Transcript Bound, resonant and continuum states in the complex

Structure of Resonance
and Continuum States
Unbound Nuclei Workshop
Pisa, Nov. 3-5, 2008
Hokkaido University
1. Resolution of Identity in Complex Scaling Method
Bound st.
 r
r


e

Spectrum of
 r




e
Resonant st. r
Hamiltonian
Continuum st.
Non-Resonant st.
Completeness Relation (Resolution of
Identity)
1  |un u~n  1  dk | k ~k |
n b
R
R.G. Newton, J. Math. Phys. 1 (1960), 319
R
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
Separation of resonant states from continuum states
1  |un u~n 
n b
Nr ( L)
~  1 dk | ~ |
|
u
u
 r r 
k
k
nr
Resonant states
L
Deformed
continuum states
T. Berggren, Nucl. Phys. A 109, 265 (1968)
Deformation of
the contour
Matrix elements of resonant states
r 2 ~ * ˆ
~
ˆ
u1 O u2  lim  dr e
u1 Ou2
 0 R
Convergence Factor
Method
Ya.B. Zel’dovich, Sov. Phys. JETP 12, 542 (1961).
N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965).
Complex scaling method
coordinate:
r  re
reiθ
i
r
 r 2 ~ * ˆ
~
ˆ
u1 O u2  lim  dr e u1 Ou2
 0 R

i ~ *
i ˆ
i
d
(
re
)
u
(
re
)
O
(

)
u
(
re
)
1
2

R
momentum:

B. Gyarmati and T. Vertse,
Nucl. Phys. A160, 523
(1971).
 i
k  ke
N
Nrr



 ~~

  ~~
~
11
11 
||uunn unn |  |uunn uun n|  k kdkdk||k k k k| |

nnbb
nnrr
LL
inclination of
the semi-circle
T. Myo, A. Ohnishi and K. Kato. Prog. Theor. Phys. 99(1998)801]
Resolution of Identity in Complex Scaling Method
E
k
k
E
 0
 0
Single Channel system
E|
 0
b3 b2 b1
r1
r2
B.Giraud and K.Kato, Ann.of
Phys. 308 (2003), 115.
E|
r3
B.Giraud, K.Kato and A.
Ohnishi, J. of Phys. A37
(2004),11575
Coupled Channel system
Three-body system
Structures of three-body continuum states
(Complex scaled)
Physical Importance
red: 0+
blue: 1-
of Resonant States
0+
M. Homma, T. Myo
and K. Kato, Prog.
Theor. Phys. 97
(1997), 561.
1－
B.S.
• Kiyoshi Kato
R.S.
Sexc=1.5
2
2
e fm MeV
Contributions from B.S. and R.S. to the Sum rule value
２．Complex Scaled COSM
(A) Cluster Orbital Shell Model (COSM)
• Y. Suzuki and K. Ikeda, Phys. Rev. C38 (1988), 410
Core+Xn system
The total Hamiltonian:

1
X 1 2
 X 
H  H C    p i  U i    vij 
pi  p j 
( Ac  1)m
i 1  i 1 2m
 i j 

X

1

2

i
where

X
HC : the Hamiltonian of the core cluster AC
X
Ui : the interaction between the core and
the valence neutron (Folding pot.)
vij : the interaction between the valence
neutrons (Minnesota force, Av8, …)
(B) Extended Cluster Model ー T-type coordinate system ー
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
(2n : [ i ]2J 0 )  P (cos ),
θ


  1
which has a peak in a region :
  1
The two-neutron distance :
dR

When R~5-7fm, to describe the short range
correlation accurately up to 0.5 fm, the
maximum  -value is 10~14.
(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
(  14) 2
Rapid
convergence!!
(p,sd)+T-base
Two-neutron density distribution of 6He
(0p3/2)
Hybrid-TV
２
S=1
Total
Hybrid-TV model (COSM 9ch + ECM 1ch)
S=0
18O
6He
H.Masui, K. Kato and K.Ikeda, PRC75 (2007), 034316.
Excitation of two-neutron halo nuclei (Borromean
nuclei)
Structure of three-body continuum
Three-body resonant states
Complex scaling method
Resonant state 
Soft-dipole mode
(divergent)
Bound state
(no-divergent)
S. Aoyama, T. Myo, K, Kato and K. Ikeda; Prog. Theor. Phys. 116, (2006) 1.
1- ( Soft Dipole Resonance) pole in 4He+n+n (CSM+ACCC)
(1   )V (r )  V (r )
Er~3 MeV
Γ~32 MeV
1- resonant state??
It is difficult to observe as an isolated resonant state!!
Y. Aoyama；Phys. Rev. C68 (2003) 034313.
7He: 4He+n+n+n
COSM
T. Myo, K. Kato and K. Ikeda, PRC76 (2007), 054309
3. Coulomb breakup reactions of Borromean systems
Structures of three-body continuum state
Coulomb breakup reaction
Strength Functions of Coulomb Breakup Reaction
T. Myo, A. Ohnishi and K. Kato, Prog. Theor. Phys. 99 (1998), 801.
in CSM
N r
1  |un u~n  | un u~n  1  k dk | k ~k |  1  k ' dk' | k' ~k' |  1  k " dk"| k" ~k"
n b
nr
Resonances
L
L
9Li+n+n
10Li(1+)+n
L
10Li(2+)+n
T. Myo, K. Kato, S.
Aoyama and K. Ikeda,
PRC63(2001), 054313
PRL 96, 252502 (2006)
coupled
channel
[9Li+n+n]
+
[9Li*+n+n]
T. Myo
4. Unified Description of Bound and Unbound States
Continuum Level Density
Definition of LD:

 (E)   (E  Ei )
H i  Ei i
 
1

 ( E )   ImTr 

   E  H  i 
1
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
11
 

 ( E )   ImTr 
RI in complex

   E  H  i 
scaling
NB
N B

1
1
1
1 
C
  Im

  dE

B
R
C
  nB E  EnB nR E  EnR L
E  E 
1
nR
Resonance:
E   i
Rotated Continuum:
E C   R  i I
R
nR
N R
R
nR
2
nR / 2
I
  (E  E )  
  dE
R 2
2
 n ( E   n )  n / 4  L
( E   R ) 2   I2
n
NB
B
B
nB
1
R
R
1
C
R
Descretization
εI
εI
E
E
2θ
2θ
Continuum Level Density:
( E)   ( E)  0 ( E)
 

1
1
( E )   ImTr 


   E  H  i E  H 0  i 
1
  ImTrG ( E )  G0 ( E )
1

N
Basis function method:
   cnn
n 1

Phase shift calculation in the complex scaled basis
function method
1


 d
( E ) 
Tr S ( E )
S ( E )
2i 
dE

In a single channel case,
S.Shlomo, Nucl. Phys.
A539 (1992), 17.
S ( E )  exp{2i ( E )}
1 d ( E )
( E ) 
 dE
E
 ( E )    dE' ( E ' )
0
Phase shift of 8Be=+ calculated with discretized app.
Base+CSM: 30 Gaussian basis and =20 deg.
Description of unbound states in the Complex Scaling Method
( H 0  V )  E
H0=T+VC
V； Short Range Interaction
H 00  E0
（Ψ0; regular at
origin）
Solutions of Lippmann-Schwinger Equation
1
  0 
V0
E  H  i
Outgoing
waves

Complex
Scaling
()
1
 0 
V0
E  H ( )
A. Kruppa, R. Suzuki and K. Kato,
phys. Rev.C75 (2007), 044602
T-matrix
Tl(k)
Tl(k)＝
Second term is approximated as
where
Lines : Runge-Kutta method
●Circles : CSM+Base
●
Complex-scaled Lippmann-Schwinger Eq.
•
CSLM solution
•
B(E1) Strength
Direct breakup
Final state interaction (FSI)
Dalitz distribution of
6He
• Decay process
– Di-neutron-like decay is not seen clearly.
2
E  E1  E2 
k 2k 2 cos 
A cm
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.
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.)