Complex absorbing potential for the continuum in real...
Transcript of Complex absorbing potential for the continuum in real...
Complex absorbing
potential for the continuum
in real-space calculations
Takashi NAKATSUKASA (RIKEN Nishina Center)
JAPAN-ITALY EFES Workshop on Correlations in Reactions and Continuum,
Torino, Italy, 6-8, Sep. 2010
L2 approximation of continuum
wave functions
Time period of emitted
particles to return
v
RT =
Corresponding to
M
E
RT
π∆E
92≈=
Therefore, in order to obtain
ΔE<1 MeV, R≥200 fm is required.
R
20 30 40 50E [ MeV ]
0
10
0
10
Monopole
Str
ength
[ fm
4 ]
0
10
0
10
20
Unperturbed
RPA
Continuum RPA
Box RPA
(R=10 fm)
[Γ=1 MeV]
(R=10 fm)
(R=100 fm)
(R=200 fm)
Test calculation:
GMR for 16O with BKN interaction
)(riη−
Continuum boundary simulated by ABC
• ABC (Absorbing boundary condition)
Particles are taken away by absorber
Outgoing waves will damp as
rkikrikr ee −→
>−−
<=
ccc
c
RrRRRri
Rrri
for )/()(
for 0)(~
00ηη
Rc R0
2/32/1
002/1
0
2/1
)8)((10
1
)8)((7 EmRR
mRR
Ec
c
−<<−
η
Criterion for a good absorber
ηiHEEG
+−=+ 1
)()(
)(~1
)()(
riHEEG
η+−=+
)(~ riη
Potential scattering
r
efee
ikriri )()(
scatt
)( Ω+ →Ψ+=Ψ ⋅∞→+⋅+ rkrk
( ) krk ViHEViHE
=Ψ+−⇒+−
=Ψ ++ )(
scatt
)(
scatt )(~1
ηη
Wave function of an outgoing scattering state V=0 (irrelevant space)
V≠0
(relevant space)All we need is the scattering wave function in the interacting
region of V≠0
0 20 40 60 80 100
θc.m.
( deg. )
10−5
10−4
10−3
10−2
10−1
100
101
dσ
el./d
σR
uth
.
exact
Wabs
= 2 MeV
Wabs
= 200 MeV
Wabs
= 2000 MeV
0 10 20 30
r ( fm )
−0.2
−0.1
0.0
0.1
0.2
Re α
L(r
)
Wabs
= 2 MeV
Wabs
= 200 MeV
Wabs
= 2000 MeV
16O+12C with optical potential
0~ ≠η0~ =η
L=30E=139.2 MeV
( ) ( ) ( )rrVerdm
frki rr
h
rr)(
22
+− Ω∫−=Ω φπ
Three-body model
(Deuteron breakup reaction)
r
R
neutron
proton
Target
L’=15, l’=2
R
r
Ueda, Yabana, TN, PRC 67, 014606 (2003)
Comparison with CDCC
d+58Ni at Ed=80 MeV
Elastic S-matrix & cross section
CDCC calculation
Yahiro et al, PTPS 89, 32 (1986)
Deuteron breakup S-matrix
( ) ( ) ( )r
eferV
HiEr
ikr
r
ikz Ω→−+
=∞→ε
χ1r
Scattering wave
( ) ikzerV
Time-dependent picture
( ) ( ) ( ) ikziHttiEiikz erVeedti
erVHiE
r hh
h
r //)(
0
11 −+∞
∫=−+
= ε
εχ
( ) ( )
( ) ( )trHtrt
i
erVtr ikz
,,
0,
rrh
r
ψψ
ψ
=∂∂
==
( ) ( )tredti
r iEt ,1 /
0
r
h
r hψχ ∫∞
=
(Initial wave packet)
(Propagation)
( ) ( ) ( )
( ) ikzikz
ikz
erVHiE
e
rrVTiE
er
−++=
−++= ++
ε
φε
φ
1
1 )()( rr
Time-dependent scattering wave
Projection on E :
( ) ( )∫=T
iEt trei
r0
/ ,1 r
h
r hψχ
( ) ( ) ( )rrVerdm
frki rr
h
rr)(
22
+− Ω∫−=Ω φπ
Time-dependent approaches
( ) ikzerV
( )riη~−
Finite time period up to T
Absorb all outgoing waves outside
the interacting region
( ) ( )( ) ( )trriHtrt
i ,~,rr
h ψηψ −=∂∂
Time evolution can stop when all the outgoing
waves are absorbed.
s-wave
absorbing potentialnuclear potential
( ) ( ) ( )krrjrVtrv ll == 0,
( ) ( )
( ) ( ) ikzerVtr
trHtrt
i
==
=∂∂
0,
,,
r
rrh
ψ
ψψ
100806040200
r (fm)
differential eq. linear eq. time-dep. Fourier transf.
( )riη~
( ) ( )tredti
r iEt ,1 /
0
r
h
r hψχ ∫∞
=
( )[ ]0Re =l
rrχ
( )[ ]0,Re =l
rtrψ
Solved for three-body systems in the time-dependent manner.
( ) ( )trredt
FeFedtidE
EFdB
F
m
F
iEt
iHt
m
iEt
,0,Re1
1Im
1),(
0
*/
0
/
0
0
/
rr
h
h
h
hh
ψψπ
φφπ
∫ ∑
∑ ∫∞
−+∞
=
−=
( )
00
00
2
0','
'
1Im
1
)'('),(
φε
φπ
φδφ
φφδ
FHiE
F
FHEF
FEEdEdE
EFdBlmE
lm
−+−=
−=
−=
+
+
∑∫
Strength function in the continuum
TN, Yabana, Ito, Eur. Phys. J. Special Topics 156, 249 (2008)
0 1 2 3
t [ /MeV ]
TDHF (TDDFT) with an absorbing potential
IS octupole resonances in 16O
0 10 20 30 40 50
[ MeV ]
0
100
200
IS octupole strength
0 1 2 3
t [ /MeV ]
Time evolution of IS
octupole moment
IS octupole strength function
[ fm6/MeV ]
Particle decay of
HEOR
Simple BKN
interaction is used.
0 10
t [ MeV−1
]
0 10 20
t [ MeV−1
]
<Ψ
(t)|
t3z |
Ψ(t
)>
TDHF for 16O
10 20 30 40
E [ MeV ]
0
50
100
σabs [
mb
]
10 20 30 40
E [ MeV ]
•SGII parameters
•Full Skyrme functional
•Γ=0.1 MeV
•T=30 ħ/MeV
Photoabsorption cross section
BBC ABC
Time evolution of E1 moment
Continuum
is discretized
pn zA
z
A
ZV ˆˆext −=
)()( extext tVktV δ=
1
Continuum is taken into account
TN and Yabana, J. Chem. Phys. 114, 2550 (2001);
Phys. Rev. C 71, 024301 (2005)
Summary
The complex absorbing potential is a simple
alternative for calculation of the scattering
properties.
The real-space representation is intuitive and
conceptually simple, but requires
considerable computational cost.
Combination with time-dependent
approaches provides us with a powerful tool.
No continuum discretization is involved.