Energy of vanishing flow in heavy-ion collisions: Role of Coulomb interactions and asymmetry of a reaction

Abstract

We aim to understand the role of Coulomb interactions as well as of different equations of state on the disappearance of transverse flow for various asymmetric reactions leading to same total mass. For the present study, the total mass of the system is kept constant (ATOT = 152) and asymmetry of the reaction is varied between 0.2 and 0.7. We find that the contribution of mean-field at low incident energies is more for nearly symmetric systems, while the trend is opposite at higher incident energies. The Coulomb interactions as well as different equations of state are found to affect the balance energy significantly for larger asymmetric reactions.

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arXiv:1009.6078v1 [nucl-th] 30 Sep 2010
APS/123-QED
Energy of vanishing flow in heavy-ion collisions: Role of Coulomb interactions and
asymmetry of a reaction
Sanjeev Kumar, Varinderjit Kaur and Suneel Kumar∗
School of Physics and Materials Science,
Thapar University Patiala-147004, Punjab (India)
(Dated: October 1, 2010)
We aim to understand the role of Coulomb interactions as well as of different equations of state
on the disappearance of transverse flow for various asymmetric reactions leading to same total mass.
For the present study, the total mass of the system is kept constant (ATOT = 152) and asymmetry
of the reaction is varied between 0.2 and 0.7. We find that the contribution of mean-field at low
incident energies is more for nearly symmetric systems, while the trend is opposite at higher incident
energies. The Coulomb interactions as well as different equations of state are found to affect the
balance energy significantly for larger asymmetric reactions.
PACS numbers: 25.70.Pq, 25.70.-z, 24.10.Lx, 25.70.Mn, 21.65.Cd
I.INTRODUCTION
The heavy-ion physics has attracted much attention
during the last three decades [1–5].
of nuclear matter under the extreme conditions of
temperature, density, angular momentum etc., is a
very important aspect of heavy-ion physics. One of the
important quantity which has been used extensively to
study this hot and dense nuclear matter is the collective
transverse in-plane flow [1, 2, 6]. This quantity has a
beauty of vanishing at a certain incident energy. This
energy is dubbed as balance energy (Ebal) or the energy
of vanishing flow (EVF) [6, 7]. This beauty is due to the
counterbalancing of attractive mean-field at low incident
energies and repulsive nucleon-nucleon (NN) collisions
at higher incident energies.
the masses ranging from C12+ C12to U238+ U238at
different colliding geometries was studied experimentally
and theoretically and found to be sensitive with the
composite mass of the system [6, 8] as well as with the
impact parameter of a reaction [6, 9, 10].
With the passage of time, isospin degree of freedom
in terms of symmetry energy and NN cross section is
found to affect the balance energy or energy of vanishing
flow and related phenomenon in heavy-ion collisions
[4, 5, 11, 12].
Experimentally, Pak et al., studied the isospin effects
on the collective flow and balance energy at central
and peripheral collision geometries [11]. On the other
hand, theoretically, this effect is studied by using the
isospin-dependent Boltzmann Uehling-Uhlenbeck model
(IBUU) [3, 5, 13], and isospin-dependent quantum
molecular dynamics (IQMD) model [9, 12, 14, 15].
As noted, balance energy is due to the counterbalancing
of the attractive mean-field and repulsive nucleon-
nucleon collisions.The Coulomb interaction in in-
The behavior
The balance energy of
∗Electronic address: suneel.kumar@thapar.edu
termediate energy heavy-ion collisions is expected to
play a dominant role in balance energy due to its
repulsive nature. These effects are supposed to be more
pronounced in the presence of isospin effects [16]. The
comparative study which will show the shift in balance
energy due to Coulomb interactions in the presence
of isospin effects by taking into account asymmetry of
reaction in a controlled fashion is still missing in the
literature. The second point is the asymmetry of the
reaction. In some of the studies, the asymmetry of a
reaction is taken into care, but not in other, which is
very important to study the isospin effects [16, 17].
The asymmetry of the reaction can be defined by the
parameter η = | (AT− AP)/(AT+ AP) |;
and AP are the masses of target and projectile. The η
= 0 corresponds to the symmetric reactions, whereas,
non-zero value of η define different asymmetry of the
reaction.It is worth mentioning that the reaction
dynamics in a symmetric reaction (η = 0) can be quite
different compared to asymmetric reaction (η ?= 0) [19].
This is due to the deposition of excitation energy in
the form of compressional energy and thermal energy
in symmetric and asymmetric reactions, respectively.
The effect of the asymmetry of a reaction on the mul-
tifragmentation is studied many times in the literature
[16, 17, 19]. Unfortunately, very little study is available
for the asymmetry of the reaction in terms of transverse
in-plane flow.
In this paper, we will perform the first ever study for the
balance energy in terms of asymmetry of the reaction
and then observe the effect of Coulomb interactions,
symmetry energy, equations of state as well as different
frame of references.The IQMD model used for the
present analysis is explained in the Sec.-II. The results
are presented in Sec.-III, leading to the conclusions in
Sec.-IV.
where AT

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II.THE MODEL
The isospin-dependent quantum molecular dynamics
(IQMD)[15] model treats different charge states of
nucleons, deltas and pions explicitly [21], as inherited
from the Vlasov-Uehling-Uhlenbeck (VUU) model [22].
The IQMD model was used successfully in analyzing
the large number of observables from low to relativistic
energies [12, 15, 21, 22].One of its version (QMD),
has been very successful in explaining the subthreshold
particle production [23], multi-fragmentation [24, 25],
collective flow [6, 26], disappearance of flow [6], and
density temperature reached in a reaction [24]. We shall
not take relativistic effects into account, since in the
energy domain we are interested, there is no relativistic
effect [27]. The isospin degree of freedom enters into
the calculations via both cross sections, mean field
and Coulomb interactions [22]. The details about the
elastic and inelastic cross sections for proton-proton and
neutron-neutron collisions can be found in Refs.[15, 27].
In this model, baryons are represented by Gaussian-
shaped density distributions
fi(r,p,t) =
1
π2?2e
−(r−ri(t))2
2L
e
−(p−pi(t))2.2L
?2
. (1)
Nucleons are initialized in a sphere with radius R =
1.12A1/3fm, in accordance with the liquid drop model.
Each nucleon occupies a volume of ?3so that phase
space is uniformly filled.
randomly chosen between 0 and Fermi momentum pF.
The nucleons of the target and projectile interact via two
and three-body Skyrme forces and Yukawa potential.
The isospin degrees of freedom is treated explicitly by
employing a symmetry potential and explicit Coulomb
forces between protons of the colliding target and pro-
jectile. This helps in achieving the correct distribution
of protons and neutrons within the nucleus.
The hadrons propagate using Hamilton equations of
motion:
The initial momenta are
d? ri
dt
=d < H >
dpi
;
d? pi
dt
= −d < H >
dri
. (2)
with
< H >=< T > + < V > is the Hamiltonian.
=
?
×fj(?r′,?p′,t)d? rd?r′d? pd?p′.
i
p2
2mi
i
+
?
i
?
j>i
?
fi(? r,? p,t)Vij(?r′,? r)
(3)
The baryon-baryon potential Vij, in the above relation,
reads as
Vij(?r′−? r)=Vij
Skyrme+ Vij
Y ukawa+ Vij
Coul+ Vij
Sym
=t1δ(?r′−? r) + t2δ(?r′−? r)ργ−1(
?r′+? r
2
)
+t3exp(|?r′−? r |/µ)
(|?r′−? r |/µ)
1
ρoTi
+
ZiZje2
|?r′−? r |
+t4
3Tj
3.δ(?r′
i− ? rj).(4)
Where µ = 0.4fm, t3= −6.66MeV and t4= 100MeV.
The values of t1 and t2 depends on the values of α, β
and γ [2]. Here Zi and Zj denote the charges of the
ithand jthbaryon, and Ti
components (i.e. 1/2 for protons and -1/2 for neutrons).
The parameters µ and t1,........,t4 are adjusted to the
real part of the nucleonic optical potential.
density dependence of the nucleon optical potential,
standard Skyrme-type parameterizations is employed.
The Skyrme energy density have been shown to be
very successful at low incident energies where fusion is
dominant channel [28, 29]. The Yukawa term is quite
similar to the surface energy coefficient used in the
calculations of nuclear potential for fusion [30].
choice of the equation of state (or compressibility) is
still a controversial one. Many studies advocates softer
matter, whereas, much more believe the matter to be
harder in nature [6, 22]. We shall use both hard (H) and
soft (S) equations of state that have compressibilities of
380 and 200 MeV, respectively.
3, Tj
3are their respective T3
For the
The
III.RESULTS AND DISCUSSIONS
As discussed earlier, asymmetry of a reaction is
found to affect the phenomena of muti-fragmentation
in intermediate energy heavy-ion collisions [17, 19]. On
the other hand, system mass dependence of balance
energy is studied many times in the literature [6]. To
check the effect of Coulomb interactions and asymmetry
of a reaction on the balance energy, we have fixed
(ATOT = AT+ AP = 152) and varied the asymmetry
of the reaction just like this: 26Fe56+44Ru96(η = 0.2),
24Cr50+44Ru102(η = 0.3),20Ca40+50Sn112(η = 0.4),
16S32+50Sn120(η = 0.5), 14Si28+54Xe124(η = 0.6),
8O16+54Xe136(η = 0.7). The asymmetry of a reaction
with multi-fragmentation in this fashion is varied many
times [17, 19]. The whole reaction dynamics is studied
at semi-central geometry by varying the incident energy
between 50 and 250 MeV/nucleon with an increment
of 50 MeV/nucleon by employing hard as well as soft
equations of state. We have checked the stability of the
reacting nuclei in laboratory (lab) as well as in center of
mass (c.m.) frame by taking into account the Coulomb
interactions. Our main purpose here is to understand
the effect of equations of state and Coulomb interactions

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FIG. 1: Asymmetry dependence of directed flow in lab as
well as center of mass frame in L.H.S, while, R.H.S for the
relative effect of Coulomb interactions. The different lines in
the figure are representing the effect of symmetry energy and
Coulomb interactions.
on the energy of vanishing flow or alternatively, on the
balance energy by taking into account the asymmetry of
a reaction.
The directed transverse flow is calculated using < Pdir
[6]
x
>
< Pdir
x
>=1
A
?
i
sgn{Y (i)}Px(i), (5)
where Y(i) and Px(i) are, respectively, the rapidity dis-
tribution and transverse momentum of the ithparticle.
To check the effect of frame of reference, we display
in Fig.1, variation of the asymmetry η on directed flow
< Pdir
x
> in lab. as well as center of mass frame at
incident energy of E = 50 MeV/nucleon. The top and
bottom panels in the right hand side of the figure are
representing the relative Coulomb effect with respect to
the asymmetry of the reaction. This relative effect is
calculated as:
< ∆Pdir
x
|Coul>=< Pdir
x
>Coul+Sym− < Pdir
x
>NoCoul+Sym
(6)
As is evident from the figure, directed flow is found to
increase in a systematic manner in C.M. as well as in
lab frame at E = 50 MeV/nucleon with asymmetry of
the reaction.The inclusion of Coulomb interactions
does not alter the conclusions. Note that the increase in
the asymmetry is related with the increase in the N/Z
ratio. Because the symmetry potential for the neutron
rich systems is stronger compared to the neutron poor
systems due to large relative neutron strength. Further-
more, the symmetry potential is repulsive for neutrons
and attractive for protons.
negative value of directed flow (dominating the mean
field) is observed in the absence of Coulomb interactions
in center of mass as well as in lab. frames. This is due
to the enhancement of the chemical and mechanical
instability domains in the absence of Coulomb interac-
tions [32]. Similar type of study and conclusion was also
performed for nuclear stopping in ref. [16].
Extensive study in the literature proves the stability
of reactions in lab. frame, but, keep in mind that the
reaction under consideration in these studies were sym-
metric in nature. As is clear from the figure, asymmetric
systems are found to be more stable in the center of
mass frame compared to the lab frame. Moreover, if one
consider lab frame, one is surely missing the effect of
asymmetry. To further strengthen the stability of center
of mass frame with asymmetry, the relative effect of
Coulomb interactions < ∆Pdir
x
frames. The relative effect of the Coulomb interactions
is found to decrease with increase in the asymmetry
of a reaction. The systematic decrease can be seen in
center of mass frame with asymmetry, while very weak
dependence of Coulomb interactions on the asymmetry
is obtained in lab frame.
have opted the center of mass frame because we want
to see the shift in the balance energy due to Coulomb
interactions, whose effect is clearly visible in center of
mass frame.
Before we proceed further, let us check the time evo-
lution of directed transverse flow. In Fig. 2, we display
the time evolution of directed flow from E = 50 to 200
MeV/nucleon in center of mass frame for Soft (L.H.S)
and Hard (R.H.S) equations of state.
compressibilities of soft and hard equations of state are
200 and 380 MeV, respectively [2]. The time evolution
is plotted in the absence of Coulomb interactions to see
the maximum effect of asymmetry of a reaction on the
directed in-plane flow. The figure reveals the following
points:
a). The quantity < Pdir
x
> is observed to be constant
throughout the whole distribution of time, while, the
large variation is observed in the value at initial and
final time steps when observed in the lab frame [6]
b). With the increase in the incident energy, the directed
flow is approaching towards more positive value. This
is due to the well known fact of increase in the frequent
NN collisions with increase in the incident energy [6].
c). The behavior of the directed flow with asymmetry
of a reaction follows the opposite trend at E = 50
MeV/nucleon as compared to other high incident ener-
On the other hand, more
|Coul> is studied in both
For the further study, we
Note that the

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FIG. 2: The time evolution of directed flow at different in-
cident energies in center of mass frame in the absence of
Coulomb interactions. The left and right panels are for soft
and hard equations of state, respectively.
gies. It has been discussed many times in literature and
also clear from the present findings that attractive mean-
field is dominating at E = 50 MeV /nucleon compared
to higher incident energies under consideration [6, 33].
This is due to the different mechanisms contributing at
low and high incident energies within isospin-dependent
quantum molecular dynamics. It was shown by us as
well as by others [4, 5, 12] that symmetry potential
dominates the physics at low incident energy, while, NN
cross sections one major driven force at higher incident
energies.Furthermore, at low incident energies, sym-
metry potential is repulsive for neutrons and attractive
for protons. With the increase in the asymmetry of a
reaction, the number of neutrons increases and hence
comparative repulsion due to neutrons also increases
leading to less attractive value of the flow. It is also clear
from ref. [17], that with the increase in asymmetry of
a reaction, the participant zone decreases and spectator
zone increases. At higher incident energies, where the
effect of symmetry energy is negligible, the contribution
of NN collisions comes from the participant zone whereas
mean-field contribution comes from the spectator zone.
Hence directed flow is found to be less positive with
increase in the asymmetry of the reaction.
d). The directed flow has less positive value with soft
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??
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FIG. 3:
asymmetries with and without Coulomb interactions at semi-
central geometry.
Excitation function of directed flow at different
equation of state compared to hard equation of state.
The less positive means the dominance of mean-field.
This is due to the different compressibilities of hard (380
MeV) and soft (200 MeV) equations of state. Naturally,
more is the compressibility, more are the number of
collisions and hence more positive is the directed flow.
This is indicating that the directed flow is sensitive
towards the equations of state.
Finally, in Figs.3 and 4, we display the excitation
function of directed flow at different asymmetries from
η = 0.2 to 0.7. The value of abcissa at zero value of
< Pdir
x
> corresponds to the energy of vanishing flow
(EVF) or alternatively, the balance energy (Ebal). The
Fig.3 shows the shift in the balance energy due to
Coulomb interactions, while, Fig. 4 is representing the
shift in the balance energy due to different equations
of state. In Fig. 3, one sees a linear enhancement in
the nuclear flow with increase in the incident energy.
This increase in the transverse flow is sharp at smaller
incident energies (upto 200 MeV/nucleon). If one goes
to higher incident energies, the value gets saturated
as discussed in ref [6].We have displayed here the
results upto 250 MeV/nucleon, since we are interested
in and around balance energy.
Coulomb interactions, more positive value of the flow
is obtained.This is due to the well known repulsive
In the presence of

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FIG. 4: Excitation function of directed flow with hard and
soft equations of state in the absence of Coulomb interactions
for different asymmetries.
nature of Coulomb interactions.
the repulsion due to Coulomb interactions is stronger
during the early phase of the reaction and transverse
momentum increases sharply. The overall effect depends
on the asymmetry of the reaction. If one looks at the
balance energy, the shift in the incident energy towards
the higher value is obtained at < Pdir
the asymmetry of the reaction.
with increase in asymmetry of the reaction and in the
absence of Coulomb interactions, attractive mean-field
is dominating the large region of incident energy. The
systematics of the balance energy with asymmetry of
the reaction is discussed in Fig. 5.
As we have seen in Fig.2, that different equations of
state show sensitivity towards the directed flow with
respect to the asymmetry of a reaction. The detailed
analysis with soft (S) and hard (H) equations of state
is displayed in Fig. 4.
directed flow follows similar trend as explained in Fig.
3. For nearly symmetric systems (η = 0.2), the balance
energy is found to be independent of the equations
of state, however, resonable differences are observed
at higher incident energies.
directed flow are obtained with hard equation of state
compared to soft equation of state.
all asymmetries from (η = 0.3 to 0.7). This is due to
At higher energies,
x
>= 0 with
This is showing that
The excitation function of
More positive values of
This is true at
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FIG. 5: Power law dependence of balance energy with asym-
metry of the reaction. The lower panel is representing the
relative % effect of Coulomb interactions on the balance en-
ergy.
different compressibilities of hard (380 MeV) and soft
(200 MeV) equations of state. With an increase in the
asymmetry of a reaction, the shift in the balance energy
towards the higher value of incident energy takes place
with soft equation of state compared to hard one. This
is consistent with the findings in the literature [11].
To sum up, in Fig.5, we have displayed the asymmetry
dependence of balance energy. We displayed the results
for hard and soft equations of state by switching off the
Coulomb interactions. In addition , for a comparative
study, the results in the presence of Coulomb interactions
with soft equation of state are also shown. All the lines
are fitted with power law of the form Ebal= C(η)τ,
where C and τ are the constants. The values of τ in
the absence of Coulomb interactions for soft and hard
equations of state are 0.375 and 0.282, respectively,
while in the presence of Coulomb interactions for soft
equation of state is τ = 0.06067.
If we compare the asymmetry dependence of balance
energy with mass dependence, the trend is opposite
[6]. It clearly indicates that if one wants to study the
isospin effect, then one has to consider the systems
with ATOT = constant, otherwise one is not able to
get the exact information about the isospin effects. It