Probing the Equation of State of Neutron-Rich Matter with Heavy-Ion Reactions

Bao-An Li

Arkansas State University

1. Equation of State and Symmetry Energy of Neutron-Rich Matter

• Current status and major issues

• Importance in astrophysics and nuclear physics

2. A Transport Model for Nuclear Reactions Induced by Radioactive Beams

• Some details of the IBUU04 model

• Momentum dependence of the isovector nucleon potential in isospin asymmetric matter

3. Determining the Density Dependence of Nuclear Symmetry Energy

• At sub-saturation densities: isospin transport in heavy-ion reactions and neutron-skin in 208Pb

• At higher densities: reactions at RIA and GSI using high energy radioactive beams

4. Summary

Collaborators:Collaborators:L.W. Chen, C.M. Ko, Texas A&M UniversityL.W. Chen, C.M. Ko, Texas A&M University

P. Danielewicz and W.G. Lynch, Michigan State UniversityP. Danielewicz and W.G. Lynch, Michigan State University Andrew W. Steiner, Los Alamos National LaboratoryAndrew W. Steiner, Los Alamos National Laboratory G.C. Yong and W. Zuo, Chinese Academy of ScienceG.C. Yong and W. Zuo, Chinese Academy of Science C.B. Das, C. Gale and S. Das Gupta, McGill UniversityC.B. Das, C. Gale and S. Das Gupta, McGill University

K. Oyamatsu, I. Tanihata, Y. Sugahara, K. Sumiyoshi and H. Toki, NPA 634 (1998) 3.

-20

-10

0

10

0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2nucleon density [fm-3]

RMFSHF (SIII)

1.0

0.6

0.4

0.2

0

0.8

1.0

0.6

0.4

0.2

0

0.8

nucleon density [fm-3]

Bin

ding

ene

rgy

per

nucl

eon

[MeV

] (a)

Two typical equation-of-states

neutron matterneutron matter

stable nucleistable nuclei

2 4( , ) ( ,0 ( )) ( )symEE E Equation of State of Neutron-Rich Matter:

N/Z N/Z

saturation linessaturation lines

(TM1)(TM1)

( ) /( )n p n p Isospin asymmetryIsospin asymmetry

Esym (ρ) predicted by microscopic many-body theories

Sym

met

ry e

ner

gy

(MeV

)

Density

Effective field theory

DBHFRMF

BHF

Greens function

Variational

A.E. L. Dieperink et al., Phys. Rev. C68 (2003) 064307

( ) ( ) ( )sym pureneutron nuclearE E E

Esym(ρ) from Hartree-Fock approach using different effective interactions

EEaa sy

msy

m

EEbbsymsym

B. Cochet, K. Bennaceur, P. Bonche, T. Duguet and J. Meyer, nucl-th/0309012B. Cochet, K. Bennaceur, P. Bonche, T. Duguet and J. Meyer, nucl-th/0309012J. Stone, J.C. Miller, R. Koncewicz, P.D. Stevenson and M.R. Strayer, PRC J. Stone, J.C. Miller, R. Koncewicz, P.D. Stevenson and M.R. Strayer, PRC 6868, 034324 (2003)., 034324 (2003).Bao-An Li, PRL Bao-An Li, PRL 88,88, 192701 (2002) (where paramaterizations of E 192701 (2002) (where paramaterizations of Eaa

symsym and E and Ebbsym sym are given)are given)

HF predictions u

sing 90

HF predictions u

sing 90

effecti

ve in

teractions s

catte

r

effecti

ve in

teractions s

catte

r

between E

between E

aa sym

sym and E and E

bb sym

sym

New Skyrme New Skyrme interactionsinteractions

Amplication around Amplication around normal densitynormal density

The multifaceted influence of symmetry energy in astrophysics and nuclear physics

J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542. A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. (2005).

Isospin physicsIsospin physicsn/p n/p

isoscaling isoscaling

isotransportisotransport isodiffusionisodiffusion

t/t/33HeHe isofractionationisofractionation

KK++/K/K00

isocorrelation isocorrelation

Expanding fireball and Expanding fireball and gamma-ray burst (GRB) from gamma-ray burst (GRB) from the superdene neutron star the superdene neutron star (magnetar) SGR 1806-20 (magnetar) SGR 1806-20 on 12/27/2004. on 12/27/2004. RAO/AUI/NSFRAO/AUI/NSF

ππ--//ππ++

In pre-supernova In pre-supernova explosion of massive explosion of massive stars stars ee p n is easier with smaller is easier with smaller symmetry energysymmetry energy

GRB and nucleosynthesis GRB and nucleosynthesis in the expanding fireball in the expanding fireball after an explosion of a after an explosion of a supermassive object supermassive object depends on the n/p ratiodepends on the n/p ratio

The proton fraction x at ß-equilibrium in proto-neutron stars is determined by

Critical points of the direct URCA process

APR

Akmal et al.

A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. (2005).

3 30 0(0.048[ / ( )] ( / )() 1 2 )symsymE Ex x

The critical proton fraction for direct URCA process to happen is The critical proton fraction for direct URCA process to happen is XXcc=1/9 =1/9

from energy-momentum conservation on the proton Fermi surfacefrom energy-momentum conservation on the proton Fermi surface

Slow cooling: modified URCA:Slow cooling: modified URCA:

( , ) ( , )

( , ) ( , )

e

e

n n p p n p e

p n p n n p e

Faster cooling by 4 to 5 orders of Faster cooling by 4 to 5 orders of magnitude: direct URCAmagnitude: direct URCA

e

e

n p e

p n e

Consequence: long surface Consequence: long surface thermal emission up to a few thermal emission up to a few million yearsmillion years

PSR J0205+6449 in 3C58 PSR J0205+6449 in 3C58 was suggested as a candidatewas suggested as a candidate

Promising Probes of the Esym(ρ) in Nuclear Reactions (an incomplete list !)

At low densities 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 Neutron-proton differential flow Neutron-proton correlation functions at low relative momenta t/3He ratio

The latest conclusion: Esym(ρ)=32(ρ/ρ0)

1.1 for ρ < 1.2 ρ0 from isospin diffusion studies L.W. Chen, C.M. Ko and B.A. Li, PRL 94 (2005) 32701.

Towards high densities reachable at RIA and GSI π – yields and π -/π + ratio Neutron-proton differential transverse flow n/p ratio at mid-rapidity

K+/K0 ratio ??

Most proposed probes in heavy-ion collisions are based on transport Most proposed probes in heavy-ion collisions are based on transport model studiesmodel studies

N

N

( , , )f r p t

������������� �

b bbr b r p b bb bm

b

f pf U f I I

t E

�������������� ������������������������������������������

m mmr m mm bm

m

f kf I I

t E

��������������

Hadronic transport equations:Hadronic transport equations:

Baryons: Baryons:

Mesons:Mesons:

Simulate solutions of the coupled transport Simulate solutions of the coupled transport equations using test-particles and Monte Carlo:equations using test-particles and Monte Carlo:

1( , , ) ( ) ( )i i

it

f r p t r r p pN

����������������������������������������������������������������� �����

The evolution of is followed The evolution of is followed

on a 6D latticeon a 6D lattice ( , , )f r p t������������� �

An example: An example:

(gain)(gain)(loss)(loss)

Mean-field potential for baryonsMean-field potential for baryons

The phase space distributionfunctions, mean fields and collisions integrals are allisospin dependent

Symmetry energy and single nucleon potential used in the IBUU04 transport code for reactions with radioactive beams

12'

'0 0 0 0

, , '3 3 '2 2 2 2

0

0

0

1 2 2,

( , , , , ) ( ) ( ) ( ) (1 ) 81

2 2( , ') ( , ')' '1 ( '

' , ( ) 121 , ( ) 96 ,

) / 1 ( ') /

2112 1 1

u l

l u

BU p A A B

C Cf r p f r pd p d p

p p

B BA A

x x x x x

xK MeVx

p

xx

p

��������������

ρ

B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).

softsoft

stiff

stiff

densitydensityHF using a modified Gogny force:HF using a modified Gogny force:

Momentum and density dependence of the symmetry potential

Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions provides a boundary condition at ρ0:

1

1

( ) / 2 ,

28 6 , 0.1 0.2

Lane n p R kin

R

U U U V E

V MeV

for Efor Ekinkin < 100 MeV < 100 MeV

P.E. Hodgson, The Nucleon Optical Model, P.E. Hodgson, The Nucleon Optical Model, World Scientific, 1994 World Scientific, 1994

G.W. Hoffmann et al., PRL, 29, 227 (1972).G.W. Hoffmann et al., PRL, 29, 227 (1972). Consistent with the Lane potential Consistent with the Lane potential below 100 MeVbelow 100 MeV

/n p isoscalar LaneU U U

momentummomentum Densit

y

Density ρρ

//ρρ 00

δδ

δδ

Neutron-proton effective k-mass splitting in neutron-rich matter

*1[1 ]Fp

m m U

m p p

With the modified Gogny effective interaction

B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).

Nucleon-nucleon cross sections and nuclear stopping power in neutron-rich matter

2*

/ NNmedium free

NN

in neutron-rich matterin neutron-rich matter/medium free

*NN is the reduced mass of theis the reduced mass of the

colliding pair NN in mediumcolliding pair NN in medium

Effects on the nuclear stopping power and Effects on the nuclear stopping power and nucleon mean free-path in n-rich matternucleon mean free-path in n-rich matterJ.W. Negele and K. Yazaki, PRL 47, 71 (1981)J.W. Negele and K. Yazaki, PRL 47, 71 (1981)V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992)V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992)M. Kohno et al., PRC 57, 3495 (1998)M. Kohno et al., PRC 57, 3495 (1998)D. Persram and C. Gale, PRC65, 064611 (2002).D. Persram and C. Gale, PRC65, 064611 (2002).

1.1. In-medium xsections are reducedIn-medium xsections are reduced2.2. nn and pp xsections are splitted nn and pp xsections are splitted due to the neutron-proton effective due to the neutron-proton effective

mass slitting in neutron-rich mattermass slitting in neutron-rich matter

Isospin transport (diffusion) in heavy-ion collisions as a probe of Esym (ρ) at subnormal densities

Particle Flux:

Isospin diffusion coefficient DI

depends on the symmetry potential

Isospin Flow:

L. Shi and P. Danielewicz, L. Shi and P. Danielewicz, Phys. Rev. CPhys. Rev. C6868, 017601 (2003)., 017601 (2003).

0 50 100 150

0.0

0.5

1.0

1.5

/0

x=-1 MDI SBKD /

0 (MDI)

/0 (SBKD)

t (fm/c)

124Sn + 112Sn

MSU data

Ri

Momentum-independentMomentum-independent Momentum-dependentMomentum-dependent

All having the same EAll having the same Esymsym ( (ρρ)=32 ()=32 (ρρ//ρρ00))1.11.1

MSU experiments: R=0.42-0.52 inMSU experiments: R=0.42-0.52 in124124Sn+Sn+112112Sn at ESn at Ebeambeam/A=50 MeV/A=50 MeV

mid-central collisions.mid-central collisions. With With 112112Sn+Sn+112112Sn and Sn and 124124Sn+Sn+124124Sn Sn as references. as references. Use X=Use X=77Li/Li/77Be or Be or δδof the projectile residue, etc.of the projectile residue, etc.

M.B. Tsang et al. PRL 92, 062701 (2004)

Extract the EExtract the Esymsym((ρρ)) from isospin transport from isospin transportA quantitative measure of the isospin non-equilibrium and stopping power A quantitative measure of the isospin non-equilibrium and stopping power In A+B using any isospin tracer X, F. Rami (FOPI), PRL, 84, 1120 (2000).In A+B using any isospin tracer X, F. Rami (FOPI), PRL, 84, 1120 (2000).

2 A B A A B BA BX A A B B

X X XR

X X

1A AXR

0A BXR

1B BXR

If complete isospin mixing/ equilibriumIf complete isospin mixing/ equilibrium

SBKDSBKD: : momentum-independent momentum-independent Soft Bertsch-Kruse-Das Gupta EOSSoft Bertsch-Kruse-Das Gupta EOS

MDI: Momentum-Dependent InteractionMDI: Momentum-Dependent Interaction

-700 -600 -500 -400 -3000.4

0.6

0.8

1.0

x=-2(=1.60)

MSU data

x=-1(=1.05)

x=0(=0.69) x=1

E/A=50 MeV and b=6 fm MDI interaction SBKD interaction

124Sn + 112Sn

1-R

i

Kasy

(MeV)

Comparing momentum-dependent IBUU04 calculations

with data on isospin transport from NSCL/MSUExperiments favors:

Esym()=32 (/ρ0 )1.1 for ρ<1.2ρ0

Kasy(ρ0)~-550 MeV

Isobaric incompressibility of asymmetric nuclear matterIsobaric incompressibility of asymmetric nuclear matter2

0( , ) ( ) ( )asyK KK

L.W. Chen, C.M. Ko and B.A. Li, L.W. Chen, C.M. Ko and B.A. Li, PRL 94, 32701 (2005).PRL 94, 32701 (2005).

0 0 002 2 2

09 ( / ) 18 ( / )( ) sym sys ma y d E d dE dK

Str

engt

h of

isos

pin

tran

spor

tS

tren

gth

of is

ospi

n tr

ansp

ort

Next step: Next step: 1. Reduce the error bars of the data 1. Reduce the error bars of the data and the calculationsand the calculations2. Compare with results using other observables2. Compare with results using other observables3. Exam effects of in-medium cross sections3. Exam effects of in-medium cross sections

Predictions for reactions with high energy radioactive beams at RIA and FAIR/GSI

Examples: • Isospin fractionation• π - yields and π -/π + ratio

Besides many other interesting physics, it allows the determination of nuclear equation of state for neutron-rich matter at high densities where it is most uncertain and most important for several key questions in astrophysics.

Formation of dense, asymmetric nuclear matter at RIA and GSI

Soft ESoft E symsym

Stiff EStiff Esymsym

n/p ratio of then/p ratio of thehigh density regionhigh density region

Isospin fractionation (distillation): at isospin equilibrium Isospin fractionation (distillation): at isospin equilibrium EOS requirement:EOS requirement: lowlow (high)(high) density region is more neutron-rich withdensity region is more neutron-rich with stiff stiff (soft)(soft) symmetry energysymmetry energy

1 1 2 2( ) ( )sym symE E

Isospin asymmetry of free nucleonsIsospin asymmetry of free nucleons

densitydensity

Symmetry enengySymmetry enengy

ρρ00

softsoft

stiff

stiff

2( , ) ( ,0) ( )symE E E

Near-threshold pion production with radioactive beams at RIA and GSINear-threshold pion production with radioactive beams at RIA and GSI

However, pion yields are also sensitive to the symmetric part of the EOSHowever, pion yields are also sensitive to the symmetric part of the EOS

yields are more sensitive to the symmetry energy yields are more sensitive to the symmetry energy EEsymsym((ρρ)) since since

they are mostly produced in the neutron-rich regionthey are mostly produced in the neutron-rich region

2 4( , ) ( ,0 ( )) ( )symEE E

ρρ

densitydensitystiffstiff softsoft

Pion ratio probe of symmetry energy

0

nn 0 1 5 a) Δ(1232) resonance model pp 5 1 0 in first chance NN scatterings: np(pn) 1 4 1 (negelect rescattering and reabsorption)

2

2

2

)(5

5ZN

NZZ

NZN

R. Stock, Phys. Rep. 135 (1986) 259. b) Thermal model: (G.F. Bertsch, Nature 283 (1980) 281; A. Bonasera and G.F. Bertsch, PLB195 (1987) 521)

exp[( ) / ]n p kT

H.R. Jaqaman, A.Z. Mekjian and L. Zamick, PRC (1983) 2782.

c) Transport models (more realistic approach): see, e.g., Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701.

31 1( ) {ln ( ) ( )}

2

m mn p mnn p asy asy Coul m T n p

mp

mV V V kT b

m

Time evolution of Time evolution of ππ--//ππ++ ratio in central reactions at RIA and GSI ratio in central reactions at RIA and GSI

softsoft

stiffstiff

( )like

0 *0

*

1 23 31 23 3

N

N

t

From the overlapping From the overlapping n-skins of the colliding nuclein-skins of the colliding nuclei

Summary

• The EOS of n-rich matter, especially the Esym(ρ) is very important for many interesting questions in both astrophysics and nuclear physics

• Transport models are invaluable tools for studying the isospin-dependence of in-medium nuclear effective interactions and properties of n-rich matter

• A transport model anaysis of recent isospin transport experiments indicates:

Esym()=32 (/ρ0 )1.1 for ρ<1.2ρ0, and Kasy(ρ0)~-550 MeV

• High energy radioactive beams at RIA and GSI will allow us to study the EOS of n-rich matter up to 2ρ0. Several sensitive probes of the Esym() are proposed.

Neutron-proton differential flow as a Neutron-proton differential flow as a probe of the symmetry energy:probe of the symmetry energy:

( )

1

1( )

( )

i N yx x

n p i ii

F y pN y

Transverse flow as a probe Transverse flow as a probe of the nuclear EOS:of the nuclear EOS:

y

px

ppxx

yy

symmetry potential is generallysymmetry potential is generallyrepulsive for neutrons and repulsive for neutrons and attractive for protonsattractive for protons

1i for n and pfor n and p