Nuclear Dynamicsin Time-Dependent Picture

Takashi Nakatsukasa

University of Tsukuba

Collaborator: Kazuhiro Yabana (U.T.)

The 6th China-Japan Joint Nuclear Physics Symposium, May 16-20, 2006

Time rep. vs Energy rep. Nuclear TDDFT (TDHF)

Giant resonances

Nuclear screening effects at drip line

Time-dependent approach to quantum mechanical problems

Basic equations• Time-dep. Schroedinger eq.• Time-dep. Kohn-Sham eq.

Energy resolution ΔE 〜 ћ/T All energies

Boundary Condition• No need for BC• Approximate BC• Easy for complex systems

Basic equations• Time-indep. Schroedinger eq. • Static Kohn-Sham eq.

(Eigenvalue equation)

Energy resolution ΔE 〜 0 A single energy point

Boundary condition• Exact scattering boundary co

ndition is possible• Difficult for complex systems

Time Domain Energy Domain

)()( tHtt

i EH

Applications of the time-dependent framework

TDHF with effective interactions

Fusion reactions:

NPA722 (2003) 261c; PTPS154 (2004) 85; nucl-th/0506073 (PLB)

Linear response in molecules:

JCP114 (2001) 2550; CPL374 (2003) 613

Linear response in nuclei:

PTPS146 (2002) 447; EPJA20 (2004) 163; PRC71 (2005) 024301

In this talk, we focus on the linear density response (RPA).

Skyrme TDHF in real space

X [ fm ]

y [

fm ]

3D space is discretized in lattice

Single-particle orbital:

N: Number of particles

Mr: Number of mesh points

Mt: Number of time slices

Nitt MtnMrknkii ,,1,)},({),( ,1

,1

rr

),()()](,,,,[),( exHF ttVthtt

i iti rJsjr �

Time-dependent Hartree-Fock equation

Spatial mesh size is about 1 fm.

Time step is about 0.2 fm/c

Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301

ri~

Real-time calculation of response functions

1. Weak instantaneous external perturbation

2. Calculate time evolution of

3. Fourier transform to energy domain

dtetFtd

FdB ti

)(ˆ)(Im1)ˆ;(

)(ˆ)( tFt

)(ˆ)(ext tFtV

)(ˆ)( tFt

ω [ MeV ]

d

FdB )ˆ;(

LEOR & HEOR in 16O

Exp for HEOR

BKN with continuum

SGII without continuum

0 10 5020 30 40E [ MeV ]

IS O

ctu

pole

Str

eng

th [

fm6/M

eV

]

300

Low-lying 3–

stateSGII int.→ E ≈ 7, 13, 14 MeV

Exp . → E ≈ 6.1, 11.6, 13, 14 MeV

Perrin et al. (1977)

E1 resonances in 16,22,28O

0 20 400

50

0

050

50

E [ MeV ]

σ [

mb

]σ

[ m

b ]

σ [

mb

]16O

22O

28O

SGII parameter set

Г=0.5 MeV

Note: Continnum is NOT taken into account !

Leistenschneider et al, PRL86 (2001) 5442

Berman & Fultz, RMP47 (1975) 713

Giant dipole resonance instable and unstable nuclei

npClassical image of GDR

Neutrons

Protons

δρ> 0

δρ< 016O

ppp tt 0)()(

nnn tt 0)()( Time-dep. transition density

28O

Skyrme HF for 8,14Be

S.Takami, K.Yabana, and K.Ikeda, Prog. Theor. Phys. 94 (1995) 1011.

8Be

14Be

Neutron Proton

x

z

x

y

x

z

Solid: K=1Dashed: K=0

nnn tt 0)()(

8Be

14Be

δρ> 0

δρ< 0

Giant dipole resonance

n

p

n

p

“Screening”N=Z nuclei Neutron-rich

E1 polarizability

Neutrons

Protons

8Be

No dynamical screening With dynamical screening

NeutronsProtons

Protons

Protons

Neutrons

Neutrons

Total Total

Total Total

1

11

Eext

iiEiE

DV

DD

14Be 14Be

8Be

Negative polarization of weakly-bound neutrons

No dynamical screening Dynamical screening

tEext

+

+

+

+

+

+

+

－

－

－

－

－

－

－

tEind

)1(ext EV

npV

n p

Electronic dynamical screening Nuclear dynamical screening

Summary• TDDFT(TDHF)+ABC to study dynamical

aspects of nuclear response in the continuum

• Neutron-proton attractive correlation leads to a complex dipole motion (“screening”) for neutron-rich nuclei

• …, though the frequency decomposition is necessary for a definite answer.

Stable (N=Z) Neutron-rich (N >> Z)

)()'(2)( VEEiS k'kk'kk'

Boundary Condition

0extV

ri~

Absorbing boundary condition (ABC)

Absorb all outgoing waves outside the interacting region

trriHtrt

i ,~,

How is this justified?

All the scattering information resides in the interacting region.

iiiE

iEt cctFtedtdE

EEdB..)(),(),(Im

1),( *)0(

0

/

rr

Localized w.f.

T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114(2001)2550.

Linear optical absorption

ExpTDDFT

Without dynamical screening(frozen Hamiltonian)

TDDFT accurately describe optical absorptionDynamical screening effect is significant

),()],([),( trtrnhtrt

i ii

),()]([),( 0 trrnhtrt

i ii

withwithout

Dynamical screening

tEext

++

+

++

+

+

－－

－－

－

－

－

tEind

PZ+LB94

Damping width of GDR near drip line

Enhancement of escape width : Г↑

Phase space

Threshold effect

Enhancement of Landau damping : ГL↓

Large diffuseness of the mean-field potential

→ Many 1p-1h states with the same symmetry

ωGDR ≈ 79 A-1/3 MeV ≈ 2 ω0

Positive-parity

1p-1h 2 ω0 excitations