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Probes of EOS in Heavy Ion Collisions : results from transport theories

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Probesof EOS in HeavyIonCollisions :

resultsfromtransporttheories

Nuclear Structure &

Astrophysical Applications

July 8-12, 2013

ECT*, TRENTO

Maria Colonna

INFN - Laboratori Nazionali del Sud (Catania)

a way to create Nuclear Matter in several conditions of ρ, T, N/Z…

Nuclear effective interaction

Equation of State (EOS)

- Self-consistent Mean Field calculations (like HF) are a powerful framework to understand the structure of medium-heavy (exotic) nuclei.

Source: F.Gulminelli

- Widely employed in the astrophysicalcontext
- (modelization of neutron stars and supernova explosion)

one-body

Mean-field

Residual interaction

TDHF

Average effect of the residual interaction

Fluctuations

Transition rate W

Interpreted in terms of

NN cross section

--Ifstatisticalfluctuationslargerthan quantum ones

Main ingredients:

Residual interaction (2-body correlations and fluctuations)

In-medium nucleon cross section

Effective interaction

(self consistent mean-field)

see BHF

Transportequationfor the one-bodydistributionfunctionf

Chomaz,Colonna, Randrup

Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro

Phys. Rep. 410, 335 (2005)

(semi-classicalanalogofWignerfunction)

Residual interaction:

Correlations, Fluctuations

k

δk

vrel

Vlasov

Boltzmann-Langevin (BL) equation

Two-body Collision Integral

2

3

4

1

(1,2) (3,4)

Stochastic Mean-Field (SMF) model

Fluctuations are projected in

coordinate space

(density profile)

Fluctuations in collision integral

Molecular Dynamics approaches (AMD, ImQMD, QMD,…)

A.Ono, Phys.Rev.C59,853(1999)

Zhang and Li, PRC74,014602(2006)

J.Aichelin, Phys.Rep.202,233(1991)

stochastic NN collisions

Collision integral fully stochastic, but approx. description of mean-field effects…

Colonna, Ono, Rizzo, Phys. Rev..C82,054613(2010)

Asystiff

Effective interactions and symmetry energy

The nuclear interaction, contained in the Hamiltonian H, is represented by

effective interactions (Skyrme, Gogny, …)

E/A (ρ) = Es(ρ) + Esym(ρ) β²

The density dependence of Esym is rather

controversial, since there exist effective interactions

leading to a variety of shapes for Esym:

β=(ρn-ρp)/ρ

Symmetry energy

around ρ0

Neutron skin

Isovector modes

Pigmy resonances

g<1 Asysoft, g>1 Asystiff

γ = L/(3S0)

γ = 0.5

γ = 1

- Investigate the sensitivity of the reaction dynamics
- to this ingredient
- Put some constraints on the effective interactions
- EOS

- Fusion, Deep-Inelastic, Chargeequilibration

- Fragmentationmechanisms at Fermi energies

- Studyof the PDR withtransporttheories

Low energy reaction mechanisms

Fusion vs Deep Inelastic

E/A = 9 MeV

Fusion or

break-up ?

Competition between reaction mechanisms: fusion vs deep-inelastic

Fusion probabilities may depend on the N/Z of the reaction partners:

- A mechanism to test the isovector part of the nuclear interaction

Important role of fluctuations

C.Rizzo et al., PRC83, 014604 (2011)

Competition between reaction mechanisms

t = 200 fm/c

t = 40

t = 100

t = 140

t = 0

36Ar + 96Zr ,

E/A = 9 MeV,

b = 6 fm

θ

β2 , β3,

E*~250 MeV, J ~70ћ

z (fm)

- Starting from t = 200-300 fm/c, solve the Langevin Equation (LE) for

selected degrees of freedom: Q (quadrupole), β3 (octupole), θ, and related velocities

Break-up times

of the orderof 500-1000 fm/c !

Examples of trajectories

Extract the fusion

cross section

soft

stiff

Break-up

configurations

l (ћ)

Shvedov, Colonna, Di Toro, PRC 81, 054605 (2010)

Chargeequilibration in fusion and D.I. collisions

Relative motion of neutron and proton centers of mass

TDHF calculations

If N1/Z1≠ N2/Z2

Simenel et al,

PRC 76, 024609 (2007)

SMF simulations

A1

A2

132Sn + 58Ni , D0 = 45 fm

E/A = 10 MeV

40Ca + 100Mo

Initial Dipole

D(t) : bremss. dipole radiationCN: stat. GDR

C.Rizzo et al., PRC 83,

014604 (2011)

40Ar + 92Zr

Dynamicaldipoleemission

Bremsstrahlung:

Quantitative estimation

V.Baran, D.M.Brink, M.Colonna, M.Di Toro, PRL.87 (2001)

Experimental evidence of the extra-yield (LNS data)

B.Martin et al., PLB 664 (2008) 47

Asysoft

larger symmetry energy,

larger effect (30 %)

stiff

soft

soft

stiff

C.Rizzo et al., PRC83, 014604 (2011)

see also A.Corsi et al., PLB 679, 197 (2009), LNL experiments

D.Pierroutsakou et al., PRC80, 024612 (2009)

2(TLF)

Ediss / Ecm

Chargeequilibration in D.I. collisions at ~ 30 MeV/A

low energy

Full N/Z equilibrationnotreached:

Degreeofeq. ruledbycontacttime and symmetryenergy

--Chargeequilibration: fromdipoleoscillationstodiffusion

Overdamped dipole oscillation

D(t)

Observable:

(N/Z)CP = N/Z of light chargeparticles

emittedby PLF

as a functionof

Dissipatedenergy:

(impact parameter, contacttime)

τd Esym

t

INDRA data, Galichet et al., PRC (2010)

“Isospindiffusion” in D.I. collisions

Mass(A) ~ Mass(B) ; N/Z(A) = N/Z(B)

(TLF)

A dominance

Isospin transport ratio R :

+1

B

(PLF)

mixing

0

A

-1

B. Tsang et al., PRL 102 (2009)

B dominance

-- AA and BB refer to two symmetric reactions between n-rich and n-poor nuclei

AB to the mixed reaction

-- X is an observable related to the N/Z of PLF (or TLF)

X = N/ZPLF

X = Y(7Li)/ Y(7Be)

stiff

soft

- More centralcollisions:
- largercontacttime
- more dissipation, smaller R
- GoodsensitivitytoAsy-EoS

ComparisonbetweenImQMDcalculations

and MSU data

SMF calculations,

124Sn +112Sn, 50 AMeV

B. Tsang et al., PRL 102 (2009)

J. Rizzo et al., NPA(2008)

“Exotic” fragments in reactions

at Fermi energies

Fragmentationmechanisms

at Fermi energies

TLF

Y.Zhang et al., PRC(2011)

dynamicalemission

IMF

124Sn + 64Ni 35 AMeV:

4π CHIMERA detector

- Fragmentemission at mid-rapidity
- (neckemission)
- neutron-enrichmentof the neckregion

Charge Z

E. De Filippo et al., PRC(2012)

PLF, TLF

neck

emitted

nucleons

asy-soft

asy-stiff

Sn112

+ Sn112

Sn124

+ Sn124

Isospin migration in neck fragmentation

- Transfer of asymmetry from PLF and TLF to
- the low density neck region: neutron enrichment
- of the neck region
- Effect related to the derivative of the symmetry
- energy with respect to density

Asymmetry flux

ρIMF < ρPLF(TLF)

stiff - full

Density gradients derivative of Esym

J.Rizzo et al.

NPA806 (2008) 79

Larger derivative withasy-stiff

largerisospinmigrationeffects

The asymmetry of the neck is larger than

the asymmetry of PLF (TLF) in the stiff case

Properties of ‘dynamically emitted’ (DE) fragments

Charge distribution

Parallel velocity distribution

Good reproduction of overall dynamics

The N/Z content of IMF’s in better reproduced

by asystiff (L =75 MeV)

DE

124Sn + 64Ni 35 AMeV

SE

E. De Filippo et al., PRC(2012)

The IsovectorDipoleResponse (DR) in neutron-rich nuclei

Pygmy dipole strength

- Giant DR
- Pygmy DR

X neutrons – protons

Xc core neutrons-protons

Y excess neutrons - core

Klimkiewicz et al.

X.Roca-Maza et al., PRC 85(2012)

Isovector dipole response

GDR

Neutron center

of mass

Proton center

of mass

PDR

- Pygmy-like initial conditions (Y)

Baran et al.

X

Y

X c

Neutronskin and core are coupled

The low- energyresponseisnot sensitive to

the asy-stiffness

-- soft

--stiff

--superstiff

see M.Urban, PRC85, 034322 (2012)

25.0

GDR-like initial conditions (X)

Fourier transform of D

Strength of the Dipole Response

E (MeV)

PDR is isoscalar-like frequency not dependent on Esym

GDR is isovector-like dependent on Esym

The strength in the PDR region depends on the asy-stiffness

(increases with L)

Larger L larger strength , but also

Larger L larger neutron skin

2.7 % soft

4.4 % stiff

4.5 % superstiff

see A.Carbone et al., PRC 81 (2010)

PDR energy and strength (a systematicstudy)

superstiff

stiff

soft

Neutronskinextensionwith soft, stiff

and superstiff

The energycentroidof the PDR as a functionof

the mass forSnisotopes, 48Ca, 68Ni, 86Kr, 208Pb

~ 41 A-1/3

superstiff

stiff

Correlationbetween the EWSR

exhaustedby the PDR

and the neutronskinextension

soft

V.Baran et al., arXiv:1306.4969

Asystiff

“Summary” of Esym constraints

- Reduce experimental error bars
- Perform more complete analyses
- Improve theoretical description
- (overcome problems related to
- model dependence)

Isospin diffusion

Galichet 2010

De Filippo 2012

A.Carbone et al., PRC 81 (2010)

V.Baran (NIPNE HH,Bucharest)

M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania)

M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich)

E.Galichet, P.Napolitani (IPN, Orsay)

Heavy Ion Collisions (HIC) allow one to explore the behavior of nuclear matter under several conditions of density, temperature,

spin, isospin, …

HIC from low to Fermi energies (~10-60 MeV/A) are a way to probe the density domain just around and below normal density.

The reaction dynamics is largely affected by surface effects, at the

borderline with nuclear structure.

Increasing the beam energy, high density regions become accessible.

Varying the N/Z of the colliding nuclei (up to exotic systems) , it becomes possible to test the isovector part of the nuclear interaction

(symmetry energy) around and below normal density.

M.B.Tsang et al., PRC (2012)

A.Carbone et al., PRC 81 (2010)

V.Baran (NIPNE HH,Bucharest)

M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania)

M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich)

E.Galichet, P.Napolitani (IPN, Orsay)

- open points higher than full points

(n-rich mid-rapidity particles)

- Isospin equilibration reached for

Ediss/Ecm = 0.7-0.8 ?

(open and full dots converge)

- Data fall between the two calculations

SMF transport calculations:

N/Z of the PLF (Quasi-Projectile)

Squares: soft Stars: stiff

After statistical decay :

N = Σi Ni ,Z = Σi Zi

Charged particles: Z=1-4

Comparison with data

-- stiff (γ=1) + SIMON

forward PLF

-- soft(γ=0.6) + SIMON

forward n-n c.m.

PLF

PLF

CP

CP

N/ZCP

PLF

PLF

CP

CP

Calculations:- N/Z increases with the centrality of

collision for the two systems and energies

(For Ni + Ni pre-equilibrium effects)

- In Ni + Au systems more isospin diffusion for asy-soft (as expected)
- - (N/Z)CP linearly correlated to (N/Z)QP

Interpretation in termsofisovector-like and

- isoscalar-likemodes in asymmetricsystems

Energy-density

variation

Curvature matrixNormalmodes

stiff

soft

Isoscalar-like

Isovector-like

α =450

for symmetric matter

Zero temperature

αincreaseswith L strength in the PDR region (isoscalar-like mode)

increaseswith L

Fragment isotopic distribution and symmetry energy

Fragmentation can beassociatedwithmean-fieldinstabilities

(growthofunstablecollectivemodes)

Oscillationsof the total (isoscalar-like density) fragmentformation

and average Z/A (seeδρn /δρp)

Oscillationsof the isovector density (ρn - ρp )isotopicvariance

and distributions (Y2 / Y1 , isoscaling)

Studyofisovectorfluctuations, link withsymmetryenergy in fragmentation

λ = 2π/k

stiff

soft

At equilibrium, according to the fluctuation-dissipation theorem

Fv coincides with the free symmetry energy

at the considered density

Isovector density ρv =ρn –ρp

What does really happen in fragmentation ?

Full SMF simulations

freeze-out

T = 3 MeV, Density: ρ1 = 0.025 fm-3 , 2ρ1 , 3ρ1

I = 0.14

T

F’ ~ T / σ

Average Z/A and isovectorfluctuations

as a functionoflocal density ρ

M.Colonna, PRL,110(2013)

ρ

ρ

soft

stiff

stiff

The isospindistillation

effectgoestogetherwith

the isovectorvariance

soft

F’ follows the localequilibriumvalue !

I

high density

low density

Dissipation and fragmentation in “MD” models

ImQMDcalculations, 112Sn +112Sn, 50 AMeV

- More ‘explosive’ dynamics:
- more fragments and
- light clusters emitted
- more ‘transparency’

Y.Zhang et al., PRC(2011)

124Sn +112Sn, 50 AMeV

Isospin transport

ratio R

- Whathappenstochargeequilibration ?
- Ratherflatbehaviorwith impact parameter b:
- - Weakdependence on bofreactiondynamics ?
- Otherdissipationsources (notnucleonexchange) ?
- fluctuations, cluster emissionweaknucleonexchange

SMF = dashedlines

ImQMD = full lines

6 fm

γ = 0.5

8 fm

- Forsemi-central impact parameters:
- Largertransparencyin ImQMD(butnot so a drasticeffect)
- Othersourcesofdissipation(in additiontonucleonexchange)
- More cluster emission

- Isospintransport R around PLF rapidity:
- Good agreement in peripheralreactions
- Elsewhere the differentdynamics
- (nucleonexchangelessimportant in ImQMD)
- leadstolessiso-equilibration

γ = 2

SMF

ImQMD

γ = 0.5

What about fragment N/Z ?

Differenttrends in ImQMD and SMF!

- Correlations are introduced in the time evolution of the one-body density: ρρ +δρ
- as corrections of the mean-field trajectory
- Correlated density domains appear due to the occurrence of mean-field (spinodal)
- instabilities at low density
- Fragmentation Mechanism: spinodal decomposition
- Is it possible to reconstruct fragments and calculate their properties only from f ?

T

gas

liquid

ρ

Extract random A nucleons among test particle

distribution Coalescence procedure

Check energy and momentum conservation

A.Bonasera et al, PLB244, 169 (1990)

Liquid phase: ρ> 1/5 ρ0 Neighbouring cells are connected (coalescence procedure)

Fragment

Recognition

Fragment excitation energy evaluated by subtracting

Fermi motion (local density approx) from Kinetic energy

- Several aspects of multifragmentation in central and semi-peripheral collisions well
- reproduced by the model
- Statistical analysis of the fragmentation path
- Comparison with AMD results

Chomaz,Colonna, Randrup Phys. Rep. 389 (2004)

Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005)

Tabacaru et al., NPA764, 371 (2006)

A.H. Raduta, Colonna, Baran, Di Toro,., PRC 74,034604(2006)

iPRC76, 024602 (2007)

Rizzo, Colonna, Ono, PRC 76, 024611 (2007)

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