Transcript Document

Determining Modern Energy Functional for Nuclei And
The Status of The Equation of State of Nuclear Matter
Shalom Shlomo
Texas A&M University
Outline
1.
Introduction.
Collective States, Equation of State,
2.
Energy Density Functional.
Hartree-Fock Equations (HF), Skyrme Interaction
Simulated Annealing Method, Data and Constraint
3.
4.
Results and Discussion.
HF-based Random-Phase-Approximation (RPA).
Fully Self Consitent HF-RPA, Hadron Excitation of Giant
Resonances, Compression Modes and the NM EOS,
Symmetry Energy Density
5.
6.
Results and Discussion.
Conclusions.
Introduction
1. Important task: Develop a modern Energy Density
Functional (EDF), E = E[ρ], with enhanced predictive
power for properties of rare nuclei.
2. We start from EDF obtained from the Skyrme N-N
interaction.
3. The effective Skyrme interaction has been used in
mean-field models for several decades. Many different
parameterizations of the interaction have been realized
to better reproduce nuclear masses, radii, and various
other data. Today, there is more experimental data of
nuclei far from the stability line. It is time to improve the
parameters of Skyrme interactions. We fit our meanfield results to an extensive set of experimental data and
obtain the parameters of the Skyrme type effective
interaction for nuclei at and far from the stability line.
Map of the existing nuclei. The black squares in the central zone are stable nuclei,
the broken inner lines show the status of known unstable nuclei as of 1986 and the
outer lines are the assessed proton and neutron drip lines (Hansen 1991).
Equation of state and nuclear matter compressibility
The symmetric nuclear matter (N=Z and no Coulomb) incompressibility
coefficient, K, is a important physical quantity in the study of nuclei, supernova
collapse, neutron stars, and heavy-ion collisions, since it is directly related to the
curvature of the nuclear matter (NM) equation of state (EOS), E = E(ρ).
2
2
f
2
f
k
fo

2
d
(
E
/
A
)
d
(
E
/
A
)
2
K

k

9
2
dk
d 
2

1
o


E
[]

E
[o
]
K


18
 o
 
E/A [MeV]


o
ρ = 0.16
   o (  
1

EANM [  o (  ]  E[  o (  ]  K (  o (  )
18

(


o


fm-3
E[o ( ]  E[o ]  J 2
K[ o ( ]  K  Kv 2
E/A = -16 MeV
ρ [fm-3]
1 d 2 ( E / A)
ESYM (  ) 
8 dy2
 , y 1 / 2
J  ESYM [ o ]
  (N  Z ) / A
yZ/A
2
Macroscopic picture of giant resonance
L=0
L=1
L=2
A Classical Picture of the Breathing Mode
In the classical description of the
breathing mode, the nucleus is modeled
after a drop of liquid that oscillates by
expanding and contracting about its
spherical shape.
We consider the isoscalar breathing
mode in which the neutrons and protons
move in phase (∆T=0, ∆S=0).
o
 (r, t )  o   (r, t )
In the scaling model, we have the matter density oscillates as
 (r )   3 (t )  ( (t )r ),
 (t )  1   cos(t ),  
2E
.
h
We consider small oscillations, so є is very close to zero (≤ 0.1).
Performing a Taylor expansion of density
 (r , t )   3 (t )  ( (t )r )   3 (t )  ( (t )r )  (t )1
( (t )3  ( (t )r ))
 [ (t )  1]
 ... ,
 (t )
 ( t ) 1
we obtain,,


 (r , t )  o (r )   cos(t ) 3o  r
do 
.

dr 
We have,
0.24
 (r, t )  o (r )   (r ) cos(t )
o (r )   (r )
0.20
o (r )
0.16
Where  (r ) is equal to


 (r )    3o  r
 (r )
o (r )   (r )
0.12
0.08
do 

dr 
0.04
0.00
0
1.5
3
4.5
6
7.5
9
10.5
12
13.5
15
-0.04
This  (r ) nicely agrees with the
0.08
transition density obtained from
microscopic (HF based RPA)
calculations
  0 .1
0.04
 (r )
0.00
0
1.5
3
4.5
6
7.5
9
-0.04
-0.08
r[ fm]
10.5
12
13.5
15
Modern Energy Density Functional
Within the HF approximation: the ground state wave function





(
r
,

,
) 
(
r
,

,
) ...

(
r
,

,
)




(
r
,

,

)

(
r
,

,

)
...

(
r
,

,

)
1


11 11
21 11
A1 11
12 2 2
22 2 2
A2 2 2




A
!




(
r
,

,

)

(
r
,

,

)
...

(
r
,

,

)
1A AA 2A AA
AA
A
A
In spherical case




R
(
r
)


i
(
r
,,)
 Y
(
r
,)m
(
)
i
jlm

r
HF equations:
minimize
ˆ 
E
H
total
Skyrme interaction

NN Coul
For the nucleon-nucleon interaction V
(
r
,
r
)

V
V
i j
ij 
ij .
2 A


ij
ij
Coule
V


,



ij
4
r
i,j
1r
i
j
2
VijNN
ij i  j
we adopt the standard Skyrme type interaction

2
V
t(
1

x
P
)

(
r

r
)

t
(
1

x
P
)[
k

(
r

r
)


(
r

r
)
k
]

0 ij
i j
1
1ij ij i j
i j ij
2



r
r

  1
 i
j


t
(
1

x
P
k

(
r
r
k

t
(
1

x
P


(
r
r

2
2
ij)
ij
i
j)
ij
3
3
ij)
i
j)


6
2
  
iW
k

(
r
r


k
,
0
ij
i
j)(
i
j)
ij
NN
0
ij
ti, xi, , W
0
  1

2 
are 10 Skyrme parameters.
The total Hamiltonian of the nucleus
A2


p
i
ˆ 


H
T

V


V
r
,
r

total 
i j
2
m
i

1

j

1
i i
where

NN Coul
V
(
r
,
r
)

V
V
i j
ij 
ij .
The total energy
2 A
 

* 
ˆ
E 
H
 
i (r)
i (r)dr
total

2mi1
 *  

 

i (r)j (r')V(r,r')i (r)j (r')drdr'
A
*
ij


 
*  * 


(
r
)

(
r
'
)
V
(
r
,
r
'
)

(
r
'
)

(
r
rd
r'
i
j )d
 i j
A
ij
The total energy
 
E   Hˆ total    T  VCoulomb  V12    H (r )dr
where




H (r )  H Kinetic(r )  HCoulomb (r )  H Skyrme (r )
2
2





H Kinetic (r ) 
 p (r ) 
 n (r )
2m p
2mn
  2 


ch (r , r ' ) 
 e
 ch (r ' ) 
H Coulomb (r )   ch (r )    dr '    
dr '
2 
r  r'
r  r'

2

H Skyrme (r )  Η0  H3  Heff  H fin  H so  H sg


 A  2

r
)



r
,

,

)


(
i(


(r)
(r)
i

1


 
J(r)
J(r)





 A 

(
r
)

(
r
,
,
)
(
r



i
i ,,)
i

1



(r)
(r)














 A*
 


J
(
r
)


i
(
r
,
)

(
r
,',
)
'

 
i ,
i
'
i

1
,


1 1

(
r
,
r
'
)


(
r
,

,)

(
r
,

,)

2
2
ch
*
i


'
'
i
i
,,'
Now we apply the variation principle to derive the Hartree-Fock
equations. We minimize the Energy E, given in terms of the energy
density functional
E   Hˆ total    H  r dr



  





d
r

i

,





E
i


E

d
r



0(*
i

,



i


,

,


,

.
where










 
2





 



E

(
r
)

U
(
r
)
(
r
)

W
(
r
)
J
(
r
)
d
r





*
2
m
(
r
)
,








(
r
,

'
,

)

(
r
,

'
,

)


i

*
i
i
,'




  *

(
r
)


(
r
,
'
,
)

(
r
,)
 
i
i ,'
i
,'
















 

*
J
(
r
)


i
(
r
,
'
,
)

(
r
,
'
'
,
)

'
"
 
i
i
i
,
'
,
'
'
After carrying out the minimization of energy, we
obtain the HF equations:
2
2
l(
l
1
)

 "
 d   '


R
(
r
) 2 R
(
r
)
 
R
(
r
)



*
*




2
m
r
)
r
m
r
)
 dr
(
(
2


3


j
(
j

1
)

l
(
l

1
)

 

2




1d 
4






U
r
) 

W
r
)
R
(
r
)
(
(

*


rdr
m
r
)
r


(
2





R
(
r
)

where m* (r ) , U (r ) , and W (r ) are the effective
mass, the potential and the spin orbit potential.
They are given in terms of the Skyrme parameters
and the nuclear densities.



2 2




1
11
1
11






t
(
1

x
)

t
(
1

x
)
(
r
)

t
(

x
)

t
(

x
)
(
r
)
 
11
2 2
11
22
*



2
m
4
22
4
22
2
m
(
r
)




 


 1
1
1
1
1  
U
(
r
)

t
(
1

x
)

(
r
)

t
(

x
)

(
r
)

t
(
1

x
)

t
(
1

x2)
(r)

0
0
0
0 
1
1
2

2
2
4
2
2 
1 1
1
1
  2

1
 t1( x
)

t
(

x
)

(
r
)

t
(
1

x
)

(r)
1
2
2 
3
3
4 2
2
12
2

 1
1 1

1
2
2 

 t3( x3)
(r)
(
r
)


(
r
)

t
(

x
)

(r)



3
3

12 2
6 2
1
1
1  2  1 1
1
2 
 3
t1(1 x
)

t
(
1

x
)


(
r
)

3
t
(

x
)

t
(

x
)

r)
1
2
2
1
1
2
2
(

8
2
2 
8 2
2




r')

1
2
ch
.(
W

J
(
r
)


J
(
r
)


e
d
r
'
 ,
0


1
,
2
rr'
2













1
1
1 


W
(
r
)

W

(
r
)


(
r
)

(
t

t
)
J
(
r
)

[
t
x

t
x
]
J
(
r
0
1
2
1
1
2
2)
2
8
8


With an initial guess of the single-particle wave functions (example;
harmonic oscillator wave functions), we can determine m*, U(r), and
W(r) and solve the HF equation to get a set of new single-particle
wave functions; then one can proceed in this way until reaching
convergence.
NOTES:
1. One should start close to the solution.
2. Accuracy and convergence in three dimension
3
Convergence of HFB equations in three dimension?
Approximations
Coulomb energy


1
3
dir
ex
H

V
(
r
)
(
r
)

V
(
r
)
(
r
)
Coulomb
Coul
p
Coul
p
2
4

(
r
')
dr
'
p
dir 2
V

e
Coul   
r
r
'
3

(
r
)3
3
p
ex
2
V


e
Coul
  



1
Note that here the direct term and exchange Coulomb
terms each include the spurious self-interaction term.
Interaction:
KDEX,
KDE0,
neglect exchange term
KDE,
include exchange term
include contributions of g.s. correlations
Center of mass correction
a). Correction to the total binding energy:
We use the harmonic oscillator approximation. The CM energy is taken as
osc 3
KCM
  but
4
 Is determined by using the mass mean-square radii r 2
 3



 2
N

i

mA
r i
 2

2

b). Correction to the charge rms radii rch
The charge mean-square radius to be fitted to the experimental data
is obtained as







22 3
2N
21
r

r


r

r

nlj
(
2
j

1
)
.
l



ch
p
lj
HF
p
n
2
A
Z
Z
mc


Simulated Annealing Method (SAM)
The SAM is a method for optimization problems of large scale, in
particular, where a desired global extremum is hidden among many
local extrema.
We use the SAM to determine the values of the Skyrme
parameters by searching the global minimum for the chi-square
function
2
exp
th
d


M

M
1N
i
i






N

N
i

1
d
p 
i


2

Nd is the number of experimental data points.
Np is the number of parameters to be fitted.
M iexp and M ith are the experimental and the corresponding
theoretical values of the physical quantities.
i
is the adopted uncertainty.
Implementing the SAM to search the global minimum of
1.
ti, xi, , W
are written in term of
0
2. Define
3. Calculate

2
function:
B
/A
, K
, 
,...
nm
nm


'
v
(
B
/
A
,
K
,
,
m
*
/
m
,
E
,
J
,
L
,
,
G
,
W
)
nm
nm
s
0
0
2
 old
for a given set of experimental data and the corresponding
HF results (using an initial guess for Skyrme parameters).
.
4. Determine a new set of Skyrme parameters by the following steps:
+ Use a random number to select a component
+ Use another random number
vr
of vector
 to get a new value of

v
vr
vr vr d
+ Use this modified vector
parameters.

v
to generate a new set of Skyrme
5. Go back to HF and calculate
2
 new
6. The new set of Skyrme parameters is accepted only if
 

2
2



2
old
new

P
()

exp

 T 



0   1
7. Starting with an initial value of
number of loops.
T  Ti , we repeat steps 4 - 6 for a large
8. Reduce the parameter T as T 
Ti
and repeat steps 1 – 7.
k
9. Repeat this until hopefully reaching global minimum of
2
Fitted data
- The binding energies for 14 nuclei ranging from normal to the exotic
(proton or neutron) ones: 16O, 24O, 34Si, 40Ca, 48Ca, 48Ni, 56Ni, 68Ni, 78Ni,
88Sr, 90Zr, 100Sn, 132Sn, and 208Pb.
- Charge rms radii for 7 nuclei: 16O, 40Ca, 48Ca, 56Ni, 88Sr, 90Zr, 208Pb.
- The spin-orbit splittings for 2p proton and neutron orbits for
(2p1/2) - (2p3/2) = 1.88 MeV (neutron)
(2p1/2) - (2p3/2) = 1.83 MeV (proton).
56Ni
- Rms radii for the valence neutron:
1
d
3
.36
fm
in the 1d5/2 orbit for 17O r
n(
5
/2)
1
f7/2)
3
.99
fm
in the 1f7/2 orbit for 41Ca r
n(
- The breathing mode energy for 4 nuclei: 90Zr (17.81 MeV),
(15.9 MeV), 144Sm (15.25 MeV), and 208Pb (14.18 MeV).
116Sn
Constraints
2

3

0 
0
cr
1. The critical density












Landau
'
'


V

F

F

G

G
r

r
p

h 
l l
1
2l1
2l1
2
1
21
2
l
Landau stability condition:
Example:
'
'
F
F
G
G

(
2
l
1
)
l,
l,
l,
l
22

k

1

F
0
K

6F


1

F
/
3
1
2
m
'
2. The Landau parameter G0 should be positive at    0
3. The quantity P3
dS
must be positive for densities up to 3  0
d
4. The IVGDR enhancement factor
0
.25
0
.5

2

NZ
ES
(
E
)
dE

(
1

)

2
m
A
T

1
L

1
v
v0
v1
d
B/A (MeV)
16.0
17.0
15.0
0.4
Knm (MeV)
230.0
200.0
300.0
20.0
ρnm (fm-3)
0.160
0.150
0.170
0.005
m*/m
0.70
0.60
0.90
0.04
Es (MeV)
18.0
17.0
19.0
0.3
J (MeV)
32.0
25.0
40.0
4.0
L (MeV)
47.0
20.0
80.0
10.0
Kappa
0.25
0.1
0.5
0.1
G’0
0.08
0.00
0.40
0.10
W0 (MeV fm5)
120.0
100.0
150.0
5.0
Variation of the average value of
2
T
as a function of the inverse of
the control parameter T for the KDE0 interaction for the two different choices
of the starting parameter.
The values of the Skyrme parameters
Parameter
KDE0
KDE
SLy7
t0 (MeV fm3)
-2526.51 (140.63)
-2532.88 (115.32)
-2482.41
t1 (MeV fm5)
430.94 (16.67)
403.73 (27.63)
457.97
t2 (MeV fm5)
-398.38 (27.31)
-394.56 (14.26)
-419.85
t3 (MeV fm3(1+α))
14235.5 (680.73)
14575.0 (641.99)
13677.0
x0
0.7583 (.0.0655)
0.7707 (0.0579)
0.8460
x1
-0.3087 (0.0165)
-0.5229 (0.0298)
-0.5110
x2
-0.9495 (0.0179)
-0.8956 (0.0270)
-1.000
x3
1.1445 (0.0882)
1.1716 (0.0767)
1.3910
128.96 (3.33)
128.06 (4.39)
126.00
0.1676 (0.0163)
0.1690 (0.0144)
0.1667
W0 (MeV fm5)
α
Nuclear matter properties
Parameter
KDE0
KDE
SLy7
B/A (MeV)
16.11
15.99
15.92
K (MeV)
228.82
223.89
229.7
ρnm
0.161
0.164
0.158
m*/m
0.72
0.76
0.69
Es (MeV)
17.91
17.98
17.89
J (MeV)
33.00
31.97
31.99
L (MeV)
45.22
41.43
47.21
қ
0.30
0.16
0.25
G’0
0.05
0.03
0.04
χ2min
1.3
2.2
Nuclei
Results for the total
binding-energy B
G. Audi et al, Nucl.
Phys. A729, 337
(2003)
∆B = Bexp -Bth
Bexp
KDE0
KDE
SLy7
-0.93
16O
127.620
0.394
1.011
24O
168.384
-0.581
0.370
34Si
283.427
-0.656
0.060
40Ca
342.050
0.005
0.252
-2.85
48Ca
415.99
0.188
1.165
0.11
48Ni
347.136
-1.437
-3.67
56Ni
483.991
1.091
1.016
1.71
68Ni
590.408
0.169
0.539
1.06
78Ni
641.940
-0.252
0.763
88Sr
768.468
0.826
1.132
90Zr
783.892
-0.127
-0.200
100Sn
824.800
-3.664
-4.928
-4.83
132Sn
1102.850
-0.422
-0.314
0.08
208Pb
1636.430
0.945
-0.338
-0.33
Results for the charge rms radii
E. W. Otten, in Treatise onn
Heavy-Ion Science, Vol 8
(1989).
H. D. Vries et al, At. Data
Nucl. Tables 36, 495
(1987).
F. Le Blanc et al, Phys.
Rev. C 72, 034305 (2005).
Nuclei
Expt.
KDE0
KDE
16O
2.730
2.771
2.769
24O
2.778
2.771
34Si
3.220
3.208
40Ca
3.490
3.490
3.479
48Ca
3.480
3.501
3.488
3.795
3.777
3.768
3.750
68Ni
3.910
3.893
78Ni
3.969
3.950
48Ni
56Ni
3.750
88Sr
4.219
4.221
4.200
90Zr
4.258
4.266
4.1005
4.480
4.457
100Sn
132Sn
4.709
4.710
4.685
208Pb
5.500
5.489
5.459
GIANT RESONANCES
• Hadron Scattering
• HF-Based RPA
• Results
Hadron excitation of giant resonances
Ψf
χf
VαN
α
χi
Nucleus
Ψi
Theorists: calculate transition strength S(E) within HF-RPA using a simple scattering
operator F ~ rLYLM:
Experimentalists: calculate cross sections within Distorted Wave Born Approximation
(DWBA):
or using folding model.
The Scattering Operator



 



  
 
  
 
 *
 
 k f , ki , r    f k f , R V R  r i ki , R dR
can be simplified in certain cases to have a simple form
 
 

A) Born Approximation: α-wave functions are plane wave:  (k , r )  exp ik .r

 
~
 

  
 

i k f  k i . R
iq .(r  r ')
  e
V R  r dR   e
V (r ' )dr '  V (q ) f (r )





 


V (q)   e V (r ' )dr ', momentum transfer q  k f  ki , f (r )  expiqr .
 
 
VN R  r  V0 R  r
~

iqr

B)

 *
  f





 

 
      

k f , R V0 R  r  i ki , R dR  V0 f (r )
     

  *
f ( r )   f ( k f , r )  i ( ki , r )
C) Coulomb excitation by a classical particle





Ze 2


V R (t )  r  

A
R
(
t
)
f
(
r
),
lm
  lm
R (t )  r
lm

flm (r )  r lY lm( , )
Thus: the cross section is directly related to the matrix element

 f O i ~  f  f (rk ) i
One defines:
S ( E )   0 F n  ( E  En ),
2
n
S(E) ≡ strength function, response function, structure function.
Example: inelastic electron scattering in Born approximation
d
 d 
dd

 S (E )
d


 Mott
General transition operator F:


S  (1;  ) 

T   (1;  ) 

F  f (r ) Y  S
L

 J

T
M 
(Electric 0+, 1-, 2-; Magnetic 0-, 1+, 2-)
(Isoscalar, Isovector)
Energy weighted sum rule (EWSR)
1
0 F , F , H  0
2
n
Giant resonance: contributes 30 - 100% to EWSR.
EWSR( F )   ES ( E )dE   En n F 0
2

DWBA-Folding model description
EWSR = energy weight
sum rule


  ES ( E )dE
0

Elastic angular distributions
for 240 MeV alpha particle.
Filled squares represent the
experimental data. Solid lines
are fit to the experimental data
using the folding model DWBA
with nucleon-alpha interaction.
History
A.
ISOSCALAR GIANT MONOPOLE RESONANCE (ISGMR):
1977 – DISCOVERY OF THE CENTROID ENERGY OF THE ISGMR IN 208Pb
•
E0 ~ 13.5 MeV (TAMU)
This led to modification of commonly used effective nucleon-nucleon interactions.
Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations, with
effective interactions (Skyrme and others) which reproduce data on masses, radii
and the ISGMR energies have:
K∞ = 210 ± 20 MeV (J.P. BLAIZOT, 1980).
A.
ISOSCALAR GIANT DIPOLE RESONANCE (ISGDR):
1980 – EXPERIMENTAL CENTROID ENERGY IN 208Pb AT
E1 ~ 21.3 MeV (Jülich), PRL 45 (1980) 337;
•
~ 19 MeV, PRC 63 (2001) 031301
HF-RPA with interactions reproducing E0 predicted E1 ~ 25 MeV.
K∞ ~ 170 MeV from ISGDR ?
T.S. Dimitrescu and F.E. Serr [PRC 27 (1983) 211] pointed out “If further measurement
confirm the value of 21.3 MeV for this mode, the discrepancy may be significant”.
→ Relativistic mean field (RMF) plus RPA with NL3 interaction predict K∞=270 MeV
from the ISGMR [N. Van Giai et al., NPA 687 (2001) 449].
Hartree-Fock (HF) - Random Phase Approximation (RPA)
In fully self-consistent calculations:
1.
Assume a form for the Skyrme parametrization (δ-type).
2.
Carry out HF calculations for ground states and determine the Skyrme
parameters by a fit to binding energies and radii.
3.
Determine the residual p-h interaction
4.
Carry out RPA calculations of strength function, transition density etc.
Green’s Function Formulation of RPA
In the Green’s Function formulation of RPA, one starts with the RPAGreen’s function which is given by

1
G

G
(
1

V
G
)
o
pho
where Vph is the particle-hole interaction and the free particle-hole
Green’s function is defined as
  
1

1
G
(
r
,
r
'
,
E
)


*
(
r
)
 
(
r
'
)

o
i
i

h


E
h


E
i
o
i
o
i


where φi is the single-particle wave function, єi is the single-particle
energy, and ho is the single-particle Hamiltonian.
The continuum effects, such as particle escape width, can
be taken into account using
r1
1
2m
r2  2 U (r )V (r ) / W
h0  Z
where r< and r> are the lesser and greater of r1 and r2
respectively, U and V are the regular and irregular solution
of (H0-Z)ψ = 0, with the appropriate boundary conditions,
and W is the Wronskian.
NOTE the two terms in the free particle-hole greens
function
A
We use the scattering operator F
Ff (ri)
i1
to obtain the strength function
 
1
S
(
E
)

0
F
n
(
E

E
)

Im[
Tr
(
f

G

f
)]

n
2
n
and the transition density




E
1
3

(
r
,
E
)
 
f
(
r
'
)

[
Im
G
(
r
,
r
'
,
E
)]
d
r
'

S
(
E
)


E
RPA
t
 RPA
is consistent with the strength in

S
(
E
)


EE/ 2
2

(
r
,
E
)
f
(
r
)
d
r
E
RPA
Relativistic Mean Field + Random Phase Approximation
The steps involved in the relativistic mean field based RPA calculations are analogous to
those for the non-relativistic HF-RPA approach. The nucleon-nucleon interaction is
generated through the exchange of various effective mesons. An effective Lagrangian
which represents a system of interacting nucleons looks like
It contains nucleons (ψ) with mass M; σ, ω, ρ mesons; the electromagnetic field; non
linear self-interactions for the σ (and possibly ω) field.
Values of the parameters for the most widely used NL3 interaction are mσ=508.194 MeV,
mω=782.501 MeV, mρ=763.000 MeV, gσ=10.217, gω=12.868, gρ=4.474, g2=-10.431 fm-1
and g3=-28.885 (in this case there is no self-interaction for the ω meson).
NL3: K∞=271.76 MeV, G.A.Lalazissis et al., PRC 55 (1997) 540.
RMF-RPA: J. Piekarewicz PRC 62 (2000) 051304; Z.Y. Ma et al., NPA 686 (2001) 173.
Are mean-field RPA calculations fully self-consistent ?
NO ! In practice, has made approximations.
A.
Mean field and Vph determined independently → no information on K∞.
B.
In HF-RPA one
1. neglects the Coulomb part in Vph;
2. neglects the two-body spin-orbit;
3. uses limited upper energy for s.p. states (e.g.: Eph(max) = 60 MeV);
4. introduces smearing parameters.
Main effects:
change in the moments of S(E), of the order of 0.5-1 MeV; note: E  K
spurious state mixing in the ISGDR;
inaccuracy of transition densities.
Commonly used scattering operators:
• for ISGMR
• for ISGDR
In fully self-consistent HF-RPA calculations the (T=0, L=1) spurious state (associated
with the center-of-mass motion) appears at E=0 and no mixing (SSM) in the ISGDR
occurs.
In practice SSM takes place and we have to correct for it.
Replace the ISGDR operator with
(prescriptions for η: discussion in the literature)
NUMERICS:
Rmax = 90 fm
Ephmax ~ 500 MeV
ω1 – ω2 ≡ Experimental range
Δr = 0.1 fm (continuum RPA)
Self-consistent calculation within constrained HF
Strength function for the spurious state and ISGDR calculated using a smearing
parameter Г/2 = 1 MeV in CRPA. The transition strengths S1, S3 and Sη correspond
to the scattering operators f1, f3 and f η, respectively. The SSM caused due to the
long tail of the spurious state is projected out using the operator f η
Isoscalar strength functions of 208Pb
for L = 0 - 3 multipolarities are
displayed. The SC (full line)
corresponds to the fully selfconsistent calculation where LS
(dashed line) and CO (open circle)
represent the calculations without
the ph spin-orbit and Coulomb
interaction in the RPA, respectively.
The Skyrme interaction SGII [Phys.
Lett. B 106, 379 (1981)] was used.
S(E) (fm4/MeV)
90Zr
116Sn
144Sm
208Pb
E (MeV)
Isoscalar monopole strength function
A. Kolomiets, O. Pochivalov, and
ISGMR, f(r) = r2Y00, Eα = 240 MeV
S. Shlomo, PRC 61 (2000) 034312
SL1 interaction, K = 230 MeV.
Reconstruction of the ISGMR EWSR
in 116Sn from the inelastic α-particle
cross sections. The middle panel:
maximum (00) double differential
cross section obtained from ρt (RPA).
The lower panel: maximum cross
section obtained with ρcoll (dashed
line) and ρt (solid line) normalized to
100% of the EWSR. Upper panel:
The solid line (calculated using RPA)
and the dashed line are the ratios of
the middle panel curve with the solid
and dashed lines of the lower panel,
respectively.
d

0


3


r
coll
0
dr
S. Shlomo and A.I. Sanzhur, Phys. Rev. C
65, 044310 (2002)
3 5 2 
r r r
Y


ISGDR f
1
M
3


SL1 interaction, K = 230 MeV, Eα = 240 MeV
Reconstruction of the ISGDR EWSR
in 116Sn from the inelastic α-particle
cross sections. The middle panel:
maximum double differential cross
section obtained from ρt (RPA). The
lower panel: maximum cross section
obtained with ρcoll (dashed line) and ρt
(solid line) normalized to 100% of the
EWSR. Upper panel: The solid line
(calculated using RPA) and the dashed
line are the ratios of the middle panel
curve with the solid and dashed lines
of the lower panel, respectively.
  5

d


10
r

3
r

r

(
r
)




3
dr


2
coll

2
0
0

Fully self-consistent HF-RPA results for ISGMR centroid energy (in MeV) with the
Skyrme interaction SK255, SGII and KDE0 are compared with the RRPA results using
the NL3 interaction. Note the corresponding values of the nuclear matter
incompressibility, K, and the symmetry energy , J, coefficients. ω1-ω2 is the range of
excitation energy. The experimental data are from TAMU.
Nucleus
ω1-ω2
90Zr
0-60
10-35
116Sn
208Pb
SGII
KDE0
18.7
18.9
17.9
18.0
18.9
17.9
18.0
17.3
16.4
16.6
17.3
16.4
16.6
16.2
15.3
15.5
16.2
15.2
15.5
14.3
13.6
13.8
14.4
13.6
13.8
15.85±0.20
16.1
15.40±0.40
0-60
10-35
SK255
17.1
0-60
10-35
NL3
17.81±0.30
0-60
10-35
144Sm
Expt.
14.2
13.96±0.30
K (MeV)
272
255
215
229
J (MeV)
37.4
37.4
26.8
33.0
Conclusions
• We have developed a new EDFs based on Skyrme type interaction
(KDE0, KDE, KDE0v1,... ) applicable to properties of rare nuclei
and neutron stars.
• Fully self-consistent calculations of the compression modes
(ISGMR and ISGDR) within HF-based RPA using Skyrme forces
and within relativistic model lead a nuclear matter incompressibility
coefficient of K∞ = 240 ± 20 MeV, sensitivity to symmetry energy.
• Sensetivity to symmetry energy: IVGDR, GR in neutron rich nuclei,
Rn – Rp, stll open problems.
• Possible improvements:
– Account for effect of correlations on B.E. Radii, S.P. energies
– Properly account for the isospin dependency of the
spin-orbit interaction
– Include additional data, such as IVGDR (J) and ISGQR (m*)
Acknowledgments
Work done at:
Work supported by:
Grant number: PHY-0355200
Grant number: DOE-FG03-93ER40773