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Dynamics and Thermodynamics with
Maria Colonna
Laboratori Nazionali del Sud (Catania)
What can we learn from reactions at intermediate energy
(30-100 MeV/A) with exotic beams ?
 Energy functional of asymmetric nuclear matter:
constrain the iso – EOS (symmetry energy)
Information on the behaviour of the symmetry energy at subsaturation 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
The density-dependent symmetry energy
and n-p effective mass splitting
Isospin Transport: the density dependent Esym

currents
drift
diffusion
jn  Dn   D I
I
n
j p  D p   D pI I
Self-consistent mean-field calculations
E/A (ρ) = Es(ρ) + Esym(ρ)I²
I=(N-Z)/A
  q 

Dq  
   I ,T
  q 
  q  n, p 
DqI  
 I   ,T
asy-stiff
n   p  4Esym   I
jn  j p  Esym (  )I 
Diffusion
Esym (  )

I
asy-soft
Drift
Direct Access to Value and Slope of the Symmetry Energy at ρ !
Symmetry Potentials and Effective Masses
Density dependence
124Sn
“asymmetry” I = 0.2
Momentum dependence

m U q 
 1  2

m   k k 
mq*
Lane Potentials
neutron
mn*  m*p
proton
Asy-stiff
Asy-soft
Phys.Rep.410(2005)335-466
mn*  m*p
1
(Un-Up)/2I
The density-dependent symmetry energy and
n-p effective mass splitting:
Observables
 Symmetry energy parameterizations are implemented into transport
codes (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
ISOSPIN DIFFUSION AT FERMI ENERGIES
124Sn
+
112Sn
at 50 AMeV
SMF - transport model
b=8fm
b=9 fm
b=10fm
120fm/c
100fm/c
80fm/c
contact time
Imbalance ratios
124124
112112
M
124124
112112
2I M

I

I
2
I

I

I
R P  P 124P124 112P112
; R T  T 124T124 112T112
IP
 IP
IT
 IT
asysoft eos
superasystiff eos
asy-soft EOS –
faster equilibration
experimental data
(B. Tsang et al.
PRL 92 (2004) )
Baran, Colonna, Di Toro, Pfabe, Wolter, PRC72(2005)
Imbalance ratios: isoscalar vs. isovector effects
MD, MI: isoscalar
effective forces
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 !
Isospin migration in neck fragmentation
 Transfer of asymmetry from PLF and TLF to
the low density neck region
Asymmetry flux
 Effect related to the derivative of the symmetry
energy with respect to density
ρ 1 < ρ2
Density gradients
derivative of Esym
b = 6 fm, 50 AMeV
PLF, TLF
asy-stiff
neck
asy-soft
Larger derivative with asy-stiff
larger isospin migration effects
emitted
nucleons
Sn112
+ Sn112
Sn124
+ Sn124
arXiv:0711.3761
A
Ares
Isospin exchange: βIMF / βres ratio
Neck mass A, asymmetry β + Δβ
Residues mass Ares, asymmetry β – Δβ A/Ares
Asymmetry flux
<0
minimizing symmetry energy variation
ρI < ρR
b = 6 fm, 50 AMeV
MD
MI
stiff
- - soft
Sn112 + Sn112
Sn124 + Sn124
This ratio depends only
on the symm. energy
variation around the
neck density
It should also be studied
as a function of dissipation
or observables connected to
the density
(IMF multiplicity …)
Mass splitting: N/Z of Fast Nucleon Emission
Gas asymmetry vs. p_t    0 / 8
124Sn+124Sn, 50 AMeV, b=2 fm
132Sn+124Sn,
100 AMeV, b=2 fm, y(0)0.3
asy-stiff
asy-stiff
n/p
3H/3He
• m*n>m*p
• m*n<m*p
Vs. Kinetic Energies
Light isobar (3H/3He) yields
High p_t “gas” asymmetry: Observable
very sensitive to the mass splitting and
not to the asy-stiffness
J.Rizzo et al., PRC 72 (2005)
→ Isotope Science Facility
at MSU, White Paper 2006
Collective flows
In-plane
Out-of-plane
y = rapidity
pt = transverse momentum
V1 ( y, pt )  p x / pt
X
y
Z
V2 ( y, pt ) 
V2
px2  p y2
px2  p y2
=  1 full out
= 0 spherical
= + 1 full in
V1pn  pt   V1p  pt  V1n  pt 
Differential flows
B-A Li et al. PRL2002
1 < V2 < +1
y
V2p-n ( pt )  V2p ( pt )  V2n ( pt )
 vDifferential ( y, pt ) 
 i  1(n),1( p)
1
N Z
 v ( y , p )
i i
t
Mass splitting: Elliptic Flow Difference
Au+Au 250 AMeV, b=7 fm
Z=1 data, M3 centrality, 6<b<7.5fm
129Xe+124Sn,100AMeV
124Xe+112Sn,100AMeV
mn*  m*p
m*n<m*p : larger neutron squeeze out
at mid-rapidity
m*n < m*p
m*p < m*n
MSU/RIA05, nucl-th/0505013 , AIP Conf.Proc.791 (2005) 70
Triton/He3 Transverse flow ratio
Phase transitions in finite systems
and isospin effects
Phase transitions in exotic systems: new effects
 Validate the mechanisms investigated and the conclusions
drawn from the study of symmetric matter (multifragmentation)
 New features: Instabilities in asymmetric systems
(phase diagram)
The width of
the spinodal zone
should depend on isospin
Temperature
τ = 100 fm/c τ = 50 fm/c
Level density, limiting temperature …
Density
Colonna et al., PRL2002
New features: Isospin distillation
Observables: isoscaling, fragment <N>/Z at break-up,
double ratios
Distillation in presence of radial flow
<N>/Z vs. Ekin
Isospin-dependent phase transition
Isospin distillation: the liquid phase is more symmetric than the gas phase
Density gradients
derivative of Esym
asy-stiff
Increased distillation
out of equilibrium
asy-soft
asy-stiff
- - -asy-soft
Spinodal decomposition in a box
β = 0.2
β = 0.1
Non-homogeneous
density
F.Matera, in preparation
Central collisions
 Sn112 + Sn112
 Sn124 + Sn124
 Sn132 + Sn132
E/A = 50 MeV, b=2 fm
Isospin distillation in presence of radial flow
Different radial flows for neutrons and protons
Fragmenting source with isospin gradient
N/Z of fragments vs. Ekin !
n
p
r
Double ratios
3≤ Zi ≤ 10
N = Σi Ni , Z = Σi Zi
asy-stiff
- - -asy-soft
 Proton/neutron repulsion:
larger negative slope in the stiff case
(lower symmetry energy)
 n-rich clusters emitted at larger
energy 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-EoS
is observed in the double N/Zs ratio
If N/Zfin = a(N/Z +b), N/Zs not affected by secondary
decay !
arXiv:0707.3416
Conclusions and Perspectives -IReactions 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.
Conclusions and Perspectives -II Need to enlarge the systematics of data (and calculations) to
validate the current interpretation and the extraction of Esym
γ
(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)
Central collisions
 Sn112 + Sn112
 Sn124 + Sn124
 Sn132 + Sn132
E/A = 50 MeV, b=2 fm
Isospin distillation in presence of radial flow
Different radial flows for neutrons and protons
Fragmenting source with isospin gradient
N/Z of fragments vs. Ekin !
n
p
r
Double ratios
3≤ Zi ≤ 10
N = Σi Ni , Z = Σi Zi
asy-stiff
- - -asy-soft
 Proton/neutron repulsion:
larger negative slope in the stiff case
(lower symmetry energy)
 n-rich clusters emitted at larger
energy 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-EoS
is observed in the double N/Zs ratio
If N/Zfin = a(N/Z +b), N/Zs not affected by secondary
decay !
Transverse flow of light clusters: 3H vs. 3He
129Xe+124Sn,
100AMeV
124Xe+112Sn,
100AMeV
m*n>m*p
m*n<m*p
Larger 3He flow
(triangles)
Coulomb effects
Larger difference
for m*n>m*p
Triton/Helium transverse flow ratio:
smaller for m*n>m*p
Good sensitivity to the mass splitting
The variance of the distribution function
spherical coordinates
 fit the Fermi sphere
 allow large volumes
p3
V  2
 cos( )
3
 2f (EF )  0.12
Δθ = 30°
p = 190 MeV/c
Clouds position
Best volume: p = 190 MeV/c, θ = 20°
Set of coordinates dp, pd, p sin( )d
t = 0 fm/c
t = 100 fm/c
p = 260 MeV/c, Δp = 10 MeV/c,   sin( )
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
124Sn
+ 64Ni
35 AMeV
Wilczynski-2 plot !
CM Vz-Vx CORRELATIONS
v_par
Sn124 + Sn124,
E/A = 50 MeV/A, b = 6 fm
Distribution after secondary
decay (SIMON)
v_z (c)
58Fe+58Fe vs. 58Ni+58Ni b=4fm
47AMeV:
Freeze-out Asymmetry distributions
Fe


Ni
Fe
Fe: fast neutron emission
Ni: fast proton emission
Ni
White circles: asy-stiff
Black circles: asy-soft
Asy-soft: small isospin migration
Angular distributions: alignment characteristics
Out-of-plane angular distributions
for the “dynamical” (gate 2) and
“statistical” (gate 1) components:
these last are more concentrated in
the reaction plane.
plane is the angle, projected into the reaction
plane, between the direction defined by the
relative velocity of the CM of the system PLFIMF to TLF and the direction defined by the
relative velocity of PLF to IMF
Dynamical Isoscaling
124
Sn
light ion yield
112
Sn
primary
50 AMeV
Z=1
(central coll.)
Z=7
final
 (   ) 

Y ( N , Z )  f ( A) exp  
2



2
N Z
 ln R21  2  2  1 


 A 
2
T.X.Liu et al.
PRC 2004
not very sensitive to Esym ?
124Sn
Carbon isotopes (primary)
Asy-stiff
Asy-soft
A
Imbalance ratios
124124
112112
M
124124
112112
2I M

I

I
2
I

I

I
R P  P 124P124 112P112
; R T  T 124T124 112T112
IP
 IP
IT
 IT
If:
I = Iin + c(Esym, tcontact) (Iav – Iin),
then:
RP = 1 – c ;
Iav = (I124 + I112)/2
RT = c - 1
50 MeV/A
• Larger isospin equilibration with MI
(larger tcontact ? )
• Larger isospin equilibration with asy-soft
(larger Esym)
• More dissipative dynamics at 35 MeV/A
35 MeV/A
N/Z vs. Alignement Correlation in semi-peripheral collisions
vtra
φ
124Sn
Experiment
Histogram:
no selection
+ 64Ni 35 AMeV ternary events
Transp. Simulations (124/64)
Asystiff
Asysoft
Chimera data: see E.De Filippo, P.Russotto
NN2006 Contr., Rio
E.De Filippo et al. , PRC71(2005)
Asystiff: more isospin migration to the neck fragments
V.Baran, Aug.06
Mass splitting: Transverse Flow Difference
Au+Au 250 AMeV, b=7 fm
mn*  m*p
Difference of n/p flows
Larger effects at high momenta
Triton vs. 3He Flows?
Z=1 data
M3 centrality
6<b<7.5fm
MSU/RIA05, nucl-th/0505013 , AIP Conf.Proc.791 (2005) 70