Probes of EOS in Heavy Ion Collisions :
results from transport theories
Nuclear Structure &
Astrophysical Applications
July 8-12, 2013
ECT*, TRENTO
Maria Colonna
INFN - Laboratori Nazionali del Sud (Catania)
Heavy Ion Collisions:
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 mediumheavy (exotic) nuclei.
Source: F.Gulminelli
 Widely employed in the astrophysical context
(modelization of neutron stars and supernova explosion)
Dynamics of many-body systems

i ρ1 (1,1' , t)  2 12 | [H, ρ 2 (t)] | 1'2
t

i ρ 2 (12,1'2' , t)  12 | [H, ρ 2 (t)] | 1'2'   O( 3 )
t
ρ2 (12,1'2')  ρ1 (1,1')ρ1 (2,2')   (12,1'2')
one-body
H  H0  v1,2
Mean-field
Residual interaction

i ρ1 (1,1' , t)  1 | [H 0 , ρ1 (t)] | 1'   K[ρ1 ]  δK[ρ 1 ,  ]
t
TDHF
K  F ( 1 , | v |2 )
K  F (v,  )
Average effect of the residual interaction
 K  0
 KK 
Fluctuations
Dynamics of many-body systems
K
f 1  f
Transition rate W
Interpreted in terms of
NN cross section
-- If statistical fluctuations larger than quantum ones
 K ( p, t )K ( p' , t ' )  C (t  t ' )
Main ingredients:
Residual interaction (2-body correlations and fluctuations)
In-medium nucleon cross section
Effective interaction
(self consistent mean-field)
see BHF
1.
Semi-classical approximation
Transport equation for the one-body distribution function
f
Chomaz,Colonna, Randrup
Phys. Rep. 389 (2004)
Baran,Colonna,Greco, Di Toro
Phys. Rep. 410, 335 (2005)
(semi-classical analog of Wigner function)
df r , p, t  f r , p, t 

  f , h  Ik
I coll
coll  f   δk
dt
t
Vlasov
Residual interaction:
Correlations, Fluctuations
Boltzmann-Langevin (BL) equation
Two-body Collision Integral
f 1  f
vrel
3
(1,2)
2
(3,4)
1
Fluctuations in collision integral
4
Stochastic Mean-Field (SMF) model
Fluctuations are projected in
coordinate space
(density profile)
2. 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)
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:
Esym  ( / 0 )g
g1 Asysoft,
around ρ0
g1 Asystiff
  0
Esym (  )  S0  L
 ...
3 0
γ = L/(3S0)
 Investigate the sensitivity of the reaction dynamics
to this ingredient
 Put some constraints on the effective interactions
EOS
β=(ρn-ρp)/ρ
Symmetry energy
Neutron skin
Isovector modes
Pigmy resonances
Asysoft
γ = 0.5
γ=1
Asystiff
 Low-energy reaction mechanisms:
Fusion, Deep-Inelastic, Charge equilibration
 Fragmentation mechanisms at Fermi energies
 Study of the PDR with transport theories
Low energy reaction mechanisms
Fusion vs Deep Inelastic
Competition between reaction mechanisms: fusion vs deep-inelastic
36Ar
+ 96Zr ,
E/A = 9 MeV
Fusion or
break-up ?
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=0
t = 40
t = 100
t = 140
t = 200 fm/c
θ
36Ar
+ 96Zr ,
E/A = 9 MeV,
b = 6 fm
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
Examples of trajectories
Break-up times
of the order of 500-1000 fm/c !
Extract the fusion
cross section
soft
stiff
Break-up
configurations
Shvedov, Colonna, Di Toro, PRC 81, 054605 (2010)
l (ћ)
Charge equilibration in fusion and D.I. collisions
A1
A2
TDHF calculations
Initial Dipole
D(t) : bremss. dipole radiation CN: stat. GDR
If N1/Z1 ≠ N2/Z2
Relative motion of neutron and proton centers of mass


NZ
1
X p (t )  X n (t )  X p ,n 
xip ,n
A
Z, N
1
DK (t )  Pp  Pn  Pp ,nSMF
 simulations
pip ,n
Z, N
D, DK   i
D(t ) 
Simenel et al,
PRC 76, 024609 (2007)
132Sn
+ 58Ni , D0 = 45 fm
E/A = 10 MeV
40Ca
C.Rizzo et al., PRC 83,
014604 (2011)
+ 100Mo
Dynamical dipole emission
B.Martin et al., PLB 664 (2008) 47
Bremsstrahlung:
Quantitative estimation
36Ar
2
2
dP
2e
 NZ  ''


 D ( )
3
dEg 3c Eg  A 
2
+ 96Zr
40Ar
+ 92Zr
V.Baran, D.M.Brink, M.Colonna, M.Di Toro, PRL.87 (2001)
Experimental evidence of the extra-yield (LNS data)
stiff
soft
Asysoft
larger symmetry energy,
larger effect (30 %)
soft
C.Rizzo et al., PRC83, 014604 (2011)
see also
A.Corsi et al., PLB 679, 197 (2009),
LNL experiments
stiff
D.Pierroutsakou et al., PRC80, 024612 (2009)
Charge equilibration in D.I. collisions at ~ 30 MeV/A
low energy
-- Charge equilibration: from dipole oscillations to diffusion
Overdamped dipole oscillation
-t/ d
D(t)
τd
D(t) D(0)e
Esym
t
1(PLF)
Full N/Z equilibration not reached:
Degree of eq. ruled by contact time and symmetry energy
2(TLF)
Observable:
(N/Z)CP = N/Z of light charge particles
emitted by PLF
as a function of
Ediss  Ecm - Ekin (PLF T LF)
Dissipated energy:
(impact parameter, contact time)
Ediss / Ecm
INDRA data, Galichet et al., PRC (2010)
“Isospin diffusion” in D.I. collisions
Mass(A) ~ Mass(B) ; N/Z(A) = N/Z(B)
(TLF)
Isospin transport ratio R :
B. Tsang et al., PRL 102 (2009)
A dominance
mixing
2 X AB  X AA  X BB
R
X AA  X BB
B dominance
B
+1
0
-1
(PLF)
A
-- 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
stiff
X = Y(7Li)/ Y(7Be)
soft
 More central collisions:
larger contact time
more dissipation, smaller R
 Good sensitivity to Asy-EoS
Comparison between
ImQMD calculations
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
Fragmentation mechanisms
at Fermi energies
Y.Zhang et al., PRC(2011)
PLF
dynamical emission
TLF
IMF
VREL / VVIOLA (1,3)
124Sn
+ 64Ni 35 AMeV:
4π CHIMERA detector
 Fragment emission at mid-rapidity
Charge Z
(neck emission)
 neutron-enrichment of the neck region
E. De Filippo et al., PRC(2012)
Isospin migration in neck fragmentation
 Transfer of asymmetry from PLF and TLF to
the low density neck region: neutron enrichment
of the neck region
Asymmetry flux
 Effect related to the derivative of the symmetry
energy with respect to density
ρIMF < ρPLF(TLF)
Density gradients
derivative of Esym
b = 6 fm, 50 AMeV
stiff - full
J.Rizzo et al.
NPA806 (2008) 79
PLF, TLF
neck
emitted
nucleons
asy-soft
asy-stiff
Larger derivative with asy-stiff
larger isospin migration effects
Sn112
Sn124
+ Sn112
+ Sn124
The asymmetry of the neck is larger than
the asymmetry of PLF (TLF) in the stiff case
Comparison with Chimera data
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
SE
124Sn
+ 64Ni 35 AMeV
E. De Filippo et al., PRC(2012)
“Exotic” collective excitations in Nuclei
The Isovector Dipole Response (DR) in neutron-rich nuclei
Isovector dipole response
Pygmy dipole strength
 Giant DR
GDR
 Pygmy DR
PDR
Klimkiewicz et al.
X.Roca-Maza et al., PRC 85(2012)
X
neutrons – protons
Xc core neutrons-protons
Y
excess neutrons - core
Neutron center
of mass
Proton center
of mass
132Sn
• Pygmy-like initial conditions (Y)
X
Y
Xc
Neutron skin and core are coupled
The low- energy response is not sensitive to
the asy-stiffness
25.0
Baran et al.
see M.Urban, PRC85, 034322 (2012)
-- soft
-- stiff
-- superstiff
• GDR-like initial conditions (X)
Fourier transform of D
Strength of the Dipole Response
E (MeV)
PDR is isoscalar-like
GDR is isovector-like
frequency not dependent on Esym
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
see A.Carbone et al., PRC 81 (2010)
2.7 % soft
4.4 % stiff
4.5 % superstiff
 PDR energy and strength (a systematic study)
superstiff
stiff
soft
The energy centroid of the PDR as a function of
the mass for Sn isotopes, 48Ca, 68Ni, 86Kr, 208Pb
~ 41 A-1/3
Correlation between the EWSR
exhausted by the PDR
and the neutron skin extension
V.Baran et al., arXiv:1306.4969
Neutron skin extension with soft, stiff
and superstiff
superstiff
stiff
soft
“Summary” of Esym constraints
Asysoft
Asystiff
 Reduce experimental error bars
Isospin diffusion
Galichet 2010
De Filippo 2012
 Perform more complete analyses
 Improve theoretical description
(overcome problems related to
model dependence)
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.
“Summary” of Esym constraints
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)
PLF
PLF
CP
CP
SMF transport
calculations:
Data:
N/Z of
the PLF
(Quasi-Projectile)
 open
points
higher
than full points
Squares:
soft Stars:
stiff
(n-rich
mid-rapidity
particles)
 IsospinAfter
equilibration
statisticalreached
decay : for
Ediss/Ecm = 0.7-0.8 ?
= Σfull
i Nidots
, Z =converge)
Σi Zi
(open N
and
N/ZCP
 Data fall
between
the
two
calculations
Charged particles: Z=1-4
Comparison with data
-- stiff (γ=1) + SIMON
-- soft(γ=0.6) + SIMON
forward PLF
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
forward n-n c.m.
 Interpretation in terms of isovector-like and
isoscalar-like modes in asymmetric systems
Energy-density
variation
   p  p   p2  2  p n  n p  n n  n2
Curvature matrix
Normal modes
stiff
soft
Isoscalar-like
Isovector-like
tan( ) 
 p
 n
α =450
for symmetric matter
c
c
tan(2 ) 

a  b 8I ( E  L  )
sym
3 0
Zero temperature
α increases with L
strength in the PDR region (isoscalar-like mode)
increases with L
nuclear matter in a box
Fragment isotopic distribution and symmetry energy
Fragmentation can be associated with mean-field instabilities
(growth of unstable collective modes)
Oscillations of the total (isoscalar-like density)
and average Z/A (see δρn /δρp )
λ = 2π/k
fragment formation
stiff
soft
Oscillations of the isovector density (ρn - ρp )
and distributions (Y2 / Y1 , isoscaling)
isotopic variance
Study of isovector fluctuations, link with symmetry energy in fragmentation
At equilibrium, according to the fluctuation-dissipation theorem
Isovector density ρv = ρn –ρp
Fv coincides with the free symmetry energy
at the considered density
What does really happen in fragmentation ?
Nuclear matter in a box
T
Full SMF simulations
freeze-out
T = 3 MeV, Density: ρ1 = 0.025 fm-3 , 2ρ1 , 3ρ1
I = 0.14
ρ
M.Colonna, PRL,110(2013)
ρ
F’ ~ T / σ
Average Z/A and isovector fluctuations
as a function of local density ρ
soft
stiff
stiff
soft
F’ follows the local equilibrium value !
The isospin distillation
effect goes together with
the isovector variance
high density low density
I
Dissipation and fragmentation in “MD” models
ImQMD calculations, 112Sn +112Sn, 50 AMeV
 More ‘explosive’ dynamics:
more fragments and
light clusters emitted
more ‘transparency’
Y.Zhang et al., PRC(2011)
124Sn
y  vz
+112Sn, 50 AMeV
Isospin transport
ratio R
What happens to charge equilibration ?
Rather flat behavior with impact parameter b:
- Weak dependence on b of reaction dynamics ?
- Other dissipation sources (not nucleon exchange) ?
fluctuations, cluster emission weak nucleon exchange
Comparison SMF-ImQMD
6 fm
8 fm
γ = 0.5
SMF = dashed lines
ImQMD = full lines
 For semi-central impact parameters:
Larger transparency in ImQMD (but not so a drastic effect)
Other sources of dissipation (in addition to nucleon exchange)
More cluster emission
 Isospin transport R around PLF rapidity:
γ=2
SMF
Good agreement in peripheral reactions
ImQMD
Elsewhere the different dynamics
(nucleon exchange less important in ImQMD)
leads to less iso-equilibration
What about fragment N/Z ?
Different trends in ImQMD and SMF!
γ = 0.5
Details of SMF model
• Correlations are introduced in the time evolution of the one-body density: ρ
ρ +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
T
gas
Is it possible to reconstruct fragments and calculate their properties only from f ?
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
Fragment
Neighbouring cells are connected
Recognition
(coalescence procedure)
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
Chomaz,Colonna, Randrup Phys. Rep. 389 (2004)
 Statistical analysis of the fragmentation path
 Comparison with AMD results
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)
PRC76, 024602 (2007)
Rizzo, Colonna, Ono, PRC 76, 024611 (2007)
i