の原子核物理 - 多様化する原子核の描像」

多体共鳴状態の境界条件によって解析した 3α 共鳴状態の構造多体共鳴状態の境界条件によって解析した 3α 共鳴状態の構造

C. Kurokawa1 and K. Kato2

Meme Media Laboratory, Hokkaido Univ., Japan1

Div. of Phys., Grad. Sch. of Sci., Hokkaido Univ., Japan2

KEK 原子核研究会 8 １ -8/3

Theoretical studies of 12CTheoretical studies of 12C

　　 D.M.Brink in Proceedings of the Fifteen Solvay Conference on Physics (19070)

○Microscopic 3α model (RGM ・ GCM ・ OCM)

　　 Y.Fukushima and M.Kamimura in Proceedings of the International Conference on Nuclear Structure (1977)

　　 M.Kamimura, Nucl. Phys. A351(1981),456

Y.Fujiwara, H.Horiuchi, K.Ikeda, M.Kamimura, K.Katō, Y.Suzuki and E.Uegaki, Prog Theor. Phys. Suppl.

　　 68 (1980)60.

E.Uegaki, S.Okabe, Y.Abe and H.Tanaka, Prog. Theor. Phys. 57(1977)1262; 59(1978)1031; 62(1979)1621.

H.Horiuchi, Prog. Theor. Phys. 51(1974)1266; 53(1975)447.

K.Fukatsu, K.Katō and H.Tanaka, Prog. Theor. Phys.81(1988)738.

○3α+p3./2Closed shell

N.Takigawa, A.Arima, Nucl. Phys. A168(1971)593.

N.Itagaki Ph.D thesis of Hokkaido University (1999)

Y.Kanada-En’yo, Phys. Rev. Lett. 24(1998)5291.

○Deformation 　（ Mean-Field ）

G.Leander and S.E.Larsson, Nucl. Phys.A239(1975)93.

○Faddeev

Y.Fujiwara and R.Tamagaki Prog. Theor. Phys. 56(1976)1503.

H.Kamada and S.Oryu, Prog. Theor. Phys 76(1986)1260. 　　　　　　　　

α

01+

02+

31－

α

α

α

α α

Γ=8.7eV

Γ=34keV

Excited states of cluster states?

3


Situation around Ex= 10 MeVSituation around Ex= 10 MeV

[Ref.] E.Uegaki et al.,PTP57(1979)1262

0+, 2+A.Tohsaki et al., PRL87(2001)192501

Alpha-condensed state

02+ :

l=0

L=0

0+ : Er=2.7+0.3 MeV, = 2.7+0.3 MeV

2+ : Er=2.6+0.3 MeV, = 1.0+0.3 MeV

[Ref.]: M.Itoh et al., NPA 738(2004)268

Can 3αModel reproduce both of the 22

+ and the 03+

states ?

What kind of structure dose the 03

+ state have ?

Why 03+ has such a large

width ?Boundary condition for three-body resonances

Analysis of decay widths

Boundary condition for three-body resonances

Analysis of decay widths

Energy level of 12CEnergy level of 12C


Our strategyOur strategy

In order to taking into account the boundary condition for three-body resonances, we adopted the methods to 3 Model;

Complex Scaling Method (CSM) [Ref.] J.Aguilar and J.M.Combes, Commun. Math. Phys., 22(1971),269 E.Balslev and J.M.Combes, Commun. Math. Phys., 22(1971),280 Analytic Continuation in the Coupling Constant [Ref.] V.I.Kukulin, V.M.Krasnopol’sky, J.Phys. A10(1977), combined with the CSM (ACCC+CSM) [Ref.] S.Aoyama PRC68(2003),034313

Both enables us to obtain not only resonance energy but also total decay width


Model : 3 Orthogonality Condition Model (OCM)Model : 3 Orthogonality Condition Model (OCM)

folding for Nucleon-Nucleon interaction(Nuclear+Coulomb)

[Ref.]: E. W. Schmid and K. Wildermuth, Nucl. Phys. 26 (1961) 463

: OCM [Ref.]: S.Saito, PTP Supple. 62(1977),11

Phase shifts and Energies of 8Be, and Ground band states of 12C

,1

2

3

1

1c=1

1

2

3c=2

221

2

3c=3 3

3

[Ref.]: M.Kamimura, Phys. Rev. A38(1988),621

μ=0.15 fm-2

, -parity )


Methods for treatment of three-body resonant statesMethods for treatment of three-body resonant states

CSM It is sometimes difficult for CSM 　 to solve states with quite large 　 decay widths due to the limitation 　 of the scaling angle and finite basis states.

In order to search for the broad 0+ state, we employed … ACCC+CSM

Exp. Broad state

2θ

Resonance

Im(k)

Re(k)

δ→0

k

: Atractive potential with < 0


Energy levels obtained by CSM and ACCC+CSMEnergy levels obtained by CSM and ACCC+CSM

E.Uegaki et al.,PTP(1979)

03+: Er=1.66 MeV, Γ=1.48 MeV

22+: Er=2.28 MeV, Γ=1.1 MeV

0+ : Er=2.7+0.3 MeV, = 2.7+0.3 MeV

2+ : Er=2.6+0.3 MeV, = 1.0+0.3 MeV

[Ref.]: M.Itoh et al., NPA 738(2004)268

(2+)

= 0.375+0.040 MeVΓ=0.12 MeV

ACCC+CSM3α Model reproduce 22+ and 03

+ in the same energy region by taking into account the correct boundary condition

3α Model reproduce 22+ and 03

+ in the same energy region by taking into account the correct boundary condition


Structures of 0+ states through AmplitudesStructures of 0+ states through Amplitudes

12CJl=0,L=0 0,0l=2,L=2 2,2l=4,L=4

4,4

l

Ll,L

2 : Channel Amplitudes

8Be l,L= [ 8Be (l) x L ]

Wave function of 0+ statesWave function of 0+ states

E [MeV] Rr.m.s. [fm] 0,02 2,2

2 4,42

Er Re. Im. Re. Im. Re. Im. Re. Im.

01+ -7.29 0 2.36 0 0.364 0

0.382 0 0.254 0

02+ 0.76 2.4x10-

3 4.29 0.29 0.775 0.033 0.149 -0.019 0.076 -0.014

04+ 4.58 1.1 3.26 0.97 0.499 0.170 0.30

7 -0.017 0.194 -0.153

Channel Amplitudes of 01+, 02

+ and 04

+


Feature of the broad 3rd 0+ stateFeature of the broad 3rd 0+ state

2

2

2

l=0

L=0

8Be

Dominated

Re(Rr.m.s) (= -140): 5.44 fm

Large component of 0,02 makes such the large width.

Wave function of 03+ shows similar properties to 02

+.

03+ is considered as an excited state of 02

+. Higher nodal state of 02+ ?

Large component of 0,02 makes such the large width.

Wave function of 03+ shows similar properties to 02

+.

03+ is considered as an excited state of 02

+. Higher nodal state of 02+ ?

Similar property to 02+

（ Rr.m.s= 4.29 fm ）

Channel amplitudes as a function of


Summary of obtained 0+ statesSummary of obtained 0+ states

I=0

L=0

L=0 but higher nodal ?

I=0

r.m.s.=4.29 fm

03+03+

02+02+

04+04+


Structure of the 04+ stateStructure of the 04+ state

4th 0+ state ; Large component of high angular momentum compared with 2nd 0+

0,02 =0.499 2,2

2 =0.307, 4,42 =0.194

Total decay width is sharp: Er=4.58 MeV, =1.1 MeV

3αOCM with SU(3) base : K.Kato, H.Kazama, H.Tanaka, PTP 77(1986),185.

Component of linear-chain configuration: 56%

AMD: Y.Kanada-En’yo, nutl-th/0605047. FMD: T.Neff, H.Feldmeier, NPA 738(2004), 357.

Linear chain like structure is found

α α α


Probability Density of 1st 0+ and 4th 0+ states (Preliminary) Probability Density of 1st 0+ and 4th 0+ states (Preliminary)

04+04+

r1 r2 r1 = r2 = r

01+01+

r [fm

]

Probability Density of ’s


Summary and Future workSummary and Future work

We solve states above 3αthresold energy taking into account the boundary condition for three-body resonant states.

Obtained resonance parameters of many J states reproduce experimental data well.

We obtained broad 3rd 0+ state near the 2nd 2+ state. The state has similar structure to the 2nd 0+ state. It is thus expected to be an excited state of 2nd 0+.

The 4th 0+ state has large component of high angular momentum channel, [8Be (2+) x L=2], and has a sharp decay width.

These features reflect the linear-chain like structure of 3αclusters. Members of rotational band built upon the 4th 0+ state ?

How do these states contribute to the real energy ? To investigate it we calculate the Continuum Level Density in the CSM and partial decay widths to 8Be(0+, 2+, 4+)+α in feature. [Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237

R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273


0 ２+

0 ２+

Probability Density of 0+ states

04+04+


Contributions from resonant states to real energy

Contributions from resonant states to real energy

Continuum Level Density (CLD) Δ(E) [Ref.] S.Shomo, NPA 539 (1992) 17.

dE

dEEEE l

1

)(),()()0( 0

N

ii

N

ii

N

EEEE

EE

1

0

1

)()(

)()(

HE

EEEi

i

1TrIm

1

)()(

TE

EEEi

i

1TrIm

1

)()(0

Discretization with a finite number N of basis functions

Smoothing technique is needed,

but results depend on smoothing parameter.

Smoothing technique is needed,

but results depend on smoothing parameter.

[Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237.

δl: phase shift


CLD in the Complex Scaling Method[Ref.] R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273CLD in the Complex Scaling Method[Ref.] R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273

　 ER, εc(θ) have complex eigenvalues in CSM

RB

R

NNN

c

N

cI

cR

c

Ic

Ic

Rc

Ic

N

R rr

r

NNN

EE

EE

EEE

2020

20

22

2

22

0

)(

1

)(

1

4/)(

2/1

)()()0(

Smoothing technique is not needed

CLD in CSM:

R RBB N

R

NNN

C CR

N

BB

N

EEEEEE

)(Im

1Im

1)()(

１１

Bound state Resonance Continuum


Application to ３ α systemApplication to ３ α system

),,()( 3213

3

1

3

13 VVTtH i

i

OCMClNG

iiB

)(3

1

)point(3

1

03 i

i

ClG

iiB VTtH

α １

α ２1

2

3

033

033

11TrIm

1

)()()(

BB

BB

HEHE

EEE

CLD of 3αsystem


Continuum Level Density: 0+ statesContinuum Level Density: 0+ states

8Be(0+) +α8Be(0+) +α

8Be(2+) +α8Be(2+) +α

E [MeV]


Subtraction of contribution from 8Be+αSubtraction of contribution from 8Be+α

022

033

022

033

'

1111TrIm

1

)()()()()(

BBBB

BBBB

HEHEHEHE

EEEEE

)()( 1)point(

81

3

12

ClBe

OCMClNG

iiB VVTtH

)()( 1)point(

81)point(

3

1

02

ClBe

ClG

iiB VVTtH

α １

α ２

α ３

1

1

8Be• α1- α2: resonance + continuum

• (α1α2)- α3: continuum

• α1- α2: continuum

• (α1α2)- α3: continuum


Contributions from 8Be+α are subtractedContributions from 8Be+α are subtracted

02+

04+

03+

‘


Subtraction of contribution from 8Be+αSubtraction of contribution from 8Be+α


Search for broad 0+ state withSearch for broad 0+ state with

δ= － 50 MeV δ= － 110 MeV δ= － 150 MeV

δ= － 200 MeV δ= － 250 MeV

04+ 04

+

03+

04+

04+

03+

05+

05+

05+


Trajectories of the broad 03+ stateTrajectories of the broad 03+ state

Obtained resonance parameter

Present calc. Exp. data

Er (MeV) 1.66 2.73 + 0.3

Γ (MeV) 1.48 2.7 + 0.3

Complex-Energy plane Complex-Momentum plane


Methods for treatment of three-body resonant statesMethods for treatment of three-body resonant states

Complex Scaling Method (CSM) It is sometimes difficult for CSM to solve state with a quite

large decay width due to the limitation of the scaling angle .

In order to search for the broad 0+ state, we employed … Analytic Continuation in the Coupling Constant combined

with the CSM (ACCC+CSM)

Resonance

ACCC+CSMACCC+CSMCSMCSM

Im(k)

Re(k)

Im(k)

Re(k)

δ→0Bound state

Anti-bound state

Branch cut

:)(U )exp( i

k k

Resonance

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多体共鳴状態の境界条件によって解析した 3α 共鳴状態の構造