11

軽い不安定核における

共鳴状態の構造

KEK 理論セミナー 　 2010.10.07

明　孝之

大阪工業大学

Outline

1. Structures of He isotopes2. “core+valence neutrons” with complex scaling 3. Results

• 7He (+3n) , 8He (+4n)4. Tensor correlation in 4,5,6He using “TOSM”

TM, K. Kato, K. Ikeda PRC76 (2007) 054309TM, K. Kato, H. Toki, K. Ikeda, PRC76 (2007) 024305TM, R. Ando, K. Kato PRC80 (2009) 014315TM, R. Ando, K. Kato PLB691(2010)150 TM, H. Toki, K. Ikeda PTP121(2009)511

3

Nuclear Chart

Observation of halo structure in 11LiI.Tanihata et al. PRL55(1985)2676.

11Li

44

Characteristics of He isotopes (expt.)

Halo

Skin

4-body resonance

5-body resonance3-body

resonance

4Cf. TUNL Nuclear Data Evaluation Golovkov et al., PLB672(2009)22

• Cluster Orbital Shell Model (COSM)Open channel effect is included. – 8He : 7He+n, 6He+2n, 5He+3n, ...

• Complex Scaling Method

Resonances with correct boundary condition as “Gamow states”

Give continuum level density (resonance+continuum)

5Y. Suzuki, K. Ikeda, PRC38(1988)410, H. Masui, K. Kato, K. Ikeda, PRC73(2006)034318

(4He)

S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116(2006)1 (review)

i ie , e r r k k

Method

E=Eri/2

6

Cluster Orbital Shell Model• System is obtained based on RGM equation

1

4 4( He) ( ) ( He) ( He) ( ) 0N

V Vji

Ai iN n H E N n C

A

4rel( He) ( He) ( )V

AH H H N n 1

4( He) ( He) ( )V

N

i ii

A C N n

A

44( He) : (0s)

• Orthogonarity Condition Model (OCM) is applied.

c

4

4He1

( )VNN

i jnncnklk

i k k lj i i jk

p pV C E E CT V

A m

PF =0 : i

1 2 3( ) : 2 Vi ii i in LN A{

i : configuration index

No explicit tensor correlation

, Gaussian expansion

Remove Pauli Forbidden states (PF)

valence neutron number

7

Hamiltonian

• V4He-n : microscopic KKNN potential

• phase shifts of 4He+n scattering

• Vn-n : Minnesota potential with slightly strengthened

A. Csoto, PRC48(1993)165. K. Arai, Y. Suzuki and R.G. Lovas, PRC59(1999)1432.TM et al. PTP113(2005)763.TM, S. Aoyama, K. Kato, K. Ikeda, PRC63(2001)054313

(4He)

Fit 6He(0+)

8

Complex scaling for 3-body case( ) : exp( ) , exp( ) , U i i r r k k

1 B B SSB

S

10 9

1

( Li+n , Li+n )

+

n

C

R

B

C

BB

R

C

R

T. Berggren, NPA109(’68)265.J.Aguilar and J.M.Combes, Commun. Math. Phys.,22(’71)269.E.Balslev and J.M.Combes, Commun. Math. Phys.,22(’71)280.

B.G. Giraud, K. Kato, A. Ohnishi

J. Phys. A 37 (‘04)11575

Completeness relation

9

Schrödinger Eq. and Wave Func. in CSM1( ) ( )U HU H T V 2 , ( )i iT e T V V e

r3/ 2 , ( ) ( )i iH E H E e e

r r

Asymptotic Condition in CSM ( ) r

( )

r

r

resr r

resr r

r r r r

exp( ) exp( )

exp( ) exp( )

exp cos( ) exp sin( )

i

iik r i e r

ik r i r e

i r r

rr r r, 0ik e

State No scaling ScalingBound

Resonance

Continuum

0 0

ie k r ie k r0

10

Treatments of the unbound states in CSM

• Exact asymptotic condition for resonances• Discretize continuum states.

1 2( )nn A{

cf. Continuum Discretized Coupled Channel (CDCC) calculation by Kyusyu Group

11 9

1( Li) ( Li) ( )

N

i ii

nn

A

2 ( )

Nnl

n lmn

a rC r Y re

Gaussian expansioni: configuration index

11

Spectrum of 6He with 4He+n+n model

A. Csoto, PRC49 (‘94) 3035, S. Aoyama et al. PTP94(’95)343, T. Myo et al. PRC63(’01)054313

Eth(4He+n+n)

4He+n+n

6He(*)

5He+n

12121212TM, K.Kato, K.Ikeda PRC76(’07)054309TM, R.Ando, K.Kato PRC80(’09)014315

4-bodyresonance

5-body resonance

3-bodyresonance

TUNL Nuclear Data Evaluation

He isotopes : Expt vs. COSM (4He:(0s)4)

TM, R.Ando, K.Kato, PLB691(‘10)150

13

Matter & Charge radii of 6,8He

[fm]

I. Tanihata et al., PLB289(‘92)261 G. D. Alkhazov et al., PRL78(‘97)2313O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05)215. P. Mueller et al., PRL99(2007)252501

TheorExptRm

Rch

14

6He=4He+n+n with ACCC+CSM

S. Aoyama (Niigata) PRC68(’03)034313

Eth(4He+n+n)

ACCC: Analytical Continuation in Coupling Constant (Niigata group)

Large decay width is obtained.

H H V

soft dipole resonance in 6He (1−). E=(3.02i15.6) MeV

15

0 (0)(0)

†

1 1Im Tr ( ) ( ) , ,

1 Tr ( ) ( ) 1 (single channel case 2

)

E G E G E GE H

dE S E S E ddEi dE

Continuum Level Density in CSM

S. Shlomo, NPA539(’92)17K. Arai and A. Kruppa, PRC60(’99)064315R. Suzuki, T. Myo and K. Kato, PTP113(’05)1273.

• CLD in CSM

00

1

1 (asymptotic)

GE H

GE H

01 Im Tr ( ) ( )E G E G E

(Kinetic)

16

4He+n scattering with complex scaling

Energy eigenvalues P3/2 scattering phase shift

30 Gaussian basis functions

17

4He+n scattering with discretized continuum

Energy eigenvaluesmeasured from Eth(4He+n)

Phase shifts(s,p-waves)

R. Suzuki, T. Myo and K. Kato, PTP113(’05)1273.

18

Strength function in CSM( )S E

† 1ˆ ˆ( ) ( ) Im ( )IIS E O O E E R E

• Strength function

• Green’s function and Response function

1( ) CB R CB RC

B B CB R C

dEE E E

G EE H E E E

( ) ( ) ( ) ( )B R CS E S E S E S E T. Berggren, NPA109(’68)265, T. Myo, A. Ohnishi and K. Kato, PTP99(’98)801

†

† †

†

ˆ( )

ˆ ˆ ˆ

ˆ ˆ

(ˆ

) II

B B I R R II I

B RB R

IC CIC

C C

R E O O

O O O OE E E E

O OdEE E

G E

Bi-orthogonal relation

19

Energy eigenvalues E1 transition

E1 of 6He into 4He+n+n(3-body breakup)

20

Coulomb breakup strength of 6He

6He : 240MeV/A, Pb Target (T. Aumann et.al, PRC59(1999)1252)

TM, K.Kato, S. Aoyama and K.IkedaPRC63(2001)054313.

6 4He (G.S.) He n n

Kikuchi, TM, Takashina, Kato, IkedaPTP122(2009)499PRC81 (2010) 044308

E1+E2Equivalent photon method

Coulomb breakup strength of 11Li

E1 strength by using the Green’s function method

+Complex scaling method+Equivalent photon method (TM et al., PRC63(’01))

• Expt: T. Nakamura et al. , PRL96,252502(2006) • Energy resolution with 　　　 =0.17 MeV.E

11 9Li (G.S.) Li n n

No three-body resonance

T.Myo, K.Kato, H.Toki, K.IkedaPRC76(2007)024305

22222222

7He (unbound) : Expt vs. Complex Scaling

1.50

1.95

Experiments TM, K.Kato, K.Ikeda PRC76(’07)054309

n n n 4(0 )s

complex scaling4-body resonance

23

Experiments of 7Hea) RIKEN p(8He,d)7He

A. A. Korsheninnikov et al., PRL82(1999)3581.

b) Berlin 9Be(15N,17F)7He G. Bohlen et al. ,PRC64(2001)024312.

c) GSI 8He breakup M. Meister et al., PRL88(2002)102501.

d) ANL 2H(6He, p)7He at 11.5 MeV/u A. H. Wuosmaa et al., PRC72(2005) 061301.

e) SPIRAL p(8He,d)7He F. Skaza et al., PRC73(2006)044301.

f) KVI, 7Li(d,2He)7He N. Ryezayeva et al., PLB639(2006)623.

2424

S-factor of 6He-n component in 7He

,'6 ' 7 2He HeJJ nlj

nlj

J JS a Bi-orthogonal relationT. Berggren,

NPA109(1968)265

TM, K.Kato, K.Ikeda, PRC76(2007)054309

7He(J)

Weak coupling of 6He(0+)+n(p1/2)

6He(halo)

2525

One-neutron removal strength in CSM( )S E

† 11

† 1 1

( ) ( )

1 Im ( )

( )

AAA A

A AA A

i

i i

i

iii

i

a a

S E a a E E

R E

R EE E

• Strength function and response function

• Complex scaled-Green’s function

1( ) i i

i iG E

E H E E

T. Berggren, NPA109(’68)265, T. Myo, A. Ohnishi and K. Kato, PTP99(’98)801

Bi-orthogonal relation

S.Aoyama, TM, K.Kato, K.Ikeda, PTP116(2006)1 (review)

complete set of (A-1) SYSTEM

Response function

energy of (A-1) SYSTEM

2626

', '

6 7 2( ) He ( ) HeJ

J J nljnlj

JS E E a

7He(3/2−)

” 4He+n+n” complete set using CSM

4He+n+n

6He(*)

5He+n

n−1

One-neutron removal strength of 7HeGSTM, Ando, KatoPRC80(2009)014315

2+1

4He+2n

27

32000 dim. Full diagonalization of complex matrix @ SX8R of NEC

Energy spectrum 8He with complex scaling

TM, R.Ando, K.Kato, PLB691(‘10)150

28

8He : 0+1 & 0+

2 states

0+1

0+2

sum=4

†lj lja a

0+1 : (p3/2)4 ~ 87%

0+2 : (p3/2)2(p1/2)2 ~ 96%

lj

8He : 0+1 & 0+

2 states

29

0+1

0+2

sum=4C2=6

J

†π πJ J(αβ) (αβ)A A

: orbit

(p3/2)4

0+ : 2+ = 1 : 5(p3/2)2(p1/2)2

0+ : 1+ : 2+ = 2 : 1.5 : 2.5

Cf. AMD by Kanada-En’yo

Monopole Strength of 8He (Isoscalar)

30

6He+2n

0+2

4He+4n7He+n

Spin flip : p3/2 → p1/2

CSM=20 deg.

Monopole Strength of 8He (Isoscalar)

31

CSM=20 deg.

6He+2n4He+4n7He+n

7He+n

2r

0+2

Spin flip : p3/2 → p1/2

32

Summary• Cluster Orbital Shell Model

+ Complex Scaling (Level density)• Coulomb breakups of 6He and 11Li

• 7He : Importance of 6He(2+1) resonance

• 8He : Five-body resonances– Differences between 0+

1 and 0+2

– Monopole strength : 8He → 7He+n → 6He+n+nCf: Coulomb breakup, Iwata et al. PRC62 (2000) 064311

2n density in 6He

33

Dineutron

Cigar

Y. Kikuchi

Lowest config.

6He(t,p)8He reaction (2n transfer)• PLB672(2009)22, JINR, Dubna

0+2