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.
