# Isospin Dynamics in Heavy Ion Collisions: EoS-sensitive Observables

**Abstract**

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 fragmentation collisions induced by neutron rich projectiles. Differential flow measurements will also shed lights on the controversial neutron/proton effective mass splitting in asymmetric 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 “direct” 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 production. 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.

arXiv:nucl-th/0609081v1 29 Sep 2006

Isospin Dynamics in Heavy Ion Collisions: EoS-sensitive Observables

M.Di Toro

a∗

, V.Baran

b

, M.Colonna

a

, G.Ferini

a

, T.Gaitanos

c

, V.Greco

a

, J.Rizzo

a

,

H.H.Wolter

c

.

a

Laboratori Nazionali del Sud INFN, I-95123 Catania, Italy,

and Physics-Astronomy Dept., University of Catania

b

Dept.of Theoretical Physics, Bucharest Univ., Magurele, Bucharest, Romania

c

Dept. f¨ur Physik, Universit¨at M¨unchen, D-8574 8 Garching, G ermany

Heavy Ion Collisions (HIC) r epresent 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 fr om 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 E

sym

appears in t he energy density ǫ(ρ, ρ

3

) ≡ ǫ(ρ)+ρE

sym

(ρ

3

/ρ)

2

+

O(ρ

3

/ρ)

4

+.., expressed in terms of total (ρ = ρ

p

+ρ

n

) and isospin (ρ

3

= ρ

p

−ρ

n

) densities.

The symmetry term gets a kinetic contribution directly from basic Pauli correlations and

a potential par t 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) t o present results that are severely

constraining the existing eﬀective interaction models. We will discuss dissipative colli-

∗

ditoro@lns.infn.it

2 M.Di Toro

-200

-150

-100

-50

0

oct(a.u.)

0

10

20

30

40

50

60

N

-200

-150

-100

-50

0

50

oct(a.u)

0

10

20

30

40

50

60

70

-90

-60

-30 0 30

60

90

oct(a.u)

0

50

100

150

200

(a)

(b)

(c)

Figure 1. Distribution of the octupole moment of primary fragments for the

132

Sn +

64

Ni

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, o ﬀ er 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 a ngular 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 f or the new RIB facility Spiral 2, [

7] we have studied the reaction

132

Sn +

64

Ni 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 smoo t h decreasing function

towards low densities, and a nother 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 P LF/T LF 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, larg er

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

Isospin Dynamics 3

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 o f the diﬀerent interaction times on the charge equilibration mechanism,

probed starting from entrance channels with large N/Z asymmetries, like

132

Sn(N/Z =

1.64) +

58

Ni(N/Z = 1.07). Moreover the equilibration mechanism is also directly driven

by the strenght of the symmetry term.

3. Isospin Dynamics in N eck Fragmentation at Fermi Energies

It is now quite well established that the la r gest part of the reaction cross section for

dissipative collisions at Fermi energies goes through the Neck Fragmentation channel, with

IMF s directly produced in t he interacting zone in semiperipheral collisions o n 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 potentia ls

is given by µ

n

− µ

p

= 4E

sym

(ρ

3

/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

IMF s. The alignment between P LF − IMF and P LF − T LF directions represents a

very convincing evidence of t he dynamical o r ig in of the mid-rapidity fragments produced

on short time scales [ 10]. The form of the Φ

plane

distributions (centroid and width)

can give a direct informatio n on the fragmentation mechanism [ 11]. Recent calculations

conﬁrm that the light fra gments 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 fr om Viola systematics observed in event-by-event ve-

locity correlations between IMF s and the P LF/T LF residues [ 10, 11, 13]. We can ﬁgure

out a continuous transition f r om 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 ana l-

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 evid ence in favor of a rel atively

larg e 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

4 M.Di Toro

[ 21] are very g oo d candidates since they are expected to be very sensitive to the mo-

mentum dependence of the mean ﬁeld, see [ 22, 1]. The transverse ﬂow, V

1

(y, p

t

) = h

p

x

p

t

i,

provides information on the anisotropy of nucleon emission on the reaction plane. Very

important for the reaction dynamics is the elliptic ﬂow, V

2

(y, p

t

) = h

p

2

x

−p

2

y

p

2

t

i. The sign

of V

2

indicates the azimuthal anisotropy of emission: on the reaction plane (V

2

> 0) or

out-of-plane (squeeze − out, V

2

< 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, p

t

) ≡ V

n

1,2

(y, p

t

) − V

p

1,2

(y, p

t

) at various rapidities and transverse momenta in

semicentral (b/b

max

= 0.5)

197

Au +

197

Au collisons at 250AMeV , where some proton data

are existing from the F OP I 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.

Figure 2. Diﬀerence between proton and neutron V

1

ﬂows in a semi-cent ral 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 V

1

proton 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 F ig. 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

∗

n

< m

∗

p

. Just to show that our simulations give

realistic results we compare in lower right panel of Fig. 2 with the pro t on data of the

F OP I collabora tion for similar selections of impact parameters rapidities and transverse

Isospin Dynamics 5

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

∗

n

< m

∗

p

, revealing a faster neutron emission and so more neutron squeeze out (more

spectator shadowing). The measurement o f n/p ﬂow diﬀerences appears essential. Due to

the diﬃculties in measuring neutrons, our suggestion is to measure the diﬀerence between

light isobar ﬂows, like

3

H vs.

3

He 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].

5. Relativistic Collisions

Finally we focus our attention on relativistic heavy ion collisions, that provide a unique

terrestrial oppo rt unity to probe the in-medium nuclear interaction at high densities. An

eﬀective Lagrangian a pproa ch 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 (a

4

param-

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