Transcript 基研研究会2005．11

E1 strength distribution in even-even nuclei
studied with the time-dependent density
functional calculations
Takashi NAKATSUKASA
Theoretical Nuclear Physics Laboratory
RIKEN Nishina Center
•Mass, Size, Shapes → DFT (Hohenberg-Kohn)
•Dynamics, response → TDDFT (Runge-Gross)
2008.9.25-26 Workshop “New Era of Nuclear Physics in the Cosmos”
Time Domain
Energy Domain
Basic equations
• Time-dep. Schroedinger eq.
• Time-dep. Kohn-Sham eq.
Basic equations
• Time-indep. Schroedinger eq.
• Static Kohn-Sham eq.
• dx/dt = Ax
• Ax=ax (Eigenvalue problem)
• Ax=b (Linear equation)
Energy resolution
ΔE〜ћ/T
All energies
Energy resolution
ΔE〜0
A single energy point
Boundary Condition
• Approximate boundary
condition
• Easy for complex systems
Boundary condition
• Exact scattering boundary
condition is possible
• Difficult for complex systems
How to incorporate scattering
boundary conditions ?
• It is automatic in real time !
• Absorbing boundary condition
γ
A neutron in the
continuum
n
Potential scattering problem
 2
 () 
2
() 




  V r 
r  E r 

 2m

e ikr
() 
ikz
 r   e  f  
r 
r
d
2
 f  
d
For spherically symmetric potential
ul r 
() 
 r   
Pl cos 
r
l
  2 d 2  2l (l  1)


 V r ul r   Eul r 

2
2
2m r
 2m dr

l


ul 0   0, ul r   sin  kr    l 
r 
2


Phase shift  l
Time-dependent picture

()

r   eikz 
1
() 
V r  r  Scattering wave
E  i  T

1
ikz





r

V
r
e
1
ikz
E  i  H
e 
V r eikz
E  i  H

 r  

eikr
 f  
r 
r
1
1
ikz
V r e   dt ei ( E i )t /  e iHt / V r eikz
E  i  H
i 0
Time-dependent scattering wave

 r , t  0  V r eikz
(Initial wave packet)

 
i  r , t   H r , t  (Propagation)
t



1
iEt / 
Projection on E :  r  
dt e  r , t 

i 0
V r eikz
Boundary Condition
Absorbing boundary condition (ABC)
Absorb all outgoing waves outside
the interacting region
V r eikz



~
i  r , t   H  i r  r , t 
t
 i~r 
How is this justified?
 ik r
m
() 
f    
d r e V r  r 
2 
2
k' S k   (k'k )  2 i ( E ' E ) k' V  (  )
Finite time period up to T
Time evolution can stop when all the outgoing
waves are absorbed.

1 T iEt /  
 r    e  r , t 
i 0
s-wave

 
i  r , t   H r , t 
t

Re r , t 0 
vl r, t  0  V r rjl kr 
nuclear potential

 r , t  0  V r eikz
~ r 
i
absorbing potential

Re r 0 
differential eq.
linear eq.
time-dep. Fourier transf.



1
iEt / 
 r    dt e  r , t 
i 0
0
20
40
60
r (fm)
80
100
3D lattice space calculation
Skyrme-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
Time-dependent Kohn-Sham equation

~ r 
 i



i  i (r , t )  hHF [  , , j, s, J ]( t )  Vext (t )  i (r , t )
t
3D space is discretized in lattice
Single-particle orbital:
Mt
i (r, t )  {i (rk , tn )}nk 11,,
Mr , i  1,, N
N: Number of particles
y [ fm ]
Mr: Number of mesh points
Mt: Number of time slices
Spatial mesh size is about 1 fm.
Time step is about 0.2 fm/c
Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301
X [ fm ]
Real-time calculation of
response functions
1. Weak instantaneous external
perturbation
(t ) Fˆ (t )
Vext (t )  Fˆ (t )
2. Calculate time evolution of
(t ) Fˆ (t )
3. Fourier transform to energy domain
dB(; Fˆ )
1
it
ˆ
  Im  (t ) F (t ) e dt
d

dB(; Fˆ )
d
ω [ MeV ]
Cross section [ Mb ]
Nuclear photoabsorption
cross section
(IV-GDR)
4He
Skyrme functional
with the SGII
parameter set
Γ=1 MeV
0
50
Ex [ MeV ]
100
12C
14C
10
20
30
Ex [ MeV ]
10
20
30
Ex [ MeV ]
40
40
18O
16O
Prolate
10
20
30
Ex [ MeV ]
40
10
20
24Mg
26Mg
Prolate
Triaxial
30
Ex [ MeV ]
40
10
20
30
Ex [ MeV ]
40
28Si
30Si
Oblate
Oblate
10
20
30
Ex [ MeV ]
10
20
30
Ex [ MeV ]
40
40
32S
34S
Prolate
Oblate
10
10
20
30
Ex [ MeV ]
40
20
30
Ex [ MeV ]
40
40Ar
Oblate
10
20
30
Ex [ MeV ]
40
44Ca
Prolate
48Ca
40Ca
10
20
30
Ex [ MeV ]
10
10
20
30
Ex [ MeV ]
40
20
30
Ex [ MeV ]
40
Cal. vs. Exp.
Electric dipole strengths
Z
SkM*
Rbox= 15 fm
G = 1 MeV
N
Numerical calculations by
T.Inakura (Univ. of Tsukuba)
Peak splitting by deformation
3D H.O. model
Bohr-Mottelson, text book.
Centroid energy of IVGDR
O
He Be C
EGDR  f ( N , Z )  g ( A)
Fe
Cr
Ne
Mg Si Si Ar
Ca
Ti
Low-energy strength
Low-lying strengths
Be C
He
O
S Ar
Mg
Si
Ne
Ca
Ti
Fe
Cr
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
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.
Theoretical Nuclear Data Tables
including nuclei far away from the stability line
→ Nuclear structure information, and a variety of
applications; astrophysics, nuclear power, etc.