Maria Colonna Laboratori Nazionali del Sud (Catania)

Dynamics and Thermodynamics with

Energy functional of asymmetric nuclear matter:constrain the iso – EOS (symmetry energy)Information on the behaviour of the symmetry energy at sub-saturation and super-saturation densities

Phase transitions in finite systems:phase diagram of exotic systems & new features of the fragmentation mechanism

Important implications in the astrophysical context:neutron star crust, supernova explosion (clustering of low-density matter)Important for studies of the structure of exotic nuclei

What can we learn from reactions at intermediate energy (30-100 MeV/A) with exotic beams ?

The density-dependent symmetry energy and n-p effective mass splitting

Isospin Transport: the density dependent Esym

I

EIEjj sym

sympn

)()(

IDDj

IDDjIppp

Innn

currents

diffusion

Diffusion Drift

drift

pnqI

D

D

T

qIq

TI

qq

,,

,

Direct Access to Value and Slope of the Symmetry Energy at ρ !

E/A (ρ) = Es(ρ) + Esym(ρ)I²I=(N-Z)/A

IEsympn 4

Self-consistent mean-field calculations

asy-soft

asy-stiff

124Sn “asymmetry” = 0.2

Symmetry Potentials and Effective Masses

neutron

proton

Asy-stiffAsy-soft

Lane Potentials

**pn mm

**pn mm

Density dependence Momentum dependence1

2

*

1

k

U

k

m

m

m qq

(Un-Up)/2I

Phys.Rep.410(2005)335-466

Symmetry energy parameterizations are implemented into transportcodes (Stochastic Mean Field - SMF)

Observables related to isospin diffusion and drift:isospin equilibration (imbalance ratio) , isospin migration(neck composition)

Observables related to n-p effective mass splitting: high pt distribution of pre-equilibrium emission,collective flows, light clusters

Disantangle isovector effects from isoscalar effects Better focus on iso-EOS

The density-dependent symmetry energy and

n-p effective mass splitting: Observables

b=8f

mb=

10fm

ISOSPIN DIFFUSION AT FERMI ENERGIES124Sn + 112Sn at 50 AMeV

SMF - transport modelb=8fmb=9 fmb=10fm

120fm/c100fm/c80fm/c

112112T

124124T

112112T

124124T

MT

T112112P

124124P

112112P

124124P

MP

PII

III2R;

II

III2R

experimental data(B. Tsang et al. PRL 92 (2004) )

asysoft eos superasystiff eos

contact time

Baran, Colonna, Di Toro, Pfabe, Wolter, PRC72(2005)

Imbalance ratios

asy-soft EOS –faster equilibration

Imbalance ratios: isoscalar vs. isovector effects

If:

then:

β = (N-Z)/A τ symmetry energy

tcontact dissipation

Kinetic energy loss as a measure of dissipation (time of contact)

R dependent only on the isovector part of the interaction !

MD, MI: isoscalar effective forces

Isospin migration in neck fragmentation

Transfer of asymmetry from PLF and TLF to the low density neck region

Effect related to the derivative of the symmetryenergy with respect to density

PLF, TLF

neck

emittednucleons

ρ1 < ρ2

Asymmetry flux

asy-stiff

asy-soft

Larger derivative with asy-stiff larger isospin migration effects

Sn112 + Sn112

Sn124 + Sn124

b = 6 fm, 50 AMeVDensity gradients derivative of Esym

arXiv:0711.3761

ρI < ρR

Asymmetry flux

A

Ares

Isospin exchange: βIMF / βres ratioNeck mass A, asymmetry β + ΔβResidues mass Ares, asymmetry β – Δβ A/Ares

minimizing symmetry energy variation

MD MI

stiff - - soft

This ratio depends onlyon the symm. energy variation around the neck density

It should also be studiedas a function of dissipationor observables connected to the density (IMF multiplicity …)

< 0

Sn112 + Sn112 Sn124 + Sn124

b = 6 fm, 50 AMeV

J.Rizzo et al., PRC 72 (2005) → Isotope Science Facility at MSU, White Paper 2006

Gas asymmetry vs. p_t 8/0

Light isobar (3H/3He) yields

Mass splitting: N/Z of Fast Nucleon Emission

High p_t “gas” asymmetry: Observable very sensitive to the mass splitting and not to the asy-stiffness

Vs. Kinetic Energies

132Sn+124Sn, 100 AMeV, b=2 fm, y(0)0.3

3H/3Hen/pasy-stiffasy-stiff

124Sn+124Sn, 50 AMeV, b=2 fm

• m*n>m*p• m*n<m*p

Collective flowsIn-plane Out-of-plane

yyx

yxt pp

pppy

22

22

2 ),(V

ytxt pppyV /),(1

tnt

pt

np pVpVpV 111 )(V)(V)(V n2

p2

n-p2 ttt ppp

1 < V2 < +1

X

Z

y = rapiditypt = transverse momentum

= 1 full outV2 = 0 spherical = + 1 full in

Differential flows)(1),(1

),(1

),(

pn

pyvZN

pyv

i

tiitalDifferenti

B-A Li et al. PRL2002

Au+Au 250 AMeV, b=7 fm

m*n<m*p : larger neutron squeeze out at mid-rapidity

Z=1 data, M3 centrality, 6<b<7.5fm

**pn mm

Mass splitting: Elliptic Flow Difference

129Xe+124Sn,100AMeV

124Xe+112Sn,100AMeV

m*n < m*p

m*p < m*n

Triton/He3 Transverse flow ratioMSU/RIA05, nucl-th/0505013 , AIP Conf.Proc.791 (2005) 70

Phase transitions in finite systems and isospin effects

Validate the mechanisms investigated and the conclusionsdrawn from the study of symmetric matter (multifragmentation)

New features: Instabilities in asymmetric systems (phase diagram)

New features: Isospin distillationObservables: isoscaling, fragment <N>/Z at break-up, double ratiosDistillation in presence of radial flow <N>/Z vs. Ekin

Phase transitions in exotic systems: new effects

Density

Tem

pera

ture

τ = 100 fm/c τ = 50 fm/c

The width of the spinodal zone should depend on isospin

Colonna et al., PRL2002Level density, limiting temperature …

Isospin-dependent phase transition

Isospin distillation: the liquid phase is more symmetric than the gas phase

β = 0.2

β = 0.1

Non-homogeneous density

Spinodal decomposition in a box asy-stiff - - -asy-soft

F.Matera, in preparation

asy-soft

asy-stiff

Density gradients derivative of Esym

Increased distillationout of equilibrium

Isospin distillation in presence of radial flow Sn112 + Sn112 Sn124 + Sn124 Sn132 + Sn132E/A = 50 MeV, b=2 fm

N = Σi Ni , Z = Σi Zi 3≤ Zi ≤ 10 asy-stiff - - -asy-soft

Proton/neutron repulsion:larger negative slope in the stiff case(lower symmetry energy) n-rich clusters emitted at largerenergy in n-rich systems

To access the variation of N/Z vs. E: “shifted” N/Z: N/Zs = N/Z – N/Z(E=0) Larger sensitivity to the asy-EoSis observed in the double N/Zs ratio If N/Zfin = a(N/Z +b), N/Zs not affected by secondary decay !

Different radial flows for neutrons and protonsFragmenting source with isospin gradient N/Z of fragments vs. Ekin !

Dou

ble

ratio

s

Central collisions

pn

r

arXiv:0707.3416

Conclusions and Perspectives -I-

Reactions with exotic beams at intermediate energy are very important for the study of fundamental properties of nuclear matter:

The “elusive” symmetry energy behaviour far from normal density

Phase diagram of finite nuclei and Phase transitions

Good observables have been proposed:

Imbalance ratio, neck neutron enrichment, isotopic content of pre-equilibrium emission (pt dependence), differential flows, isoscaling, isospin distillation, N/Z vs. Ekin.Isospin effects are enhanced by increasing the system asymmetry.

Need to enlarge the systematics of data (and calculations) to validate the current interpretation and the extraction of Esym (consensus on Esym~(ρ/ρ0) with γ~0.7-1 at low density)Still large uncertainty at high density

It is important to disantangle isovector from isoscalar effects. Cross-check of “isoscalar” and “isovector” observables

V.Baran (NIPNE HH,Bucharest) M.Di Toro, J. Rizzo (LNS-Catania)F. Matera (Florence) M. Zielinska-Pfabe (Smith College) H.H. Wolter (Munich)

Conclusions and Perspectives -II-

γ

Isospin distillation in presence of radial flow Sn112 + Sn112 Sn124 + Sn124 Sn132 + Sn132E/A = 50 MeV, b=2 fm

N = Σi Ni , Z = Σi Zi 3≤ Zi ≤ 10 asy-stiff - - -asy-soft

Proton/neutron repulsion:larger negative slope in the stiff case(lower symmetry energy) n-rich clusters emitted at largerenergy in n-rich systems

To access the variation of N/Z vs. E: “shifted” N/Z: N/Zs = N/Z – N/Z(E=0) Larger sensitivity to the asy-EoSis observed in the double N/Zs ratio If N/Zfin = a(N/Z +b), N/Zs not affected by secondary decay !

Different radial flows for neutrons and protonsFragmenting source with isospin gradient N/Z of fragments vs. Ekin !

Dou

ble

ratio

s

Central collisions

pn

r

Transverse flow of light clusters: 3H vs. 3He

m*n>m*p m*n<m*p

129Xe+124Sn, 100AMeV

124Xe+112Sn, 100AMeV

Larger 3He flow (triangles) Coulomb effects

Larger differencefor m*n>m*p

Triton/Helium transverse flow ratio:smaller for m*n>m*p

Good sensitivity to the mass splitting

dppddp )sin(,,Set of coordinates

)sin( p = 260 MeV/c, Δp = 10 MeV/c,

t = 0 fm/c t = 100 fm/c

)cos(3

23

p

V

The variance of the distribution function

p = 190 MeV/c Δθ = 30°

spherical coordinates fit the Fermi sphere allow large volumes

Clouds position

Best volume: p = 190 MeV/c, θ = 20°

12.0)(2 Ff E

DEVIATIONS FROM VIOLA SYSTEMATICS

r - ratio of the observed PLF-IMF relative velocity to the corresponding Coulomb velocity;

r1- the same ratio for the pair TLF-IMF

The IMF is weakly correlated with both PLF and TLF

Wilczynski-2 plot !

124Sn + 64Ni 35 AMeV

v_z (c)

v_x

(c)

Distribution after secondary decay (SIMON)

Sn124 + Sn124, E/A = 50 MeV/A, b = 6 fm

CM Vz-Vx CORRELATIONS

v_par

58Fe+58Fe vs. 58Ni+58Ni b=4fm 47AMeV:Freeze-out Asymmetry distributions

Fe

Ni

Fe Ni

White circles: asy-stiffBlack circles: asy-soft

Asy-soft: small isospin migration

Fe: fast neutron emission

Ni: fast proton emission

Angular distributions: alignment characteristicsAngular distributions: alignment characteristics

plane is the angle, projected into the reaction plane, between the direction defined by the relative velocity of the CM of the system PLF-IMF to TLF and the direction defined by the relative velocity of PLF to IMF

Out-of-plane angular distributions for the “dynamical” (gate 2) and “statistical” (gate 1) components: these last are more concentrated in the reaction plane.

Dynamical Isoscaling

Z=1

Z=7

primary

final

yieldionlightSn

Sn112

124

A

ZNR

AfZNY

12221

2

2

2ln

)(exp)(),(

not very sensitive to Esym ?

124Sn Carbon isotopes (primary)

AAsy-soft

Asy-stiffT.X.Liu et al.

PRC 2004

50 AMeV

(central coll.)

I = Iin + c(Esym, tcontact) (Iav – Iin), Iav = (I124 + I112)/2

RP = 1 – c ; RT = c - 1

112112T

124124T

112112T

124124T

MT

T112112P

124124P

112112P

124124P

MP

PII

III2R;

II

III2R

Imbalance ratios

If:

then:

50 MeV/A 35 MeV/A

• Larger isospin equilibration with MI

(larger tcontact ? ) • Larger isospin equilibration with asy-soft

(larger Esym)• More dissipative dynamics at 35 MeV/A

124Sn + 64Ni 35 AMeV ternary events

N/Z vs. Alignement Correlation in semi-peripheral collisions

Experiment Transp. Simulations (124/64)

Chimera data: see E.De Filippo, P.Russotto NN2006 Contr., Rio

Asystiff

Asysoft

V.Baran, Aug.06

Asystiff: more isospin migration to the neck fragments

Histogram: no selection

E.De Filippo et al. , PRC71(2005)

φ

vtra

Au+Au 250 AMeV, b=7 fm

Z=1 dataM3 centrality6<b<7.5fm

Difference of n/p flows

Larger effects at high momenta

Triton vs. 3He Flows?

**pn mm

Mass splitting: Transverse Flow Difference

MSU/RIA05, nucl-th/0505013 , AIP Conf.Proc.791 (2005) 70