We aim to see the sensitivity of collective transverse in-plane flow to
symmetry energy at low as well as high densities and also to see the effect of
different density dependencies of symmetry energy on the same.Comment: 3 pages, 2 figs, DAE BRNS Symposium on Nuclear Physics Dec 26-30,
  (2011), Visakhapatnam A.P. INDI

Sood, Aman D.

English

arXiv

Aman D Sood

CiteSeerX

Proceedings of the DAE Symp. on Nucl. Phys. 56 (2011) 834
Available online at www.sympnp.org/proceedings
Observable to explore high density behaviour of
symmetry energy
Aman D. Sood1∗
1SUBATECH, Laboratorie de Physique Subatomique et des Technologies Associées,
Université de Nantes-IN2P3/CNRS-EMN 4 rue Alfred Kastler, F-44072 Nantes, France.
Introduction
The nuclear matter equation of state (EOS)
of asymmetric nuclear matter has attracted
a lot of attention recently [1]. The EOS of
asymmetric nuclear matter can be described
approximately by
E(ρ, δ) = E0(ρ, δ = 0) + Esym(ρ)δ
2 (1)
where δ =
ρn−ρp
ρn+ρp
is isospin asymmetry, E0(ρ,
δ) is the energy of pure symmetric nuclear
matter, and Esym(ρ) is the symmetry energy.
The symmetry energy is important not only
to the nuclear physics community as it sheds
light on the structure of radioactive nuclei,
reaction dynamics induced by rare isotopes,
but also to astrophysicists since it acts as a
probe for understanding the evolution of mas-
sive stars and the supernova explosions. Al-
though Esym at saturation density is known
to be around 30 MeV its value at higher densi-
ties is poorly known. Experimentally, symme-
try energy is not a directly measurable quan-
tity and has to be extracted from observables
which are related to symmetry energy. There-
fore, a crucial task is to find such observables
which can shed light on symmetry energy. In
this paper, we aim to see the sensitivity of
collective transverse in-plane flow to symme-
try energy at low as well as high densities and
also to see the effect of different density de-
pendencies of symmetry energy on the same.
The various forms of symmetry energy used in
present study are: Esym ∝ (u), Esym ∝ (u)
0.4,
and Esym ∝ (u)
2, where u = ρ
ρ0
. The differ-
ent density dependencies of symmetry energy
∗
Electronic address: amandsood@gmail.com
are shown in Fig. 1. The present study is car-
ried out using IQMD model which is described
briefly in the next section.
The model
In IQMD [2] model the hadrons propagate
using Hamilton equations of motion:
d~ri
dt
=
d〈H〉
d~pi
;
d~pi
dt
= −
d〈H〉
d~ri
where (2)
〈H〉 = 〈T 〉+ 〈V 〉
=
∑
i
p2i
2mi
+
∑
i
∑
j>i
∫
fi(~r, ~p, t)V
ij(~r ′, ~r)
×fj(~r
′, ~p ′, t)d~r d~r ′ d~p d~p ′. (3)
The baryon potential Vij , in the above rela-
tion, reads as
V ij(~r ′ − ~r) = V ijSkyrme + V
ij
Y ukawa + V
ij
Coul
+V ijmdi + V
ij
sym. (4)
Where the potential terms in eq. 4 represent,
respectively, Skyrme, Yukawa, Coulomb, mo-
mentum dependent interaction, and symme-
try potential.
Results and Discussion
There are several methods used in the liter-
ature to define the nuclear transverse in-plane
flow. In most of the studies, one uses (px/A)
plots where one plots (px/A) as a function of
Yc.m./Ybeam. Using a linear fit to the slope,
one can define the so-called reduced flow (F).
Alternatively, one can also use a more inte-
grated quantity “directed transverse in-plane
flow 〈pdirx 〉” which is defined as [3]:
〈pdirx 〉 =
1
A
∑
i
sign{Y (i)} px(i), (5)
Proceedings of the DAE Symp. on Nucl. Phys. 56 (2011) 835
Available online at www.sympnp.org/proceedings
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
50
100
150
200
250
300
    = 0.4
    = 1.0
    = 2.0
 
 
E s
ym
(M
eV
)
/
0
E
sym
 = 32( /
0
)
FIG. 1: Different forms of symmetry energy.
where Y (i) and px(i) are the rapidity distribu-
tion and transverse momentum of the ith par-
ticle. In this definition, all rapidity bins are
taken into account. It therefore presents an
easier way of measuring the in-plane flow than
complicated functions such as (px/A) plots.
In Fig. 2 we display < px
A
> as a function of
Yc.m./Ybeam at final time (left panels) and the
time evolution of < pdirx > (right panels) cal-
culated at 100 (top panel), 400 (middle) and
800 MeV/nucleon (bottom) for different den-
sity dependencies of symmetry energy. Solid,
dash dotted, and dotted lines represent the
symmetry energy proportional to ρ, ρ0.4 and
ρ2, whereas dashed line represents calculations
without symmetry energy. Comparing the left
and right panels in Fig. 2, we find that both
the methods show similar behavior to symme-
try energy. For example, at incident energy of
100 MeV/nucleon for Esym ∝ ρ
0.4, < pdirx >
= 0. Similarly, the slope of < px
A
> at midra-
pidity is zero. We also find that the trans-
verse momentum is sensitive to symmetry en-
ergy and to its density dependences F1(u),
F2(u) and F3(u) in the low energy region (100
MeV/nucleon) only. At energies above Fermi
energy, both the methods show insensitivity to
the different symmetry energies. This could be
because (i) the repulsive nn scattering domi-
nates the mean field at high energies. (ii) As
explained in ref [3] that the effect of Esym on
nucleons during the initial stages of the reac-
tion (0-30 fm/c) decide the fate of final state
< pdirx >. At low energies duration of high
dense phase will prevail for a longer duration
-10
-5
0
5
10
-15
-10
-5
0
5
-60
-30
0
30
60
-10
0
10
20
30
40
50
-2 -1 0 1 2
-60
-30
0
30
60
0 20 40 60 80
-10
0
10
20
30
40
50
   0.4
    2
  
 
E
sym
E=800 MeV/nucleon
E=400 MeV/nucleon
 
 
E=100 MeV/nucleon
   
   off
<p
xd
ir >
 (M
eV
/c
)
 
<p
x/A
> 
(M
eV
/c
)
 
 
(f)(e)
(d)(c)
(b)
Y
c.m.
/Y
beam
 
Time (fm/c)
(a)
FIG. 2: Left panel: < px
A
> as a function of
Yc.m./Ybeam and < p
dir
x > at different incident
energies. Panels and lines are explained in the
text.
alllowing the Esym to affect the flow. Whereas
at high energies, although maximum density
reached will be larger but the duration of high
density phase will be very small not providing
Esym with sufficient time to make an impact
on flow.
Acknowledgments
This work has been supported by a grant
from Indo-French Centre for the Promotion of
Advanced Research (IFCPAR) under project
no 4104-1.
References
[1] B. A. Li et. al., Phys. Rep. 464, 113 (2008).
[2] C. Hartnack et. al., Eur. Phys. J. A 1, 151
(1998).
[3] S. Gautam et. al., Phys. Rev. C 83, 034606
(2011).


Observable to explore high density behaviour of symmetry energy

3 pages, 2 figs, DAE BRNS Symposium on Nuclear Physics Dec 26-30, (2011), Visakhapatnam A.P. INDIAWe aim to see the sensitivity of collective transverse in-plane flow to symmetry energy at low as well as high densities and also to see the effect of different density dependencies of symmetry energy on the same

Observable to explore high density behaviour of symmetry energy

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