1. 軽い不安定核における
共鳴状態の構造
明　孝之
大阪工業大学 1
KEK 理論セミナー 　

2. Outline Structures of He isotopes
“core+valence neutrons” with complex scaling
Results
7He (a+3n) , 8He (a+4n)
Tensor correlation in 4,5,6He using “TOSM”
TM, K. Kato, K. Ikeda PRC76 (2007)
TM, K. Kato, H. Toki, K. Ikeda, PRC76 (2007)
TM, R. Ando, K. Kato PRC80 (2009)
TM, R. Ando, K. Kato PLB691(2010)150
TM, H. Toki, K. Ikeda PTP121(2009)511

3. 11Li Nuclear Chart Observation of halo structure in 11Li
I.Tanihata et al. PRL55(1985)2676. 3 3

4. Characteristics of He isotopes (expt.)
4-body
resonance
5-body
resonance
3-body
resonance
Halo
Skin
Cf. TUNL Nuclear Data Evaluation
Golovkov et al., PLB672(2009)22 4 4 4

5. Method (4He) Cluster Orbital Shell Model (COSM) Complex Scaling Method
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)
E=Er- iG/2
(4He)
Y. Suzuki, K. Ikeda, PRC38(1988)410, H. Masui, K. Kato, K. Ikeda, PRC73(2006)034318 5
S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116(2006)1 (review)

6. Cluster Orbital Shell Model
System is obtained based on RGM equation
valence neutron number
i : configuration index
 No explicit tensor correlation
, Gaussian expansion
Orthogonarity Condition Model (OCM) is applied.
Remove Pauli Forbidden states (PF) 6

7. Hamiltonian (4He) V4He-n : microscopic KKNN potential
phase shifts of 4He+n scattering
Vn-n : Minnesota potential with slightly strengthened
Fit 6He(0+)
(4He)
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

8. Completeness relation
Complex scaling for 3-body case
Completeness relation
B.G. Giraud, K. Kato, A. Ohnishi J. Phys. A 37 (‘04)11575
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. 8

9. Schrödinger Eq. and Wave Func. in CSM
Asymptotic Condition in CSM
State
No scaling
Scaling
Bound
Resonance
Continuum

10. Treatments of the unbound states in CSM
i: configuration index
Gaussian expansion
Exact asymptotic condition for resonances
Discretize continuum states.
cf. Continuum Discretized Coupled Channel (CDCC) calculation by Kyusyu Group 10

11. Spectrum of 6He with 4He+n+n model
Eth(4He+n+n)
4He+n+n
6He(*)
5He+n
A. Csoto, PRC49 (‘94) 3035,
S. Aoyama et al. PTP94(’95)343, T. Myo et al. PRC63(’01)054313 11

12. He isotopes : Expt vs. COSM (4He:(0s)4)
3-body
resonance
4-body
resonance
5-body
resonance a
TM, K.Kato, K.Ikeda PRC76(’07)054309
TM, R.Ando, K.Kato PRC80(’09)014315
TM, R.Ando, K.Kato, PLB691(‘10)150 12 12 12 12
TUNL Nuclear Data Evaluation

13. Matter & Charge radii of 6,8He
[fm]
Expt
Theor Rm
Rch
I. Tanihata et al., PLB289(‘92) G. D. Alkhazov et al., PRL78(‘97)2313
O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05) P. Mueller et al., PRL99(2007)252501 13

14. 6He=4He+n+n with ACCC+CSM
Eth(4He+n+n)
soft dipole resonance in 6He (1−).
E=(3.02i15.6) MeV
ACCC:
Analytical Continuation
in Coupling Constant
(Niigata group)
Large decay width
is obtained.
S. Aoyama (Niigata)
PRC68(’03)034313 14

15. Continuum Level Density in CSM
S. Shlomo, NPA539(’92)17
K. Arai and A. Kruppa, PRC60(’99)064315
R. Suzuki, T. Myo and K. Kato, PTP113(’05)1273.
CLD in CSM
(Kinetic) 15

16. 4He+n scattering with complex scaling
Energy eigenvalues
P3/2 scattering phase shift
30 Gaussian basis functions 16

17. 4He+n scattering with discretized continuum
Energy eigenvalues
measured from
Eth(4He+n)
Phase shifts
(s,p-waves)
R. Suzuki, T. Myo and K. Kato, PTP113(’05)1273.

18. Strength function in CSM
Bi-orthogonal relation
Green’s function and Response function 18
T. Berggren, NPA109(’68)265, T. Myo, A. Ohnishi and K. Kato, PTP99(’98)801

19. E1 of 6He into 4He+n+n (3-body breakup) Energy eigenvalues
E1 transition

20. Coulomb breakup strength of 6He
E1+E2
Equivalent photon method
TM, K.Kato, S. Aoyama and K.Ikeda
PRC63(2001)
Kikuchi, TM, Takashina, Kato, Ikeda
PTP122(2009)499
PRC81 (2010)
6He : 240MeV/A, Pb Target (T. Aumann et.al, PRC59(1999)1252) 20

21. Coulomb breakup strength of 11Li
No three-body
resonance
E1 strength by using the Green’s function method
+Complex scaling method
+Equivalent photon method
(TM et al., PRC63(’01))
T.Myo, K.Kato, H.Toki, K.Ikeda
PRC76(2007)024305
Expt: T. Nakamura et al. , PRL96,252502(2006)
Energy resolution with 　　　=0.17 MeV.

22. 7He (unbound) : Expt vs. Complex Scaling
Experiments
TM, K.Kato, K.Ikeda PRC76(’07)054309 22 22 22 22
4-body resonance
complex scaling

23. Experiments of 7He
a) RIKEN p(8He,d)7He A. A. Korsheninnikov et al., PRL82(1999)3581.
b) Berlin Be(15N,17F)7He G. Bohlen et al. ,PRC64(2001)
c) GSI He breakup M. Meister et al., PRL88(2002)
d) ANL H(6He, p)7He at 11.5 MeV/u A. H. Wuosmaa et al., PRC72(2005)
e) SPIRAL p(8He,d)7He F. Skaza et al., PRC73(2006)
f) KVI, Li(d,2He)7He N. Ryezayeva et al., PLB639(2006)623. 23

24. S-factor of 6He-n component in 7He
Bi-orthogonal relation
T. Berggren, NPA109(1968)265
TM, K.Kato, K.Ikeda,
PRC76(2007)054309
Weak coupling of 6He(0+)+n(p1/2)
6He(halo)
7He(Jp) 24 24

25. complete set of (A-1) SYSTEM
One-neutron removal strength in CSM
Bi-orthogonal
relation
Strength function and response function
energy of (A-1) SYSTEM
Response function
complete set of (A-1) SYSTEM
Complex scaled-Green’s function
T. Berggren, NPA109(’68)265, T. Myo, A. Ohnishi and K. Kato, PTP99(’98)801 25 25
S.Aoyama, TM, K.Kato, K.Ikeda, PTP116(2006)1 (review)

26. One-neutron removal strength of 7HeGS
TM, Ando, Kato
PRC80(2009)014315
” 4He+n+n” complete set using CSM
7He(3/2−)
n−1
6He(*)
5He+n
4He+n+n
4He+2n
2+1 26 26

27. Energy spectrum 8He with complex scaling
32000 dim.
Full diagonalization of
complex SX8R of NEC
TM, R.Ando, K.Kato, PLB691(‘10)150 27

28. a 8He : 0+1 & 0+2 states lj 0+1 0+2 0+1 : (p3/2)4 ~ 87% sum=4
0+2 : (p3/2)2(p1/2)2 ~ 96%

29. a 8He : 0+1 & 0+2 states Jp 0+1 0+2 (p3/2)4 0+ : 2+ = 1 : 5
Cf. AMD by
Kanada-En’yo
a,b : orbit
0+1 a
0+2 Jp
(p3/2)4
0+ : 2+ = 1 : 5
(p3/2)2(p1/2)2
0+ : 1+ : 2+ = 2 : 1.5 : 2.5
sum=4C2=6

30. Monopole Strength of 8He (Isoscalar)
0+2
6He+2n
Spin flip :
p3/2 → p1/2
CSM
q=20 deg.
4He+4n
7He+n 30

31. Monopole Strength of 8He (Isoscalar)
7He+n
0+2
6He+2n
Spin flip :
p3/2 → p1/2
CSM
q=20 deg.
4He+4n
7He+n 31

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+n
Cf: Coulomb breakup, Iwata et al. PRC62 (2000) 32

33. 2n density in 6He
Y. Kikuchi a
Dineutron a
Lowest config.
Cigar a

34. 6He(t,p)8He reaction (2n transfer)
PLB672(2009)22, JINR, Dubna
0+2