Probing the nuclear EOS with fragment production

Maria Colonna Laboratori Nazionali del Sud (Catania)

Fragmentation events: IMF properties in central and semi-peripheral collisions (neutron-rich systems)• Kinematical properties• Size and asymmetry (N/Z)Insight into the reaction mechanism responsible forfragment emission and isospin transport: density vs N/Z concentration gradients Dependence of the results on the asy – EOS

A new method to incorporate full fluctuations into a transport treatment (Boltzmann-Langevin theory)

Conclusions and perspectives

( , , ) ( , , ) ( , , ) ( ) ( )K r p t K r p t p p r r r t t

Instantaneous equilibrium

)1()(2 ffpf

),()()(),( cov2 ppppppp f

fWWfdt

df

Ensemble average

Langevin: randomwalk in phase-space

Semi-classical approach to the many-body problem

Time evolution of the one-body distribution function ( , , )f r p t

Boltzmann

),,()(),(),(),( tprKfKprffhprft

LangevinVlasov

Vlasov Boltzmann Langevin

)(2

)(2

fUm

pfh

i

i

Vlasov: mean field

Boltzmann: average collision term

( ) ( ) NNf i f i

dp p E E

d

3 3 32 1 2

2 1' 2'3 3 3( , ) (12 1 2 )

d p d p d pW r p f f f w

h h h Loss term

Correlation function

)( WW

dt

d

Focus on the variance σ2f = <δfδf>

D(p,p’,r)

Stochastic mean field (SMF) calculations

b = 4 fm b = 6 fm

Sn124 + Sn124, E/A = 50 MeV/A

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

Central collisions

Ni + Au, E/A = 45 MeV/A

(fluctuations projected on ordinary space)

Isospin Transport and Chemical Potentials

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 !

asy-soft

asy-stiff

E/A (ρ) = Es(ρ) + Esym(ρ)I²I=(N-Z)/A

IMF properties in central collisions

Sn124 + Sn124, E/A = 50 MeV/A, b = 2 fm

• bubble-like configuration• radial flow• correlations:Size, N/Z vsradial distance - velocity

Primary fragment propertiesX.T.Liu et al, PRC69(2004) V.Baran et al, NPA703(2002)

N/Z vs charge

1200 events for each reaction

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

N/Z vs fragment energy (1) Sn112 + Sn112 Sn124 + Sn124 Sn132 + Sn132E/A = 50 MeV, b=2 fm

“gas” phase(pre-equilibrium)

asy-stiff asy-soft

“liquid”

asy-stiff asy-soft

N = Σi Ni , Z = Σi Zi 1.64

Double ratio = (N/Z)2/(N/Z)1

Δ

Δ’

3≤ Zi ≤ 10

N/Z vs fragment energy (2)

To combine the two effects:Different slope vs. n-rich cluster emission “shifted” N/Z: N/Zs = N/Z – N/Z(E=0)Larger sensitivity to the asy-EoSis observed in the double N/Zs ratio

IMF emission

Double ratio R = (N/Z)2/(N/Z)1

Large double ratio in the asy-stiff case !(opposite to pre-equ. emission)

Famiano et al. PRL 06

Pre-equilibrium emission

IMF emission

asy-stiff

asy-soft

3≤ Zi ≤ 10

Sn124 – Sn112

(BUU calculations)

Δ

Δ’

Still arbitrary:•r-space distance

Procedure easily applicable to nuclear reactions Careful check of Pauli-blockingTake into account possible nucleon deformations

in p space

I J

J’

I’

p-space

Our improved method

• test particles i and j cells I and J

•

• Spherical search around I and J

•“clouds” rotated to final states

),,,min(, JIJItransCM nnnnndrr

x Pauli violations: average trajectory altered

x Shape of more similar to classical than to quantum case

)(2 pf

Fluctuations in phase space

Original procedure by Bauer et al.

• Cross section reduction

1 nucleon = NTEST nearest neighbours

• Phase-space distance

• < pi > and < pj > ; Δp assigned to each “cloud”

•Clouds translated to final states ( no rotation)

• Pauli blocking checked only for i and j t.p.

-Δp

Δp

TEST

NNNN N

2222 )()/()( kiFkiik rrRpppd

Bauer, Bertsch, Das Gupta, PRL58 (1987) 58

(Improve the treatment of fluctuations in p space)

r-space: periodic 3-D box, l = 26 fm;

ρ = 0.16 fm-3 (2820 nucleons); 500 test particles

p-space: Fermi-Dirac configuration , kT = 5 MeV

Illustrative results

t =0t = 100fm/c

1,)1(2

V

VN N

N

ffV

dppddp )sin(,,Set of coordinates

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

)sin( p = 260 MeV/c, Δp = 10 MeV/c,

Check of the <f> profile

Check of the fluctuation varianceon the Fermi surface

3)/30( cMeVVp

Our result: 3)/40( cMeVVcloud

In this case: ‘nucleon’ volume

Δp

Propagation of fluctuations by the unstable mean-field

Box calculations : ρ = 0.05 fm , T = 3 MeV

Fourier analysis ofthe density variance <δρδρ> :rapid growth of density fluctuations

Fragment multiplicity and charge distributions (300 nucleons)

-3

CONCLUSIONS

Study of IMF’s emitted in heavy ion collisions:

• Correlations between N/Z and velocity

• Fragments with smaller kinetic energy are more neutron rich (asy – stiff)

• Double ratios as sensitive observables to the asy-EOS

Development of a full 3D treatment of the BL theory :

•Improvement of the treatment of fluctuations in p space (thermal fluctuations)

V.Baran (NIPNE HH,Bucharest) M.Di Toro, J. Rizzo (LNS-Catania)Ph. Chomaz (GANIL, France) H.H. Wolter (Munich)

124Sn + 64Ni 35 AMeV ternary events

N/Z-IMF 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)

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

b=8f

mb=

10fm

ISOSPIN DIFFUSION AT FERMI ENERGIES124Sn + 112Sn at 50 AMeV

BNV - 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

Elab = 30 Mev/A, b = 4 fm

a) softb) stiff

Competition between reaction mechanisms: fusion vs deep-inelastic

M.Colonna et al., PRC57(1998)1410

neutron-rich

proton-rich

Comparison with INDRA data

-- stiff

-- soft

forward c.m.

forward QP

I = Iin + c(Esym) (Iav – Iin)

RP = 1 – c ; RT = c - 1

softstiff

b

20% difference in the slopebetween stiff and soft

INDRA data: Ni + Ni, Ni + Au @ 52, 74 MeV/A: N/Z vs b

E.Galichet et al., Nucl.Phys.A submittedIPN, Orsay

if

112112T

124124T

112112T

124124T

MT

T112112P

124124P

112112P

124124P

MP

PII

III2R;

II

III2R

112112T

124124T

112112T

124124T

MT

T112112P

124124P

112112P

124124P

MP

PII

III2R;

II

III2R

H

LH LH

H LL

asy-soft

asy-stiff

Sn + Ni Elab = 10 MeV/Ab = 6,7,8 fm, t = 500 fm/c

More dissipative neck dynamics with asy-stiff !

64132

asy-stiff asy-soft

Octupole distribution

Competition between deep-inelastic and neck emission

V.Baran, Aug.06SPIRAL2 proposal

INDRA data: Ni + Ni, Ni + Au @ 52, 74 MeV/A

Isospin effects on dissipation

soft stiff

E.Galichet et al., Nucl.Phys.A submittedIPN, Orsay

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.)