takashi nakatsukasa theoretical nuclear physics laboratory riken nishina center

E1 strength distribution in even-even nuclei studied with the time-dependent density

functional calculations

Takashi NAKATSUKASA

Theoretical Nuclear Physics Laboratory

RIKEN Nishina Center

2008.9.25-26 Workshop “New Era of Nuclear Physics in the Cosmos”

•Mass, Size, Shapes → DFT (Hohenberg-Kohn)

•Dynamics, response → TDDFT (Runge-Gross)

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

• dx/dt = Ax

Energy resolution ΔE 〜 ћ/T All energies

Boundary Condition• Approximate boundary conditi

on• Easy for complex systems

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

• Ax=ax (Eigenvalue problem)• Ax=b (Linear 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

How to incorporate scattering boundary conditions ?

• It is automatic in real time !

• Absorbing boundary condition

γ

n

A neutron in the continuum

2

)(

)()(22

2

fd

dr

efer

rErrVm

ikrikz

r

For spherically symmetric potential

lr

ll

ll

ll

l

lkrruu

rEururVmr

ll

dr

d

m

Pr

rur

2sin,00

2

)1(

2

cos

2

2

2

22

)(

Phase shift l

Potential scattering problem

r

eferV

HiEr

ikr

r

ikz

1Scattering wave

ikzerV

Time-dependent picture

ikziHttiEiikz erVeedti

erVHiE

r

//)(

0

11

trHtrt

i

erVtr ikz

,,

0,

tredti

r iEt ,1 /

0

(Initial wave packet)

(Propagation)

ikzikz

ikz

erVHiE

e

rrVTiE

er

1

1 )()(

Time-dependent scattering wave

Projection on E :

T iEt trei

r0

/ ,1

rrVerdm

f rki

)(

22

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

Boundary Condition

ikzerV

ri~

Finite time period up to T

Absorbing boundary condition (ABC)

Absorb all outgoing waves outside the interacting region

trriHtrt

i ,~,

How is this justified?

Time evolution can stop when all the outgoing waves are absorbed.

s-wave

absorbing potentialnuclear potential

krrjrVtrv ll 0,

ikzerVtr

trHtrt

i

0,

,,

Re u(r)

100806040200

r ( f m)

di ff erent i al eq. l i near eq. t i me-dep. Fouri er transf .

ri~

tredti

r iEt ,1 /

0

0Re r

0,Re tr

3D lattice space calculationSkyrme-TDDFT

Mostly the functional is local in density →Appropriate for coordinate-space representation

Kinetic energy, current densities, etc. are estimated with the finite difference method

Skyrme TDDFT 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 Kohn-Sham 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;(

Nuclear photo-absorption

cross section(IV-GDR)

4He

Ex [ MeV ]

0 50 100

Skyrme functional with the SGII parameter set

Γ=1 MeV

Cro

ss s

ectio

n [

Mb

]

Ex [ MeV ]10 4020 30

Ex [ MeV ]10 4020 30

14C12C

Ex [ MeV ]

10 4020 30

18O16OProlate

Ex [ MeV ]10 4020 30

Ex [ MeV ]10 4020 30

24Mg 26MgProlate Triaxial

Ex [ MeV ]10 4020 30

Ex [ MeV ]10 4020 30

28Si 30SiOblateOblate

Ex [ MeV ]10 4020 30 Ex [ MeV ]

10 4020 30

32S 34SProlate Oblate

Ex [ MeV ]

104020 30

40ArOblate

40Ca

44Ca

48Ca

Ex [ MeV ]10 4020 30 Ex [ MeV ]

10 4020 30

Ex [ MeV ]10 20 30

Prolate

Cal. vs. Exp.

Electric dipole strengths

SkM*Rbox= 15 fm = 1 MeV

Numerical calculations by T.Inakura (Univ. of Tsukuba)

Z

N

Peak splitting by deformation 3D H.O. model

Bohr-Mottelson, text book.

He

O

SiBe NeSi

CMg

Centroid energy of IVGDR

Ar

Fe

Ti

Cr

Ca

)(),( AgZNfEGDR

Low-energy strength

Low-lying strengths

Be CHe

ONe Mg Si S Ar Ca

Ti

CrFe

Low-energy strengths quickly rise up beyond N=14, 28

Summary

Small-amplitude TDDFT with the continuum Fully self-consistent Skyrme continuum RPA for deformed nuclei

Theoretical Nuclear Data Tables including nuclei far away from the stability line → Nuclear structure information, and a variety of applications; astrophysics, nuclear power, etc.

Photoabsorption cross section for light nuclei Qualitatively OK, but peak is slightly low, high energy tail is too low For heavy nuclei, the agreement is better.