Dedicated to Professor Apolodor Aristotel Răduţă’s 70th Anniversary
FROM NEUTRON SKIN TO PYGMY DIPOLE RESONANCE: THE ROLES OF
SYMMETRY ENERGY IN A TRANSPORT APPROACH
V. BARAN1,* , B. FRECUS2 , M. COLONNA3 , M. DI TORO3,4 , A. CROITORU5 , D. DUMITRU2
1
Physics Faculty, University of Bucharest, Romania
E-mail∗ : [email protected]
2
Doctoral School, Physics Faculty, University of Bucharest, Romania
3
Laboratori Nazionali del Sud INFN, I-95123 Catania, Italy
4
Physics and Astronomy Dept., University of Catania, Italy
5
”Theoretical Physics” Research Center, Physics Faculty, University of Bucharest, Romania
Received June 5, 2013
We investigate the effects of the symmetry energy on several properties of neutron rich nuclei, including the neutron skin thickness, polarizability and low energy
dipole response, within a microscopic transport model based on Landau-Vlasov kinetic equation. In an energy density functional approach we employ three different
parametrizations with density for the isovector part of the mean-field and study the evolution with mass of these properties. While the neutron skin and the polarizability are
directly correlated with the slope parameter L of the symmetry energy, for the pygmy
resonance a more careful discussion is required to characterize the role of the symmetry
energy on its properties.
1. INTRODUCTION
With the development of advanced experimental facilities at GSI, MSU, or
RIKEN, the detailed study of exotic nuclear systems, in extreme conditions of density and/or isospin has become possible. The recent experimental data are imposing
more stringent tests for the models aiming to describe the energy density functional
of asymmetric nuclear matter. Symmetry energy is influencing several static and dynamical properties of nuclei, including the collective modes. One of the important
issues in this context is to provide an appropriate density dependence of the symmetry energy which is able to explain in a unified manner phenomena at densities below
saturation as well as at large compressions of nuclear matter. This dependence plays
a crucial role in the development of the neutron skin of neutron rich nuclei as well
as in determining the structure of neutron stars, being intensively investigated in the
last decade [1, 2].
A useful method aiming to identify the role of the symmetry energy and to
provide constraints on its density dependence is based on the study of the correlations between different observables. In recent years there were explored some possible correlations between the properties of the Pygmy Dipole Resonance (PDR) and
the neutron skin of heavy nuclei [3], between PDR and the features of symmetry
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Vol. 1208–1220
58, Nos. 9-10,(2013)
P. 1208–1220,
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From neutron skin to pygmy dipole resonance
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energy [4, 5], or between various properties of finite nuclei including nuclear dipole
polarizability and the neutron skin thickness [6, 7].
The goal of this work is to investigate some of these correlations within a microscopic transport model based on Landau-Vlasov equations.
After a brief introduction to our approach and a description of the choices regarding the isovector sector of the nuclear equation of state we study the sensitivity of
the neutron skin to the value of the slope parameter L, which characterizes the density dependence of the symmetry energy around saturation. We consider the systems
48 Ca, 68 Ni, 86 Kr, 132 Sn and 208 Pb. Then we calculate the dipole strength function
and investigate the evolution with mass of the dipole polarizability for the three asyEOS. For all nuclei mentioned above we were able to identify a low energy collective
dipole response in the structure of strength function, which we associate with Pygmy
Dipole Resonance. We also discuss the mass and the symmetry energy dependence
of the energy centroid of PDR as well as the Energy Weighted Sum Rule (EWSR)
exhausted by this mode.
2. A SEMICLASSICAL APPROACH TO NUCLEAR DYNAMICS
The well known many-body methods, in particular Hartree-Fock theory and
its descendants, are searching for an appropriate many-body wave function of the
physical system. Once the Hamiltonian is introduced, a variational principle problem
is solved within a defined trial space. However when considering realistic systems,
these formalisms can become difficult to implement, the computational requirements
being one of the reasons.
For nuclear systems, due to the presence of the short-range repulsive core, such
approaches may also lead to quite inaccurate results since the nuclear many-body
problem manifests non-perturbative features with respect to the independent particle
models. Even for the soft-core interactions, which allows for a perturbative treatment of the nuclear many-body problem, the lowest order is not sufficient to reach
a convergent solution. Therefore, in nuclear physics a different direction, namely
the nuclear energy density functional (EDF), was adopted to treat still rigorously the
problem of strongly interacting nucleons. This approach has many common features
with the density functional theory, which was proposed initially for the description of
the electronic systems. The ideas behind density functional theory can be extended to
nucleons interacting through the strong force, but in contrast to electronic systems,
the self bound nature of the nuclei makes the original Hohenberg-Kohn work not
entirely applicable (here there is no external potential that confines the nucleons).
Nuclear EDT were introduced starting from effective interactions. For a Skyrme
interaction including effects from three-body forces, within a Hartree-Fock approximation, a functional depending on local proton and neutron densities was obtained
RJP 58(Nos. 9-10), 1208–1220 (2013) (c) 2013-2013
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already in the 70’s [8]. In this case the ground state energy of the system characterized by the effective Hamiltonian H = T + Vef f , considering the wave function to
be a single Slater determinant, can be derived using the Hartree-Fock approximation
as:
Z
E = hψ|(T + Ve f f )|ψi =
d3 r E(r),
(1)
where E(r) = Ekin (r) + Epot (r) represents the energy density functional. Even if
the functional is obtained from an effective interaction and a many-body variational
method, there are several subtle differences between HF and EDF approach. The
coefficients factorizing the terms which are functions of the local density and of the
density gradient, are adjusted by requiring that a set of experimental properties such
as the saturation density, the binding energy at saturation, the incompressibility modulus, the symmetry energy at saturation and possibly other features, are reproduced.
Therefore, these coefficients encode more physics than those provided within HF theory. By employing the energy density functional technique one can also determine
the self-consistent nuclear mean-field as U (ρ) = δEpot (r)/δρ(r), (ρ = A/V ). This
quantity is one of the main ingredients required in our transport model devoted to
describe the nuclear dynamics.
Nuclear matter is a two components system and to describe its EDF it is possible to introduce as degrees of freedom proton and neutron densities ρp ρn or
total, isoscalar density ρisoscalar = ρ = ρn + ρp and isospin or isovector density
ρisovector = ρi = ρn − ρp . The total energy per nucleon can be written as [1]:
Esym
E
E
(ρ, I) = (ρ) +
(ρ)I 2 ,
(2)
A
A
A
ρi N − Z
E
where the isospin parameter is I = =
while (ρ) describes the properties
ρ
A
A
of symmetric nuclear matter, for which I = 0. The energy density functional is
E
E
= E(ρ, ρi ) = Ekin + Epot = ρ (ρ, ρi ). An important thermodynamic response
V
A
1 ∂V
1 ∂ρ
function is the compressibility β = −
=
which characterizes the relative
V ∂P
ρ ∂P
variation of the nuclear density with the pressure. Here the pressure of the system,
∂E
dε
at zero temperature, is given by P = −
= ρ2
with ε = E/A = εkin + εpot . In
∂V
dρ
equilibrium conditions, at saturation density, the pressure is equal to zero, P (ρ0 ) = 0.
A related quantity is the nuclear incompressibility modulus, K defined as:
9
18
d2 ε
K = β −1 = P (ρ) + 9ρ2 2 ,
ρ
ρ
dρ
(3)
which, at saturation, is a measure of the curvature parameter of the EOS as a function
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From neutron skin to pygmy dipole resonance
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of density. The nuclear incompressibility is linked to the velocity of the sound which
can propagates in nuclear matter and to the frequency of giant monopole resonances.
Taking into account the experimental information we shall require that at saturation
the symmetric nuclear matter has the following properties: (i) the equilibrium density
ρ0 = 0.16 f m−3 , (ii) the energy per nucleon E/A(ρ0 ) = −16 MeV/nucleon, while
(iii) the incompressibility modulus, corresponding to a soft-EOS, K(ρ0 ) = 201 MeV.
For the isovector sector, from the symmetry energy coefficient definition asym =
Esym /A = εsym
in the Bethe-Weizsäcker mass formula [9], we can write asym =
1 ∂2 E
(ρ, I) . We shall obtain a kinetic contribution, associated with Pauli
2
2 ∂I A
I=0
correlations, as well as a contribution from the nuclear interaction:
Esym
1
εsym ≡
(ρ) = εsym kin + εsym pot = εF (ρ) + εsym pot ,
(4)
A
3
~2 3π 2 2/3 2/3
where εF (ρ) =
ρ
is the Fermi energy. From asym provided by the
2m 2
Bethe-Weizsäcker formula we notice that the kinetic contribution represents less than
a half, the remaining part being a consequence of the properties of the nucleonnucleon interaction.
Around the saturation point the symmetry energy dependence on density is
characterized by the values of the slope L and curvature Ksym parameters, appearing
in the expansion up to the second order of εsym (ρ) [1, 2]:
Ksym ρ − ρ0 2
L ρ − ρ0
+
,
(5)
εsym (ρ) = εsym (ρ0 ) +
3
ρ0
18
ρ0
2
dεsym 2 d εsym Then L = 3ρ0
and
K
=
9ρ
. To exemplify, for a simsym
0
dρ ρ=ρ0
dρ2 ρ=ρ0
plified Skyrme-like effective interaction [9]:
r + r σ−1
1
i
j
)
δ(ri − rj ) , (6)
Vij = t0 (1 + x0 Pσ )δ(ri − rj ) + t3 (1 + x3 Pσ ) ρ(
6
2
with the coefficients t0 = −2973 M eV · f m3 , t3 = 19034 M eV · f m(3×σ) , x0 =
7
0.025, x3 = 0, σ = , the interaction contribution to the energy density functional
6
has the form:
A ρ2
B ρσ+1 C(ρ) ρ2i
+
+
,
(7)
2 ρ0 σ + 1 ρσ0
2 ρ0
i
3
σ+1 σ
1 h
t3
where A = t0 ρ0 , B =
t3 ρ0 , C(ρ) = − ρ0 t0 (1 + 2x0 ) + (1 + 2x3 )ρσ−1 .
4
16
4
6
In the spirit of EDF formalism, in the following we shall resume to a potential
energy density defined by Eq. (7). The mean-field potentials for protons and neutrons
Epot (ρ, ρi ) =
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are the functional derivatives of Epot with respect to the proton (neutron) density [9]:
ρ σ
δEpot
ρ
ρi 1 dC(ρ) ρ2i
Up =
= A +B
− C(ρ) +
,
(8)
δρp
ρ0
ρ0
ρ0 2 dρ ρ0
ρ σ
δEpot
ρ
ρi 1 dC(ρ) ρ2i
= A +B
+ C(ρ) +
.
(9)
δρn
ρ0
ρ0
ρ0 2 dρ ρ0
However, the knowledge of the equation of state for asymmetric nuclear matter beyond normal conditions still remains poor. Based on the philosophy of EDF, while
keeping the value of symmetry energy at saturation almost the same, we shall allow
for three different dependences with density away from equilibrium. For the asysoft
EOS we imply a Skyrme-like, SKM*, parametrisation, the symmetry term being almost flat around saturation density and slowly decreasing towards higher densities (a
small slope parameter, L = 14.4M eV ). For the asystiff EOS C(ρ) is density independent (C(ρ) = 32M eV , L = 72M eV ), and thus the potential part of the symmetry
E
energy sym
A will increase linearly with respect to the nuclear density. Lastly, for the
asysuperstiff EOS, the symmetry term increases rapidly around saturation density,
ρ2
. It becomes clear that away from the satL = 96.6M eV , proportional to
ρ0 (ρ + ρ0 )
uration density, ρ0 = 0.16 fm−3 , the mean fields perceived by protons and neutrons
are quite different for the three equations of state introduced above.
We shall consider a semi-classical transport approach based on two coupled
Landau-Vlasov kinetic equations for protons and neutrons valid in a large domain of
energies, from Coulomb barrier until hundreds of MeV per nucleon. These equations
determine the time dependence of neutron and proton one-body distribution functions
fq (r, p, t) with q = n, p:
Un =
∂fq p ∂fq ∂Uq ∂fq
+
−
= Icoll [fn , fp ] .
(10)
∂t
m ∂r
∂r ∂p
in the presence of mean-field and two-body collisions. If the effects of collision integral are neglected is arrived at Vlasov equation. This can be obtained from the
Wigner transform of the Time Dependent Hartree-Fock equations by considering the
classical limit [10]. We neglect here the collision integral Icoll [f ] for fermionic systems since the excitation energies are quite small. The values of the coefficients
appearing in Eqs. (8,9) are A = −356 MeV, B = 303 MeV, σ = 7/6. The effective
nucleon mass is equal to the bare mass, 940 MeV. Then, the saturation density of
symmetric nuclear matter is 0.16f m−3 , the binding energy is EB = −16M eV /n,
while the compressibility modulus takes the value K = 200 MeV, [1]. The numerical procedure to integrate the transport equations is based on pseudo-particle (or
test-particle, t.p.) method. To accommodate a reasonable running time and a good
spanning of phase-space we work with 1300 t.p. per nucleon.
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Once the ground state was prepared we focus first on the predictions of our
model regarding the proton and neutron distributions, see also [11]. An efficient method to determine the mean-square radius of protons and neutrons is to observe their
small amplitude oscillations around the equilibrium values after a weak perturbation.
In Fig.1 we show the time evolution of proton and neutron mean square radii, Rp and
Rn , defined as:
q
p
Rn = hrn2 i ; Rp = hrp2 i .
(11)
where:
Z
1
=
d3 r r2 ρp (r, t) ,
Np
Z
1
2
hrn i =
d3 r r2 ρn (r, t) .
Nn
hrp2 i
(12)
(13)
We remark the stability of the evolution for more than 1800f m/c. Using this procedure we obtain for charge mean square radius of 208 Pb a value around Rp = 5.45 fm,
to be compared with the experimental value Rp,exp = 5.50 fm.
7
6.5
6
Rn,p(fm)
5.5
5
4.5
4
3.5
3
2.5
0
250
500
750
1000
1250
1500
1750
t(fm/c)
p
2
Fig. 1 – The time evolution of the
qneutron mean square radius Rn = hrn i (thick lines) and of the
proton mean square radius Rp = hrp2 i (thin lines) after a weak perturbation of the ground state. From
the top the pairs of lines correspond to 208 Pb (red), 132 Sn (blue), 68 Ni (green) and 48 Ca (maroon). The
asystiff EOS case.
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In Figure 2 is represented the isospin dependence of neutron skin thickness
∆Rnp :
q
p
∆Rnp = Rn − Rp = hrn2 i − hrp2 i ,
(14)
predicted by our model for the three asy-EOS introduced above. A quite linear variation of ∆Rnp with I 2 is observed. We see that the neutron skin size increases with
the slope parameter L, a more pronounced variation being evidenced when heavier
systems are considered. This effect is related to the tendency of the nuclear systems
to stay more symmetric even at lower densities if the symmetry energy varies slowly,
as is the case of asysoft EOS. Our results are consistent with the relation expected
between the neutron skin thickness and the values of isovector Landau parameter F 0
below saturation, [12].
0.35
0.35
(a)
(b)
0.3
∆Rnp(fm)
∆Rnp(fm)
0.3
0.25
0.2
0.15
0.1
0.25
0.2
0.15
0.015 0.03 0.045 0.06 0.075
I
0.1
0
20 40 60 80 100 120
2
L(MeV)
Fig. 2 – (a) The neutron skin thickness as a function of isospin parameter I: asysoft EOS (red, triangles),
asystiff EOS (blue, squares), asysuperstiff EOS (black, circles). (b) The neutron skin size dependence
on slope parameter L. From the top the lines correspond to 132 Sn (blue, triangles left), 208 Pb (maroon,
diamonds), 68 Ni (red, squares), 86 Kr (green, circles) and 48 Ca (black, triangles down).
3. DIPOLE STRENGTH FUNCTION
Since one of our goals is to explore the possible correlations between the neutron skin and the pygmy mode we shall investigate in this section the dipolar response
within a Vlasov approach. We consider the case of ”GDR-like” initial excitation,
boosting at t = t0 all neutrons along z direction and in opposite direction all protons,
while keeping the center of mass of the nucleus at rest. In this case one can relate the
strength function to the imaginary part of the Fourier transform of the total dipole,
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From neutron skin to pygmy dipole resonance
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D(ω), [13]. The total dipole momentum is expressed as [14]:
NZ
X;
(15)
A
where X is the neutron-proton relative coordinate. The conjugate momentum to X
is:
NZ 1
1
P=
( PZ − PN ) .
(16)
A Z
N
We approximate the excitation process at t = t0 with an instantaneous perturbation:
D=
NZ
X,
(17)
A
with η being the perturbation strength, and D the dipole operator. The effect of
this perturbation corresponds indeed to a dipolar velocity field superimposed on the
ground state, |Φ0 i, so that the new state is:
Vext = ηδ(t)D ≡ ηδ(t − t0 )
|Φ(0)i = eiηD |Φ0 i .
(18)
Then the expectation value of the collective momentum P at t = t0 becomes:
NZ
,
(19)
A
since the expectation value of the collective momenta in the ground state, hΦ0 |P|Φ0 i,
should be zero.
The strength function:
X
S(E) =
|hn|D|0i|2 δ(E − (En − E0 )),
(20)
hΦ(0)|P|Φ(0)i = hΦ0 |e−iηD PeiηD |Φ0 i = ~η
n6=0
where En are the excitation energies of the states |ni while E0 is the energy of
the ground state |0i = |Φ0 (t0 )i, is obtained from the imaginary part of the Fourier
transform
of the time-dependent expectation value of the dipole momentum D(ω) =
Z
D(t)eiωt dt as:
S(E) =
Im(D(ω)) N Z Im(D(ω))
=
.
πη~
A πhΦ(0)|P|Φ(0)i
(21)
In the last equality we used Eq. 19.
In the numericalZsimulations the Fourier transform was obtained over a finite
tmax
time domain, D(ω) =
t0
D(t)eiωt dt where tmax = 1830 fm/c and t0 = 30 fm/c.
In order to eliminate the artefacts appearing from a finite time domain analysis of
the signal, a filtering procedure, as described in [15], was applied. Consequently a
πt
smooth cut-off function was introduced such that D(t) → D(t)cos2 (
). In Fig.
2tmax
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3 we report the dipole strength functions for 132 Sn for the three asy-EOS.
We tested
Z
∞
ES(E) dE
the accuracy of this procedure by comparing the numerical value
0
2
S(E)(fm /MeV)
~2 N Z
with the expected theoretical value
and concluded that the differences are
2m A
of only few percentages.
30
30
30
25
25
25
20
20
20
15
15
15
10
10
10
5
5
5
0
0
0
0 3 6 9 12 15 18
0 3 6 9 12 15 18
E(MeV)
E(MeV)
0 3 6 9 12 15 18
E(MeV)
Fig. 3 – (Color online) The strength function for 132 Sn: asysoft (right), asystiff (center) and asysuperstiff (left) EOS.
We notice that the energy centroid situated at higher energies is symmetry
energy dependent, having larger value for asysoft EOS, while the peak at lower energies, around 7.5 − 8.0 MeV, is very weakly dependent on the parametrization with
density of the isovector mean field.
Once the strength function is known we can also calculate the nuclear dipole
polarizability as:
Z
2
αD = 2e
0
∞
S(E)
dE.
E
(22)
This quantity is plotted as a function of mass and asy-EOS in Fig. 4 for 208 Pb,
132 Sn, 86 Kr, 68 Ni and 48 Ca. In the case of 208 Pb for example, our predictions are
between 21 fm3 and 28 fm3 when we pass from asysoft to asysuperstiff EOS. The
increase with mass of dipole polarizability can be traced back to the corresponding
decrease of GDR energy centroid with mass. The correlation which manifests in
our approach is that for a given system, the greater the skin thickness, the larger the
dipole polarizability.
RJP 58(Nos. 9-10), 1208–1220 (2013) (c) 2013-2013
From neutron skin to pygmy dipole resonance
(a)
3
30
27
24
21
18
15
12
9
6
3
0
0
αD(fm )
3
αD(fm )
10
50
100 150 200 250
30
27
24
21
18
15
12
9
6
3
0
0
1217
(b)
20 40 60 80 100 120
A
L(MeV)
Fig. 4 – (a) The dipole polarizability as a function of mass number: asysoft EOS (red, triangles) asystiff
EOS (blue, squares) and asysuperstiff EOS (black, circles). (b) The dipole polarizability as a function
of slope parameter L. From the top the lines correspond to the systems 208 Pb (maroon, diamonds),
132
Sn (blue), 86 Kr (green), 68 Ni (red) and 48 Ca (black).
4. PYGMY DIPOLE RESONANCE
We notice that from our calculations, as is also observed for 132 Sn in Fig. 3,
results that the dipolar response presents two peaks for all studied systems. Higher
energy centroid is associated with Giant Dipole Resonance, while the peak at lower
energies, around 10 MeV in the case of 68 Ni, 8 MeV in the case of 132 Sn and 7.3 MeV
for 208 Pb, corresponds to Pygmy Dipole Resonance. The nature and the properties
of these states were of great interest during the last few years, see [16], [17], [18] for
detailed discussions from theoretical and experimental perspectives. In our approach
the low energy dipole response manifests collective features and the peak position
is weakly dependent on symmetry energy parametrization with density. In Figure
5 we represented the mass dependence of the centroid energy of PDR considering
again the nuclei 48 Ca, 68 Ni, 86 Kr, 132 Sn and 208 Pb. We conclude that a very good
description with mass for these systems is provided by the parametrization EP DR =
41A−1/3 . Quite recently, using the virtual photon scattering technique, Wieland et
al. [19] reported the existence of a peak at approximately 11 MeV, attributed to the
low lying dipole response of the neutron rich nickel isotope, 68 Ni. This peak is
found to be well below the GDR centroid, whose energy lies near ∼ 17 MeV. With
our semi-classical investigation we report the values ∼ 9.8 MeV and ∼ 16 MeV for
PDR and GDR respectively. For the same system, by means of a random phase
approximation approach with various types of interactions, Roca-Maza et al. [20]
reported the pygmy centroid to range from 9.3 MeV (using the SLy5 force) to 10.45
MeV (using the SkI3 force). In the same study, the energy centroid corresponding to
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V. Baran et al.
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0.05
(a)
16
(b)
0.04
14
0.03
12
fy
Epygmy(MeV)
11
10
0.02
8
0.01
6
4
0
40 80 120 160 200 240
0
0
A
50
100 150 200
A
Fig. 5 – (a) The mass dependence of the energy centroid of Pygmy Dipole Resonance. The blue line
corresponds to the parametrization 41A−1/3 . (b) The EWSR fraction exhausted by the PDR: asysoft
EOS (squares, maroon) asystiff EOS (red, squares) and asysuperstiff EOS (blue,circles).
the PDR for tin, 132 Sn, and lead, 208 Pb, isotopes was reported. It ranges from 8.52
MeV (using the SGII force) to 9.23 MeV (with SkI3) for tin, and from 7.61 MeV
(SGII) to 8.01 MeV (SkI3) for lead. In our model we find the PDR centroid at 7.5 −
8.0 MeV for 132 Sn and 7.3 MeV for 208 Pb observing a quite reasonable agreement
between these approaches. Similar results are obtained within the relativistic RPA
framework. Piekarewicz [3] reported a pygmy centroid around ∼ 8 MeV (employing
various interactions, FSUGold, FSUGold’, NL3 and NL3’) for tin nucleus. These
theoretical predictions seems to be close to experimental data for 132 Sn reported by
Adrich et al. [21], based on Coulomb dissociation technique following an in-flight
fission of a 238 U beam, which obtained a PDR peak at 11 MeV. For 208 Pb, Tamii et
al. [22], within a polarized proton inelastic scattering at very forward angles method,
observed a low energy resonance-like structure at around 7.37 MeV.
From the Fig. 5 (b) we see that it is difficult to establish a direct correlation
between the EWSR fraction exhausted by PDR and the mass of the system. However
the results clearly show that EWSR is influenced by the parametrization with density
of symmetry energy. The fraction fy is enhanced when we pass from asysoft to
asysuperstiff EOS, i.e. with the increase of the slope L. Since we also observed that
the neutron skin thickness depends on L a correlation between the fraction fy and
the slope parameter L is expected. Indeed, a quite linear dependence is observed
in Fig. 6 between the two quantities for a given system. Moreover, it seems that
with an appropriate rescaling for each nucleus, all points can collapse on a unique
line. However this observation requires a more detailed analysis which should also
be extended to other systems and will be presented elsewhere [23].
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0.05
0.04
fy
0.03
0.02
0.01
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
∆Rnp(fm)
Fig. 6 – The correlation between the EWSR fraction exhausted by PDR and the neutron skin thickness.
The results are represented for the three asy-EOS considering the systems: 132 Sn (violet, triangles
down), 208 Pb (blue, triangles up), 68 Ni (green, diamonds), 86 Kr (red, squares) and 48 Ca (black, triangles down).
5. CONCLUSIONS
In this work we studied various correlations between the features of symmetry
energy and several observables characterizing the properties of atomic nuclei, within
a microscopic transport model and an energy density functional approach. We notice that the neutron skin size, the dipole polarizability as well as the EWSR fraction
exhausted by pygmy mode are proportional with the value of slope parameter L characterizing the density dependence of symmetry energy around saturation. Our model
predict, for all studied systems, a collective response below Giant dipole Resonance,
whose energy centroid is well described by the dependence EP DR = 41A−1/3 and
very weakly influenced by the symmetry energy. Based on an improved numerical
implementation of the transport model we can extend to other systems this analysis
aiming to uncover additional correlation between the properties of symmetry energy
and the dynamics of collective modes, including Pygmy Dipole Resonance. A systematic investigation through periodic system of the properties of PDR, in connection also with experimental analysis, may provide new insights regarding its nature
as well as additional constraints on the density dependence of symmetry energy.
Acknowledgements. This work for V. Baran and A. Croitoru was supported by a grant of the
Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-IDPCE-2011-3-0972.
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