1. XII Convegno su Problemi di Fisica Nucleare Teorica
Nuclear Matter and Nuclear Dynamics
Maria Colonna
Laboratori Nazionali del Sud (Catania)
XII Convegno su Problemi di Fisica Nucleare Teorica
Cortona, 8-10 Ottobre 2008

2. The EOS of symmetric and neutron matter from many-body theories:
the energy functional is calculated from the bare nucleon-nucleon interaction
Information on Esym behavior from Heavy Ion Collisions
Transport theories
High density EOS: implications on the structure of neutron stars
Transition to the QGP ? Role of isospin

3. Li, Lombardo, Schulze, Zuo, PRC77(08)
BBG calculations with two- and three-body forces
The energy functional is calculated from the bare nucleon-nucleon interaction
Microscopic three-body force(TBF) exchange diagrams on the basis of mesons, incorporating Δ, Roper,
nucleon-antinucleon excitations
TBF consistent with the underlying two-nucleon
One Boson Exchange potential
Results for EOS and symmetry energy
Bonn B
Nijmejen potential
Argonne v18 potential
phenomenological
Urbana type TBF
EOS symm. matter
Phenomenological Urbana type TBF
Similar EOS
v18
Stiffer EOS with TBF
Bonn B
Constraints on pressure from
nuclear flow data analysis
Baldo,Shaban, PLB661(08)
The overall effect of the same TBF
on the EOS can be different according
to the two-body force adopted
Li, Lombardo, Schulze, Zuo, PRC77(08)
Li, Lombardo, Schulze, Zuo, PRC 2008

4. Astrophysical problems
EOS of Symmetric and Neutron Matter
Symmetric Matter | Symmetry Energy | Neutron Matter
asy-soft
asy-stiff
BOB
NLρδ
Urbana
NLρ
DD-F
Effective parameterizations
of symmetry energy
Slope at normal density:
Isospin transport at Fermi energies
Transport codes
Nuclear Dynamics
Astrophysical problems
Dirac-Brueckner
RMF
Density-Dependent couplings
Constraints from compact stars & heavy ion data
T.Klaehn et al. PRC 74 (2006)
AFDMC
S.Gandolfi et al. , PRL98(2007)102503

5. Extracting information on the symmetry energy
from terrestrial lab.s
Nuclear Dynamics
Transport equations
Fermi energies, MeV/A (below and around normal density):
GDR
Charge equilibration
Fragmentation in exotic systems
Intermediate energies, GeV/A (above normal density):
Meson production (pions, kaons)
Collective response (flows)
Phys. Rep. 389 (2004)
Phys.Rep.410(2005)335
High density behavior Neutron stars

6. Semi-classical approach to the many-body problem
Vlasov
Boltzmann
Langevin
Time evolution of the one-body distribution function
Vlasov
Boltzmann
Langevin
Vlasov: mean field
Boltzmann: average collision term
Loss term
Langevin: random walk in phase-space
D(p,p’,r)
D(p,p’,r) w
Ensemble average
Fluctuation variance:
σ2f = <δfδf>
SMF model : fluctuations projected onto ordinary space density fluctuations δρ

7. Collective excitations
Charge equilibration

8. Relativistic nuclear excitation of GDR
in the target in semi-peripheral collisions
Dasso,Gallardo,Lanza,Sofia, NPA801(2008)129
Equations of motion for n and p centroids
obtained from Einstein’s set
- Restoring force
- Coulomb + nuclear excitation (Wood-Saxon)
Zrel = zn – zp
Xrel = xn - xp
(neutron skin)
Larger amplitude due to nuclear field
one-phonon
two-phonon
P(b): probability for a given reaction channel
T(b): attenuation factor due to depopulation of reaction channels

9. Pre-equilibrium Dipole Radiation
Charge Equilibration Dynamics:
Stochastic → Diffusion
vs.
Collective → Dipole Oscillations of the Di-nuclear System Fusion Dynamics
Initial Dipole
D(t) : bremss. dipole radiation CN: stat. GDR
36Ar + 96Zr
40Ar + 92Zr
- Isovector Restoring Force
- Neutron emission
Neck Dynamics (Mass Asymmetry)
Anisotropy
Cooling on the way to Fusion
Symmetry energy
below saturation
Experimental evidence of the extra-yield LNS data
B.Martin et al., PLB 664 (2008) 47

10. Larger restoring force with asy-soft larger strength !
Isospin gradients: Pre-equilibrium dipole emission
SPIRALS → Collective Oscillations!
TDHF: C.Simenel, Ph.Chomaz, G.de France
132Sn + 58Ni
124Sn + 58Ni
Bremsstrahlung:
Quantitative estimations
V.Baran, D.M.Brink, M.Colonna, M.Di Toro, PRL.87(2001)
arXiv:
Larger restoring force with asy-soft larger strength !

11. Imbalance ratios ISOSPIN DIFFUSION AT FERMI ENERGIES
124Sn Sn at 50 AMeV
SMF - transport model
calculations
b=8fm
Imbalance ratios
Time
L Sn Sn
H Sn Sn
M Sn Sn
experimental data (B. Tsang et al. PRL 92 (2004) )
x = β = (N-Z)/A
M Sn Sn
H Sn Sn
L Sn Sn
Several isoscalar
interactions
τ symmetry energy Esym
Smaller R for larger Esym
tcontact energy dissipation
Kinetic energy loss
Rizzo, Colonna, Baran, Di Toro, Pfabe, Wolter, PRC72(2005) and
J.Rizzo et al. NPA806 (2008) 79

12. Unstable dynamics Liquid-gas phase transition
Fragmentation in exotic systems

13. Stochastic mean field (SMF) calculations
(fluctuations projected on ordinary space)
b = 4 fm
b = 6 fm
Central collisions
Ni + Au, E/A = 45 MeV/A
Sn124 + Sn124, E/A = 50 MeV/A

14. asy-stiff - - -asy-soft
Isospin-dependent liquid-gas phase transition
Isospin distillation: the liquid phase is more symmetric than the gas phase
Density gradients derivative of Esym
asy-soft
asy-stiff
asy-stiff asy-soft
arXiv:
Spinodal decomposition in a box (quasi-analytical calculations)
Non-homogeneous
density
N/Z and variance decrease
in low-density domains
Isospin “tuning”
β = 0.2
asy-soft
β = 0.1
β = 0.1
asy-stiff
β = 0.2
Correlations of N/Z vs. Ekin
Cluster density
Colonna & Matera, PRC77 (08)
arXiv:

15. Isospin migration in neck fragmentation
ρ1 < ρ2
Asymmetry flux
Transfer of asymmetry from PLF and TLF to
the low density neck region
Effect related to the derivative of the symmetry
energy with respect to density
Experimental evidence of
n-enrichment of the neck:
Correlations between N/Z
and deviation from Viola
systematics
LNS data – CHIMERA coll.
Vrel/VViola (IMF/PLF)
(IMF/TLF)
b = 6 fm, 50 AMeV
Density gradients derivative of Esym
PLF, TLF
neck
emitted
nucleons
asy-stiff
asy-soft
E.De Filippo et al.,
PRC71, (2005)
E.De Filippo et al. NUFRA 2007
Sn112
+ Sn112
Sn124
+ Sn124
Larger derivative with asy-stiff
larger isospin migration effects
J.Rizzo et al. NPA806 (2008) 79

16. Reactions at intermediate energies:
Information on high density behavior
of Esym

17. Relativistic structure also
Quantum Hadrodynamics (QHD) → Relativistic Transport Equation (RMF)
NN scattering nuclear interaction from meson exchange:
main channels (plus correlations)
s(0+,0) w(1-,0) d(0+,1) r(1-,1)
OBE
Scalar
Vector
Scalar
Vector
Isoscalar
Isovector
Nuclear interaction by Effective Field Theory
as a covariant Density Functional Approach
Attraction & Repulsion
Saturation
Relativistic structure also
in isospin space !
Esym= kin. + (r-vector) – ( d-scalar)

18. RBUU transport equation
Wigner transform ∩ Dirac + Fields Equation
Relativistic Vlasov Equation
+ Collision Term…
Vector field
Scalar field
drift
mean field
Non-relativistic Boltzmann-Nordheim-Vlasov
“Lorentz Force”→ Vector Fields
mean-field + pure relativistic term
Collision term:
Self-Energy contributions to the inelastic channels!

19. Effects on particle production
Au+Au central: π and K yield ratios vs. beam energy
NLρδ
NLρ
RMF Symmetry Energy: the δ -mechanism NL
Effects on particle production
132Sn+124Sn
Kaons:
~15% difference between
DDF and NLρδ
Pions: large effects at lower energies
Inclusive multiplicities
G.Ferini et al.,PRL 97 (2006)

20. Isospin Collective (elliptic) flow Out-of-plane Differential flows
= full out
V = spherical
= full in
High pT selection
Isospin
Differential flows
V.Giordano, Diploma Thesis
m*n<m*p : larger neutron squeeze out
at mid-rapidity
B-A Li et al. PRL2002
Measure of effective masses in high density – highly asymmetric matter !

21. Neutron stars as laboratories for the study
of dense matter

22. Density dependent quark mass
Facts about Neutron Stars :
M ~ 1 to 2M0 ( M0=1.998·1033g)
R ~ 10 Km
N obs. Pulsars
P > 1.58 ms (630 Hz)
B = 108 ÷ 1013 Gauss
Peng,Li,Lombardo, PRC77 (08)
inner core:
 = 1/fm3  dNN=1 fm
hadron-to-quark transition !
Gibbs equilibrium
condition +
CDDM model
Bonn B
density and charge
conservation
Density dependent quark mass
Tolmann-Oppenheimer-Volkov equation
Conclusions:
transition to quark phase reduces the maximum mass to values similar to data
results very sensitive to the confinement parameter D
Neutron Star Mass-Radius Diagram

23. Nicotra, Baldo, Burgio, Schulze, PRD74(06)123001
Hybrid stars
Tolmann-Oppenheimer-Volkov equation
Schulze et al.
Baryonic EOS including hyperons
soft EOS
MG/M0 about 1.5 at finite T
too small masses for NS at T = 0
Metastability of hot PNS
Including quarks:
MG/M0 about 1.5
no metastable hybrid PNS
Rather low limiting masses of PNS
NJL: the onset of the pure quark phase in the
inner core marks an instability
No transition to quarks with hyperons
smaller masses than NS, no metastability
masses around 1.8
Serious problems for our understanding of
the EOS if large masses (about 2) are observed !
Nicotra, Baldo, Burgio, Schulze, PRD74(06)123001
Burgio & Plumari, PRD77(08)085022

24. EOS of low-density neutron matter
Inner crust of NS: nuclear lattice permeated by a gas of neutrons
At a given point nuclei merge and form more complicated structures
Study of EOS of pure neutron matter
QMC
M.Baldo & C. Maieron, PRC 77, (2008)
- Only s-wave matters, but the “unitary limit” is actually
never reached. Despite that the energy is ½ the kinetic energy
in a wide range of density (for unitary from QMC).
- The dominant correlation comes from the Pauli operator
- Both three hole-line and single particle potential effects are small
and essentially negligible. Three-body forces negligible.
- Scattering length and effective range determine
completely the G-matrix.
- Variational calculations are slightly above BBG.
Good agreement with QMC.
In this density range one can get the “exact” neutron matter EOS

25. Gas Liquid Plasma of Quarks and Gluons Temperature Density r/r0
Phases of Nuclear Matter
Plasma of Quarks and Gluons
Isospin ?
MeV
Collisions
Heavy
Ion
Mixed Phase
In terrestrial
Labs.?
Temperature
1: nuclei 5?
Density r/r0
Philippe Chomaz artistic view

26. T.Gaitanos, RBUU calculations
Mixed phase in
terrestrial labs ?
In a C.M. cell
Exotic matter over 10 fm/c ?
T.Gaitanos, RBUU calculations

27. Crucial role of symmetry energy
Testing deconfinement with RIB’s?
NLρ
NLρδ
GM3
B1/4 =150 MeV
1 AGeV
300 AMeV
symmetric
neutron
Quark-
Bag model
(two flavors)
Trajectories of 132Sn+124Sn, semicentral
M. Di Toro
Hadron-RMF
Symmetry energies:
rtrans onset of the mixed phase
→ decreases with asymmetry
- Large variation for hadron EOS
- Quark matter: Fermi contribution only
Signatures?
Crucial role of symmetry energy
in quark matter !
Neutron migration to the quark clusters (instead of a fast emission)
Drago,Lavagno, Di Toro, NPA775(2006)

28. QGP dynamics

29. Quark dynamics in the QGP phasex y z
RHICS discoveries: We have not just a bunch of particles, but a transient state of high energy plasma with Strong collective phenomena (elliptic flow v2) in condition similar to those 10-5 s after
the Big Bang
~15 GeV/fm3 >> ec T~ 350 MeV
(according to hydrodynamical calculations) px py
- The plasma is not a so perfect fluid …
(hydrodynamical) scaling of v2 not observed
- Importance of parton coalescence
Perform a Fourier expansion of the
momentum space particle distributions
But finite mean free path
call for a transport approach!
Parton cascade

30. v2(pT) as a measure of /s
Finite cross section calculations
corresponding to constant finite shear viscosity
(quantum limit) can reproduce experimental features
No freeze-out
No freeze-out
/s=1/4
Quantum mechanism h/s > 1/15 :
Kinetic Theory
v2/ε scaling broken, v2/<v2> scaling reproduced
what about v2 absolute value?
v2(pT) as a measure of /s
h/s  freeze-out
Open the room to need
coalescence in the region
of Quark Number Scaling
Ferini et al., [nucl-th]

31. Calculations for nuclear matter inside a box
Kinetic approach to relativistic heavy ion collisions
Ab initio partonic transport code p-p collisions
Predictions for rapidity distributions at LHC
Total cross section
…with the possibility to include an LQCD inspired mean-field based on the Bag model
Calculations for nuclear matter inside a box

32. Conclusions and Perspectives
Reactions with exotic beams at intermediate energy are
important for the study of fundamental properties of
nuclear matter:
The “elusive” symmetry energy behavior far from normal density
(consensus on Esym~(ρ/ρ0) with γ~ at low density)
Evidences from Giant Monopole Resonance in Sn isotopes
T.Li et al, PRL99(2007)162503
Still large uncertainties at high density
Cross-check with the predictions of BBG theory
High density behavior neutron stars
Transition to the quark phase ? Role of isospin to be investigated
Quark dynamics in the QGP phase, collective flows and
hadronization mechanisms in UrHIC γ

33. Rotation on the Reaction Plane of the Emitting Dinuclear System
All probed
Rotating angles Φi Φf
Dynamical-dipole emission
Charge equilibrium
Beam Axis
θγ : photon angle vs beam axis
Average over reaction planes:
ΔΦ=2 → x=0 → a2=-1/4 : Statistical result,
Collective Prolate on the Reaction Plane
ΔΦ=0 → Φi =Φf = Φ0
No rotation: Φ0=0 → sin2θγ pure dipole
36Ar+96Zr vs. 40Ar+92Zr: 16AMeV Fusion events: same CN selection
Angular distribution of the
extra-yield (prompt dipole):
anisotropy !
Simulations
Martin et al.
Accurate Angular Distrib. Measure:
Dipole Clock!

34. asy-stiff - - -asy-soft
Isospin distillation in presence of radial flow
Central collisions
Sn112 + Sn112
Sn124 + Sn124
Sn132 + Sn132
E/A = 50 MeV, b=2 fm p n
Different radial flows for neutrons and protons
Fragmenting source with isospin gradient
N/Z of fragments vs. Ekin ! r
N = Σi Ni , Z = Σi Zi
3≤ Zi ≤ 10
asy-stiff asy-soft
Double ratios (DR)
DR = (N/Z)2 / (N/Z)1
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:

35. Conclusions: optimistic?
Chimera-LAND
at GSI ?
Samurai Int. Coll.
at RIKEN?
Exotic Beams at FAIR?
Last page (252!) of the review
“Recent Progress and New Challenges in Isospin Physics with HIC”
Bao-An Li, Lie-Wen Chen, Che Ming Ko
ArXiv: , 22 Apr 2008 (Phys. Rep. 464 (2008) )

36. Conclusions and Perspectives -II-
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~ 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 γ
V.Baran (NIPNE HH,Bucharest) M.Di Toro, J. Rizzo (LNS-Catania)
F. Matera (Florence) M. Zielinska-Pfabe (Smith College)
H.H. Wolter (Munich)

37. asy-stiff - - -asy-soft
Isospin distillation in presence of radial flow
Central collisions
Sn112 + Sn112
Sn124 + Sn124
Sn132 + Sn132
E/A = 50 MeV, b=2 fm p n
Different radial flows for neutrons and protons
Fragmenting source with isospin gradient
N/Z of fragments vs. Ekin ! r
N = Σi Ni , Z = Σi Zi
3≤ Zi ≤ 10
asy-stiff asy-soft
Double ratios
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 !
Proton/neutron repulsion:
larger negative slope in the stiff case
(lower symmetry energy)
n-rich clusters emitted at larger
energy in n-rich systems

38. Transverse flow of light clusters: 3H vs. 3He
129Xe+124Sn, 100AMeV
124Xe+112Sn, 100AMeV
Larger 3He flow (triangles)
Coulomb effects
Larger difference
for m*n>m*p
m*n>m*p
m*n<m*p
Triton/Helium transverse flow ratio:
smaller for m*n>m*p
Good sensitivity to the mass splitting

39. The variance of the distribution function
spherical coordinates
fit the Fermi sphere
allow large volumes
Best volume: p = 190 MeV/c, θ = 20°
p = 190 MeV/c
Δθ = 30°
Clouds position
Set of coordinates
t = 0 fm/c
t = 100 fm/c
p = 260 MeV/c, Δp = 10 MeV/c,

40. r - ratio of the observed PLF-IMF relative velocity to
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 !

41. v_par CM Vz-Vx CORRELATIONS Sn124 + Sn124, E/A = 50 MeV/A, b = 6 fm
Distribution after secondary
decay (SIMON)
v_z (c)

42. 58Fe+58Fe vs. 58Ni+58Ni b=4fm 47AMeV:
Freeze-out Asymmetry distributions Fe Ni Ni Fe
White circles: asy-stiff
Black circles: asy-soft
Fe: fast neutron emission
Ni: fast proton emission
Asy-soft: small isospin migration

43. Details of SMF model ρ Fragment Recognition T gas liquid
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
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 ρ 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, (2007)
Rizzo, Colonna, Ono, PRC 76, (2007)

44. 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 PLF-IMF to TLF and the direction defined by the relative velocity of PLF to IMF

45. not very sensitive to Esym ?
Dynamical Isoscaling
primary
Z=1
Z=7
50 AMeV
(central coll.)
final
not very sensitive to Esym ?
124Sn Carbon isotopes (primary) A
Asy-soft
Asy-stiff
T.X.Liu et al.
PRC 2004

46. Imbalance ratios
If:
I = Iin + c(Esym, tcontact) (Iav – Iin), Iav = (I124 + I112)/2
then:
RP = 1 – c ; RT = c - 1
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

47. N/Z vs. Alignement Correlation in semi-peripheral collisions
vtra
124Sn + 64Ni 35 AMeV ternary events φ
Experiment
Transp. Simulations (124/64)
Histogram: no selection
Asystiff
Asysoft
Asystiff: more isospin migration to the neck fragments
Chimera data: see E.De Filippo, P.Russotto
NN2006 Contr., Rio
V.Baran, Aug.06
E.De Filippo et al. , PRC71(2005)

48. Mass splitting: Transverse Flow Difference
Au+Au 250 AMeV, b=7 fm
Difference of n/p flows
Larger effects at high momenta
Z=1 data
M3 centrality
6<b<7.5fm
Triton vs. 3He Flows?
MSU/RIA05, nucl-th/ , AIP Conf.Proc.791 (2005) 70