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arXiv:nucl-th/0609081v1 29 Sep 2006
Isospin Dynamics in Heavy Ion Collisions: EoS-sensitive Observables
M.Di Toroa∗, V.Baranb, M.Colonnaa, G.Ferinia, T.Gaitanosc, V.Grecoa, J.Rizzoa,
H.H.Wolterc.
aLaboratori Nazionali del Sud INFN, I-95123 Catania, Italy,
and Physics-Astronomy Dept., University of Catania
bDept.of Theoretical Physics, Bucharest Univ., Magurele, Bucharest, Romania
cDept. f¨ ur Physik, Universit¨ at M¨ unchen, D-85748 Garching, Germany
Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium nuclear
interaction in regions away from saturation and at high nucleon momenta. In this report
we present a selection of reaction observables particularly sensitive to the isovector part
of the interaction, i.e. to the symmetry term of the nuclear Equation of State (EoS) At
low energies the behavior of the symmetry energy around saturation influences dissipation
and fragment production mechanisms. Predictions are shown for deep-inelastic and frag-
mentation collisions induced by neutron rich projectiles. Differential flow measurements
will also shed lights on the controversial neutron/proton effective mass splitting in asym-
metric matter. The high density symmetry term can be derived from isospin effects on
heavy ion reactions at relativistic energies (few AGeV range), that can even allow a “di-
rect” study of the covariant structure of the isovector interaction in the hadron medium.
Rather sensitive observables are proposed from collective flows and from pion/kaon pro-
duction. 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.
1. Introduction
The symmetry energy Esymappears in the energy density ǫ(ρ,ρ3) ≡ ǫ(ρ)+ρEsym(ρ3/ρ)2+
O(ρ3/ρ)4+.., expressed in terms of total (ρ = ρp+ρn) and isospin (ρ3= ρp−ρn) densities.
The symmetry term gets a kinetic contribution directly from basic Pauli correlations and
a potential part from the highly controversial isospin dependence of the effective interac-
tions [ 1]. Both at sub-saturation and supra-saturation densities, predictions based of the
existing many-body techniques diverge rather widely, see [ 2]. We take advantage of new
opportunities in theory (development of rather reliable microscopic transport codes for
HIC) and in experiments (availability of very asymmetric radioactive beams, improved
possibility of measuring event-by-event correlations) to present results that are severely
constraining the existing effective interaction models. We will discuss dissipative colli-
∗ditoro@lns.infn.it
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-90 -60 -30 0 30 60 90
oct(a.u)
0
50
100
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200
(a)
(b)
(c)
Figure 1. Distribution of the octupole moment of primary fragments for the132Sn+64Ni
reaction at 10 AMeV (impact parameters (a):b = 6fm, (b):7fm, (c):8fm). Solid lines:
asysoft. Dashed lines: asystiff
sions in a wide range of energies, from just above the Coulomb barrier up to a few AGeV .
The transport codes are based on mean field theories, with correlations included via hard
nucleon-nucleon elastic and inelastic collisions and via stochastic forces, selfconsistently
evaluated from the mean phase-space trajectory, see [ 1, 3, 4, 5]. Stochasticity is essential
in order to get distributions as well as to allow the growth of dynamical instabilities.
2. Isospin effects on Deep-Inelastic Collisions
Dissipative semi-peripheral collisions at low energies, including binary and three-body
breakings, offer a good opportunity to study phenomena occurring in nuclear matter under
extreme conditions with respect to shape, excitation energy, spin and N/Z ratio (isospin).
In some cases, due to a combined Coulomb and angular momentum (deformation) effect,
some instabilities can show up [ 6]. This can lead to 3-body breakings, where a light cluster
is emitted from the neck region. Three body processes in collisions with exotic beams
will allow to investigate how the development of surface (neck-like) instabilities, that
would help ternary breakings, is sensitive to the structure of the symmetry term around
(below) saturation. In order to suggest proposals for the new RIB facility Spiral 2, [
7] we have studied the reaction132Sn +64Ni at 10AMeV in semicentral events, impact
parameters b = 6,7,8fm, where one observes mostly binary exit channels, but still in
presence of large dissipation. Two different behaviors of the symmetry energy below
saturation have been tested: one (asysoft) where it is a smooth decreasing function
towards low densities, and another one (asystiff) where we have a rapid decrease, [
1]. The Wilczynski plots, kinetic energy loss vs. deflection angle, show slightly more
dissipative events in the asystiff case, consistent with the point that in the interaction
at lower densities in very neutron-rich matter (the neck region) we have a less repulsive
symmetry term. In fact the neck dynamics is rather different in the two cases, as it can be
well evidenced looking at the deformation of the PLF/TLF residues. The distribution
of the octupole moment over the considered ensemble of events is shown in Fig.1 for
the three considered impact parameters. Except for the most peripheral events, larger
deformations, strongly suggesting a final 3-body outcome, are seen in the asystiff case.
Now, due to the lower value of the symmetry enrgy, the neutron-rich neck connecting
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the two systems survives a longer time leading to very deformed primary fragments,
from which eventually small clusters will be dynamically emitted. Finally we expect
to see effects of the different interaction times on the charge equilibration mechanism,
probed starting from entrance channels with large N/Z asymmetries, like132Sn(N/Z =
1.64) +58Ni(N/Z = 1.07). Moreover the equilibration mechanism is also directly driven
by the strenght of the symmetry term.
3. Isospin Dynamics in Neck Fragmentation at Fermi Energies
It is now quite well established that the largest part of the reaction cross section for
dissipative collisions at Fermi energies goes through the Neck Fragmentation channel, with
IMFs directly produced in the interacting zone in semiperipheral collisions on very short
time scales [ 8]. We can predict interesting isospin transport effects for this new frag-
mentation mechanism since clusters are formed still in a dilute asymmetric matter but
always in contact with the regions of the projectile-like and target-like remnants almost at
normal densities. Since the difference between local neutron-proton chemical potentials
is given by µn− µp= 4Esym(ρ3/ρ), we expect a larger neutron flow to the neck clusters
for a stiffer symmetry energy around saturation, [ 1, 9]. The isospin dynamics can be
directly extracted from correlations between N/Z, alignement and emission times of the
IMFs. The alignment between PLF − IMF and PLF − TLF directions represents a
very convincing evidence of the dynamical origin of the mid-rapidity fragments produced
on short time scales [ 10]. The form of the Φplane distributions (centroid and width)
can give a direct information on the fragmentation mechanism [ 11]. Recent calculations
confirm that the light fragments are emitted first, a general feature expected for that
rupture mechanism [ 12]. The same conclusion can be derived from direct emission time
measurements based on deviations from Viola systematics observed in event-by-event ve-
locity correlations between IMFs and the PLF/TLF residues [ 10, 11, 13]. We can figure
out a continuous transition from fast produced fragments via neck instabilities to clusters
formed in a dynamical fission of the projectile(target) residues up to the evaporated ones
(statistical fission). Along this line it would be even possible to disentangle the effects of
volume and shape instabilities. A neutron enrichment of the overlap (”neck”) region is
expected, due to the neutron migration from higher (spectator) to lower (neck) density
regions, directly related to the slope of the symmetry energy [ 12]. A very nice new anal-
ysis has been presented on the Sn+Ni data at 35 AMeV by the Chimera Collab., Fig.2
of ref.[ 14]. A strong correlation between neutron enrichemnt and alignement (when the
short emission time selection is enforced) is seen, that can be reproduced only with a stiff
behavior of the symmetry energy. This is the first clear evidence in favor of a relatively
large slope (symmetry pressure) around saturation.
4. Effective Mass Splitting and Collective Flows
The problem of Momentum Dependence in the Isovector channel (Iso − MD) is still
very controversial and it would be extremely important to get more definite experimen-
tal information, see the recent refs. [ 15, 16, 17, 18, 19, 20]. Intermediate energies are
important in order to have high momentum particles and to test regions of high baryon
(isoscalar) and isospin (isovector) density during the reactions dynamics. Collective flows
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[ 21] are very good candidates since they are expected to be very sensitive to the mo-
mentum dependence of the mean field, see [ 22, 1]. The transverse flow, V1(y,pt) = ?px
provides information on the anisotropy of nucleon emission on the reaction plane. Very
important for the reaction dynamics is the elliptic flow, V2(y,pt) = ?p2
of V2indicates the azimuthal anisotropy of emission: on the reaction plane (V2> 0) or
out-of-plane (squeeze − out, V2< 0) [ 21, 22]. We have then tested the Iso − MD of
the fields just evaluating the Difference of neutron/proton transverse and elliptic flows
V(n−p)
1,2
(y,pt) ≡ Vn
semicentral (b/bmax= 0.5)197Au+197Au collisons at 250AMeV , where some proton data
are existing from the FOPI collaboration at GSI [ 23, 24]. The transport code has been
implemented with a BGBD − like [ 25, 26] mean field with a different (n,p) momentum
dependence, see [ 16, 17, 18], that allow to follow the dynamical effect of opposite n/p ef-
fective mass splitting while keeping the same density dependence of the symmetry energy.
pt?,
x−p2
p2
t
y
?. The sign
1,2(y,pt) − Vp
1,2(y,pt) at various rapidities and transverse momenta in
Figure 2. Difference between proton and neutron V1 flows in a semi-central reaction
Au+Au at 250 AMeV for three rapidity ranges. Upper Left Panel: |y(0)| ≤ 0.3; Upper
Right: 0.3 ≤ |y(0)| ≤ 0.7; Lower Left: 0.6 ≤ |y(0)| ≤ 0.9. Lower Right Panel: Comparison
of the V1proton flow with FOPI data [ 23] for three rapidity ranges. Top: 0.5 ≤ |y(0)| ≤
0.7; center: 0.7 ≤ |y(0)| ≤ 0.9; bottom: 0.9 ≤ |y(0)| ≤ 1.1.
For the difference of nucleon transverse flows, see Fig. 2, the mass splitting effect is
evident at all rapidities, and nicely increasing at larger rapidities and transverse mo-
menta, with more neutron flow when m∗
realistic results we compare in lower right panel of Fig. 2 with the proton data of the
FOPI collaboration for similar selections of impact parameters rapidities and transverse
n< m∗
p. Just to show that our simulations give
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momenta. The same analysis has been performed for the difference of elliptic flows, [
17]. Again the mass splitting effects are more evident for higher rapidity and tranverse
momentum selections. In particular the differential elliptic flow becomes negative when
m∗
spectator shadowing). The measurement of n/p flow differences appears essential. Due to
the difficulties in measuring neutrons, our suggestion is to measure the difference between
light isobar flows, like3H vs.3He and so on. We expect to clearly see the effective mass
splitting effects, maybe even enhanced due to larger overall flows shown by clusters, see [
1, 27].
n< m∗
p, revealing a faster neutron emission and so more neutron squeeze out (more
5. Relativistic Collisions
Finally we focus our attention on relativistic heavy ion collisions, that provide a unique
terrestrial opportunity to probe the in-medium nuclear interaction at high densities. An
effective Lagrangian approach to the hadron interacting system is extended to the isospin
degree of freedom: within the same frame equilibrium properties (EoS, [ 28]) and trans-
port dynamics [ 29, 30] can be consistently derived. Within a covariant picture of the
nuclear mean field, for the description of the symmetry energy at saturation (a4param-
eter of the Weizs¨ aecker mass formula) (a) only the Lorentz vector ρ mesonic field, and
(b) both, the vector ρ (repulsive) and scalar δ (attractive) effective fields [ 31, 32] can
be included. In the latter case the competition between scalar and vector fields leads to
a stiffer symmetry term at high density [ 31, 1]. The presence of the hadronic medium
leads to effective masses and momenta M∗= M +Σs, k∗µ= kµ−Σµ, with Σs, Σµscalar
and vector self-energies. For asymmetric matter the self-energies are different for protons
and neutrons, depending on the isovector meson contributions. We will call the corre-
sponding models as NLρ and NLρδ, respectively, and just NL the case without isovector
interactions.
For the description of heavy ion collisions we solve the covariant transport equation
of the Boltzmann type [ 29, 30] within the Relativistic Landau Vlasov (RLV ) method,
using phase-space Gaussian test particles [ 33], and applying a Monte-Carlo procedure
for the hard hadron collisions. The collision term includes elastic and inelastic processes
involving the production/absorption of the ∆(1232MeV ) and N∗(1440MeV ) resonances
as well as their decays into pion channels, [ 34, 35]. A larger repulsive vector contribution
to the neutron energies is given by the ρ-coupling. This is rapidly increasing with density
when the δ field is included [ 31, 1]. As a consequence we expect a good sensitivity to
the covariant structure of the isovector fields in nucleon emission and particle production
data. Moreover the presence of a Lorentz magnetic term in the relativistic transport
equation [ 29, 30, 1] will enhance the dynamical effects of vector fields [ 36].
Differential flows will be directly affected. In Fig.3 transverse and elliptic differential
flows are shown for the132Sn+124Sn reaction at 1.5 AGeV (b = 6fm), [ 36]. The effect of
the different structure of the isovector channel is clear. Particularly 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 ρ-field in the compression stage.
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-1
-0,5
0
0,5
1
y/yproj
-40
-20
0
20
40
Fpn (y)
0 0,20,4
pT/pproj
0,6
0,81
-0,06
-0,04
-0,02
0
0,02
<V2
p-n>
Figure 3. Differential neutron-proton flows for the132Sn +124Sn reaction at 1.5 AGeV (b =
6fm) from the two different models for the isovector mean fields. Left: Transverse Flows. Right:
Elliptic Flows. Full circles and solid line: NLρδ. Open circles and dashed line: NLρ.
0 1020 30 40
50 60
time (fm/c)
0
5
10
15
20
25
multiplicity
0 10 2030 40
5060
time (fm/c)
0
0,02
0,04
0,06
0,08
NL
NLρ
NLρδ
DDF
5 10
time (fm/c)
15 20
1
1,5
2
2,5
π-
π0
π+
∆-
∆0
∆+
∆++
K0
K+
dK0/dK+
Figure 4. Time evolution of the ∆±,0,++resonances and pions π±,0(left), and kaons (K+,0
(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 shown.
6. Isospin effects on sub-threshold kaon production at intermediate energies
Kaon production has been proven to be a reliable observable for the high density EoS
in the isoscalar sector [ 37, 38] Here we show that the K0,+production (in particular the
K0/K+yield ratio) can be also used to probe the isovector part of the EoS.
Using our RMF transport approach we analyze pion and kaon production in central
197Au +197Au collisions in the 0.8 − 1.8 AGeV beam energy range, comparing models
giving the same “soft” EoS for symmetric matter and with different effective field choices
for Esym[ 35]. Here we also use a Lagrangian with density dependent couplings (DDF, [
32]), recently suggested for better nucleonic properties of neutron stars [ 39]. In the DDF
model the ρ-coupling is exponentially decreasing with density, resulting in a rather ”soft”
symmetry term at high density. The hadron mean field propagation, which goes beyond
the “collision cascade” picture, is essential for particle production yields: in particular the
isospin dependence of the self-energies directly affects the energy balance of the inelastic
channels.
Fig. 4 reports the temporal evolution of ∆±,0,++resonances, pions (π±,0) and kaons
(K+,0) for central Au+Au collisions at 1AGeV . 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 densities), kaon production occur within the very early (compression) stage, and the
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yield saturates at around 20fm/c. From Fig. 4 we see that the pion results are weakly
dependent on the isospin part of the nuclear mean field.
(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.
When isovector fields are included the symmetry potential energy in neutron-rich matter
is repulsive for neutrons 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 fractionation [ 1], is responsible for a reduction of the neutron to proton ratio
during the high density phase, with direct consequences on particle production in inelastic
NN collisions. Threshold effects represent a more subtle point. The energy conservation
in a hadron collision is expressed in terms of the canonical momenta, i.e. for a reaction
1 + 2 → 3 + 4 as sin= (kµ
with effective (kinetic) momenta and masses, an equivalent relation should be formulated
starting from the effective in-medium quantities k∗µ= kµ−Σµand m∗= m+Σs, where Σs
and Σµare the scalar and vector self-energies. The self-energy contributions will influence
the particle production at the level of thresholds as well as of the phase space available
in the final channel. In neutron-rich colliding systems Mean field and threshold effects
are acting in opposite directions. At low energies, around the production threshold,, the
energy conservation (i.e. the self energy contributions) is dominant, as we see from Fig.
4, in particular for kaons.
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 interaction appears
much larger (see Fig.8 of ref.[ 34]). 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 almost absent at equilibrium due to the inverse transitions. 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.
However, a slight increase
1+ kµ
2)2= (kµ
3+ kµ
4)2= sout. Since hadrons are propagating
7. Testing Deconfinement at High Isospin Density
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 critical densities are expected to depend on the isospin of the system, but
no experimental tests of this dependence have been performed so far. In order to check
the possibility of observing some precursor signals of some new physics even in collisions
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 reaction238U +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 region. After about
10 fm/c in the overlap region a nice local equilibration is achieved. A rather exotic
nuclear matter is formed in a transient time of the order of 10 fm/c, with baryon density
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around 3−4ρ0, temperature 50−60 MeV , energy density ≈ 500 MeV fm−3and proton
fraction between 0.35 and 0.40, likely inside the estimated mixed phase region [ 40].
We can study the isospin dependence of the transition densities [ 41] in a systematic way.
Concerning the hadronic phase, we use the relativistic non-linear model of Glendenning-
Moszkowski (in particular the “soft” GM3 choice) [ 42], where the isovector part is treated
just with ρ meson coupling, and the iso-stiffer NLρδ interaction [ 40]. 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 [
43], i.e. parameter sets for which the so-called Witten-Bodmer hypothesis is satisfied [
44, 45]. 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 [ 40]. The structure of the mixed phase is
obtained by imposing the Gibbs conditions [ 46] for chemical potentials and pressure and
by requiring the conservation of the total baryon and isospin densities
µ(H)
B
ρB= (1 − χ)ρH
= µ(Q)
B , µ(H)
3
= µ(Q)
3
,P(H)(T,µ(H)
B,3) = P(Q)(T,µ(Q)
3+ χρQ
B,3),
B+ χρQ
B, ρ3= (1 − χ)ρH
3, (1)
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 [ 46, 41]. For a
fixed value of the conserved charge ρ3we will study the boundaries of the mixed phase
region in the (T,ρB) plane. In the hadronic phase the charge chemical potential is given
by µ3= 2Esym(ρB)ρ3
channel in the hadronic EoS.
In Fig. 5 we show the crossing density ρcrseparating nuclear matter from the quark-
nucleon mixed phase, as a function of the proton 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. 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. At 300 AMeV we are just reaching the border of the mixed phase, and we are
well inside it at 1 AGeV . Statistical fluctuations could help in reducing the density at
which drops of quark matter form. The reason is that a small bubble can be energetically
favored if it contains quarks whose Z/A ratio is smaller than the average value of the
surrounding region [ 40]. This corresponds to a neutron trapping effect, 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 protons 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 [ 40]. 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.
ρB. Thus, we expect critical densities rather sensitive to the isovector
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Figure 5.
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-central132Sn+132Sn
collision at 1 AGeV (circles) and at 300 AMeV (crosses).
Variation of the transition density with proton fraction for various hadronic EoS
8. Perspectives
We have shown that violent collisions of n-rich heavy ions from low to relativistic en-
ergies can bring new information on the isovector part of the in-medium interaction,
qualitatively different from equilibrium EoS properties. We have presented quantitative
results in a wide range of beam energies. At low energies we have shown isospin effects
on the dissipation in deep inelastic collisions, at Fermi energies the Iso-EoS sensitivity of
the isospin transport in fragment reactions and finally at intermediate the dependence of
differential flows on the Iso−MD and effective mass splitting. In relativistic collisions we
have shown the possibility of a direct measure of the Lorentz structure of the isovector
fields at high baryon density, from differential collective 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, cen-
tral or semi-central, of neutron-rich heavy ions in the energy range of a few AGeV . As
signatures we suggest to look at the isospin structure of hadrons produced at high trans-
verse momentum, as a good indicator of the neutron trapping effect. In conclusion the
results presented here appear very promising for the possibility of exciting new results
from dissipative collisions with radioactive beams.
Acknowledgements
We warmly thanks A.Drago and A.Lavagno for the intense collaboration on the mixed
hadron-quark phase transition at high baryon and isospin density.
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