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Constraints to Universal Energy Density Functionals by Giant Resonances
FIRST FIDIPRO-JSPS WORKSHOP
Jyvaskyla、October 25-27, 2007
H. Sagawa, University of Aizu
1. Introduction
2. Incompressibility and ISGMR
3. Isotope Dependence of ISGMR and symmetry term
of Incompressibility
4. Summary
nuclear energy density functional
Density functional theory
E   Ĥ    Ĥ eff   E̂ 

Slater determinant
 ̂
  A (1 (r1 )     A (rA ))
Mean field:
E
ˆ
h
 ̂
Eigenfunctions:
hˆ i  i i
Single-particle states
density matrix
A
ˆ (r, r' )   i (r ) i (r' )
i 1
Interaction:
2

E
ˆ
V 
̂ ̂
p-h interactions
Collective Excitations
Theoretical Mean Field Models
Skyrme HF model
Gogny HF model
+tensor correlations
（Abe,Satula)
RMF model
RHF model
pion-coupling, rho-tensor coupling (Meng)
Many different parameter sets make possible to do systematic
study of nuclear matter properties and EOS.
Hamiltonian density for infinite nuclear matter : Hnm   n ,  p 
 Isoscalar part
 Isovector part
h     lim
Hnm
I 0
1
 2  Hnm 
      lim


2 I 0 I 2   
Physical properties of the infinite symmetric nuclear matter
 Saturation density :  nm
0
 h
 
   
   nm
 Symmetry energy : J
J      nm   asym
 Saturation energy per nucleon : E 0  1st derivative of  : L
 E0 
h   nm 
( SHF ),
 nm
h   nm 
 E0 
 M ( RMF )
 nm
 Incompressibility : K
2  h 
K  9
 
 2   
2
L  3
    

   nm
 2nd derivative of  : Ksym
 2    
Ksym  9 
 2
2
   nm
   nm
n   p 

I   


Nuclear Matter
SHF
RMF
Nuclear Matter
EOS

Supernova Explosion
Isoscalar Giant Monopole
Resonances
Isoscalar Compressional Dipole
Resonances
Incompressibility K
N Z

A
Self consistent HF+RPA calculations
Self consistent RMF+RPA (TD Hatree) calculations
( , ) experimtent
,
Self-consistent HF+RPA theory with Skyrme Interaction
1. Direct link between nuclear matter properties and collective
excitations
2. The coupling to the continuum is taken into account properly
by the Green’s function method.
3. The sum rule helps to know how much is the collectiveness of
obtained states.
4. Numerical accuracy will be checked also by the sum rules.
RPA Green’s Function Method
Unperturbed Green’s function
1
1
G (r , r ';  )    (r ) r

r ' i (r ')
  i  h0   i   i  h0   i
ioccupied
(0)
*
i
where i (r ) and  i are the HF single particle wave function and energy
h0i (r )   ii (r )
The inverse operator equation can be solved as
1
r
r '   Y *ljm (r ,  )Yljm (r ',  ) glj (r , r ')
h0   i    i
ljm
where
glj (r , r ')  
and
2m* (r )
2
1
u(r )v(r )
W (u, v)
r  r , r  r ' if r  r '

 r  r ', r  r if r  r '
u 
(h0   i   )    0
v
where u (v) is the regular (irregular) solution at the origin and
u (v) behaves like a standing (outgoing) wave at infinity.
RPA Green's function is then given by
G
RPA
G
(0)
 v RPA
(0)  v 1 (0)
G
G
 (1  G
) G


(0)
Strength function
S ( E )   n | f (r ) | 0  ( En  E0  E )
2
n

1

' *
'
d
r
d
r
f
(
r
)
Im{G(
r
,
r
';

)}
f
(
r
)
 
(355MeV)
(217MeV)
(256MeV)
Youngblood, Lui et al.,（２００２）、Gogny RPA
Nuclear Matter EOS

Isoscalar Monopole Giant Resonances
Isoscalar Compressional Dipole Resonances
Incompressibility K
K  (230  10) MeV for Skyrme
 (230  10) MeV for Gogny
 (250  10) MeV for RMF
What does make this difference ?
（G. Colo ,2004）
(Lalazissis,2005)
Science 298 (2002) 1592-1596
Constraints to Universal Energy Density Functionals by Giant Resonances
FIRST FIDIPRO-JSPS WORKSHOP
Jyvaskyla、October 25-27, 2007
H. Sagawa, University of Aizu
1. Introduction
2. Incompressibility and ISGMR
3. Isotope Dependence of ISGMR and symmetry term
of Incompressibility

4. Summary
N Z
A
Isovector properties of energy density functional
by extended Thomas-Fermi approximation
n   p 

I


 

J      nm 
Parameter sets of SHF and relativistic mean field (RMF) model
Notation for the Skyrme interactions
Notation for the RMF parameter sets
1
SI
2
SIII
3
SIV
14
NL3
15 NLC
4
SVI
5
Skya
6
SkM
16
NLSH
17 TM1
7
SkM*
8
SLy4
9
MSkA
18
TM2
19 DD-ME1
20
DD-ME2
10 SkI3
13 SGII
11 SkI4
12 SkX
Correlation among nuclear matter properties

2
3

2
3

N Z
A
Parameter sets of SHF and relativistic mean field (RMF) model
Notation for the Skyrme interactions
Notation for the RMF parameter sets
1
SI
2
SIII
3
SIV
15
NLSH
16 NL3
4
SVI
5
Skya
6
SkM
17
NLC
18 TM1
7
SkM*
8
SLy4
9
MSkA
19
TM2
20 DD-ME1
21
DD-ME2
10 SkI3
11 SkI4
13 SGII
14 SGI
12 SkX
-200
FSUGold
SG2
-300
SkM*
K(MeV)
-400
SkI4
SG1
SLy4
SkM
SkI3
NLC
Sa
-500
MSkA
TM1
DDME2
DDME1
S3
SkX
S1
S6
S4
-600
NL3
-700
200
TM2
220
240
260
280
300
KOO (MeV)
320
340
360
380
Correlation among nuclear matter properties
J.M. Lattimer and M. Prakash, Sience 304 (2004)
Summary
1. The pressure and incompressibility of RMF is higher than that of
SHF in general.
2. Nuclear incompressibility K is determined empirically to be
K~230MeV(Skyrme,Gogny), K~250MeV(RMF).
3. K  (500  50)MeVIs extracted from isotope dependence of
GMR of Sn
4. Then it turns out to be J=(32+/-2)MeV, L=(５0+/-１０)MeV,
Ksym= -(100+/-40)MeV
5. A clear correlation between neutron skin thickness and neutron
matter EOS, and volume symmetry energy.
6． Neutron skin thickness can be obtained by the sum rules of
charge exchange SD and also spin monopole excitations.
7．The SD strength gives a critical information both on the neutron
EOS and mean field models. 90Zr  np  0.07  0.04 fm
 np  rn2  rp2
Model independent observation of neutron skin
Electron scattering parity violation experiments
Polarized electron beam experiment at Jefferson Lab.
---- scheduled in summer 2008 ---
Sum Rule of Charge Exchange Spin Dipole Excitations
Oˆ   r[Yl 1  ] t
S  S = f | Oˆ  | i
f

2
  f | Oˆ  | i
2  1

N r2
4
2
f
n
 Z r2
p

Results of MDA for 90Zr(p,n) & (n,p) at 300 MeV
(K.Yako et al.,PLB 615, 193 (2005))
• Multipole Decomposition (MD)
Analyses
– (p,n)/(n,p) data have been
analyzed with the
same MD technique
– (p,n) data have been
re-analyzed up to 70 MeV
• Results
– (p,n)
• Almost L=0 for GTGR
region
(No Background)
• Fairly large L=1 strength
up to 50 MeV excitation at
around (4-5)o
– (n,p)
• L=1 strength up to 30MeV
at around (4-5)o
L=0
L=1
L=2
Neutron skin thickness


9
S  S 
N r2  Z r2
n
p
4
Sum rule value ⇒ N r 2  Z r 2  207  17 fm 2
n
p
Neutron thickness
r2
p
 4.19 fm
 np  0.07  0.04 fm
e scattering &
proton form factor
 np
method
nucleus
(fm) Ref.
p elastic scatt.
90Zr
IVGDR by α scatt.
116,124Sn
… ±0.12
Krasznahorkay,
PRL66(1991)1287
SDR by (3He,t)
114--124Sn
… ±0.07
Krasznahorkay,
PRL82(1999)3216
Yako(2006)
90Zr
0.09±0.07
0.07±0.04
Ray, PRC18(1978)1756
Collaborators
Theory: Satoshi Yoshida, Guo-Mo Zeng, Jian-Zhong Gu,
Xi-Zhen Zhang
Publications
S. Yoshida and H.S., Phys. Rev. C69, 024318 (2004), C73,024318(2006).
H.S., S. Yoshida, G.M.Zeng, J.Z. Gu, X.Z. Zhang, PRC76,024301(2007).
Theory: roadmap
126
Nuclear
matter
protons
82
82
28
20
2
A  
50
50
8
20
28
neutrons
28
Neutron
matter
Nuclear Landscape
Z=113
RIKEN
126
superheavy
nuclei
stable nuclei
82
protons
known nuclei
unknown region
50
proton halo
82
p-n pairing
28
20
50
neutron halo, skin, di-neutrons
Clustering, BCS-BEC crossover
8
28
2
20
2 8
neutrons
neutron stars
J=
Volume symmetry energy J=asym as well as the neutron matter pressure
acts to increase linearly the neutron surface thickness in finite nuclei.
Pigmy GDR
GDR
(p,p)
Multipole decomposition analysis
MDA
 exp ( cm , E x )   a J   calc
( cm , E x )
ph; J 
J

L  0, 1, 2, 3
DWIA
90Zr(n,p)
angular dist.
ω= 20 MeV
0-, 1-, 2-: inseparable
(1g 7 / 2 , 1g 9 /21 )
DWIA inputs
• NN interaction:
t-matrix by Franey & Love
• optical model parameters:
Global optical potential
(Cooper et al.)
• one-body transition density:
pure 1p-1h configurations
• n-particle
1g7/2, 2d5/2, 2d3/2, 1h11/2, 3s1/2
• p-hole
1g9/2, 2p1/2, 2p3/2, 1f5/2, 1f7/2
radial wave functions … W.S. / RPA
(1g 9 / 2 , 1g 9 /21 )
Neutron Matter
AV14+3body
Isoscalar and Isovector nuclear matter properties and Giant Resonances
Trento，Italy, October 8, 2007
---nuclear structure from laboratory to stars----
H. Sagawa, University of Aizu
1. Introduction
2. Incompressibility and ISGMR
3. Neutron Matter EOS and Neutron Skin Thickness
4. Isotope Dependence of ISGMR and symmetry term
of Incompressibility
5. Summary