Plasma Science and Technology, Vol.14, No.7, Jul. 2012
Theoretical Study of the Nuclear Charge Distributions of Tin
Isotopes∗
LIU Jian (刘健)1 , CHU Yanyun (褚衍运)1 , REN Zhongzhou (任中洲)1,2
1
2
School of Physics, Nanjing University, Nanjing 210008, China
Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator,
Lanzhou 730000, China
Abstract
Nuclear binding energies, charge radii and the charge distributions of even-even tin
(Sn) isotopes are calculated using relativistic mean field theory, and the theoretical results are
found to be in accordance with the experimental data. The nuclear charge form factors for Sn
isotopes are calculated using the phase-shift analysis method. It is shown that the minima of the
charge form factors shift upward and inward with an increase in the neutron number of the Sn
isotopes.
Keywords: elastic electron-nucleus scattering, relativistic mean-field theory, phase-shift
analysis method, nuclear charge form factor
PACS: 21.10.Ft, 25.30.Bf, 27.60.+j
DOI: 10.1088/1009-0630/14/7/11
1
Introduction
It has been important and exciting to investigate the
properties of exotic nuclei in both experimental and
theoretical nuclear physics. With the development of
radioactive ion beam facilities, it is possible to produce
short-lived exotic nuclei and investigate their properties
in a laboratory [1∼4] . In the past several decades, many
exotic nuclear properties, such as nuclear bubbles, nuclear skins and nuclear halos, have been predicted or
discovered, all of which are relevant to nuclear matter distributions [5∼11] . Until now, the charge density
distributions of exotic nuclei are mainly deduced from
nucleus-nucleus collisions, and explanations of such experimental data are model-dependent. To get the precise nuclear charge densities of unstable nuclei, experiments on the elastic electrons scattering off these nuclei
should be carried out. Electron scattering experiments
for stable nuclei have been a powerful tool to investigate
nuclear charge densities [12∼21] , and it is only natural
to apply this experimental method to studying unstable
nuclei. Experimental facilities for this purpose are now
under construction in RIKEN and GSI [22∼25] , and a
demonstration experiment of electron scattering off unstable nuclei has been implemented in RIKEN [26] .
In recent years, much theoretical and experimental research into tin (Sn) isotopes has been carried
out [27∼32] . The neutron skin thickness of Sn isotopes is
an important property for understanding nuclear structure and the nuclear equation of state [33,34] . With neutron numbers increasing in Sn isotopes, neutron skins
or neutron halos may appear. For the neutron skin
type distribution, the half-density radius of a neutron
is larger than that of a proton. For the halo type dis∗ supported
tribution, the half-density radius of a neutron is nearly
equal to that of a proton, but the diffuseness parameter, which is used to describe the neutron distribution,
is larger than that of the proton [35,36] . The Sn isotopes have the same proton number, but the different
neutron properties can result in different proton distributions (and charge distributions) along the Sn isotopic
chain, and the information about the nuclear charge
distributions can be extracted from the corresponding
elastic electron scattering experiments. Many elastic
electron-nucleus scattering experiments for Sn isotopes
were performed during the 1970s [37∼41] , but they only
covered the stable isotopes near the nuclear stability
line.
Among Sn isotopes, the doubly-magic nucleus 132 Sn
plays an important role in the study of nuclear physics.
For a long time our knowledge about the doubly-magic
nuclei has been limited to the stable ones near the stability line, such as 16 O, 40 Ca, 48 Ca, 56 Ni and 208 Pb.
During the past several decades there has been substantial progress in the experimental study of nuclei far
from the stability line, and the development of radioactive nuclei beams is currently giving a strong impetus
to the study of the exotic nuclei 100 Sn and 132 Sn. These
experiments will provide new data on the single-particle
energies and spin-parities of nuclear levels. Precise determination of the nuclear skin thickness in 132 Sn and
208
Pb can set strong constraints on the nuclear equation of state. Therefore, the accurate nuclear charge
density distributions of these two nuclei are important
in understanding neutron skin thickness and neutron
properties.
In this article we study the properties of Sn iso-
by National Natural Science Foundation of China (Nos. 10535010, 10675090, 10775068, 10735010, 10975072 and 11035001),
by the 973 National Major State Basic Research and Development of China (2007CB815004), by CAS Knowledge Innovation Project
(KJCX2-SW-N02), and by the Research Fund of Doctoral Point (RFDP) (Nos. 20070284016, 20100091110028)
LIU Jian et al.: Theoretical Study of the Nuclear Charge Distributions of Tin Isotopes
topes ranging from the proton-rich isotope 100 Sn to
the neutron-rich isotope 132 Sn. The basic properties
of these nuclei (including binding energies, charge radii
and charge distributions) are generated from the relativistic mean-field (RMF) theory, and the theoretical
results of elastic electron scattering off these nuclei are
calculated with the phase-shift analysis method. This
article is organized as follows. Section 2 briefly introduces the theoretical methods used in the current study,
Section 3 presents the numerical results and discussion,
and the last section provides a summary.
2
Theory
RMF theory has achieved great success in describing
different nuclear properties, including binding energies
and matter distributions. Here we use the RMF model
with the following effective Lagrangian [42∼45]
L = Ψ̄(iγ µ ∂µ − M )Ψ − gσ Ψ̄σΨ − gω Ψ̄γ µ ωµ Ψ
1
1
1
1
−gρ Ψ̄γ µ ρaµ τ a Ψ+ ∂ µ σ∂µ σ− m2σ σ 2 − g2 σ 3 − g3 σ 4
2
2
3
4
1 2 µ
1
1 ~ µν ~
1 µν
µ 2
− Ω Ωµν + mω ω ωµ + c3 (ωµ ω ) − R · Rµν
4
2
4
4
1 2 µ
1 µν
1
µ
+ mρ ρ
~ ρ
~µ − F Fµν − eΨ̄γ Aµ (1 − τ 3 )Ψ, (1)
2
4
2
with the tensor field
Ωµν = ∂ µ ω ν − ∂ ν ω µ ,
(2)
~ µν = ∂ µ ρ
R
~ν − ∂ ν ρ
~µ ,
(3)
µν
µ ν
ν µ
F =∂ A −∂ A .
(4)
This model takes into account the fields of nucleons (Ψ),
photons (Aµ ), σ mesons, ω mesons and ρ mesons. The
masses of all the field particles and coupling constants
can be determined according to some well-known particle masses and by fitting some experimental nuclear
properties.
The proton and neutron density distributions can be
obtained by summing up all the single proton distributions and all the single neutron distributions, respectively. The charge density distribution is assumed to
be totally contributed by the protons, and can be obtained in the following way
Z
ρ(r) = ρp (r0 )ρp (r − r0 )dr0 ,
(5)
where ρp (r) is the nuclear proton distribution and ρp (r)
is the single proton charge distribution. The electrostatic potential for the electrons is expressed in the following way
Z
ρ(r0 )
e2
dr0 ,
(6)
Vc (r) = −
4πε0
|r − r0 |
Since ρ(r) is spherical, Vc (r) is also spherical [46] .
The electrons are scattered in a modified Coulomb
field of nuclei:
V (r) = Vc (r) + Vsr (r),
(7)
where Vsr (r) is a short range potential that vanishes for
r> rc .
To calculate the inner phase shifts, d, the radial equations are integrated from r=0 up to a certain distance
rm beyond the range rc of V sr (r). For r> rm , the
field is purely Coulombian, and the normalized uppercomponent radial Dirac function can be expressed as:
u
u
PEk (r) = cos dk fEk
(r) + sin dk gEk
(r),
µ
fEk
(r)
(8)
µ
gEk
(r)
where
and
are regular and irregular
Dirac-Coulomb functions for infinity. As usual, the
phase shift dk is determined by matching this outer
analytical form to the inner numerical solution at rm ,
requiring continuity of the radial function PEk (r) and
its derivative.
The elastic electron-nucleus scattering cross section
is [47∼49]
dσ
= |f (θ)|2 + |g(θ)|2 ,
(9)
dΩ
with
∞
+
−
1 X
f (θ) =
[(l + 1)(e2iδl − 1) + l(e2iδl − 1)]Pl (cos θ),
2ik
l=0
(10)
and
g(θ) =
∞
−
1 X 2iδ+
[e l − e2iδl ]Pl1 (cos θ),
2ik
(11)
l=0
in which P l and P 1l are the Legendre function and associated Legendre function, respectively, and δ ± are the
spin-up and spin-down phase shifts of the partial waves
of the electrons moving in the field V (r). The charge
form factor is defined as
dσ/dΩ
|F (q)|2 =
,
(12)
dσM /dΩ
where dσM /dΩ is the Mott cross section and q denotes
the momentum transfer.
3
Numerical results and discussion
We choose the force parameter sets NL1 and NL3
for investigating Sn isotopes with the RMF model.
The pairing gaps for the Sn isotopes are included
by the BCS √treatment with the neutron pairing gap
∆n = 11.2/ A MeV. Detailed results of the isotopes
100
Sn, 112 Sn, 116 Sn, 118 Sn, 124 Sn and 132 Sn are presented in Table 1. In this table, B/A (MeV), R c (fm)
and S p (MeV) are, respectively, the binding energy
per nucleon, root-mean-square (RMS) radius of the
charge distribution, and the one-proton separation energy. The experimental data for the binding energies
per nucleon and the one-proton separation energies are
taken from Ref. [50], and the experimental charge RMS
radii are taken from Ref. [28]. It can be seen from Table 1 that the RMF model can well describe the groundstate properties of Sn isotopes both near and far from
the stability line. According to the calculation, the
experimental binding energies per nucleon and charge
RMS radiiof the Sn isotopes are quantitatively repro615
Plasma Science and Technology, Vol.14, No.7, Jul. 2012
Table 1.
The RMF results with the NL1 and NL3 parameter sets, and the corresponding experimental data
100
B/A (expt.) (MeV)
B/A (NL1.) (MeV)
B/A (NL3.) (MeV)
R c (expt.) (fm)
R c (NL1.) (fm)
R c (NL3.) (fm)
S p (expt.) (MeV)
S p (NL1.) (MeV)
S p (NL3.) (MeV)
Sn
8.248
8.350
8.284
4.47
4.47
2.80
3.75
3.62
112
Sn
8.513
8.508
8.475
4.59
4.60
4.59
7.55
9.19
8.81
duced with two parameter sets, so it may be reasonable
and reliable for us to use the RMF model to calculate
the charge density distributions of the Sn isotopes. We
can also see from Table 1 that the one-proton separation energies, which denote the single particle energies
of the last proton, become smaller when the neutron
number of the Sn isotope gradually decreases. This indicates weak binding of the outermost protons in the
proton-rich isotopes, and weak binding of the last proton in proton-rich isotopes result in a spread-out charge
density distribution.
The theoretical nuclear charge density distributions
of 100 Sn, 112 Sn, 124 Sn and 132 Sn are calculated with the
NL1 parameter set and presented in Fig. 1. In this figure, the X-axis is the radial coordinate and the Y -axis
is the charge density ρ(r). It can be seen from Fig. 1
that the weak binding of the protons in the last nuclear
orbit of 100 Sn leads to an extended tail charge density
distribution. In Fig. 1 we also note that the central nuclear charge densities gradually decrease as the neutron
number increases. Similar results were also obtained
and studied in our previous research [51] . The theoretical nuclear charge density distributions of 100 Sn, 112 Sn,
124
Sn and 132 Sn with the NL3 parameter set are presented in Fig. 2. The results with the two different
parameter sets are in accordance with each other.
116
Sn
8.523
8.526
8.482
4.63
4.62
4.61
9.28
10.27
9.83
118
Sn
8.516
8.532
8.461
4.64
4.64
4.63
10.00
10.89
10.52
124
Sn
8.467
8.467
8.448
4.68
4.67
4.66
12.10
13.47
12.89
132
Sn
8.355
8.324
8.361
4.72
4.71
15.71
16.93
16.02
of even-even Sn isotopes are 0+ , the elastic scattering
cross sections and form factors are determined only by
their static charge distributions rather than their nuclear current distributions.
Fig.2 The charge density distributions of 100 Sn, 118 Sn,
124
Sn and 132 Sn from RMF theory with the NL3 parameter set
Before we study the electron scattering off the unstable Sn isotopes, it is necessary for us to check the validity of phase-shift analysis method. It is shown in Fig. 3
that the theoretical cross sections are in good agreement
with the experimental ones for the stable Sn isotopes
112
Sn, 116 Sn, 118 Sn and 124 Sn. The amplitudes, minima and maxima of the experimental scattering cross
sections are well reproduced. So we can draw the
Fig.1 The charge density distributions of 100 Sn, 118 Sn,
Sn and 132 Sn from RMF theory with the NL1 parameter set
124
In order to find out the regularities in elastic electron
scattering off different isotopes along the Sn isotopic
chain, we calculate the nuclear charge form factors and
cross sections for the unstable exotic nuclei 100,132 Sn
and the stable isotopes 116,118,124 Sn with the phase-shift
analysis method. Since the ground-state spin-parities
616
Fig.3 Cross sections of elastic electron scattering off stable Sn isotopes 112 Sn, 116 Sn, 118 Sn and 124 Sn at 225 MeV.
The corresponding charge density distributions are calculated with the NL1 parameter set, and the experimental
data are taken from Refs. [28,31]
LIU Jian et al.: Theoretical Study of the Nuclear Charge Distributions of Tin Isotopes
conclusion that the phase-shift analysis method is a precise method to calculate the elastic scattering cross sections. And the stability and validity of the combination
of the RMF model with the phase-shift analysis method
to study elastic electron-nucleus scattering are also supported by the results in Fig. 3.
In view of the validity of the combination of the RMF
model with the phase-shift analysis method to study
elastic electron-nucleus scattering off the stable Sn isotopes 112 Sn, 116 Sn, 118 Sn and 124 Sn, we now investigate
the charge form factors of the unstable isotopes 100 Sn
and 132 Sn. With the nuclear charge densities presented
in Fig. 1, the charge form factors for 100 Sn and 132 Sn
are calculated using the phase-shift analysis method.
According to the charge form factors shown in Fig. 4,
we can see that the minima shift upward and inward
with an increasing neutron number. This trend change
is due to the variation in the nuclear charge densities,
especially the details of the outer parts.
Fig.4 Nuclear charge form factors for the charge densities
in Fig. 1, with the electron beam energy at 225 MeV
Fig. 5 shows the nuclear charge form factors of Sn
isotopes whose corresponding charge densities are presented in Fig. 2. And the two figures calculated with
different parameter sets are in accordance with each
other. The experimental charge form factors of unstable nuclei can be obtained in future electron-nucleus
scattering experiments. The electron scattering cross
sections of some stable Sn isotopes have been measured
Fig.5 Nuclear charge form factors for the charge densities
in Fig. 2, with the electron beam energy at 225 MeV
to a high precision, while such measurements for the
unstable nuclei, such as 100 Sn and 132 Sn, have not
been performed. The half-life of the isotope 132 Sn is
39.7±0.5 s, and the experimental study for such shortlived isotopes is supported by the demonstration experiments with 133 Cs in RIKEN [22] . Therefore, we believe that it will be possible to carry out precise measurement for short-lived nuclei in the next-generation
facilities in the near future. Once the experiments of
electron-nucleus scattering off the unstable Sn isotopes
can be carried out, the nuclear charge density can be
deduced in a model-independent way. In addition, the
theoretical and experimental results can be compared
with each other in order to assess the validity of the
various types of nuclear models.
4
Summary
The binding energies, charge radii and charge distributions of Sn isotopes are studied using the RMF
model. The NL1 and NL3 parameter sets are used during the calculations, and the theoretical results are in
good agreement with the experimental data. The isotopic chain ranges from the proton-rich nuclei 100 Sn to
the neutron-rich nuclei 132 Sn. We all know that 100 Sn
and 132 Sn are doubly-magic nuclei, and their properties
are important for us to understand nuclear structure
and constrain the parameters in the RMF model. In
addition, the neutron skin thickness in 132 Sn can also
be used to determine the nuclear equation of state.
The proton number of Sn isotopes is a magic number,
and many stable Sn isotopes exist in nature. Elastic
electron scattering experiments on some stable Sn isotopes were performed in the 1970s, and their charge
density distributions can be deduced precisely from
those experiments. In this paper the RMF model and
the phase-shift analysis method are combined to calculate the cross sections of elastic electron-nucleus scattering and the corresponding charge form factors for
both stable and unstable Sn isotopes. From the calculations it can be seen that the RMF model can reproduce
the ground properties of Sn isotopes well. The experimental and theoretical cross sections of stable Sn isotopes 112 Sn, 116 Sn, 118 Sn and 124 Sn coincide with each
other. Then the Sn isotopic chain is investigated from
the proton-rich nuclei to the neutron-rich ones. Nuclear charge density distributions of Sn isotopes with
two parameter sets are calculated, respectively. The
corresponding charge form factors are then investigated
by the phase-shift analysis method. It is evident in
Figs. 4 and 5 that the minima of the charge form factors shift upward and inward with an increase in the
neutron number of the Sn isotopes. Results with different parameter sets coincide with each other. The
charge form factors for unstable nuclei should be obtained in the future, and our results will provide useful
references for future experiments.
617
Plasma Science and Technology, Vol.14, No.7, Jul. 2012
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(Manuscript received 17 May 2011)
(Manuscript accepted 8 September 2011)
E-mail address of corresponding author CHU Yanyun:
[email protected]