Constraining the Nuclear Equation of State via
Nuclear Structure observables
曹李刚
中科院近物所
第十四届全国核结构大会，湖州，2012.4.12-16
The Nuclear equation of state
E
(  , I )  ESNM (  )  S 2 (  ) I 2  S 4 (  ) I 4  O(6)
A
E
(  , I )  ESNM (  )  S 2 (  ) I 2
A
2
3
K 0     0  Q0     0 

 

  O(4)
ESNM (  )  E0 
2  3 0 
6  3 0 
E0  ESNM (   0 )
2

ESNM (  )
2
K 0  90
 2
 
IS Monopole giant resonance
0
3

ESNM (  )
3
Q0  27  0
 3
 
0
The Nuclear equation of state
E
(  , I )  ESNM (  )  S 2 (  ) I 2  S 4 (  ) I 4  O(6)
A
E
(  , I )  ESNM (  )  S 2 (  ) I 2
A
2
3
    0  K sym     0  Qsym     0 
 

 

  O(4)
S 2 (  )  Esym  L
2  3 0 
6  3 0 
 3 0 
S 2 (  )
L  3 0
   0
K sym
2

S2 ( )
2
 90
 2   
IV dipole giant resonance
Qsym
0
3

S2 ( )
3
 27  0
 3   
0
K
The incompressibility of nuclear matter
Nuclear structure, Heavy ion collision, Physics of neutron star
The incompressibility of nuclear matter can not be measured
directly, it can be deduced from the distribution Of ISGMR in heavy
nuclei, such as 208Pb.
Fraction E0 EWSR/MeV
0.4
Exp.
SKI3(258)
SLy5(230)
SKP(201)
0.3
208
0.2
Pb
0.1
0.0
5
10
15
E(MeV)
20
25
K ?
is found around 230 MeV by using non relativistic
interaction, such as Skyrme and Gogny
In the case of relativistic interaction, It is around 260
MeV.
Both of them can produce very well the ISGMR
energy in Pb208
Using different density dependent Skyrme force, Colo
point out that it is possible to reproducing the ISGMR
energy in Pb208 for some Skyrme interaction Which
have 250 MeV of incompressibility.
Piekarewicz built non-linear relativistic Lagrangian
with 230 MeV incompressibility, which can produce
the ISGMR energy in Pb208.
K  230MeV both non relativistic and relativistic.
New problem is appeared.
Why Tin is so soft?
Or
Why pb is so hard?
How to understand this discrepancy in Sn isotopes and Pb208,
Several theoretical works have been done to try to explain it.
Since the Sn isotopes are open shell nuclei, the pairing shall play
A role both in ground states and excited states. Using the
Constraint HF(HFB) and energy-weighted sum rule approach, one
can get the constrained energy of ISGMR, it is found that pairing
Has an important effect in producing the ISGMR energies in Sn
Isotopes. E. Khan PRC80, 011307(R)(2008).
Based on the HFB+QRPA calculation, the ISGMR energies in Sn
Isotopes are obtained using different Skyrme interaction, but
There is No satisfied conclusion according to those calculation
Because the calculations are not fully self-consistent, such as
The spin-orbit interaction is dropped . J. Li et.al.,78,064304(2008)
Some groups try to solve this problem by introducing a isospin
dependent Incompressibility, they can get better description in Sn
isotopes, but fails in Pb208. J. Piekarewicz, PRC79, 054311 (2009)
The problem is still open.
In this talk, I will present our recent work on ISGMR in Cd, Sn and
Pb isotopes. It is based on the HF+BCS+QRPA.


A
B



X 

 X

    E   
 B * A * Y 
Y 
(1)
Amin j  ( m   i ) ij mn  mj Vres in
 ( m   i ) ij mn  Vmjin
Bmin j  mn Vres ij  Vmnij
Vmjin includes：full Skyrme force, spinorbit, coulomb, and also the pp channel
The strength function is
2
S ( E )   0 Fˆ n  ( E  En )
n
The various moments are defined as
mk   E S ( E )dE
k
And various energies are defined as
Econ
m3
m1
m1

, Ecen 
, Es 
m1
m0
m1


  (r )   
 
 (r1  r2 )
V pair (r1 , r2 )  V0 1   
  0 

equals to 1, 0.5,0 corresponding to surface, mixed, and volume
Pairing.
 n  1.334MeV
 n  1.485MeV
 n  0.841MeV
14.5
14.5
(a)
m1/m0(MeV)
13.5
13.5
Pb isotopes SLy5
13.0
12.0
m1/m0(MeV)
14.0
14.0
12.5
(b)
Exp. RCNP
Exp. KVI
volumn p
mixed p
204
13.0
Exp. Lui.
filling approx
surface p
12.0
206
A
Pb isotopes SKM*
12.5
208
204
206
A
14.5
(c)
SLy5 230MeV
SKM *
217 MeV
SKP 202MeV
m1/m0(MeV)
14.0
13.5
13.0
12.5
12.0
Pb isotopes SKP
204
206
A
208
208
HFBCS  QRPA
SLy5 230MeV
mixed
SKM *
pairing
217 MeV
SKP 202MeV
17.5
17.5
(a)
16.5
16.0
15.5
15.0
Exp. RCNP
volumn p
mixed p
106
108
112
114
116
SKP 202MeV
m1/m0(MeV)
217 MeV
15.5
14.5
104
118
106
108
110
112
114
17.5
17.0
SKM *
16.0
A
A
SLy5 230MeV
16.5
15.0
filling appro
surface p
110
Cd isotopes
SKM*
17.0
m1/m0(MeV)
m1/m0(MeV)
17.0
14.5
104
(b)
Cd isotopes
SLy5
Cd isotopes
SKP
(c)
16.5
16.0
15.5
15.0
14.5
104
106
108
110
112
A
114
116
118
116
118
SLy5 230MeV
SKM *
217 MeV
SKP 202MeV
17.0
17.0
(d)
Sn isotopes
SLy5
15.5
15.5
15.0
Exp. RCNP
filling appro
surface P
112
114
116
118
120
17.0
SKM
217 MeV
SKP 202MeV
122
(c)
124
14.5
110 112 114 116 118 120 122 124
A
Sn isotopes
SKP
m1/m0(MeV)
16.5
SLy5 230MeV
15.0
Exp. TAM
volumn p
mixed p
A
*
16.5
16.0
16.0
14.5
110
Sn isotopes
SKM*
m1/m0(MeV)
m1/m0(MeV)
16.5
(b)
16.0
15.5
15.0
14.5
110 112 114 116 118 120 122 124
Summary I




We have studied the ISGMR in Cd, Sn and Pb isotopes based on the
fully self-consistent HF+BCS plus QRPA calculations. The SLy5,
SKM*, and SKP and different pairing interactions are used in our
work.
We found that the pairing does play a role in producing the ISGMR
properties.
The SLy5 interaction together with the effect of pairing can give better
description on ISGMR both in Cd and Pb isotopes, but it still can not
get better results in Sn isotopes, while SKM* can produce the
experimental data in Cd and Sn isotopes, but fails in Pb isotopes, for
SKP, it fails for all isotopes because the incompressibility is too low.
For future work, the calculations based on fully self-consistent
HFB+QRPA may give more satisfied description, or other effect will be
investigated, such as isospin-dependent incompressibility, surface effect,
isospin-dependent pairing interaction.
Ni68
50.3-89.4
Sn132
29.0-82.0
L( 0 )  64.8  15.7MeV
S ( 0 )  32.3  1.3MeV
Summary II


We have constrained the slope of symmetry energy at saturation
density using the recent experimental results on pygmy dipole
resonance.
L( 0 )  64.8  15.7MeV , S ( 0 )  32.3  1.3MeV
Thank You!