congratulations and thanks, joe!

Congratulations and Thanks, Joe!

The density curvature parameter and high density behavior of the symmetry energy

Lie-Wen Chen ( 陈列文 )Department of Physics and Astronomy, Shanghai Jiao Tong University

(lwchen@sjtu.edu.cn)

“International Workshop on Nuclear Dynamics and Thermodynamics”, in Honor of Prof. Joe Natowitz, TAMU,

College Station, USA, August 19-22, 2013

The symmetry energy Esym and its current constraints Systematics of the density dependence of the Esym Information on the density curvature Ksym and the high

density Esym from constraints at subsaturation densities Summary

Outline

The symmetry energy Esym and its current constraints Systematics of the density dependence of the Esym Information on the density curvature Ksym and the high

density Esym from constraints at subsaturation densities Summary

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 Symmetry Energy

Symmetry energy term(poorly known)

Symmetric Nuclear Matter(relatively well-determined)

Isospin asymmetry

0

symsym 0sym

2

0 0

0 0

sym

sym 0

0

0, ( )3 18

30 MeV (LD mass formula: )

( )3 (Many- 50 ~B 2

(

0ody 0 Me Theo

( )

ry: : ; Ex

)

~

V p

( )

E My

E

ers & Swiatecki, NPA81; Pomorski & D

KLE

udek, P

E

R 7

L

C6

L

0

sy

2sym2

0 sym2m

: ???)

( )9 (Many-Body Theory: : ; Exp: ?700 ~ 4 ??V )66 MeK

EK

p. 1

Facilities of Radioactive Beams Cooling Storage Ring (CSR) Facility at HIRFL/Lanzhou in China (2008) up to 500 MeV/A for 238U http://www.impcas.ac.cn/zhuye/en/htm/247.htm

Beijing Radioactive Ion Facility (BRIF-II) at CIAE in China (2012) http://www.ciae.ac.cn/

Radioactive Ion Beam Factory (RIBF) at RIKEN in Japan (2007) http://www.riken.jp/engn/index.html

Texas A&M Facility for Rare Exotic Beams -T-REX (2013) http://cyclotron.tamu.edu

Facility for Antiproton and Ion Research (FAIR)/GSI in Germany (2016) up to 2 GeV/A for 132Sn (NUSTAR - NUclear STructure, Astrophysics and Reactions ) http://www.gsi.de/fair/index_e.html

SPIRAL2/GANIL in France (2013) http://pro.ganil-spiral2.eu/spiral2

Selective Production of Exotic Species (SPES)/INFN in Italy (2015) http://web.infn.it/spes

Facility for Rare Isotope Beams (FRIB)/MSU in USA (2018) up to 400(200) MeV/A for 132Sn http://www.frib.msu.edu/

The Korean Rare Isotope Accelerator (KoRIA-RAON(RISP Accelerator Complex) (Starting) up to 250 MeV/A for 132Sn, up to 109 pps …… p. 2

Esym at low densities: Clustering Effects

p. 3

Current constraints (An incomplete list) on Esym (ρ0) and L from terrestrial experiments and astrophysical observations

Esym ： Around saturation density

L.W. Chen, arXiv:1212.0284B.A. Li, L.W. Chen, F.J. Fattoyev, W.G. Newton, and C. Xu, arXiv:1212.1178

Esym(ρ0) = 32.5±2.5 MeV, L = 55±25 MeV

p. 4

IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102,062502(2009)

A Quite Soft Esym at supra-saturation densities ???

ImIQMD, Feng/Jin, PLB683, 140(2010)

Softer

Stiffer Pion Medium Effects?Xu/Ko/Oh

PRC81, 024910(2010)

Threshold effects?Δ resonances?

……

ImIBLE, Xie/Su/Zhu/Zhang,PLB718,1510(2013)

High density Esym: pion ratio

Softer

p. 5

PRC87, 067601 (2013)

The pion in-meidum effects seem comparable to Esym effects in the thermal model !!!But how about in more realistic dynamical model ???

How to treat self-consistently the pion in-medium effects in transport model remains a big challenge !!!

High density Esym: pion ratio

p. 6

J. Hong and P. Danielewicz, arXiv:1307.7654

High density Esym: pion ratio

No Esym effects ! Esym effects show up for squeeze-out pions !

p. 7

A Soft or Stiff Esym at supra-saturation densities ???P. Russotto,W. Trauntmann, Q.F. Li et al.,

PLB697, 471(2011) (UrQMD)

High density Esym: n/p v2

M.D. Cozma, W. Trauntmann, Q.F. Li et al., arXiv:1305.5417 (Tubingen QMD - MDI)

Moderately stiff to roughly linear density dependence !

p. 8

Esym ： at supra- and saturation density

At very low density (less than about ρ0/10), the clustering effects are very important, and the mean field model significantly under-predict the symmetry energy.

Cannot be that all the constraints on Esym (ρ0) and L are equivalently reliable since some of them don’t have any overlap. However, all the constraints seem to agree with:

Esym(ρ0) = 32.5±2.5 MeV L = 55±25 MeV

All the constraints on the high density Esym come from HIC’s, and all of them are based on transport models. The constraints on the high density Esym are elusive and controversial for the moment !!!

p. 9

Outline

The symmetry energy Esym and its current constraints Systematics of the density dependence of the Esym Information on the density curvature Ksym and the high

density Esym from constraints at subsaturation densities Summary

So far (most likely also in future), essentially all the constraints on Esym have been obtained based on some energy density functionals or phenomenological parameterizations of Esym. Are there some universal laws (systematics) for the density dependence of Esym within these functionals or parameterizations? While more high quality data and more reliable models are in progress to constrain the high density Esym, can we find other ways to get some information on high density Esym? Can we get some information on high density Esym from the knowledge of Esym around saturation density?

Esym systematics and high density Esym

sym 0 sym, , a( ) nd E L K sym 0 up to 2 or even higher densities!!!E

p. 10

Systematics of the densiy dependence of Esym

L.W. Chen, Sci. China Phys. Mech. Astron. 54, suppl. 1, s124 (2011) [arXiv:1101.2384]

sym 0 sy

0

m

sym

th

( ) up to about 2 is essentially

determined by characteristic parameters: ( ) ,

red

e, an E L

E

K

sym 0 sym, , a( ) nd E L K

sym 0(2 ) ?E

s

sym 0

ym sym

The higher-order chracteristic parameters et al seem only have tiny effects

on ( ) below about 2 Based o( n S ) F

,

H

J I

E

p. 11

sym 0 sym 0 sym(2 ) ( ) / 3 / 18E E L K

Roca-Maza et al., PRL106, 252501 (2011)46 interactions +BSK18-21+MSL1+SAMi +SV-min+UNEDF0-1+TOV-min+IU-FSU+BSP+IU-FSU*+TM1* (Totally 60 interactions in our analysis)

sym 0 sym, , a( ) nd E L K sym 0(2 ) ?E

Systematics of the densiy dependence of Esym

p. 12

2/3

sym0 0

( ) 12.3 20E

sym 0 sym 0 sym(2 ) ( ) / 3 / 18E E L K

Systematics of the densiy dependence of Esym

Phenomenological parameterizations in transport models for HIC’s

p. 13

s

2/3

sym0

ym0

0

( )

( ) 12.3 0

32.3

2

s

E

E

sym 0 sym 0 sym(2 ) ( ) / 3 / 18E E L K

Systematics of the densiy dependence of Esym

Phenomenological parameterizations in transport models for HIC’s

p. 13

2/3

sy

s

m0 0

(

ym

2/3

sym0 0

)

0

0

( ) 13 ( )

(18.6 ( ))

(Chen/Ko/Li, PRL94, 03MDI in

2t

701(200era

( )

ct

32.

ion

( ) 12.3 20

5)

3

,)

s

G x

E F x

F x

E

E

sym 0 sym 0 sym(2 ) ( ) / 3 / 18E E L K

Systematics of the densiy dependence of Esym

Phenomenological parameterizations in transport models for HIC’s

p. 13

sym sym 0

2sym

( ) ( )

/ 2

E E L

K

sy

sym

m sym

0 0

sym

0.2

Good linear relationship between

( ) ( )

(Linear correlation coeficient is lar

( ) and ( ) :

ger 3

than 96% for )

E A

E E

BE

Systematics of the densiy dependence of Esym

Linear correlation at different densities

p. 14

sym

sym0 0

''( ) 3

dEL

d

L K

0 0

( ) '( )(Linear correl

Good linear relations

ation coeficient is larger than 93% for

( ) and

)

hip be

0.

t'(

w:

5 3

)een

L A BLL L

Systematics of the densiy dependence of Esym

Density slope L:Linear correlation at different densities

p. 15

sym 0 sym, , a( ) nd E L K0

sym 0 0

0

( ) ( ) or

( ) (

0.2 3

0. )5 3

E

L

sym

sym 0 0 0 0

sym

0 sy

0 0 0 0

m

of ( ) (0.2 3 ) or ( ) (0.5 3 )

essentailly determine as well as

( ) (0.2 3 ) and ( ) (0.5 3 )

THREE values

( ), , and E

L

L

E

E

K

L

Systematics of the densiy dependence of Esym

sym sym

sym sym

Note: and are usually not zero, and are usua

( ) ( )

lly not 1(Correct

ions from

( ) '( )

Higher-or

der , , . )..

L

L

L

L

E A BE

L A B L

J I

A AB B

sym sym 0

2sym

( ) ( )

/ 2

E E L

K

sym

sym0 0

''( ) 3

dEL

d

L K

p. 16

Outline

The symmetry energy Esym and its current constraints Systematics of the density dependence of the Esym Information on the density curvature Ksym and the high

density Esym from constraints at subsaturation densities Summary

Three values of Esym(ρ) and L(ρ)

sym 0 0 0 0of ( ) (0.THREE va 2 3 ) orlue ( ) (0.5 3 )s E L

The neutron skin of heavy nuclei L(ρr) at ρr =0.11 fm-3

Binding energy difference of heavy isotope pair Esym(ρc) at ρc =0.11 fm-3

Binding energyEsym(ρc) at ρc = ρ0

3sym

3

(Binding energy difference of heavy isotop

Z. Zhang/L.W. Chen, arXiv :1302.5327 (PLB, in pr

(0.11 fm )

(0.11 fm )

26.65 0.2 MeV

46.0 4.5 M

e pairs)

(The neutron skin of Sn

ess):

P. isotopes)

MeV

o

E

L

sym 0 ( )ller et al., PRL

32.5 0.5 MeV108, 0525

(Binding energy - FRDM) 01 (2012):

E

p. 17

High density Esym and Ksym parameter3 3

sym sym 0(0.11 fm ) (0.11 fm26.65 0.2 MeV, 46.0 4.5 MeV, 32.5 0.5 M) ( V) eE L E

sym 00

0

0 sym 0

sym 0 0

At 32.5 0.5 MeV, 46.7 13.4 MeV, 167.1 185.3 MeV

At 2 40.2

: ( ) ( ) ( )

: (2 ) (2 )

Soft to linear dens

14.

ity

7 MeV,

depen

8.8 1

d

56.6

ence of the symmetry energy is favore

MeV

E L K

E L

sym,pot 0( ) ~ (d: / ) <th 1 wiE

p. 18

The value of Ksym from SHF

L.W. Chen, Sci. China Phys. Mech. Astron. 54, suppl. 1, s124 (2011) [arXiv:1101.2384]

L.W. Chen, PRC83, 044308(2011)

Based on SHF !

Esym systematics:Ksym= -167.1±185.3 MeV

p. 19

00.7 (for 2 )

0Soft symmetry energy ( 2 ) is favored !!! P. Russotto,W. Trauntmann, Q.F. Li et al., PLB697, 471(2011)

High density Esym : Esym(2ρ0) from HIC’s

sym 0(2 ) [25.5,54.9] MeVE

p. 20

Outline

The symmetry energy Esym and its current constraints Systematics of the density dependence of the Esym Information on the density curvature Ksym and the high

density Esym from constraints at subsaturation densities Summary

The symmetry energy Esym(ρ) and its density slope L(ρ) from sub- to supra-saturation density can be essentially determined by three parameters defined at saturation density, i.e., Esym(ρ0), L(ρ0) , and Ksym(ρ0) , implying that three values of Esym(ρ) or L(ρ) can essentially determine Esym(ρ) and L(ρ).

Using Esym (0.11 fm-3) =26.65±0.2 MeV and L(0.11 fm-3) =46.0±4.5 MeV extracted from isotope binding energy difference and neutron skin of Sn isotopes, together with Esym(ρ0) =32.5±0.5 MeV extracted from FRDM analysis of nuclear binding energy, we obtain: L(ρ0) =46.7±13.4 MeV and Ksym(ρ0) = -167.1±185.3 MeV favoring soft to roughly linear density dependence of Esym(ρ).

Accurate determination of Esym(ρ) and L(ρ) around saturation density can be very useful to extract information on high density Esym(ρ).

Summary

p. 21

谢 谢！Thanks!

Esym(ρc) and L(ρc) at ρc =0.11 fm-3

Three values of Esym(ρ) and L(ρ)

ΔE always decreases with Esym(ρr) , but it can increase or decrease with L(ρr) depending on ρr

When ρr =0.11 fm-3, ΔE is mainly sensitive to Esym(ρr) !!!

Binding energy difference of heavy isotope pair Esym(ρc) at ρc =0.11 fm-3

What really determine ΔE?Zhen Zhang and Lie-Wen Chen ,

arXiv:1302.5327Skyrme HF calculations with MSL0

 2 2 2 2=op EB R N kc S in

2exp122

1

thi i

dEi i

E E

Determine Esym(0.11 fm-3) from ΔE

3sym2 : E (0.11 fm ) 26.65 0.20 MeV

2

2Here, the is not real since we use a theoretical error (The model is not

select a theoretical error (23%)

good)

to sa

.

Our strateg tisfy /y: ~ 1 dof

Zhen Zhang and Lie-Wen Chen ,arXiv:1302.532719 data of Heavy Isotope Pairs

(Spherical even-even nuclei)

What really determine NSKin?Zhen Zhang and Lie-Wen Chen ,arXiv:1302.5327

Neutron skin always increases with L(ρr) , but it can increase or decrease with Esym(ρr) depending on ρr

When ρr =0.11 fm-

3, the neutron skin is essentailly only sensitive to L(ρr) !!!The neutron skin of heavy nuclei L(ρr) at ρr =0.11 fm-3

Skyrme HF calculations with MSL0

 2 2 2 2=op EB Rc dE

Determine L(0.11 fm-3) from NSkin

32 : (0.11 fm ) 46.0 4.5 MeVL

Zhen Zhang and Lie-Wen Chen ,arXiv:1302.5327 21 data of NSKin of Sn Isotope

p-scattering, IVGDR, IVSDR, pbar Atomic, PDR, p-elastic scattering

The globally optimized parameters (MSL1)

3sym (0.11 fm ) 26.65 0.2 MeVE

Symmetry energy around 0.11 fm-3

3(0.11 fm ) 46.0 4.5 MeVL

The neutron skin of Sn isotopes

Binding energy difference of heavy isotope pairs

Zhen Zhang and Lie-Wen ChenarXiv:1302.5327

3sym (0.11 fm )

Wang/Ou/Liu, PRC87, 0(Fermi En

26.2

ergy34

D327

iffe

1.0

renc

MeV

e of Nucle2

i)( 013)

E

Extrapolation to ρ0

A fixed value of Esym(ρc) at ρc =0.11 fm-3 leads to a positive Esym(ρ0) -L correlation A fixed value of L(ρc) at ρc =0.11 fm-3 leads to a negative Esym(ρ0) -L correlation

sym 0 032.3 1.0 MeV, 45.2 10.0 M( ) ) eV(E L

Zhen Zhang and Lie-Wen Chen, arXiv:1302.5327

Nicely agree with the constraints from IAS+NSKin by P. Danielewicz; IsospinD+n/p by Y Zhang and ZX Li

sym 0 032.3 1.0 MeV, 45.2 10.0 M( ) ) eV(E L

Correlation analysis using macroscopic quantity input in Nuclear 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 literatureAgrawal/Shlomo/Kim Au

PRC72, 014310 (2005)

Yoshida/SagawaPRC73, 044320 (2006)

Chen/Ko/Li/XuPRC82,

024321(2010)

Extrapolation to ρ0

A fixed value of Esym(ρc) at ρc =0.11 fm-3 leads to a positive Esym(ρ0) -L correlation A fixed value of L(ρc) at ρc =0.11 fm-3 leads to a negative Esym(ρ0) -L correlation

sym 0

0

( )

(

32.3 1.0 MeV

45.2 10. ) 0 MeV

E

L

Zhen Zhang and Lie-Wen ChenarXiv:1302.5327

Nicely agree with the constraints from IAS+NSKin by P. Danielewicz; IsospinD+n/p by Y Zhang and ZX Li

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 Green’s Function Monte Carlo Calculation Vlowk + Renormalization Group Effective Field Theory Density Functional Theory (DFT) Chiral Perturbation Theory (ChPT) QCD-based theory Phenomenological Approaches Relativistic mean-field (RMF) theory Quark Meson Coupling (QMC) Model Relativistic Hartree-Fock (RHF) Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock) Thomas-Fermi (TF) approximations

The nuclear EOS cannot be measured experimentally, its determination thus depends on theoretical approaches

Nuclear Matter Symmetry Energy

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

Solve the Boltzmann equation using test particle method (C.Y. Wong) 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 sym1 (1 )2 z CV V V V

Phase-space distributions ( , , ) satify the Boltzmann equation( , , ) ( , )p r r p c NN

f r p tf r p t f f I f

t

Isospin-dependent BUU (IBUU) model Transport model for HIC’s

EOS

Optimization

Experimental data Binding energy per nucleon and charge rms radius of 25 spherical

even-even nuclei (G.Audi et al., Nucl.Phy.A729 337(2003), I.Angeli, At.Data.Nucl.Data.Tab

87 185(2004))

2exp2

thNi i

opi i

M M

The simulated annealing method (Agrawal/Shlomo/Kim Au, PRC72, 014310 (2005))

OptimizationConstraints:

The neutron 3p1/2-3p3/2 splitting in 208Pb lies in the range of 0.8-1.0 MeVThe pressure of symmetric nuclear matter should be consistent with

constraints obtained from flow data in heavy ion collisions

The binding energy of pure neutron matter should be consistent with constraints obtained the latest chiral effective field theory calculations with controlled uncertainties

The critical density ρcr, above which the nuclear matter becomes unstable by the stability conditions from Landau parameters, should be greater than 2 ρ 0

The isoscalar nucleon effective mass m*s0 should be greater than the isovector effective mass m*v0, and here we set m*s0 − m*v0 = 0.1m (m is nucleon mass in vacuum) to be consistent with the extraction from global nucleon optical potentials constrained by world data on nucleon-nucleus and (p,n) charge- exchange reactions and also dispersive optical model for Ca, Ni, Pb

P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002)

I. Tews, T. Kruger, K. Hebeler, and A. Schwenk, PRL 110, 032504 (2013)

C. Xu, B.A. Li, and L.W. Chen, PRC82, 054607 (2010); Bob Charity, DOM (2011)

Determine Esym(0.11 fm-3) from ΔE

2exp122

1

thi i

dEi i

E E

19

23%