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.