Structure of Resonance and Continuum States

1
Structure of Resonance and Continuum States
Unbound Nuclei Workshop Pisa, Nov. 3-5, 2008

• Hokkaido University

2
1. Resolution of Identity in Complex Scaling
Method
Bound st.
Spectrum of Hamiltonian
Resonant st.
Continuum st.
Non-Resonant st.
Completeness Relation (Resolution of Identity)
R
R.G. Newton, J. Math. Phys. 1 (1960), 319
3
Among the continuum states, resonant states are
considered as an extension of bound states
because they result from correlations and
interactions.
From this point of view, Berggren said In
the present paper,) we investigate the
properties) of resonant states and find them
in many ways quite analogous to those of the
ordinary bound states. ) NPA 109 (1968),
265. ) orthogonality and completeness
4
Separation of resonant states from continuum
states
Deformed continuum states
Resonant states
T. Berggren, Nucl. Phys. A 109, 265 (1968)
Deformation of the contour
Matrix elements of resonant states
Convergence Factor Method
Ya.B. Zeldovich, Sov. Phys. JETP 12, 542
(1961). N. Hokkyo, Prog. Theor. Phys. 33, 1116
(1965).
5
Complex scaling method
rei?
coordinate
r
B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523
(1971).
momentum
inclination of the semi-circle
T. Myo, A. Ohnishi and K. Kato. Prog. Theor.
Phys. 99(1998)801
6
Resolution of Identity in Complex Scaling Method
k
E
E
k
Single Channel system
B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115.
E
E
b3
b2
b1
r2
r3
r1
B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37
(2004),11575
Coupled Channel system
Three-body system
7
Structures of three-body continuum states

(Complex scaled)
8
Physical Importance of Resonant States
red 0
blue 1-
0
1-
M. Homma, T. Myo and K. Kato, Prog. Theor. Phys.
97 (1997), 561.
9
B.S.

• Kiyoshi Kato

R.S.
Sexc1.5
e2fm2MeV
Contributions from B.S. and R.S. to the Sum rule
value
10
(A) Cluster Orbital Shell Model (COSM)
2.Complex Scaled COSM

• Y. Suzuki and K. Ikeda, Phys. Rev. C38 (1988), 410

CoreXn system The total Hamiltonian
where HC the Hamiltonian of the core cluster
AC Ui the interaction between the core and
the valence neutron (Folding pot.) vij
the interaction between the valence
neutrons (Minnesota force, Av8, )
X
11
(B) Extended Cluster Model ? T-type coordinate
system ?
Y. Tosaka, Y Suzuki and K. Ikeda Prog. Theor.
Phys. 83 (1990), 1140. K. Ikeda Nucl. Phys. A538
(1992), 355c.
The di-neutron like correlation between valence
neutrons moving in the spatially wide region
?
which has a peak in a region The two-neutron
distance
When R5-7fm, to describe the short range
correlation accurately up to 0.5 fm, the maximum
-value is 1014.
12
(C) Hybrid-TV Model

• S. Aoyama, S. Mukai, K. Kato and K. Ikeda, Prog.
  Theor. Phys. 94, 343-352 (1995)

(p3/2)2
(p1/2)2
Rapid convergence!!
(p,sd)T-base
13
Two-neutron density distribution of 6He
(0p3/2) 2
Hybrid-TV
S0 S1 Total
Harmonic oscillator (0p3/2 only)
Hybrid-TV model (COSM 9ch ECM 1ch)
14
18O
6He
H.Masui, K. Kato and K.Ikeda, PRC75 (2007),
034316.
15
Excitation of two-neutron halo nuclei (Borromean
nuclei)
Structure of three-body continuum
Three-body resonant states
Complex scaling method Resonant state ?
Bound state (divergent)
(no-divergent)
Soft-dipole mode
S. Aoyama, T. Myo, K, Kato and K. Ikeda Prog.
Theor. Phys. 116, (2006) 1.
16
(No Transcript)
17
1- ( Soft Dipole Resonance) pole in 4Henn
(CSMACCC)
Er3 MeV G32 MeV
1- resonant state??
It is difficult to observe as an isolated
resonant state!!
Y. AoyamaPhys. Rev. C68 (2003) 034313.
18
(No Transcript)
19
7He 4Hennn COSM
T. Myo, K. Kato and K. Ikeda, PRC76 (2007), 054309
20
3. Coulomb breakup reactions of Borromean systems
Structures of three-body continuum state


Coulomb breakup reaction
21
Strength Functions of Coulomb Breakup Reaction
22
T. Myo, A. Ohnishi and K. Kato, Prog. Theor.
Phys. 99 (1998), 801.
in CSM

10Li(1)n
10Li(2)n
9Linn
Resonances
23
T. Myo, K. Kato, S. Aoyama and K. Ikeda,
PRC63(2001), 054313
24
(No Transcript)
25
PRL 96, 252502 (2006)
coupled channel 9Linn 9Linn
T. Myo
26
4. Unified Description of Bound and Unbound States
Continuum Level Density
Definition of LD
A.T.Kruppa, Phys. Lett. B 431 (1998),
237-241 A.T. Kruppa and K. Arai, Phys. Rev. A59
(1999), 2556 K. Arai and A.T. Kruppa, Phys. Rev.
C 60 (1999) 064315
27
1
RI in complex scaling
Resonance Rotated Continuum
Descretization
28
eI
eI
E
E
2?
2?
29
Continuum Level Density
Basis function method
30
Phase shift calculation in the complex scaled
basis function method
S.Shlomo, Nucl. Phys. A539 (1992), 17.
In a single channel case,
31
Phase shift of 8Be?a calculated with
discretized app. BaseCSM 30 Gaussian basis and
?20 deg.
32
Description of unbound states in the Complex
Scaling Method
H0TVC
V Short Range Interaction
(?0 regular at origin)
Solutions of Lippmann-Schwinger Equation
Complex Scaling
Outgoing waves
A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75
(2007), 044602
33
T-matrix
Tl(k)
Tl(k)
Second term is approximated aswhere
34

• Lines Runge-Kutta method
• Circles CSMBase

35
Complex-scaled Lippmann-Schwinger Eq.

• CSLM solution

• B(E1) Strength

36
Dalitz distribution of 6He

• Decay process
• Di-neutron-like decay is not seen clearly.

37
6. Summary and conclusion

• It is shown that resonant states play an
  important role in the continuum phenomena.
• The resolution of identity in the complex
  scaling method is presented to treat the
  three-body resonant states in the same way as
  bound states.
• The complex scaling method is shown to describe
  not only resonant states but also non-resonant
  continuum states on the rotated branch cuts.
• We presented several applications of the
  extended resolution of identity in the complex
  scaling method sum rule, break-up strength
  function and continuum level density.

38
Collaboration S. Aoyama(Niigata Univ.), H.
Masui(Kitami I. T.), T. Myo (Osaka Tech. Univ.),
R. Suzuki(Hokkaido Univ.), C.
Kurokawa(Juntendo Univ.), K. Ikeda(RIKEN)
Y. Kikuchi(Hokkaido Univ.)