26-Mar-14 RIKEN Nishina Center for Acceletor based Science 1
Rabindranath Tagor C. V. Raman
26-Mar-14 RIKEN Nishina Center for Acceletor based Science
2
Bijay K. Agrawal Saha Institute of Nuclear Physics
Kolkata, India
26-Mar-14 RIKEN Nishina Center for Acceletor based Science 3
Outline of talk
Introduction /Motivation Nuclear energy density functional Probes for nuclear symmetry energy Results Conclusions
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ZNZNI
AIAsJJAZca/AsaAvaB
...2)3/1(3/1232
Giant resonances Neutron rich nuclei Neutron drip line
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Neutron star internal structure
13 21
411
2222 rc
M
Mccr
Μ GPrPG
dr
dP
2
24c
rdr
d
M
Structure Equations
Equation of State (EOS) for Asymmetric nuclear matter
...)0(
03
0)0()(
...32
1)0,()0,(
...)()0,(),(
2
0
00
2
LJJ
KEE
IJEIE
MeV220K
MeV16)0,(
fm16.0
0
30
E
?MeV120-20)(L
MeV32)(
0
0
J
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6
Knowledge of J(ρ0) and L(ρ0) is indispensable !!
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Proton fraction at β equilibrium xβ
)()21(4)3( 3/12 Jxxc
Neutron star properties
Core-crust transition properties Cooling Composition Mass-radius relation ship
Status of L
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)(
)(
3)(
parameter Slope
)0,()1,(
),(
2
1)(
energySymmetry
0
0
0
2
2
LL
JJ
d
dJL
EE
I
IEJ
I
Definitions
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Nuclear energy density functional
1. Non-Relativistic mean field model
2. Relativistic mean field model
3. Density dependant meson exchange model
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Non-Relativistic mean field model : Skyrme type
.properties nuclei finite from determined 3)and0,1,2(ixt
, mass nucleon m
density number density,energkinetic
)()12()12(8
1)2()2(
8
1
))(12()2(24
1
))(12()2(4
1
2
ii
11222211
223
2333
220
2000
2
30
y
xtxtxtxt
xxt
xxt
m
npeff
np
np
kin
effkin
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22222
2
22222
2
1
222222
2
2221
22
420
22
42
2
432322
21
,
~
4
~
4
~
242
~
2
1
24
1
2
1
246
.2
1)(
mgM
mgM
mgM
mgM
mgM
m
gm
mM
kmg
M
km
ggMgi
pnJ
Jlin
linkin
Relativistic mean field model
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1)1(
2)(1
2)(1)(
,
)]1(exp[)()(
,ifor )()()(
if
idx
ic
idx
ib
iaxif
sat
x
xasatgg
xifsatigig
Density dependant meson exchange model DDME
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Density dependence of symmetry energy
J can not be measured directly !!
Supra saturation densities Supernovae,
Nucleosynthesis, Rel. HIC, Neutron Stars
Isospin Transfer Properties,
Multi-fragmentation
Nuclear Structure:
Neutron-skin, Giant resonance 26-Mar-14 RIKEN Nishina Center for Acceletor based Science 15
ρ = ρ0
structure and low energy excitations of asymmetric nuclei, skin thickness, Isovector giant resonances
Probes for the nuclear symmetry energy
Neutron skin thickness
Giant dipole resonance i)Pigmy dipole resonance ii)Dipole polarizability
Giant quadrupole resonance
Nuclear binding energies
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Covariance Analysis
Consider a model described by coupling constants: (P1,P2,....,PN)
At the optimum parameter set P0 2min0
2 )( P
Reasonable domain of parameter satisfy:
1)()( 022 PP
1)( 02 P
)( 02 P
0P26-Mar-14 RIKEN Nishina Center for Acceletor based Science 17
reasonable domain
P
)(2 P
0
0
2
12
1
22
)(
)(
P
pi
Ppjpiij
pjij
ij
pi
pjij
ij
pi
AB
pi
AA
M
AAA
BABA
BA
BAc
Μ
Μ
|cAB|= 1 : Fully correlated
cAB= 0 : Uncorrelated or Statistically independent
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Neutron skin thickness
npq
rdr
rdrrr
rrR
q
q
q
pnskin
,
)(
)(
3
32
2
2/12
2/12
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M. Centelles et.al PRL 102,122502 (2009)
L= 75±25MeV
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fm33.0 16.018.0
skinR
First PREX data PRL 108, 112502 (2012)
PREX II DATA awaited!!
A. Carbon et.al PRC 81,041301 (2010)
Pigmy Dipole Resonance (PDR)
PDR
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2
1
)(
E
E
dEEESEWSR
3.13.32)(
7.158.64)(
0
0
J
L
A. Carbon et.al PRC 81,041301 (2010)
Pigmy Dipole Resonance (PDR)
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L=64.8 ± 15.7 MeV
Skyrmee
RMF
High resolution (p,p’) measurement of D.
Tamii et. al PRL 107 (2011)
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MeV1441
fm0.156 R 0.0250.021-skin
L
Reinhard et.al, PRC 81, 051303 (R) 2010
L and Rskin from Diapole polarizability
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Rskin= 0.168 ±0.022 fm L= 47 ±12 MeV
Piekarewicz , B.K. Agrawal, et.al PRC (R) 2012
L and Rskin from Diapole polarizability
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neutron skin (fm) neutron skin (fm)
208Pb
Using αD = 20.1 ±0.6 fm3 † in 208Pb and J= 31± 2 MeV††
Neutron skin = 0.165 ± 0.043 fm L= 43 ± 26 MeV
Agrawal, et. al PRC 88, 024316 (2013) † . A. Tamii et. al, PRL 107, 062502 (2011) †† Lattimer et. al, Annu.Rev. Nucl.Part.Sci., 62,485(2012)
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SAMi-m
SAMi-j
L and Rskin from Giant Quadrupole Resonances
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Rskin = 0.14 ± 0.03 fm L = 37 ± 18 MeV
Xavi , Marco, Agrawal et.al PRC 87, 034301 (2013)
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])1(2
931)[()()(
))(()()(
)(3,))(()(
20
3/10
0
00
0
0
AAAs
Asym
JJAJJ
JJAa
JLJJ
A
densitynucleoneffectiveA
...3/1
223/132
A
Zc
aAIAs
JJ/As
A-av
aB
08.191.58
31.010.32
sJ
JArima et, al. PRC 85,2012
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modelsfieldmean(r)
fm008.0155.0
,)(
,
3-0
3
)()(
)()(
0
0
0
A
pn
pn
A
rr
rr
ZNZN
rX
X
21 )()()(320
rXrrrdXA
])1([)(3 2
233/1
0 AAs AJJ
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Agrawal ,De and Samaddar PRL 109,2012
Rskin= 0.188 ± 0.014 fm L = 64 ±5 MeV
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I))(()(0
0
JJ
II)))()((
)()(
0
0
0
3/2
k
k
CJ
CJ
*
3/510
13/2
13
III))()((
)()()(
0
00
m
mC
CCJ
CCJ
k
k
k
Rskin= 0.190 ± 0.016 fm L = 56 ±12 MeV
B. K. Agrawal, et. al PRC 87, 051306 (R) (2013)
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Jian Liu, et. Al Phys. Rev. C 88, 024324 (2013)
L = 66 ±7 MeV
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Our results
L = 47± 9 MeV Rskin
= 0.17± 0.02 fm
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Large no of the nuclear energy density functional are employed to understand the density dependence of the nuclear symmetry energy around the saturation density. Neutron-skin in 208Pb nucleus and the symmetry energy slope parameter L(ρ0) are extracted using experimental data on giant resonances and the nuclear masses.
Our results: Rskin = 0.17 ± 0.02 fm L(ρ0) = 47± 9MeV
More accurate data for the dipole polarizability and the neutron-skin thickness in 208Pb needed to constraint in a narrow window the value of the symmetry energy slope parameter.
S. K. Samaddar (SINP) J.N.De (SINP) P. -G. Reinhard (Inst. Theoretical Phys. Germany) J. Piecarewicz (Florida State University, USA) W. Nazarewicz (ORNL, USA) G. Colo (INFN, Italy) X. Roca-Maza (INFN, Italy) D. Vretenar (Phys. Dept. Croatia) N. Paar (Phys. Dept. Croatia) M. Brenna (INFN, Italy) Li-Gang Cao (IMP China)
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based Science 39