Probing the Equation of State of Neutron-Rich Matter with Heavy-Ion ReactionsBao-An Li

Arkansas State University

1. EOS and symmetry energy of neutron-rich matter• Current status, importance for nuclear physics and astrophysics

2. Major physical issues in modeling nuclear reactions induced by neutron-rich nuclei• Momentum dependence of the symmetry potential

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

• Isospin dependence of in-medium nucleon-nucleon cross sections

3. Experimental probes of the symmetry energy

•• AnAn example:example: isospin diffusion/transport in heavy-ion reactions

4. Impacts of the symmetry energy constrained by available data on the isospin diffusion

and the size of neutron-skin in 208Pb • Nuclear effective interactions

• Cooling mechanisms of proto-neutron stars

• Radii of neutron stars

CollaboratorsCollaborators::LieLie--WenWen Chen, Shanghai Jiao Tong UniversityChen, Shanghai Jiao Tong UniversityChampak B. Das, Subal Das Gupta and Champak B. Das, Subal Das Gupta and Charles Gale, McGill UniversityCharles Gale, McGill UniversityCheChe Ming Ko, Texas A&M UniversityMing Ko, Texas A&M UniversityAndrew W. Steiner, Los Alamos National LaAndrew W. Steiner, Los Alamos National LaboratoryboratoryGaoGao--Chan Yong and Wei Zuo, Chinese Academy of Science Chan Yong and Wei Zuo, Chinese Academy of Science

Esym (ρ) predicted by microscopic many-body theories

Sym

met

ry e

nerg

y (M

eV)

Density

Effective field theory of

N. Kaiser and W. Weise Dirac Brueckner Hartree-Fock

Relativistic Mean Field

Brueckner HF

Greens function

Variationalmany-body theory

A.E. L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier and V. Rodin, Phys. Rev. C68 (2003) 064307

pure neutron matter symmetric nuclear matte

2

r

4( )

( ) ( ) ( )

EOS ( , ) ( ,0) ( ), where ( ) /: sym

sym

n pE EE

E E E

ρ δ ρ δ ο δ δ ρ ρ ρρ

ρ ρ ρ

= + +

−

≡ −

≈

Components of the symmetry energy

Total symmetry

energy

Isosinglet contribution to the interaction

isotripletKinetic

Interaction

Intermediate energy Intermediate energy heavyheavy--ion reactions ion reactions are especially useful for are especially useful for studying the studying the isospinisospindependence of nuclear dependence of nuclear effective interactions effective interactions

For For ρρ ≤≤ 22ρρ00, potential , potential contribution dominates contribution dominates over the kinetic one over the kinetic one ((~ ~ ρρ2/32/3)) !!

High energyHeavy-ion reactions

Much more challening

For For ρρ > 2> 2ρρ0 0 kinetic contributionkinetic contributiondominatesdominates

A.E. L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier and V. Rodin, Phys. Rev. C68 (2003) 064307

Esym(ρ) from Hartree-Fock approach using different effective interactionsB. B. CochetCochet, K. , K. BennaceurBennaceur, P. , P. BoncheBonche, T. , T. DuguetDuguet and J. Meyer, and J. Meyer, NuclNucl. Phys. . Phys. A731A731, 34 (2004)., 34 (2004).J. Stone, J.C. Miller, R. J. Stone, J.C. Miller, R. KoncewiczKoncewicz, P.D. Stevenson and M.R. , P.D. Stevenson and M.R. StrayerStrayer, PRC , PRC 6868, 034324 (2003)., 034324 (2003).BaoBao--An Li, PRL An Li, PRL 88,88, 192701 (2002) (where 192701 (2002) (where paramaterizationsparamaterizations of of EEaa

symsym and and EEbbsymsym are given)are given)

EEaa symsym

EE bbsymsym

HF predictions HF predictions using 90 effective using 90 effective interactions scatter interactions scatter between between EEaa

symsym and and EEbb

symsym

New New SkyrmeSkyrmeinteractions withinteractions withmore than one more than one densitydensity--dependentdependenttermsterms

HF calculations using 90Skyrme and Gogny

effective interactions scatter between Ea

sym and Ebsym

Sym

met

ry e

nerg

y

0 0 0 0 0

2 2 2 2 2 200 0( )( ) 9 ( / ) and 9 ( / ) 18 ( / )asy sym symK d E d dK E d dE dρ ρ ρρ ρ ρ ρ ρ ρ ρρ∞ ≡ ≡ −

Liquid-drop formula for finite nuclei: KA = m<r2>0E2ISGMR = K∞ (1+cA-1/3) + Kasy δ2 + KCoul Z2 A-4/3

The major remaining uncertainty in determining theThe major remaining uncertainty in determining the EOS of EOS of symmetric symmetric nuclear nuclear matter is due to the poor knowledge about matter is due to the poor knowledge about EEsymsym((ρρ))

Example 1: On extracting the incompressibility K∞(ρ0) from isoscalar giant monopole resonance, J. Piekarewicz, PRC 69, 041301 (2004); G. Colo, et al., PRC 70, 024307 (2004).

data rangedata rangeThe extraction of K∞depends on Kasy(ρ0)=?

KKasyasy((ρρ00))

k∞=250

k∞=240

k∞=230

(Sly4)

2Isobaric incom pressib ility of asym m etric m atter: ( , ) ( ( )) asyKK Kρ δ ρ δρ∞ ∞= +

--566 566 < < KKasyasy < 139 < 139 MeVMeVShlomo and Youngblood, Shlomo and Youngblood, PRC 47, 529 (1993)PRC 47, 529 (1993)

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. 411, 325 (05)

Isospin Dependence of Strong Interactions

Nuclear MassesNeutron Skin Thickness

Isovector Giant Dipole ResonancesFission

Heavy Ion FlowsMulti-Fragmentation

Nuclei Far from StabilityRare Isotope Beams

Many-Body TheorySymmetry Energy

(Magnitude and Density Dependence)

SupernovaeWeak Interactions

eνEarly Rise of LBounce Dynamics

Binding Energy

Proto-Neutron Stars Opacitiesν

EmissivitiesνSN r-ProcessMetastability

Neutron StarsObservational

Properties

Binary MergersDecompression/Ejectionof Neutron-Star Matter

r-Process

QPO’sMass

Radius

NS CoolingTemperature

, z∞RDirect Urca

Superfluid Gaps

X-ray Bursters, z∞R

Gravity WavesMass/Radius

dR/dM

PulsarsMasses

Spin RatesMoments of Inertia

Magnetic FieldsGlitches - CrustMaximum Mass, Radius

Composition:Hyperons, Deconfined Quarks

Kaon/Pion Condensates

IsospinIsospin physicsphysicsn/pn/p

isoscalingisoscaling

isotransportisotransportisodiffusionisodiffusion

t/t/33HeHeisofractionationisofractionation

KK++/K/K00

isocorrelationisocorrelation

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

ππ--//ππ++

γ

In preIn pre--supernova supernova explosion of massive explosion of massive stars stars ee p n ν− + → +is easier with smaller is easier with smaller symmetry energysymmetry energy

GRB and GRB and nucleosynthesisnucleosynthesisin the expanding fireball in the expanding fireball after an explosion of a after an explosion of a supermassivesupermassive object object depends on the depends on the n/pn/p ratioratio

inTerrestrial Labs

The proton fraction x at ß-equilibrium in proto-neutron stars is determined by3 3

0 0(0 .048[ / ( )] ( / )() 1 2 )symsymE Ex xρ ρ ρ ρ −The critical proton fraction for direct URCA process to happen iThe critical proton fraction for direct URCA process to happen is s XXpp=0.14 for =0.14 for npenpeµµmatter obtained matter obtained from energyfrom energy--momentum conservation on the proton Fermi surfacemomentum 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

ν

ν

−

+

+ → + + +

+ → + + +

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 ep n e

ν

ν

−

+

→ + +

→ + +

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

B.A. Li, Nucl. Phys. A708, 365 (2002).

Direct URCA kaon condensation allowed

Neutron bubbles formationtransition to Λ-matter

Isospinseparationinstability

Probing the EOS of neutron-rich matter in terrestrial labs

Bao-An Li, Phys. Rev. Lett. 88, 192701 (2002).

J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536

A road map towards determining the nature of neutron-rich nucleonic matter

Goal

IWM2005, IWM2005, CataniaCatania, Italy, Italy

Nπ∆ → +

Nπ + →∆

( , , )f r p tπ

b bbr b r p b bb bmb

b

f p f fU I It E

∂+ ∇ − ∇ ∇ = +

∂i i

m mmr m m m b m

m

f k f I It E

∂+ ∇ = +

∂i

HadronicHadronic transport equations for the reaction dynamics of nucleustransport equations for the reaction dynamics of nucleus--nucleus collisions:nucleus collisions:

Baryons: Baryons:

Mesons:Mesons:

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

1( , , ) ( ) ( )i iit

f r p t r r p pN

δ δ≅ − −∑

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)

UUbb is the meanis the mean--field potential field potential for baryonsfor baryons

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

Bao-An Li and Wolfgang Bauer, Phys. Rev. C44, (1991) 450.

Isospin splitting and momentum dependence of nucleon mean field in neutron-rich matter

The momentum dependence of the nucleon potential is a result of the non-localityof nuclear effective interactions and the Pauli exclusion principle

Predictions of Brueckner-Hartree-Fock including 3-body forces

Un(k) is shifted upwards compared to Up(k) due to

the positive/negative symmetry potential for n/p

δ =0.2

δ =0.8

δ =0.2

δ =0.8

protons

neutrons

Un/p=U0 ± ULane• δ

I. Bombaci and U. Lombardo, Phys. Rev. C44, 1892 (1991);W. Zuo, L.G. Gao, B.A. Li, U. Lombardo and C.W. Shen, Phys. Rev. C72, 014005 (2005).

Symmetry energy and single nucleon potential used in the IBUU04 transport model

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 pp p

B BA A

x x x x x

x K MeVx

pxx

p

σστ τ

τσ

τ τ τ ττ τ

ρ ρ ρ ρρ δ τ δ τ δρρ ρ ρ

τ τ

ρ

σ

σ

σ

ρ

ρ

−

= + + − −+

+ ++

= ± = − + = − − =+ +

− Λ + − Λ∫ ∫

ρ

C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003).B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).

softsoft

stiff

stiff

Single nucleon potential within the HF approach using a modifiedSingle nucleon potential within the HF approach using a modified GognyGogny force:force:Density ρ/ρ0

NucleonNucleon--nucleon cross section in freenucleon cross section in free--spacespace

n+p

p+p(n+n is the same negelecting

charge symmetry breaking)

Neutron-proton effective k-mass splitting in neutron-rich matter at zero temperature

*1[1 ]Fp

m m Um p p

ττ

τ

τ

−∂= +

∂

With the modified Gogny effective interaction

Isospin-dependence of nucleon-nucleon cross sectionsin neutron-rich matter

2*

/ NNmedium free

NN

µσ σµ

⎛ ⎞≈ ⎜ ⎟

⎝ ⎠

in neutronin neutron--rich matter rich matter at zero temperatureat zero temperature

/medium freeσ σ

*N Nµ is the reduced effective mass of theis the reduced effective mass of the

colliding nucleon pair NNcolliding nucleon pair NN

Application in neutronApplication in neutron--rich matter:rich matter:nnnn and pp and pp xsectionsxsections are are splittedsplitted due todue tothe neutronthe neutron--proton effective mass slittingproton effective mass slitting

The effective mass scaling model:

0valid for 2 and relative momenta 240 MeV/c

Applications in symmetric nuclear matter:Applications in symmetric nuclear matter:J.W. J.W. NegeleNegele and K. and K. YazakiYazaki, PRL 47, 71 (1981), PRL 47, 71 (1981)V.R. V.R. PandharipandePandharipande and S.C. Pieper, PRC 45, 791 (1992)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. D. PersramPersram and C. Gale, PRC65, 064611 (2002).and C. Gale, PRC65, 064611 (2002).

ρ ρ≤ ≤according to Dirac-Brueckner-Hatree-Fock calculations F. Sammarruca and P. Krastev, nucl-th/0506081Phys. Rev. C (2005) in press.

BaoBao--An Li and LieAn Li and Lie--WenWen Chen, nuclChen, nucl--th/0508024, th/0508024, Phys. Rev. C (2005) in press.Phys. Rev. C (2005) in press.

Nucleon effective masses during heavy-ion reactionsB.A. Li and L.W. Chen, nucl-th/0508024, Phys. Rev. C (2005) in press.

Effective massdistributions

Instant of maximum compression

No. of potential colliding nucleon-nucleon pairs during heavy-ion collisions

Reduction factor of their corresponding cross sections in the medium

σmedium=σfree Х Rmedium

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

At sub-saturation 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/transport • Neutron-proton differential flow • Neutron-proton correlation functions at low relative momenta • t/3He ratio

Towards high densities reachable at FAIR/GSI, RIA, RIKEN and CSR/China• π -/π + ratio, K+/K0 ? • Neutron-proton differential transverse flow • n/p ratio at mid-rapidity • Nucleon elliptical flow at high transverse momenta

(1)Multi-observable correlations are important

(2) Detecting neutrons simultaneously with charged particles is critical

Extract the Extract the EEsymsym((ρρ)) at subnormal densities from at subnormal densities from isospinisospin diffusion/transportdiffusion/transport

A quantitative measure:

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

X X XRX X

+ + ++

+ +

− −≡

−

1 and 1A A B BX XR R+ += = −

0A BXR + ≈ for complete for complete isospinisospin mixingmixing

X is an X is an isospinisospin--sensitive sensitive obseravbleobseravble, , F. Rami et al. (FOPI/GSI), PRL, 84, 1120 (2000).F. Rami et al. (FOPI/GSI), PRL, 84, 1120 (2000).

int41 ( )np symmI np

np

ED dt F

mµ

σ δ∂

∝ • ∝ • +∂

The degree of isospin transport/diffusion dependson both the symmetry potential and the in-mediumneutron-proton scattering cross section. For near-equilibrium systems, the mean-field contributes:

During heavy-ion reactions, the collisional contribution to DI is expected to be proportional to σnp

Nucleon flux:

i i iVρΓ =

Isospin transport/diffusion:

n p I rDρ δΓ −Γ ∝ ∇

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

Isospin transport/diffusion experiments at the NSCL/MSUIsospinIsospin transport in transport in 124124Sn+Sn+112112Sn using Sn using 112112Sn+Sn+112112Sn and Sn and 124124Sn+Sn+124124Sn as referencesSn as referencesEEbeambeam/A=50 /A=50 MeVMeV, use X=, use X=77Li/Li/77Be or Be or isospinisospin--asymmetry of the projectileasymmetry of the projectile--like residuelike residue..M.B. Tsang et al., Phys. Rev. Lett. 92, 062701 (2004)

0 0 0

2 2 20 0( ) 9 ( / ) 18 ( / )asy sym symK d E d dE dρ ρρ ρ ρ ρ ρ≡ −

σσ

L.W. Chen, C.M. Ko and B.A. Li, L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. Phys. Rev. LettLett. 94, 32701 (2005);. 94, 32701 (2005);BaoBao--An Li and LieAn Li and Lie--WenWen Chen, Chen, Phys. Rev. C (2005) in press.Phys. Rev. C (2005) in press.

in-medium NN xsection

free-space NN xsection

MSU data range

γ

γ

γ

0.7 1.10 0

0

32( / ) ( ) 32( / )

( ) 500 50 MeV

Conservative Conclusions:

sym

asy

E

K

ρ ρ ρ ρ ρ

ρ

≤ ≤

≈ − ±

Strength of the symmetry potential at sub-normal densities

ρ ρ

δ δρ ρδ

X=1

X=0

X=-1

X= -2

Impacts of the symmetry energy constrained by the isospin diffusion data

(2) Cooling mechanisms of proto-neutron stars

0.5

1

1.5

M/M

0.1

0.2

0.3

prot

on fr

actio

n

equilibriumβNeutron stars in

=0.14px

Direct URCA limit

0 1 2 3 4 5

50

100

150

0ρ/ρ

(MeV

)sy

mE x=0

x=-1x=-2

APR

0

0

Direct URCA processes are possible for neutron stars

with central densities ρc> 3.5 ρ0and masses M>1.39M⊙

(npeµ matter)

A. Akmal, V.R.Pandharipandeand D.G. RavenhallPhys. Rev. C58, 1804 (1998)

Mass-radius correlation of neutron stars and their EOS

Different EOS predicted by various theories

EOSsEssentially, none ofthem was ever testedagainst reaction data

J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542.

Constraining the radii of neutron stars with terrestrial nuclear laboratory data

10 12 14 160

0.5

1

1.5

2

2.5

)M

(M

R (km)

x=0

x=-1

x=-2APR

z=0.35Causality

I/I=0.014

∆

11.5<R<13.6

<16.3∞14.4<Rfor 1.4M⊙

2(1 ) / 1 2 /R R z R GM Rc∞ = + = −

Graviational redshift

For EXO0748 J. Cottam et al, Nature 420, 51 (2002)

“radiation radius”seen by observer at infinity

matter radiuswhere P®=0.

APR: A. Akmal, V.R.Pandharipandeand D.G. RavenhallK0=269 MeV

K0=211 MeV for x=0, -1, and -2

Vela glitch: B. Link, et al, PRL 83, 3362 (1999).

Nuclear lim

its

Bao-An Li and Andrew W. Steiner, nucl-th/0511064

Masses of canonical neutron stars

<Mneutron star>=1.35±0.04 M⊙S.E. Thorsett and D. Charkrabarty, Astrophysics J. 512, 288 (1999).

Measuring Neutron Star Radii• Determine luminosity, temperature and deduce surface area. If black

body L=4π R∞2σT4.

– T from X-ray spectrum, such as those from Chandra satellite– Need distance to star from parallax to get L.– Deduce R from surface area (~30% corrections from curvature of space-

time). The radiation radius for an observer at infinity:

z is the gravitational redshift and R the matter radius where the pressure vanishes

• Complications– Spectrum peaks in UV and this is heavily absorbed by interstellar H.– Not a black body: often black body fit to X-ray does not fit visible spectra.– Model NS atmospheres (composition uncertain) to correct black body.

• Current status: available estimates give a wide range“Although accurate masses of several neutron stars are available, a precisemeasurement of the radius does not exist yet”……Lattimer and Prakash, Science Vol. 304 (2004) 536

2(1 ) / 1 2 /R R z R GM Rc∞ = + = −

Intro

The Chandra X-Ray Observatorylunched on July 23, 1999

The Chandra X-Ray Observatorylunched on July 23, 1999

Latest status of the radiation radius R∞ measured on space x-ray observatories (Chandra and XMM-Newton) --- Bob Rutledge, Oct. 2005

∞

Chandra image of Cas A

The radii are estimated assuming the masses are all 1.4M⊙ without actually measuring the massessimultaneously becausesome of them are isolatedneutron stars instead ofbeing in a binary

The radii of neutron stars are primarily determined by

Why ?The radius and mass are determined by the Tolman-Oppenheimer-Volkoff eq. for the gravitational equilibrium

r2 3 22

2 20

( ( ) ( ) / )( ( ) 4 ( ) / ) , m(r)= 4 r' ( ') ', ( ) is the energy density at r(1 2 ( ) / )

the radius is determined by P(R)=0, the pressure P( ) at -equilibrium is obtained from

P(

dP G e r P r c m r r P r c e r dr e rdr r Gm r rc

π π

ρ β

+ += −

− ∫

0

2 2 ' ' 2nuclear electron 0 nuclear matter sym

e e

1 1, )=P P ( ) ( ( ) ( ) ) + (1- ) E ( )4 2

with determined by the chemical equilibrium condition 4 ( ) and the charge neutrality

e e sym

n p sym

E E E

E

δρ δ ρ ρ µ ρ ρ ρ δ δ δ ρ ρρ

δ µ µ µ δ ρ ρ

∂+ = + = +

∂= − =

i

nuclear matter 0

2 2 ' 2 2 '0

0 sy

0 0 sym 0 0 0

' 'sy

0

0

m

m

E' ( ) 01

Example

P( ) ( ) (1- ) E ( ) ( )2

E ( ) ( ) for 5

Sinc

: at , 0.86 with E ( ) 32 MeV, and

thus

Most models predic

e the radius is

t

sym sym

sym

p

E E

E

ρ

ρ ρ

ρ ρ δ ρ δ δ ρ ρ ρ δ

δ

ρ

ρ ρ

ρ

ρ ρ

=

≈ = =

= + ≈

≤

i

∼

r

s 02

0

'0 ym

determined by P(R)=0, the R is thus very sensitive to the

behavior of P( ) E ( while the mass m(r)= 4 r' ( ') ' is sensitive to the EOS at all densitie s) e r drρ πρ∝ ∫

First found empirically then proven analytically by Prakash and Lattimer,Astro. Phys. Jour. 550, 426 (2001)

0

( )|symdE

d ρ

ρρunlike other properties, such as the masses

Esym (ρ) predicted by microscopic many-body theories

Sym

met

ry e

nerg

y (M

eV)

Density

Effective field theory of

N. Kaiser and W. Weise Dirac Brueckner Hartree-Fock

Relativistic Mean Field

Brueckner HF

Greens function

Variationalmany-body theory

A.E. L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier and V. Rodin, Phys. Rev. C68 (2003) 064307

pure neutron matter symmetric nuclear matte

2

r

4( )

( ) ( ) ( )

EOS ( , ) ( ,0) ( ), where ( ) /: sym

sym

n pE EE

E E E

ρ δ ρ δ ο δ δ ρ ρ ρρ

ρ ρ ρ

= + +

−

≡ −

≈

Neutron-skin in 208Pb and dEsym/dρ

Pressure forces neutrons out against surface tension. Large pressure gives large neutron skin. dEsym/dρ determines the size of n-skin andpressure of neutron matter at ρ0

B.A. Brown PRL85, 5296 (2000)

Rn-

Rp

(fm) f

or 20

8 Pb0 0 0

/ | / | 0 / |, n p np nuneutron nuclear sy clem ar sym symE E R R p dE d E dE dE d dρ ρ ρρ ρ ρ− ∝∆ = + = += +

B.A. Brown, C. Horowitz, J. Piekarewicz, R.J. Furnstahl, J.R. Stone, et al.

For the same reason, it is harder for neutrons and protons to mix-up with alarger dEsym/dρ, thus a smaller/slower isospin diffusion

correlation of neutron-skin and isospin diffusionAndrew W. Steiner and Bao-An Li, Phys. Rev. C72, 041601 (2005).

C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001)

Neutron-skin

Constraining the dEsym/dρ with data on both isospin diffusion and n-skin in 208Pb

ρ ρ

ρ

Isospinfractionation

Neutron-rich cloud

Neutron skin is calculated with Skyrme HF with interactions parameters adjusted such that the same EOS as used in calculating the isospin diffusion is obtained for a given x

Neutron-skin data: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994);

B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)

Constraining the radii of neutron stars with terrestrial nuclear laboratory data

10 12 14 160

0.5

1

1.5

2

2.5

)M

(M

R (km)

x=0

x=-1

x=-2APR

z=0.35Causality

I/I=0.014

∆

11.5<R<13.6

<16.3∞14.4<Rfor 1.4M⊙

2(1 ) / 1 2 /R R z R GM Rc∞ = + = −

Graviational redshift

For EXO0748 J. Cottam et al, Nature 420, 51 (2002)

“radiation radius”seen by observer at infinity

matter radiuswhere P®=0.

APR: A. Akmal, V.R.Pandharipandeand D.G. RavenhallK0=269 MeV

K0=211 MeV for x=0, -1, and -2

Vela glitch: B. Link, et al, PRL 83, 3362 (1999).

Nuclear lim

its

Bao-An Li and Andrew W. Steiner, nucl-th/0511064

SummaryIntermediate energy heavy-ion collisions are a unique tool to study the isospindependence of strong interactions, it is a fundamental quantity for understanding the nature of neutron-rich matter and has significant ramifications in astrophysics(Hope the director, Prof. E. Migneco will consider this in making the future plan for the INFN-LNS)

0.7 1.10 0

0

32( / ) ( ) 32( / )

( ) 500 50 MeV

Transport model anslyses of available isospin diffusion data from MSU lead to

sym

asy

E

K

ρ ρ ρ ρ ρ

ρ

≤ ≤

≈ − ±

Important constraints are put on

• EOS of neutron-rich matter• Nuclear effective interactions• Cooling mechanisms of n-stars• Radii of n-stars INFN-LNS Nov. 29, 2005

Looking forward and driving away from the valley of stability EOS of dense n-rich matter

• How supernovae explode• How heavy elements are formed in the universe

• What are the properties of n-stars• What is the critical M/R ratio of

a neutron star before it becomes a black hole when the neutron degenerate pressure in the star can no longer support its gravity

• Isospin dependence of in-mediumnuclear effective interactions

Universal multifragmentation

Coalescence

Star formation

Cataniaorange

Secondary beam

Thermal expansion