The National Superconducting Cyclotron Laboratory

@Michigan State University

Exploring the symmetry energyin nuclear equation of state

with heavy ions

Betty Tsang

Transverse and elliptical flow constraints the EoS of symmetric nuclear matter.

The density dependence of asymmetry term is largely unconstrained.

55oo

Nuclei

+drip

ped

n +e

Nuclei

+drip

ped

n +e

--

EOS ? ?EOS ? ?

00 oo

exoticsexotics10 km10 km

Nuclei

+e

Nuclei

+e--

Relevance to dilute and dense n-rich objectsRelevance to dilute and dense n-rich objectsRelevance to dilute and dense n-rich objectsRelevance to dilute and dense n-rich objects

Sizes of nuclei with n-halo and n-skin

Stability of Neutron Star and its structure

P T

Multifragmentation Scenario--consistent with mixed phase

Time Dependence--Initial compression and energy deposition-- Expansion-- Cooling-- Fragments form (freeze-out)-- Fragments decouple

Different ApproachesDynamical (AMD,BNV); Rate equations (EES); Equilibrium at freeze-out density (BUU-SMM)

Isospin mixing

PT

PT

PT

PRL, 85, 716 (2000)

Varying isospin degree of freedom

en

Enrichment of n in the gas

ep

Depletion of p in the gas

124Sn+124Sn

112Sn+112Sn1ˆ

1ˆ112

112

p

n

903.0ˆ

078.1ˆ124

124

p

n

Gain 24 n’s

Measured Isotopic yields

Shapes are similarMore n-rich isotopes from more n-rich systemIs Sequential decay effect isospin independent?

Isotopic effects are small how to quantify them?

Isoscaling from Relative Isotope Ratios

R21=Y2/ Y1

pn ZNe Zp

Nn

Factorization of yields into p & n densities

Cancellation of effects from sequential feedings

Robust observables to study isospin effects

Compact representation of isoscaling

Isoscaling observed in many reactions

Y2/ Y1

TZN pne/)(

PRL, 86, 5023 (2001)

b

P T

P T

P T

Q Value, Sep. E

ECoul Esym

Separation Energy

ECoul Esym

Chemical Potentials

ECoul Esym p n

R21exp[(-Sn·N- Sp·Z)/T]

R21exp[((-Sn+ fn*)·N+(-Sp +fp

*+ )·Z)/T]

R21exp[(-n·N- p·Z)/T]

Origin of isoscaling

Isoscaling disappears when the symmetry energy is set to zero

Provides an observable to study symmetry energy

Role of density dependent asymmetry term-Where do the fragments originate?-

• Various models predict different dependence on density dependence of asymmetry term.– Equilibrium models: fragments originate in interior.– EES model: fragments emitted from surface.

more symmetricdense region

asy-stiff (F1) asy-soft (F3)

more symmetric emitted particles

neutron-rich emitted particles

neutron-richdense regionSMM

EES

Expanding Emitting Source modelExpanding Emitting Source model W. Friedman PRC42, 667 (1990)

n, lcpfr

agm

ents

15 MeV 14 MeV

13 MeV

11 MeV

multifragmentation

evap.

vaporization

Thermal instabilityat low density.

Density dependence of asymmetry energy

S()=23.4(/o)

Strong influence

of symmetry term on

fragment isotopic ratios

EES_fragment

=0.36 =2/3

Consistent with many body calculations

with nn interactions

K(F1) = +61 MeV

K(F3) = -69 MeV

Affect neutron star radii, moments of inertia, central densities.

sym(

Symmetry Terms

RadiiM

ass

Sensitivity to the isospin terms in the EOS

Data

Y(N,Z)R21(N,Z)

SMMBUUF1,F3

N/ZP

T Freeze-out source

F1 agrees with data better

PRC, C64, 051901R (2001).

Isospin diffusion

PT

PT

Iso-scaling in projectile fragmentation• Fixed Projectile:

no diffusion: =0=Y(112+124)/Y(112+112)=0.157=Y(124+124)/Y(124+112)=0.126

• Fixed Target:

=Y(124+124)/Y(112+124)=0.401=Y(124+112)/Y(112+112)=0.448

is target dependent

Isospin diffusion between target and projectile

Experimental Determination of Isospin diffusion

4Csym[(Z2/A2)2-(Z1/A1)

2]/T -- statistical models

112+112; 124+124 no isospin gradient, no diffusion

124+112 n-diffusion from proj. to targ.

112+124 n-diffusion from targ. to proj.

[(Zb/Ab)2-(Z3/A3)

2]/ [(Za/Aa)2-(Zb/Ab)

2]

[(Z4/A4)2-(Zb/Ab)

2]/ [(Za/Aa)2-(Zb/Ab)

2]

Experimental results 3 nucleons exchange between 112 and 124 (equilibrium=6 nucleons)

Comparison with model (BUU)

[(Zb/Ab)2-(Za/Aa)

2]/ [(Z3/A3)2-(Za/Aa)

2]

[(Zb/Ab)2-(Za/Aa)

2]/ [(Zb/Ab)2-(Z4/A4)

2]

Diffusion : determined from 2 symmetric systems and 1 mixed system

BUU : parameters determined from transverse and elliptical flow data.

sym(

Initial results show sensitivity to isospin part of the EOS

Further work is needed

Summary

• Conclusions from multi fragmentation work are model dependent:– SMM favors 2 dependence of

S().– EES favors 2/3 dependence of

S(). • Isospin Diffusion from projectile

fragmentation data – test the transport model parameters

•Density dependence of symmetry energy can be examined experimentally.•Existence of isoscaling relations