arXiv:nucl-th/0607035v1 17 Jul 2006
Isospin Dynamics in Heavy Ion Collisions
Laboratori Nazionali del Sud INFN,
Via S.Sofia 62, 95123 Catania, Italy,
and Physics-Astronomy Dept., University of Catania,
Some isospin dynamics results in heavy ion collisions from low to relativistic energies obtained
through transport approaches, largely inspired by David M.Brink, are rewieved. At very low ener-
gies, just above the Coulomb barrier, the stimulating implications of the prompt dipole radiation in
dissipative collisions of ions with large isospin asymmetries are discussed. We pass then to the very
rich phenomenology of isospin effects on heavy ion reactions at intermediate energies (few A GeV
range). We show that it can allow a “direct” study of the covariant structure of the isovector in-
teraction in the hadron medium. We work within a relativistic transport frame, beyond a cascade
picture, consistently derived from effective Lagrangians, where isospin effects are accounted for in
the mean field and collision terms. Rather sensitive observables are proposed from collective flows
(“differential” flows) and from pion/kaon production (π−/π+, K0/K+yields). For the latter point
relevant non-equilibrium effects are stressed. The possibility of the transition to a mixed hadron-
quark phase, at high baryon and isospin density, is finally suggested. Some signatures could come
from an expected “neutron trapping” effect. The importance of violent collision experiments with
radioactive beams, from few AMeV to few AGeV , is stressed.
I.INTRODUCTION: THE DAVID LEGACY
It is a great pleasure and honor to contribute to this
special day devoted to celebrate the 75th anniversary of
David Brink. After many years of acquaintance and col-
laboration I would like to state a few points I mostly
got from him: 1.) The Mean Field “moves” the world,
particularly true in nuclear dynamics, as I will show in
examples discussed here; 2.)
nuclear structure and reactions; 3.) Play always a deep
attention to the suggestions of young people; 4.) Fol-
low a “sensible” behaviour towards the political choices,
i.e. try to change something only when you can count
on something better. Here of course I will focus on the
physics part showing a series of results broadly inspired
by the David ideas.
The role of relativity in
In the last years the isospin dynamics has gained a lot
of interest, as we can see from the development of new
heavy ion facilities (radioactive beams), for the unique
possibilities of probing the isovector in medium interac-
tion far from saturation, relevant for the structure of un-
stable elements as well as for nuclear astrophysics see the
recent reviews [1, 2].
Here I will show some selected results of the mean
field transport approach in a wide energy range, from
few AMeV to few AGeV , in non-relativistic and rela-
tivistic frames. At low energies I will discuss the isospin
equilibration in dissipative collisions, fusion and deep-
inelastic, through a related observable, the Prompt Col-
lective Dipole Radiation. At high energies I will shortly
present isospin effects on collective flows, on particle pro-
duction and finally on the transition to a mixed hadron-
quark phase at high baryon density.
II.THE PROMPT DIPOLE γ-RAY EMISSION
The possibility of an entrance channel bremsstrahlung
dipole radiation due to an initial different N/Z distri-
bution was suggested at the beginning of the nineties
[3, 4], largely inspired by David discussions.
time a large debate was present on the disappearing of
Hot Giant Dipole Resonances in fusion reactions. David
was suggesting the simple argument that a GDR needs
time to be built in a hot compound nucleus, meanwhile
the system will cool down by neutron emission and the
GDR photons will show up at lower temperature. The
natural consequence suggested in  was that we would
expect a new dipole emission, in addition to the statisti-
cal one, if some pre-compound collective dipole mode is
present. After several experimental evidences, in fusion
as well as in deep-inelastic reactions [5, 6, 7, 8, 9] we have
now a good understanding of the process and stimulating
new perspectives from the use of radioactive beams.
During the charge equilibration process taking place in
the first stages of dissipative reactions between colliding
ions with different N/Z ratios, a large amplitude dipole
collective motion develops in the composite dinuclear sys-
tem, the so-called dynamical dipole mode. This collec-
tive dipole gives rise to a prompt γ-ray emission which
depends: i) on the absolute value of the intial dipole mo-
D(t = 0) =NZ
|RZ(t = 0) − RN(t = 0)| =
being RZ and RN the center of mass of protons and of
neutrons respectively, while Rpand Rtare the projectile
and target radii; ii) on the fusion/deep-inelastic dynam-
ics; iii) on the symmetry term, below saturation, that is
D(t) (solid lines) and p−space DK(t) (dashed lines, in fm−1)
and the correlation DK(t) − D(t) at incident energy of
6AMeV and 9AMeV for b = 2fm.
The time evolution of the dipole mode in r−space
acting as a restoring force.
A detailed description is obtained in a microscopic
approach based on semiclassical transport equations, of
Vlasov type, introduced in the nuclear dynamics in col-
laboration with David , where mean field and two-
body collisions are treated in a self-consistent way, see
details in . Realistic effective interactions of Skyrme
type are used. The numerical accuracy of the transport
code has been largely improved in order to have reliable
results also at low energies, just above the threshold for
fusion reactions [12, 13]. The resulting physical picture
is in good agreement with quantum Time-Dependent-
Hartree-Fock calculation . In particular we can study
in detail how a collective dipole oscillation develops in
the entrance channel .
First, during the approaching phase, the two partners,
overcoming the Coulomb barrier, still keep their own re-
sponse. Then follows a dinuclear phase where the relative
motion energy, due to the nucleon exchange, is converted
in thermal motion and in the collective energy of the
dinuclear mean field. In fact the composite system is not
fully equilibrated and manifests, as a whole, a large am-
plitude dipole collective motion. Finally thermally equi-
librated reaction products are formed, with consequent
statistical particle/radiation emissions.
We present here some results for the
(N/Z asymmetric) reaction at 6 and 9 AMeV , recently
studied vs. the “symmetric”36S +96Mo counterpart in
ref.. In Fig.1 (left columns) we draw the time evolu-
tion for b = 2fm of the dipole moment in the r-space
(solid lines), D(t) =
AX(t) and in p−space (dashed
lines), DK(t) = Π/?, where Π =
Pp(Pn) center of mass in momentum space for protons
(neutrons), is just the canonically conjugate momentum
FIG. 2: Density plots of the neck dynamics for the32S +100Mo
system at incident energy of 6AMeV and 9AMeV .
of the X coordinate, see [13, 14, 15].
hand side columns we show the corresponding correla-
tion DK(t) − D(t) in the phase space. We choose the
origin of time at the beginning of the dinuclear phase.
The nice ”spiral-correlation” clearly denotes the collec-
tive nature of the mode. From Fig.1 we note that the
”spiral-correlation”starts when the initial dipole moment
D(t = 0), the geometrical value at the touching point, is
already largely quenched. This is the reason why the
dinucleus dipole yield is not simply given by the ”static”
estimation but the reaction dynamics has a large influ-
ence on it.
A clear energy dependence of the dynamical dipole
mode is evidenced with a net increase when we pass from
6AMeV to 9AMeV . A possible explanation of this ef-
fect is due to the fact that at lower energy, just above
the Coulomb barrier, a longer transition to a dinuclear
configuration is required which hinders the isovector col-
lective response. From Fig.2 a slower dynamics of the
neck during the first 40fm/c − 60fm/c from the touch-
ing configuration is observed at 6 AMeV .
collective dipole response sets in the charge is already
partially equilibrated via random nucleon exchange.
The bremsstrahlung spectra shown in Fig.3 support
In fact from the dipole evolution given from the
Vlasov transport we can directly apply a bremsstrahlung
(”bremss”) approach  to estimate the “direct” photon
emission probability (Eγ= ?ω):
On the right
where D′′(ω) is the Fourier transform of the dipole accel-
eration D′′(t). We remark that in this way it is possible
to evaluate, in absolute values, the corresponding pre-
equilibrium photon emission. In the same Fig.3 we show
statistical GDR emissions from the final excited residue.
We see that at the higher energy the prompt emission
represents a large fraction of the total dipole radiation.
In the Table we report the present status of the Dy-
namical Dipole data, obtained from fusion reactions.. We
TABLE I: The percent increase of the intensity in the linearized γ-ray spectra at the compound nucleus GDR energy region
(the energy integration limits are given in the parenthesis), the compound nucleus excitation energy, the initial dipole moment
D(t = 0) and the initial mass asymmetry ∆ for each reaction.
D(t = 0) (fm)
1.6 ±2.0 (8,21)
FIG. 3: The bremsstrahlung spectra for the32S +100Mo sys-
tem at incident energy of 6AMeV and 9AMeV (solid line)
and the first step statistical spectrum (dashed line) for three
note the dependence of the extra strength on the in-
terplay between initial dipole moment and initial mass
asymmetry: this clearly indicates the relevance of the
We must add a couple of comments of interest for the
experimental selection of the Dynamical Dipole: i) The
centroid is always shifted to lower energies (large defor-
mation of the dinucleus); ii) A clear angular anisotropy
should be present since the prompt mode has a definite
axis of oscillation (on the reaction plane) at variance with
the statistical GDR. At higher beam energies we expect
a further decrease of the direct dipole radiation for two
main reasons both due to the increasing importance of
hard NN collisions: i) a larger fast nucleon emission that
will equilibrate the isospin before the collective dipole
starts up; ii) a larger damping of the collective mode.
This has been observed in ref. and more exps. are
Before closing I would like to note two interesting
developments for future experiments with radioactive
• The prompt dipole radiation represents a nice cool-
ing mechanism on the fusion path. It could be a
way to pass from a warm to a cold fusion in the
synthesis of heavy elements with a noticeable in-
crease of the survival probability, .
• The use of unstable neutron rich projectiles would
largely increase the effect allowing a detailed study
of the symmetry potential, below saturation, re-
sponsible of the restoring force of the dipole oscil-
III. ISOSPIN PHYSICS IN A COVARIANT
We move now to a relativistic framework for the de-
scription of the isovector part of the effective interac-
tion. I will focus then the attention on relativistic heavy
ion collisions, that provide a unique terrestrial opportu-
nity to probe the in-medium nuclear interaction in high
density and high momentum regions. An effective La-
grangian approach to the hadron interacting system is ex-
tended to the isospin degree of freedom: within the same
frame equilibrium properties (EoS, ) and transport
dynamics ([20, 21]) can be consistently derived. Within
a covariant picture of the nuclear mean field, for the de-
scription of the symmetry energy at saturation (a4 pa-
rameter of the Weizs¨ aecker mass formula) (in a sense
equivalent to the a1 parameter for the iso-scalar part),
extracted in the range from 28 to 36 MeV, there are dif-
ferent possibilities: (a) considering only the Lorentz vec-
tor ρ mesonic field, and (b) both, the vector ρ (repulsive)
and scalar δ (attractive) effective fields [22, 23, 24]. The
latter corresponds to the two strong effective ω (repul-
sive) and σ (attractive) mesonic fields of the iso-scalar
sector. We get a transparent form [2, 23]:
Once the a4 empirical value is fixed from the ρ − δ
balance, important effects at supra-normal densities ap-
pear due to the introduction of the effective δ field. In
fact the presence of an isovector scalar field is increasing
the repulsive ρ-meson contribution at high baryon densi-
ties [2, 23] via a pure relativistic mechanism, due to the
different Lorentz properties of these fields (the vector ρ
field grows with baryon density whereas the scalar δ field
is suppressed by the scalar density). Dynamical, non-
equilibrium, effects can be more sensitive to such “fine
structure” of the isovector interaction. We will see the
vector couplings give γ-boosted Lorentz forces, and so
we expect a larger isospin dependence of the high energy
nucleon propagation. Moreover, the scalar δ field natu-
rally leads to an effective (Dirac) mass splitting between
protons and neutrons [2, 22, 23, 24, 25, 26], with influ-
ence on nucleon emissions and flows [27, 28]. In order
to explore the symmetry energy at supra-normal densi-
ties one has to select signals directly emitted from the
early non-equilibrium high density stage of the heavy ion
collision. A transverse momentum analysis is important
in order to select the high density source [29, 30]. The
description of the mean field is important, since nucleons
and resonances are dressed by the self-energies. This will
directly affect the energy balance (threshold and phase
space) of the inelastic channels.
The starting point is a simple phenomenological ver-
sion of the Non-Linear (with respect to the iso-scalar,
Lorentz scalar σ field) Walecka model which corresponds
to the Hartree or Relativistic Mean Field (RMF) approx-
imation within the Quantum-Hadro-Dynamics . Ac-
cording to this model the baryons (protons and neutrons)
are described by an effective Dirac equation (γµk∗µ−
M∗)Ψ(x) = 0, whereas the mesons, which generate the
classical mean field, are characterized by correspond-
ing covariant equations of motion. The presence of the
hadronic medium modifies the masses and momenta of
the hadrons, i.e.M∗= M + Σs (effective masses),
k∗µ= kµ− Σµ(kinetic momenta), where we have in-
troduced the scalar and vector self-energies Σs, Σµ.
For asymmetric matter the self-energies are different for
protons and neutrons, depending on the isovector me-
son contributions. We will call the corresponding models
as NLρ and NLρδ, respectively, and just NL the case
without isovector interactions. We will show also some
results with Density Dependent couplings, in order to
probe effects which go beyond the RMF picture. For
the more general NLρδ case the self-energies of protons
and neutrons read:
Σs(p,n) = −fσσ(ρs) ± fδρs3,
Σµ(p,n) = fωjµ∓ fρjµ
(upper signs for neutrons), where ρs= ρsp+ ρsn, jα=
isospin scalar densities and currents and fσ,ω,ρ,δare the
coupling constants of the various mesonic fields. σ(ρs)
is the solution of the non linear equation for the σ field
For the description of heavy ion collisions we solve
the covariant transport equation of the Boltzmann type
[20, 21] within the Relativistic Landau Vlasov (RLV )
method, phase-space Gaussian test particles , and
applying a Monte-Carlo procedure for the hard hadron
collisions. The collision term includes elastic and inelas-
tic processes involving the production/absorption of the
∆(1232MeV) and N∗(1440MeV) resonances as well as
their decays into pion channels, [32, 33].
A relativistic kinetic equation can be obtained from
nucleon Wigner Function dynamics derived from the ef-
fective Dirac equation . The neutron/proton Wigner
functions are expanded in terms of components with def-
inite transformation properties. Consistently with the
effective fields included in our minimal model one can
limit the expansion to scalar and vector partsˆF(i)(x,p) =
algebra a transport equation of Vlasov type for the
scalar part fi(x,p∗µ) ≡ Fi
obey to the following equation of motion:
n,ρs3= ρsp− ρsn, jα
nare the total and
S(x,p) + γµF(i)µ(x,p),i = n,p. We get after some
i)]∂µp∗}fi(x,p∗µ) = 0, with the field tensors
i. The trajectories of test particles
(xi(τ)) + ∂µM∗
In order to have an idea of the dynamical effects of
the covariant nature of the interacting fields, we write
down, with some approximations, the “force” acting on
a particle. Since we are interested in isospin contributions
we will take into account only the isovector part of the
?∇ρ3∓ fδ?∇ρS3, (p/n)
The Lorentz force (first term of Eq.(6) shows a γ =
boosting of the vector coupling, while from the second
term we expect a γ-quenched δ contribution. We remark
that the Lorentz-like force is absent in the non-relativistic
Vlasov transport equation discussed before. This nicely
shows the qualitative different dynamics of a fully rela-
tivistic approach. You cannot get it just inserting a rel-
ativistic kinematics in the classical transport equations.
FIG. 4: Differential neutron-proton flows for the132Sn +124
Sn reaction at 1.5 AGeV (b=6fm) from the three different
models for the isovector mean fields. Top: Transverse Flows.
Bottom: Elliptic Flows. Full circles and solid line: NLρδ.
Open circles and dashed line: NLρ.
The flow observables can be seen respectively as the
first and second coefficients of a Fourier expansion of the
azimuthal distribution :
mentum and y the rapidity along beam direction. The
transverse flow can be also expressed as: V1(y,pt) = ?px
The sideward (transverse) flow is a deflection of forwards
and backwards moving particles, within the reaction
plane. The second coefficient of the expansion defines
the elliptic flow given by V2(y,pt) = ?p2
the competition between in-plane and out-of-plane emis-
sions. The sign of V2indicates the azimuthal anisotropy
of emission: particles can be preferentially emitted ei-
ther in the reaction plane (V2 > 0) or out-of-plane
(squeeze − out, V2 < 0) [35, 36]. The pt-dependence
of V2is very sensitive to the high density behavior of the
EoS since highly energetic particles (pt≥ 0.5) originate
from the initial compressed and out-of-equilibrium phase
of the collision. For the isospin effects the neutron-proton
differential flows V(n−p)
(y,pt) ≡ Vn
have been suggested as very useful probes of the isovec-
tor part of the EoS since they appear rather insensitive
to the isoscalar potential and to the in medium nuclear
cross sections, [37, 38].
In heavy-ion collisions around 1AGeV with radioac-
tive beams, due to the large counterstreaming nuclear
currents, one may exploit the different Lorentz nature of
a scalar and a vector field, see the different γ-boosting
in the local force, Eq.(6). In Fig.4 transverse and el-
liptic differential flows are shown for the132Sn +124Sn
reaction at 1.5 AGeV (b = 6fm), that likely could be
studied with the new planned radioactive beam facilities
at intermediate energies, . The effect of the different
structure of the isovector channel is clear. Particularly
dφ(y,pt) ≈ 1+ 2V1cos(φ) +
yis the transverse mo-
2V2cos(2φ) where pt =
?. It measures
evident is the splitting in the high ptregion of the elliptic
flow. In the (ρ + δ) dynamics the high-ptneutrons show
a much larger squeeze−out. This is fully consistent with
an early emission (more spectator shadowing) due to the
larger repulsive ρ-field. The V2 observable, which is a
good chronometer of the reaction dynamics, appears to
be particularly sensitive to the Lorentz structure of the
effective interaction. We expect similar effects, even en-
hanced, from the measurements of differential flows for
light isobars, like3H vs.3He.
V.ISOSPIN EFFECTS ON SUB-THRESHOLD
KAON PRODUCTION AT INTERMEDIATE
Particle production represent a useful tool to constrain
the poorly known high density behaviour of the nuclear
equation of state (EoS) [39, 40]. In particular pion and
(subthreshold) kaon productions have been extensively
investigated leading to the conclusion of a soft behaviour
of the EoS at high densities, [41, 42]. Kaons (K0,K+)
appear as particularly sensitive probes since they are pro-
duced in the high density phase almost without subse-
quent reabsorption effects . At variance, antikaons
(¯K0,¯K−) are strongly coupled to the hadronic medium
through strangeness exchange reactions [41, 43]. Here we
show that the isospin dependence of the K0,+production
can be also used to probe the isovector part of the EoS:
we propose the K0/K+yield ratio as a good observable
to constrain the high density behavior of the symmetry
energy, Esym, [1, 2].
Using our RMF transport model we analyze pion and
kaon production in central197Au +197Au collisions in
the 0.8 − 1.8 AGeV beam energy range, with different
effective field choices for Esym. We will compare results
of three Lagrangians with constant nucleon-meson cou-
plings (NL... type, see before) and one with density de-
pendent couplings (DDF, see ), recently suggested
for better nucleonic properties of neutron stars . In
the DDF model the fρis exponentially decreasing with
density, resulting in a rather ”soft” symmetry term at
high density. In order to isolate the sensitivity to the
isovector components we use models showing the same
”soft” EoS for symmetric matter.
Pions are produced via the decay of the ∆(1232) reso-
nance and can contribute to the kaon yield through colli-
sions with baryons: πB −→ Y K. All these processes are
treated within a relativistic transport model including an
hadron mean field propagation. The latter point, which
goes beyond the “collision cascade” picture, is essential
for particle production yields since it directly affects the
energy balance of the inelastic channels.
Fig. 5 reports the temporal evolution of ∆±,0,++res-
onances and pions (π±,0) and kaons (K+,0) for central
Au+Au collisions at 1 AGeV . It is clear that, while the
pion yield freezes out at times of the order of 50fm/c,
i.e. at the final stage of the reaction (and at low densi-
FIG. 5: Time evolution of the ∆±,0,++resonances and pi-
ons π±,0(left), and kaons K0,+(right) for a central (b = 0
fm impact parameter). Au+Au collision at 1 AGeV incident
energy. Transport calculation using the NL,NLρ,NLρδ and
DDF models for the iso-vector part of the nuclear EoS are
ties), kaon production occur within the very early stage of
the reaction, and the yield saturates at around 20fm/c.
Kaons are then suitable to probe the high density phase
of nuclear matter. From Fig. 5 we see that the pion re-
sults are weakly dependent on the isospin part of the nu-
clear mean field. However, a slight increase (decrease) in
the π−(π+) multiplicity is observed when going from the
NL (or DDF) to the NLρ and then to the NLρδ model,
i.e. increasing the vector contribution fρin the isovector
channel. This trend is more pronounced for kaons, see
the right panel, due to the high density selection of the
source and the proximity to the production threshold.
The results for the DDF model, density dependent cou-
plings with a large fρdecrease at high density, are fully
consistent. They are always closer to the NL case (with-
out isovector interactions) but the difference, still seen
for π+,−, is completely disappearing for K0,+, selectively
produced at high densities.
When isovector fields are included the symmetry po-
tential energy in neutron-rich matter is repulsive for neu-
trons and attractive for protons. In a HIC this leads
to a fast, pre-equilibrium, emission of neutrons. Such a
mean field mechanism, often referred to as isospin frac-
tionation [1, 2], is responsible for a reduction of the neu-
tron to proton ratio during the high density phase, with
direct consequences on particle production in inelastic
Threshold effects represent a more subtle point. The
energy conservation in a hadron collision in general has
to be formulated in terms of the canonical momenta, i.e.
for a reaction 1 + 2 → 3 + 4 as sin = (kµ
effective (kinetic) momenta and masses, an equivalent
relation should be formulated starting from the effective
4)2= sout. Since hadrons are propagating with
FIG. 6: π−/π+(upper) and K+/K0(lower) ratios as a func-
tion of the incident energy for the same reaction and models
as in Fig. 5. In addition we present, for Ebeam = 1 AGeV ,
NLρ results with a density dependent ρ-coupling (triangles),
see text. The open symbols at 1.2 AGeV show the corre-
sponding results for a132Sn +124Sn collision, more neutron
in-medium quantities k∗µ= kµ− Σµand m∗= m + Σs,
where Σsand Σµare the scalar and vector self-energies,
Eqs.(4).In reactions where nucleon resonances, espe-
cially the different isospin states of the ∆ resonance, and
hyperons enter, also their self energies are relevant for
energy conservation. We specify them in the usual way
according to the light quark content and with appro-
priate Clebsch-Gordon coefficients . The self-energy
contribution to the energy conservation in inelastic chan-
nels will influence the particle production in two differ-
ent ways. On one hand it will directly determine the
thresholds and thus the multiplicities of a certain type of
particles, in particular of the sub-threshold ones, as here
for the kaons. Secondly it will modify the phase space
available in the final channel.
In fact in neutron-rich systems mean field and thresh-
old effects are acting in opposite directions on particle
production and might compensate each other. As an ex-
ample, nn collisions excite ∆−,0resonances which decay
mainly to π−. In a neutron-rich matter the mean field
effect pushes out neutrons making the matter more sym-
metric and thus decreasing the π−yield. The threshold
effect on the other hand is increasing the rate of π−’s
due to the enhanced production of the ∆−resonances:
now the nn → p∆−process is favored (with respect to
pp → n∆++) since more effectively a neutron is con-
verted into a proton. Such interplay between the two
mechanisms cannot be fully included in a non-relativistic
dynamics, in particular in calculations where the baryon
symmetry potential is treated classically [46, 47].
In the 0.8 − 1.8 AGeV range the sensitivity is larger
for the K0/K+compared to the π−/π+ratio, as we can
see from Fig.6. This is due to the subthreshold produc-
tion and to the fact that isospin effects enter twice in
the two-steps production of kaons, . Between the two
extreme DDF and NLρδ isovector interaction models,
the variations in the ratios are of the order of 14 − 16%
for kaons, while they reduce to about 8 − 10% for pions.
Interestingly the Iso-EoS effect for pions is increasing at
lower energies, when approaching the production thresh-
old. We have to note that in a previous study of kaon
production in excited nuclear matter the dependence of
the K0/K+yield ratio on the effective isovector interac-
tion appears much larger (see Fig.8 of ref.). The point
is that in the non-equilibrium case of a heavy ion collision
the asymmetry of the source where kaons are produced
is in fact reduced by the n → p “transformation”, due
to the favored nn → p∆−processes. This effect is al-
most absent at equilibrium due to the inverse transitions,
see Fig.3 of ref.. Moreover in infinite nuclear matter
even the fast neutron emission is not present. This result
clearly shows that chemical equilibrium models can lead
to uncorrect results when used for transient states of an
open system. In the same Fig. 6 we also report results
at 1.2 AGeV for the132Sn +124Sn reaction, induced
by a radioactive beam, with an overall larger asymmetry
(open symbols). The isospin effects are clearly enhanced.
VI.TESTING DECONFINEMENT AT HIGH
The hadronic matter is expected to undergo a phase
transition into a deconfined phase of quarks and gluons
at large densities and/or high temperatures. On very
general grounds, the transition’s critical densities are ex-
pected to depend on the isospin of the system, but no ex-
perimental tests of this dependence have been performed
so far. Moreover, up to now, data on the phase transi-
tion have been extracted from ultrarelativistic collisions,
when large temperatures but low baryon densities are
reached. In order to check the possibility of observing
some precursor signals of some new physics even in col-
lisions of stable nuclei at intermediate energies we have
performed some event simulations for the collision of very
heavy, neutron-rich, elements. We have chosen the reac-
tion238U +238U (average proton fraction Z/A = 0.39)
at 1 AGeV and semicentral impact parameter b = 7 fm
just to increase the neutron excess in the interacting re-
gion. To evaluate the degree of local equilibration and
the corresponding temperature we have followed the mo-
mentum distribution in a space cell located in the c.m.
of the system; in the same cell we report the maximum
mass density evolution. Results are shown in Fig. 7. We
see that after about 10 fm/c a nice local equilibration is
achieved. We have a unique Fermi distribution and from
a simple fit we can evaluate the local temperature. At
this beam energy the maximum density coincides with
the thermalization, then the system is quickly cooling
while expanding. In Fig.7, lower panel, we report the
010 20 30 40 50 60
ρB,res [ρsat] , Q
010 20 30 40 50 60
010 20 30 40 50 60
FIG. 7: Uranium-Uranium 1 AGeV semicentral. Correlation
between density, temperature, momentum thermalization in-
side a cubic cell 2.5 fm wide, located in the center of mass of
the system. Lower panel: density, temperature, energy den-
sity, momentum, proton fraction. Curves in the upper-left:
black dots - baryon density in ρ0 units; grey dots - quadrupole
moment in momentum space; squares - resonance density.
time evolution of all physics parameters inside the c.m.
cell in the interaction region.. We note that a rather ex-
otic nuclear matter is formed in a transient time of the or-
der of 10 fm/c, with baryon density around 3−4ρ0, tem-
perature 50 − 60 MeV , energy density 500 MeV fm−3
and proton fraction between 0.35 and 0.40, well inside
the estimated mixed phase region, see the following..
A study of the isospin dependence of the transition
densities has been performed up to now, to our knowl-
edge, only by Mueller . The conclusion is that, mov-
ing from symmetric nuclei to nuclei having Z/A ∼ 0.3,
the critical density is reduced by roughly 10%. Here we
explore in a more systematic way the model parameters
and we estimate the possibility of forming a mixed-phase
of quarks and hadrons in experiments at energies of the
order of a few GeV per nucleon.
Concerning the hadronic phase, we use the rela-
tivistic non-linear Walecka-type model of Glendenning-
Moszkowski (GM...) , where the isovector part is
treated just with ρ-meson couplings, and the iso-stiffer
NLρδ interaction .
For the quark phase we consider the MIT bag model
 with various bag pressure constants. In particular we
are interested in those parameter sets which would allow
the existence of quark stars [53, 54], i.e. parameters sets
for which the so-called Witten-Bodmer hypothesis is sat-
isfied [55, 56]. One of the aim of our work it to show that
if quark stars are indeed possible, it is then very likely
to find signals of the formation of a mixed quark-hadron
phase in intermediate-energy heavy-ion experiments .
The scenario we would like to explore corresponds to
the situation realized in experiments at moderate energy,
in which the temperature of the system is at maximum of
the order of a few ten MeV . In this situation, only a tiny
amount of strangeness can be produced and therefore we
can only study the deconfinement transition from hadron
matter into up and down quark matter. Since there no
time for weak decays the environment is rather different
from the neutron star case.
The structure of the mixed phase is obtained by impos-
ing the Gibbs conditions [59, 60] for chemical potentials
and pressure and by requiring the conservation of the
total baryon and isospin densities
B,3) = P(Q)(T,µ(Q)
ρB= (1 − χ)ρH
ρ3= (1 − χ)ρH
where χ is the fraction of quark matter in the mixed
phase. In this way we get the binodal surface which
gives the phase coexistence region in the (T,ρB,ρ3) space
[49, 60]. For a fixed value of the conserved charge ρ3,
related to the proton fraction Z/A ≡ (1 + ρ3/ρB)/2, we
will study the boundaries of the mixed phase region in the
(T,ρB) plane. We are particularly interested in the lower
baryon density border, i.e. the critical/transition den-
sity ρcr, in order to check the possibility of reaching such
(T,ρcr,ρ3) conditions in a transient state during an HIC
at relativistic energies. In the hadronic phase the charge
chemical potential is given by µ3= 2Esym(ρB)ρ3
we expect critical densities rather sensitive to the isovec-
tor channel in the hadronic EoS.
We compare the predictions on the transition to a de-
confined phase of the two effective Lagrangians GM3 and
NLρδ. The isoscalar part is very similar while the isovec-
tor EoS is different, because in GM3 we only have the
coupling to the vector ρ-field. In Fig. 8 we show the
crossing density ρcr separating nuclear matter from the
quark-nucleon mixed phase, as a function of the proton
FIG. 8: Variation of the transition density with proton frac-
tion for various hadronic EoS parameterizations. Dotted line:
GM3 parametrization; dashed line: NLρ parametrization;
solid line: NLρδ parametrization. For the quark EoS, the
MIT bag model with B1/4=150 MeV . The points represent
the path followed in the interaction zone during a semi-central
132Sn+132Sn collision at 1 AGeV (circles) and at 300 AMeV
(crosses), see text.
fraction Z/A. We can see the effect of the δ-coupling
towards an earlier crossing due to the larger symmetry
repulsion at high baryon densities. The δ-exchange po-
tential provides an extra isospin repulsion of the hadron
EoS, and its effect shows up in a further reduction of the
In the same figure we report the paths in the (ρ,Z/A)
plane followed in the c.m. region during the collision of
the n-rich132Sn+132Sn system, at different energies. We
see that already at 300 AMeV we are reaching the border
of the mixed phase, and we are well inside it at 1 AGeV .
Statistical fluctuations could help in reducing the den-
sity at which drops of quark matter form. The reason is
that a small bubble can be energetically favored if it con-
tains quarks whose Z/A ratio is smaller than the average
value of the surrounding region. This is again due to the
strong Z/A dependence of the free energy, which favors
clusters having a small electric charge. These configura-
tions can easily transform into a bubble of quarks having
the same flavor content of the original hadrons, even if
the density of the system is not large enough to allow
deconfinement in the absence of statistical fluctuations.
Moreover, since fluctuations favor the formation of bub-
bles having a smaller Z/A, neutron emission from the cen-
tral collision area should be suppressed, what could give
origin to specific signatures of the mechanism described
in this paper. This corresponds to a neutron trapping ef-
fect, supported also by a symmetry energy difference in
the two phases. In fact while in the hadron phase we have
a large neutron potential repulsion (in particular in the
NLρδ case), in the quark phase we only have the much
smaller kinetic contribution. If in a pure hadronic phase
neutrons are quickly emitted or “transformed” in pro-
tons by inelastic collisions, when the mixed phase starts
forming, neutrons are kept in the interacting system up to
the subsequent hadronization in the expansion stage .
Observables related to such neutron “trapping” could be
an inversion in the trend of the formation of neutron rich
fragments and/or of the π−/π+, K0/K+yield ratios for
reaction products coming from high density regions, i.e.
with large transverse momenta. In general we would ex-
pect a modification of the rapidity distribution of the
emitted “isospin”, with an enhancement at mid-rapidity
joint to large event by event fluctuations. A more de-
tailed analysis is clearly needed.
We have shown in few examples the richness of the
physics we can describe using mean field transport equa-
tions, inspired by the pioneering works of David M.
Brink. We have seen that collisions of n-rich heavy ions
from low to intermediate energies can bring new infor-
mation on the isovector part of the in-medium interac-
tion in different regions of high baryon densities, quali-
tatively different from equilibrium EoS properties. We
have presented quantitative results for charge equilibra-
tion in fusion/deep inelastic reactions, differential col-
lective flows and yields of charged pion and kaon ratios.
Important non-equilibrium effects for particle production
are stressed. Finally our study supports the possibility
of observing precursor signals of the phase transition to a
mixed hadron-quark matter at high baryon density in the
collision, central or semi-central, of neutron-rich heavy
ions in the energy range of a few GeV per nucleon. As
signatures we suggest to look at observables particularly
sensitive to the expected different isospin content of the
two phases, which leads to a neutron trapping in the
quark clusters.The isospin structure of hadrons pro-
duced at high transverse momentum should be a good
indicator of the effect.
Many new ideas for fundamental experiments with ra-
dioactive beams are emerging. My picture of David Brink
like a tree always starting new braches is more and more
I would like to warmly mention the great experience
of collaborating with exceptional people on the top-
ics and projects shortly discussed here.
to list some of them: V.Baran, M.Colonna, A.Drago,
H.H.Wolter, S.Yildirim and the essential young con-
tributors L.Bonanno, G.Ferini, R.Lionti, N.Pellegriti,
V.Prassa, J.Rizzo and E.Santini.
I will try
The interaction with experimental groups has been es-
sential, in particular I like to thank the DIPOLE Col-
lab.(D.Pierroutsakou et al.), the CHIMERA Collab.
(A.Pagano et al.), and the FOPI Collab. (W.Reisdorf
et al.) for the intense discussions and the access to the
 B.A. Li, W.U. Schroeder (Eds.), Isospin Physics in
Heavy-Ion Collisions at Intermediate Energies, Nova Sci-
ence, New York, 2001.
 V.Baran, M.Colonna, V.Greco, M.Di Toro, Phys. Rep.
410 (2005) 335.
 P. Chomaz, M. Di Toro, A.Smerzi, Nucl. Phys. A563
 P. F. Bortignon et al., Nucl. Phys. A583 (1995) 101c.
 S. Flibotte et al., Phys. Rev. Lett. 77 (1996) 1448.
 M. Cinausero et al., Nuovo Cimento 111 (1998) 613.
 D. Pierroutsakou et al., Eur. Phys. Jour. A16 (2003)
423, Nucl. Phys. A687 (2003) 245c.
 F.Amorini et al., Phys. Rev. C69 (2004) 014608.
 D. Pierroutsakou et al., Phys. Rev. C71 (2005) 054605.
 D. M. Brink and M. Di Toro,, Nucl. Phys. A372 (1981)
 V. Baran et al., Nucl. Phys. A600 (1996) 111.
 M.Cabibbo et al., Nucl. Phys. A637 (1998) 374.
 V. Baran et al., Nucl. Phys. A679 (2001) 373.
 C. Simenel et al., Phys. Rev. Lett. 86 (2000) 2971.
 V. Baran, D. M. Brink, M. Colonna, M. Di Toro, Phys.
Rev. Lett. 87 (2001) 182501
 D.Pierroutsakou et al., LNS exp. proposal 2006, PAC
 L. Bonanno, Effetti di radiazione diretta dipolare sulla
sintesi degli elementi superpesanti, Master Thesis, Cata-
nia Univ. 2004.
 A letter of intent for the new SPIRAL2 facility at GANIL
is in preparation.
 B. D. Serot, J. D. Walecka, Advances in Nuclear Physics,
16, 1, eds. J. W. Negele, E. Vogt, (Plenum, N.Y., 1986).
 C. M. Ko, Q. Li, R. Wang, Phys. Rev. Lett. 59 (1987)
 B. Bl¨ attel, V. Koch, U. Mosel, Rep. Prog. Phys. 56 (1993)
 S. Kubis, M. Kutschera, Phys. Lett. B399 (1997) 191.
 B. Liu, V. Greco, V. Baran, M. Colonna, M. Di Toro,
Phys. Rev. C65 (2002) 045201.
 T. Gaitanos, M. Di Toro, S. Typel, V. Baran, C. Fuchs,
V. Greco, H.H. Wolter, Nucl. Phys. A732 (2004) 24.
 F. de Jong, H. Lenske, Phys. Rev. C57 (1998) 3099;
E.N.E. van Dalen, C. Fuchs, A. Faessler, Nucl. Phys.
A744 (2004) 227.
 E.N.E. van Dalen, C. Fuchs, A. Faessler, Phys. Rev. Lett.
95 (2005) 022302.
 M. Di Toro, M. Colonna, J. Rizzo, AIP Conf. Proc.,
Vol.791 (2005) 70-83.
 J. Rizzo, M. Colonna, M. Di Toro, Phys. Rev. C72 (2005)
 P. Danielewicz, Roy A. Lacey, et al., Phys. Rev. Lett. 81,
C. Pinkenburg et al., Phys. Rev. Lett. 83, (1999) 1295.
 T. Gaitanos, C. Fuchs, H.H. Wolter, A. Faessler, Eur.
Phys. J. A 12 (2001) 421;
T. Gaitanos, C. Fuchs, H.H. Wolter, Nucl. Phys. A741
 C. Fuchs. H.H. Wolter, Nucl. Phys. A589 (1995) 732.
 H. Huber, J. Aichelin, Nucl. Phys. A573 (1994) 587.
 G. Ferini, M. Colonna, T. Gaitanos, M. Di Toro, Nucl.
Phys. A762 (2005) 147.
 V. Greco, V. Baran, M. Colonna, M. Di Toro, T. Gai-
tanos, H.H. Wolter, Phys. Lett. B562 (2003) 215.
 J.Y. Ollitrault, Phys. Rev. D46 (1992) 229.
 P. Danielewicz, Nucl. Phys. A673 (2000) 375.
 B.A. Li and A.T. Sustich, Phys. Rev. Lett. 82 (1999)
 B.A. Li, Phys. Rev. Lett. 85 (2000) 4221.
 R. Stock, Phys. Rep. 135 (1986) 259.
 J. Aichelin, C.M. Ko, Phys. Rev. Lett. 55 (1985) 2661.
 C. Fuchs, Prog.Part.Nucl.Phys. 56 1-103 (2006).
 C.Hartnack, H.Oeschler, J.Aichelin, Phys. Rev. Lett. 96
 W. Cassing, L.Tolos, E.L. Bratkovskaya, A. Ramos, Nucl.
Phys. A727 59 (2003).
 H. Weber, E.L. Bratkovskaya, W. Cassing, H. St¨ ocker,
Phys. Rev. C67 014904 (2003).
 T.Kl¨ ahn et al. Constraints on the high-density nuclear
equation of state ..., arXiv:nucl-th/0602038.
 B.A. Li, G.C. Yong, W. Zuo, Phys. Rev. C71 014608
 Q. Li et al., Phys. Rev. C72 034613 (2005).
 In the energy range explored here, the main contribution
to the kaon yield comes from the pionic channels, in par-
ticular from πN collisions, and from the N∆ channel,
which together account for nearly 80% of the total yield,
 H.Mueller, Nucl. Phys. A618 (1997) 349.
 N.K.Glendenning, S.A.Moszkowski, Phys. Rev. Lett. 67
 M. Di Toro, A. Drago, T. Gaitanos, V. Greco, A.
Lavagno, Testing Deconfinement...,
Nucl. Phys. A (2006) in press.
 A.Chodos et al., Phys. Rev. D9 (1974) 3471.
 P.Haensel, J.L.Zdunik, R.Schaeffer, Astron. Astrophys.
160 (1986) 121.
 A.Drago, A.Lavagno, Phys. Lett. B511 (2001) 229.
 E.Witten, Phys. Rev. D30 (1984) 272.
 A.R.Bodmer, Phys. Rev. D4 (1971) 1601.
 M.Alford, S.Reddy, Phys. Rev. D67 (2003) 074024.
 A.Drago, A.Lavagno, G.Pagliara, Phys.Rev. D69 (2004)
 L.D.Landau, L.Lifshitz, Statistical Physics, Pergamon
Press, Oxford 1969.
 N.K.Glendenning, Phys. Rev. D46 (1992) 1274.