The Nuclear Symmetry Energy and Neutron Skin Thickness of Finite Nuclei
Lie-Wen Chen ( 陈列文 )(INPAC and Department of Physics, Shanghai Jiao Tong Un
iversity. [email protected])
第十三届全国核结构研讨会暨第九次全国“核结构与量子力学”专题讨论会 ,2010 年 7 月 24-30 日 , 赤峰 , 内蒙古
Collaborators :Che Ming Ko and Jun Xu (TAMU)Bao-An Li (TAMU-Commerce) Xin Wang (SJTU)Bao-Jun Cai, Rong Chen, Peng-Cheng Chu, Zhen Zhang (SJTU)
Outline
EOS of asymmetric nuclear matter and the symmetry energy
Constraints on density dependence of symmetry energy from nuclear structure and reactions – Present status
Constraining the symmetry energy with the neutron skin thickness of heavy nuclei in a novel correlation analysis
Symmetry energy and nuclear effective interactions
Summary and outlook
Main References: B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113-281 (2008)L.W. Chen, B.J. Cai, C.M. Ko, B.A. Li, C. Shen, and J. Xu, PRC80, 014322 (200
9)L.W. Chen, C.M. Ko, B.A. Li and J. Xu, arXiv:1004.4672, 2010
Density Dependence of the Nuclear Symmetry Energy
Reactions & Structures of Neutron-Rich Nuclei (CSR/Lanzh
ou, FRIB, GSI,
RIKEN……)
Most uncertain property of an asymmetric
nuclear matter
Isospin Nuclear PhysicsWhat is the isospin dependence of the in-medium nuclear effective interac
tions???
Neutron Stars …
Structures of Neutron-rich Nuclei, …
Isospin Effects in HIC’s …
Many-Body Theory
Many-Body Theory
Transport Theory General Relativity
Nuclear Force
EOS for Asymmetric
Nuclear Matter
On Earth!!! In Heaven!!!
Isospin in Nuclear Physics
EOS of asymmetric nuclear matter and the symmetry energy
EOS of Nuclear Matter
-
The in a nuclear matter with density , temperature , and
isospin asymmetry
T
energ
( ) can be expressed as
T
/
y of per nucleon
( , Nuclear Ma, ) ( tter EO )S
he
n p
E A T
2
, constant
pressure of the nuclear matter can be expresse P
incompessibi
d as
The of the nucl
( , , )
ear mattlty K er can be expressed a
s
T N
P T
, const
300 0
ant
Empirical values about the nuclear mat
Saturation density ( ( ) 0
(
of symmetric nuclear matter a
, ,
t ) T=
ene
ter EOS:
) 9
0.16 fm0 MeV:
The rg
T N
PK T
P
0
0
0
16 MeV/nu
y of per nucleon
Incompessibilty
of symmetric nuclear matter at and T=0 MeV:
of symmetric nuclear matter at T=0 MeV:
cleon
200 400 MeVK
Liquid-drop model
(Isospin) Symmetry energy term
W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,……
Symmetry energy including surface diffusion effects (ys=Sv/Ss)
The Nuclear Symmetry Energy
EOS of Isospin Asymmetric Nuclear Matters
2 4ym ( )( , ) ( ), ( ),0) /( n pE OE E
(Parabolic law)
The Nuclear Symmetry Energy2
sym 2
1 ( , )( )
2
EE
The Nuclear Matter Symmetry Energy
Symmetry energy term(poorly known)
Symmetric Nuclear Matter(relatively well-determined)
0
sym
sy
2
0 0
0 0
s
sy
0
msym 0
0
0
y
m
m
, ( )3 18
30 MeV (LD mass formula: )
( )3 (Many-Body Theory:
( )
: ; Exp: ???
( )
50 200 e )M
( )
V
E Myers & Swiatecki, NPA81; Pomorski & Du
EL
KL
dek, PRC67
L
K
E E
0
sy
2sym2
0sy
m 0
m ym
0
s2
(Sharma et
isobaric incompressibli
( )9 (Many-Body Theory: : ; Exp: ???)
The isospin part of the of
700 466 Me
6
t
V
320 180
asymmetric nuclear matter
( : / GMR MeV
y K
EK
K K L J K L
Shlomo&Youngblood,PRC47,529(93);
al., PRC38, 2562 (88));
566 1 350 ( 34 159 MeV (T. Li et al, PRL99,162503(2007))550 100 Me )V
The Symmetry Energy
The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).
The symmetry energy is also related to some issues of fundamental physics:1. The precision tests of the SM through atomic parity violation observables (Sil et al., PRC05)2. Possible time variation of the gravitational constant (Jofre etal. PRL06; Krastev/Li, PRC07)3. Non-Newtonian gravity proposed in grand unification theories (Wen/Li/Chen, PRL10)
Nuclear Matter EOS: Many-Body Approaches
Microscopic Many-Body Approaches Non-relativistic Brueckner-Bethe-Goldstone (BBG) Theory Relativistic Dirac-Brueckner-Hartree-Fock (DBHF) approach Self-consistent Green’s Function (SCGF) Theory Variational Many-Body (VMB) approach …… Effective Field Theory Density Functional Theory (DFT) Chiral Perturbation Theory (ChPT) …… Phenomenological Approaches Relativistic mean-field (RMF) theory Relativistic Hartree-Fock (RHF) Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock) Thomas-Fermi (TF) approximations Phenomenological potential models ……
Chen/Ko/Li, PRC72, 064309(2005) Chen/Ko/Li, PRC76, 054316(2007)
Z.H. Li et al., PRC74, 047304(2006) Dieperink et al., PRC68, 064307(2003)
BHF
Esym: Many-Body Approaches
At sub-saturation densities (亚饱和密度行为) Sizes of n-skins of unstable nuclei from total reaction cross sections Proton-nucleus elastic scattering in inverse kinematics Parity violating electron scattering studies of the n-skin in 208Pb n/p ratio of FAST, pre-equilibrium nucleons Isospin fractionation and isoscaling in nuclear multifragmentation Isospin diffusion/transport Neutron-proton differential flow Neutron-proton correlation functions at low relative momenta t/3He ratio Hard photon production
Towards high densities reachable at CSR/Lanzhou, FAIR/GSI, RIKEN, GANIL and, FRIB/MSU (高密度行为) π -/π + ratio, K+/K0 ratio? Neutron-proton differential transverse flow n/p ratio at mid-rapidity Nucleon elliptical flow at high transverse momenta n/p ratio of squeeze-out emission
Promising Probes of the Esym(ρ)(an incomplete list !)
Symmetry energy around saturation density
Li/ Chen, PRC72, 064611(2005)
Symmetry energy, isospin diffusion, in-medium cross section
Isospin Diffusion Data Esym(ρ0)=31.6 MeVL=88±25 MeV
0
0
( ) 31.6( / ) MeV
(From 0 ), we ob
Fit the symmetry ener
tain
0.69 for1. 005
gy with
for 1 n
:
a d
symE
x x
Chen/Ko/Li, PRC72,064309 (2005)
Esym: Isospin Diffusion in HIC’s
Chen/Ko/Li, PRL94,032701 (2005)Isospin dependent BUU transport model
Consistent with isospin diffusion data!
Constraining Symmetry Energy by Isocaling: TAMU DataShetty/Yennello/ Souliotis, PRC75,034602(2007); PRC76, 024606 (2007)
Esym: Isoscaling in HIC’s
Isoscaling Data Esym(ρ0)=31.6 MeVL=65 MeV
ImQMD: n/p ratios and two isospin diffusion measurementsTsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009)
Esym: Isospin diffusion and double n/p ratio in HIC’s
ImQMD: Isospin Diffusion and double n/p ratio Esym(ρ0)=28 - 34 MeVL=38 - 103 MeV
Esym: Nuclear Mass in Thomas-Fermi Model
Thomas-Fermi Model + Nuclear Mass Esym(ρ0)=32 .65 MeV L=49.9 MeV
Myers/Swiatecki, NPA 601, 141 (1996)Thomas-Fermi Model analysis of 1654 ground state mass of nuclei with N,Z≥8
Esym: Pygmy Dipole Resonances
Pygmy Dipole Resonances of 130,132Sn Esym(ρ0)=32 ± 1.8 MeV L=43.125 ± 15 MeV
Pygmy Dipole Resonances of 68Ni and 132Sn Esym(ρ0)=32.3 ± 1.3 MeV, L=64.8 ± 15.7 MeV
Esym from Isobaric Analog States + Liquid Drop model with surface symmetry energy
IAS+Liquid Drop Model with Surface Esym Esym(ρ0)=32.5 ± 1 MeV L=94.5 ± 16.5 MeV
Danielewicz/Lee, NPA 818, 36 (2009)
Esym: IAS+LDM
Esym: Droplet Model Analysis on Neutron Skin
Droplet Model + N-skin Esym(ρ0)=31.6 MeV, L=66.5 ± 36.5 MeV
Droplet Model + N-skin Esym(ρ0)=28 - 35 MeV, L=55 ± 25 MeV
Esym: Droplet Model Analysis on Neutron Skin
Esym around normal density
Esym(ρ0)=28 - 35 MeVL=28 - 111 MeV
9 constraints on Esym (ρ0) and L from nuclear reactions and structures
Still within large uncertain region !!
The Nuclear Neutron Skin
For heavier stable nuclei: N>Z 1/22 1/22
n pr r
1/2 1/222 -np pnr rr Neutron Skin Thickness:
Bodmer, Nucl. Phys. 17, 388 (1960)
Sprung/Vallieres/Campi/Ko,
NPA253, 1 (1975)
Shlomo/Friedman, PRL39, 1180 (1977)
……
The Esym vs. Nuclear Neutron Skin
2 2
Neutron-Skin Thickness:
(fm)n pr rS 208( Pb) varies from 0.04 fm to 0.24 fm
depending on the Skyrme interaction !
S
Good linear Correlation: S-L
Chen/Ko/Li, PRC72,064309 (2005)
The Esym vs. Nuclear Neutron Skin
Chen/Ko/Li, PRC72,064309 (2005)
For heavier nuclei: Still good linear correlation between S-L
Neutron-Skin Thickness:
Pressure difference between n and p:
Slope of the Symmetry Energy:
n p
S
p p
L
( )n pS p p L
B.A. Brown, PRL85,5296 (2000)
The Skyrme HF Energy Density Functional
Standard Skyrme Interaction:
_________
9 Skyrme parameters:
9 macroscopic nuclear properties:
There are more than 120 sets of Skyrme- like Interactions in
the literature
Chen/Ko/Li/XuarXiv:1004.4672
Agrawal/Shlomo/Kim AuPRC72, 014310 (2005)
Yoshida/SagawaPRC73, 044320 (2006)
The Skyrme HF Energy Density Functional
Chen/Cai/Ko/Li/Shen/Xu, PRC80, 014322 (2009): Modified Skyrme-Like (MSL) Model
Chen/Ko/Li/Xu, arXiv:1004.4672
The Skyrme HF with MSL0
Chen/Ko/Li/Xu, arXiv:1004.4672
Correlations between Nuetron-Skin thickness and macroscopic Nuclear Properties
For heavy nuclei 208Pb and 120Sn: Δrnp is strongly correlated with L, moderately with Esym(ρ0), a little bit with m*s,0
For medium-heavy nucleus 48Ca:Δrnp correlation with Esym is much weaker; It further depends on GV and W0
Important Terms
Constraining Esym with Neutron Skin Data
Neutron skin constraints on L and Esym(ρ0) are insensitive tothe variations of other macroscopic quantities.
(~independent of Esym(ρ0))
A quite stringent constraint on Δrnp of 208Pb:
Constraining Esym with Neutron Skin Data and Heavy-Ion Reactions
N-Skin + HIC
Core-Crust transition density in
Neutron stars:
Global nucleon optical potential Esym(ρ0)=31.3 ± 4.5 MeV, L=52.7 ± 22.5 MeV
Xu/Li/Chen, arXiv:1006.4321v1, 2010
Esym: Global nucleon optical potential
Consistent with Sn neutron skin data!
Symmetry energy and Nuclear Effective Interaction
Chen/Ko/Li, PRC72,064309 (2005) Chen/Ko/Li, PRC76, 054316(2007)
L=58 ± 18 MeV: only 32/118 L=58 ± 18 MeV: only 8/23
We have proposed a novel method to explore transparently the correlation between observables of finite nuclei and nuclear matter properties.
The neutron skin thickness of heavy nuclei provides reliable information on the symmetry energy. The existing neutron skin data of Sn isotopes give important constraints on the symmetry energy and the neutron skin of 208Pb
Combining the constraints on Esym from neutron skin with that from isospin diffusion and double n/p ratios in HIC’s impose quite accurate constraint of L=58±18 MeV approximately independent of Esym
Our correlation analysis method can be generalized to other mean- field models (e.g., RMF) or density functional theories and a number of other correlation analyses are being performed (giant resonance, shell structure,,…… )
IV. Summary and Outlook
谢 谢!
(1) EOS of symmetric matter around the saturation density ρ0
GMR 0Frequency f K
Giant Monopole Resonance
K0=231±5 MeVPRL82, 691 (1999)Recent results:K0=240±10 MeVG. Colo et al. U. Garg et al.S. Shlomo et al.
__
0
22
0 0 2Incompressibility: K =9 ( )
d E
d
EOS of Symmetric Nuclear Matter
(2) EOS of symmetric matter for 1ρ0< ρ < 3ρ0 from K+ production in HIC’sJ. Aichelin and C.M. Ko,
PRL55, (1985) 2661C. Fuchs,
Prog. Part. Nucl. Phys. 56, (2006) 1
Transport calculations indicate that “results for the K+ excitation function in Au + Au over C + C reactions as measured by the KaoS Collaboration strongly support the scenariowith a soft EOS.”
C. Fuchs et al, PRL86, (2001) 1974
See also: C. Hartnack, H. Oeschler, and J. Aichelin,
PRL96, 012302 (2006)
EOS of Symmetric Nuclear Matter
(3) Present constraints on the EOS of symmetric nuclear matter for 2ρ0< ρ < 5ρ0 using flow data from BEVALAC, SIS/GSI and AGS
Use constrained mean fields to predict the EOS for symmetric matter
• Width of pressure domain reflects uncertainties in comparison and of
assumed momentum dependence.
P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002)
2Pressure P( )s
E
The highest pressure recorded under laboratory controlled conditions in nucleus-nucleus collisions
y
px
High density nuclear matter2 to 5ρ0
EOS of Symmetric Nuclear Matter
Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin- (momentum-) dependent mean field potential
Isospin-dependent N-N cross sections a. Experimental free space N-N cross section σexp
b. In-medium N-N cross section from the Dirac-Brueckner approach based on Bonn A potential σin-medium
c. Mean-field consistent cross section due to m* Isospin-dependent Pauli Blocking
0 sym
1(1 )
2 z CV V V V
Phase-space distributions ( , , ) satify the Boltzmann equation
( , , ) ( , )p r r p c NN
f r p t
f r p tf f I f
t
Isospin-dependent BUU (IBUU) model
Transport model for HIC’s
EOS
Isospin- and momentum-dependent potential (MDI)
30
0
0
0
0.16 fm
( ) / 16 MeV
MDI Interaction
( ) 31.6 MeV
211 MeV
*/ 0.6
g( o )
8
G ny
sym
E A
E
K
m m
Chen/Ko/Li, PRL94,032701
(2005)Li/Chen, PRC72, 064611
(2005)
Das/Das Gupta/Gale/Li,
PRC67,034611 (2003)
Transport model: IBUU04
Esym: Isospin Diffusion in HIC’s
How to measure Isospin Diffusion?
PRL84, 1120 (2000)
______________________________________
A+A,B+B,A+BX: isospin tracer
Isospin Diffusion/Transport
Isoscaling in HIC’sIsoscaling observed in many reactions
2 1
( ) /
Y / Yn pN Z Te
M.B. Tsang et al. PRL86, 5023 (2001)
Esym: Isoscaling in HIC’s
High density behaviors of Esym
n/p ratio of the high density region
Li/Yong/Zuo, PRC 71, 014608 (2005)Isospin fractionation!
Heavy-Ion Collisions at Higher Energies
Subthreshold K0/K+ yield may be a sensitive probe of the symmetry energy at high densities
Aichelin/Ko, PRL55, 2661 (1985): Subthreshold kaon yield is a sensitive probe of the EOS of nuclear matter at high densities
Theory: Ferini et al., PRL97, 202301 (2006) Exp.: Lopez et al. FOPI, PRC75, 011901(R) (2007)
96 9644 44
96 9640 40
Ru+ Ru and
Zr+ [email protected] AGeV
K0/K+ yield is not so sensitive to the symmetry energy! Lower energy and more neutron-rich system???
High density behaviors of Esym: kaon ratio
IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102,062502(2009)
High density behaviors of Esym: pion ratio
A Quite Soft Esym at supra-saturation densities ???Zhang et al.,PRC80,034616(2009)
IDQMD, Feng/Jin, PLB683, 140(2010)
Pion Medium Effects?Xu/Ko/Oh
PRC81, 024910(2010)
Threshold effects?……
High density behaviors of Esym: n/p v2
A Stiff Esym at supra-saturation densities ???
W. Trauntmann et al., arXiv:1001.3867
Esym at very low densities: Clustering effects
S. Kowalski, et al., PRC 75 (2007) 014601.
Horowitz and Schwenk, Nucl. Phys. A 776 (2006) 55
Esym at very low densities: Clustering effects
J. B. Natowitz et al., arXiv:1001.1102
PRL, 2010