# Properties of superheavy nuclei with Z = 124

M. S. Mehta1 Harvinder Kaur1,2 Bharat Kumar3 S. K. Patra3 1Department of
Applied Sciences, Punjab Technical University, Kapurthala 144 601, India
2Department of Physics, Rayat Bahra University, Mohali 140 104, India
3Institute of Physics, Sachivalaya Marg, Bhubaneswar 751 005, India

###### Abstract

We employ Relativistic Mean Field (RMF) model with NL3 parametrization to
investigate the ground state properties of superheavy nucleus, Z = 124. The
nuclei selected (from among complete isotopic series) for detailed
investigation show that the nucleon density at the center is very low and
therefore, these nuclei can be treated as semi-bubble nuclei. The considerable
shell gap appears at neutron numbers N = 172, 184 and 198 showing the magicity
corresponding to these numbers. The results are compared with the macro-
microscopic Finite Range Droplet Model (FRDM) wherever possible.

###### pacs:

21.10.Dr, 21.10.Ft, 21.10.Tg, 21.10.Gv

11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
##  I Introduction

The location of the center of ’island of stability’ and hence the next magic
number for proton beyond 208Pb (Z = 82, N = 126) in superheavy mass region is
debated since the prediction of the existence of long-lived superheavy nuclei
in sixties by sobi66 ; meld67 ; myer67 ; nils69 ; mosel69 ; seab69 . Since
then a significant progress has been made in the discovery of superheavy
nuclei hofm96 ; hofm00 ; kumar89 . Experimentally, the elements up to Z = 118
have been synthesized to-date, with half-lives varying from a few minutes to
milliseconds hofm00 . Recently, the nuclei with Z = 104 - 118 with mass number
A = 266 - 294 have been detected at Dubna ogan99 ; ogan00 ; yu01 ; ogan04 ;
ogan07 ; eich07 ; ogan10 ; yu11  using “hot fusion” reactions with the
neutron-rich 48Ca beam on actinides targets. These measurements show the
increase in half-lives with in neutron number towards N = 184 give indication
of stable center. In more detail, the cold fusion reactions involving a doubly
magic spherical target and a deformed projectiles was used at GSI hofm00 ;
hof95 ; hofm95 ; hofm96 ; hofm98 ; hofm99  to produce heavy elements upto Z =
110 - 112. At the production time of Z = 112 nucleus at GSI the fusion cross-
section was extremely small (1 pb), which led to the conclusion that reaching
still heavier elements will be very difficult. At this time, the emergence of
hot fusion reactions using 48Ca projectiles at Dubna has drastically changed
the situation and nuclei with Z = 114 - 118 were synthesized and also observed
their $\alpha$-decay chains. The element Z = 113 was first reported by
Oganessian et al. ogan04  and then using cold fusion reaction confirmed by
Morita et al. morita04 ; morita12 .

But theoretically, the studies of the shell structure of superheavy nuclei in
different approaches show that the magic shells beyond the spherical double-
magic number 208Pb (N = 126 and Z = 82), in superheavy mass region are isotope
(combination of Z and N) as well as parameter dependent. For example,
recently, more microscopic calculations have predicted various other regions
of stability, such as Z = 114, N = 184 rutz97 ; Z = 120, N = 172 or 184 gupt97
; patr99  and Z = 124 or 126, N = 184 cwio96 ; krup00 ; cwio05 . In the fram-
work of relativistic continuum Hartree-Bogoliubov theory, Zhang et al. zhan05
predicted Z = 120, 132 and 138 with neutron number N = 172, 184, 198, 228, 238
and 258 as the next nucleon shell gaps. However, in experiments, the heaviest
nucleus that could be studied so far is 254No (Z = 102, N = 152) herz06 . In
an effort in this direction, using inductively coupled plasma-sector field
mass spectroscopy, Marinov et al. mari07  have observed some neutron-deficient
Th-isotopes in naturally occurring thorium substances. The long-lived isomeric
states, with estimated half-lives T${}_{1/2}\geqslant 10^{8}$ y, have been
identified in the neutron-deficient 211,213,217,218Th isotopes, which are
associated with the super-deformed (SD) or hyper-deformed (HD) states (minima)
in potential energy surfaces (PES). In our earlier investigation patr09  of Z
= 122 isotopes (N = 160 - 198), using relativistic mean field (RMF) and Skyrme
Hartree Fock (SHF) models, we find the ground state solutions of some nuclei
are super deformed and/or even hyper-deformed. Of course, the SD ground state
strcture of superheavy nuclei are reported earlier by Ren et al. ren01 ,
within the theoretical framework of RMF calculations. Recently, Marinov et al.
mari09  obtained a possible evidence for the existence of a long-lived
superheavy nucleus with mass number A = 292 and atomic number Z = 122 or 124
in natural thorium. The half-life is again estimated to be the same as
T${}_{1/2}\geqslant 10^{8}$ y and the abundance is (1 - 10) $\times$ 10−12 as
compared to 232Th. This makes it interesting to make detailed investigation of
the properties of nuclei in this mass region.

In extreme superheavy mass region, it is difficult to identify the nuclei by
their $\alpha$-decay chains unless a proper combination of neutron and proton
close shell are located. Therefore, the identification of nuclei can be made
through the comparison with theoretical calculations. In the present
investigation we calculate the bulk properties of Z = 124 nucleus within the
framework of RMF model. Here, we choose NL3 parameter set lala97  for isotopic
chain with neutron number N = 158 to N = 220, which encompasses the neutron
numbers N = 172 and 184. Also, for the consistency of our results we calculate
the similar quantities for isotopic chain of Z = 120 nucleus.

##  II Formalism

It has now been well established that the RMF models involving sigma, omega,
rho and photon along with the self-interactions among various mesons, i.e.,
the effective field theory is very successful in explaining the structure of
nuclei throughout the nuclear landscape reinhard89 ; ring96 ; vret05 ; meng06
; lian15 . The RMF model has been proved to be a very powerful tool to explain
the properties of finite nuclei and infinite nuclear matter mach89 ; patr91 ;
patr01  for the last three decades. We start with the modified relativistic
Lagrangian density of $\sigma-\omega$ model sero86  for a nucleon-meson many-
body system, which describes the nucleons as Dirac spinors interacting through
the exchange of scalar mesons ($\sigma$), isoscalar vector mesons ($\omega$)
and isovector mesons ($\rho$). The scalar mesons cause attraction and the
vector mesons produce repulsion, whereas the charge protons generate
electromagnetic interaction.

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\overline{\psi_{i}}\\{i\gamma^{\mu}\partial_{\mu}-M\\}\psi_{i}+{1\over{2}}\partial^{\mu}\sigma\partial_{\mu}\sigma-{1\over{2}}m_{\sigma}^{2}\sigma^{2}-{1\over{3}}g_{2}\sigma^{3}$
(1) $\displaystyle-$
$\displaystyle{1\over{4}}g_{3}\sigma^{4}-g_{s}\overline{\psi_{i}}\psi_{i}\sigma-{1\over{4}}\Omega^{\mu\nu}\Omega_{\mu\nu}+{1\over{2}}m_{w}^{2}V^{\mu}V_{\mu}$
$\displaystyle+$
$\displaystyle{1\over{4}}c_{3}(V_{\mu}V^{\mu})^{2}-g_{w}\overline{\psi}_{i}\gamma^{\mu}\psi_{i}V_{\mu}-{1\over{4}}\vec{B}^{\mu\nu}.\vec{B}_{\mu\nu}$
$\displaystyle+$
$\displaystyle{1\over{2}}m_{\rho}^{2}{\vec{R}^{\mu}}.{\vec{R}_{\mu}}-g_{\rho}\overline{\psi}_{i}\gamma^{\mu}\vec{\tau}\psi_{i}-{1\over{4}}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle-$ $\displaystyle
e\overline{\psi}_{i}\gamma^{\mu}{\left(1-\tau_{3i}\right)\over{2}}\psi_{i}A_{\mu}.$

The field for the $\sigma$-meson is denoted by $\sigma$, that for the
$\omega$-meson by $V_{\mu}$ and for the isovector $\rho$-meson by
$\vec{R}_{\mu}$. $A^{\mu}$ denotes the electromagnetic field. The $\psi_{i}$
are the Dirac spinors for the nucleons whose third component of isospin is
denoted by $\tau_{3i}$. Here $g_{s}$, $g_{w}$, $g_{\rho}$ and
${e^{2}\over{4\pi}}={1\over{137}}$ are the coupling constants for $\sigma$,
$\omega$, $\rho$ mesons and photon, respectively. $g_{2}$, $g_{3}$ and $c_{3}$
are the parameters for the nonlinear terms of $\sigma$\- and $\omega$-mesons.
M is the mass of the nucleon and $m_{\sigma}$, $m_{\omega}$ and $m_{\rho}$ are
the masses of the $\sigma$, $\omega$ and $\rho$-mesons, respectively.
$\Omega^{\mu\nu}$, $\vec{B}^{\mu\nu}$ and $F^{\mu\nu}$ are the field tensors
for the $V^{\mu}$, $\vec{R}^{\mu}$ and the photon fields, respectively gamb90
.

From the relativistic Lagrangian, we get the field equations for the nucleons
and mesons. These equations are solved by expanding the upper and lower
components of Dirac spinors and the Boson fields in a deformed harmonic
oscillator basis with an initial deformation. The set of coupled equations is
solved numerically by a self-consistent iteration method. The center of mass
motion is estimated by the usual harmonic oscillator formula
$E_{c.m.}=\frac{3}{4}(41A^{-1/3})$ MeV. The quadrupole deformation parameter
$\beta_{2}$ is evaluated from the resulting quadrupole moment gamb90  using
the formula,

$\displaystyle Q=Q_{n}+Q_{p}={\sqrt{{9\over{5\pi}}}}AR^{2}{\beta_{2}},$ (2)

where $R=1.2A^{1/3}$ fm. The total binding energy of the system is,

$\displaystyle
E_{total}=E_{part}+E_{\sigma}+E_{\omega}+E_{\rho}+E_{c}+E_{pair}+E_{c.m.},$
(3)

where $E_{part}$ is the sum of the single-particle energies of the nucleons
and $E_{\sigma}$, $E_{\omega}$, $E_{\rho}$, $E_{c}$ and $E_{pair}$ are the
contributions of the mesons fields, the Coulomb field and the pairing energy,
respectively.

For the open shell nuclei, the effect of pairing interactions is added in the
BCS formalism. We consider only T=1 channel of pairing correlation, i.e.,
pairing between proton-proton and neutron-neutron. In such case, a nucleon of
quantum state $|j,m_{z}\rangle$ pairs with another nucleons having same
$I_{z}$ value with quantum state $|j,{-m_{z}}\rangle$, which is the time
reversal partner of other. The RMF Lagrangian density only accommodates term
like ${\psi}^{\dagger}{\psi}$ (density) and no term of the form
${\psi}^{\dagger}{\psi}^{\dagger}$ or $\psi\psi$. The inclusion of pairing
correlation of the form $\psi\psi$ or ${\psi}^{\dagger}{\psi}^{\dagger}$
violates the particle number conservation patra93 . Thus, a constant gap BCS-
type simple prescription is adopted in our calculations to take care of the
pairing correlation for open shell nuclei. The general expression for pairing
interaction to the total energy in terms of occupation probabilities
$v_{i}^{2}$ and $u_{i}^{2}=1-v_{i}^{2}$ is written as pres82 ; patra93 :

$E_{pair}=-G\left[\sum_{i>0}u_{i}v_{i}\right]^{2},$ (4)

with $G=$ pairing force constant. The variational approach with respect to the
occupation number $v_{i}^{2}$ gives the BCS equation pres82 :

$2\epsilon_{i}u_{i}v_{i}-\triangle(u_{i}^{2}-v_{i}^{2})=0,$ (5)

with $\triangle=G\sum_{i>0}u_{i}v_{i}$.

The densities with occupation number is defined as:

$n_{i}=v_{i}^{2}=\frac{1}{2}\left[1-\frac{\epsilon_{i}-\lambda}{\sqrt{(\epsilon_{i}-\lambda)^{2}+\triangle^{2}}}\right].$
(6)

For the pairing gap ($\triangle$) of proton and neutron is taken from the
phenomenological formula of Madland and Nix madland :

$\displaystyle\triangle_{n}=\frac{r}{N^{1/3}}exp(-sI-tI^{2})$ (7)
$\displaystyle\triangle_{p}=\frac{r}{Z^{1/3}}exp(sI-tI^{2})$ (8)

where, $I=(N-Z)/A$, $r=5.73$ MeV, $s=0.117$, and $t=7.96$.

The chemical potentials $\lambda_{n}$ and $\lambda_{p}$ are determined by the
particle numbers for neutrons and protons. The pairing energy of the nucleons
using equation (7) and (8) can be written as:

$E_{pair}=-\triangle\sum_{i>0}u_{i}v_{i}.$ (9)

In constant pairing gap calculation, for a particular value of pairing gap
$\triangle$ and force constant $G$, the pairing energy $E_{pair}$ diverges, if
it is extended to an infinite configuration space. In fact, in all realistic
calculations with finite range forces, the contribution of states of large
momenta above the Fermi surface (for a particular nucleus) to $\triangle$
decreases with energy. Therefore, the pairing window in all the equations are
extended upto the level $|\epsilon_{i}-\lambda|\leq 2(41A^{-1/3})$ as a
function of the single particle energy. The factor 2 has been determined so as
to reproduce the pairing correlation energy for neutrons in 118Sn using Gogny
force gamb90 ; patra93 ; dech80 .

##  III Results and Discussions

The superheavy nucleus Z = 124 with neutron number N = 158 - 220 are studied
for the investigation of ground state properties. The results are compared
with other models of previous works including the Finite Range Droplet Model
(FRDM), as the experimental observations could not be made at such a high Z
region so far. In numerical calculations, the number of oscillator shell for
Fermions and Bosons NF = NB = 20 are used to evaluate the physical observables
with the pairing gaps of eqns. (7) and (8) in the BCS pairing scheme.

###  III.1 Binding Energy

The binding energy of the isotopic chain of Z = 124 is calculated for mass
number A = 282-384. Since there is no experimental observation for such a
large Z number so far, therefore, the only comparison can be made with
theoretical models such as macroscopic-microscopic model. We compare our
calculations with finite range droplet model (FRDM) moel97 . Here in upper
panel of Fig. 1, we compare the results (binding energy) with available FRDM
results, which seem to be in good agreement. A small difference in binding
energy at N = 198 region can be seen in the upper panel of Fig. 1. For
example, the RMF results of binding energy and quadrupole deformation
parameter for 334124 nucleus are 2284.71 MeV and $\beta_{2}$ = 0.128, whereas
the FRDM calculations are 2286.75 MeV and $\beta_{2}$ = 0.335, respectively.
Similarly for 312124, the RMF binding energy is 2166.19 MeV and FRDM value is
2163.84 MeV with a discrepancy of 2.35 MeV. In Z = 124 isotopes, we get a
maximum difference in binding energy is 7.43 MeV for 320124 nucleus, which is
about 0.3 % discrepancy. In general, the difference in binding energy with
FRDM and RMF is $\sim$3 - 4 MeV, which is reasonable in the order of two
thousands magnitude. For consistency of our results, we also calculate the
binding energy of the isotopic chain of Z = 120 and is displayed at the lower
panel of Fig. 1. In this case the difference in binding energy is very small.
For example, the RMF result of BE and $\beta_{2}$ are 2026.51 MeV and
$\beta_{2}$ = -0.049 compared to the FRDM,binding enrgy BE = 2023.06 MeV and
$\beta_{2}$ = -0.104 for 288120\. Similarly, maximum discrepancy between RMF
and FRDM bunding energy is $\sim$5.83 MeV for 320120.

Figure 1:  The binding energy of isotopic series of nuclei Z = 120, 124 nuclei
with NL3 parameter set.

###  III.2 Separation Energy

The magic numbers in nuclei are characterized by the large shell gap in single
particle energy levels. This means the nucleon in lower level has
comparatively large value of energy than that in higher level giving rise to
the more stability. The extra stability corresponding to certain numbers can
be estimated from the sudden fall in the neutron separation energy. The
separation energy is calculated by the difference in binding energies of two
isotopes using relation:

$S_{2n}(N,Z)=BE(N,Z)-BE(N-2,Z)$

The two neutron separation energy (S2n) for the isotopic series of nuclei
Z=124 and 120 (282-344124 and 278-340120) is shown in Fig. 2. The sudden fall
in separation energy at N = 172, 184 and 198 can clearly be seen in both the
cases confirming the magic character rutz97 ; gupt97 ; patr99 ; meht02
predicted in earlier studies. Although, N = 172 is not that much pronounced in
our earlier investigation of odd nuclei meht02 , here the magicity at N = 172
increases as we move to extreme of superheavy mass region sald09 . Contrary to
some earlier literature, there is no signature of sudden change in separation
energy at deformed magic number N = 162 rutz97  in the present calculations.
The decrease in energy at N = 172 and 184 is $\sim$ 3.5 MeV whereas $\sim$2
MeV at N = 198, for Z = 124 nuclei. In case of Z = 120 isotopes the decrease
in energy is $\sim$ 5.0 MeV at N = 172 and $\sim$ 3.0 MeV and 3.5 MeV at N
=184 and 198 respectively. Such decrease at N = 198 in our calculation is
nearly same as in FRDM value. However, in FRDM the sudden decrease in
separation energy appears at N = 180 and 200 for Z = 124. Except the values at
these numbers, in general all other energies from our present calculations are
in good agreement (within $\sim 2$ MeV accuracy) with macro-microscopic
calculations (FRDM). We observed a couple of abnormal increase in $S_{2n}$ at
(N=194, Z=124) and (N=196, Z=120), which are not seen in the present RMF
calculations.

Figure 2:  The two neutron separation energy as a function of neutron for
series of Z = 120 and 124 nuclei.

###  III.3 Quadrupole Deformation Parameter

The quadrupole deformation parameter $\beta_{2}$ gives the shape of nuclei in
ground state. The value of $\beta_{2}$ is positive, negative and zero for
prolate, oblate and spherical respectively. In our calculation shown in Fig.
3. except for few nuclei all the isotopes of Z = 124 are either spherical or
near spherical. The results compared with FRDM moel97  agree for nuclei having
N = 176, 182 - 192 as shown in the upper panel of Fig. 3. At N = 176 and N =
184 the nuclei are complete spherical. There is the least agreement beyond N =
196 for Z = 124. From the figure it is clear that NL3 parameter set predicts
the deformation parameter $\beta_{2}$ very close to FRDM at middle mass region
i.e., from neutron number N = 176 to 192.

Figure 3:  The quadrupole deformation parameter $\beta_{2}$ for the isotopic
series of Z = 120, 124 nuclei.

###  III.4 Qα Energy and Half-Life (Tα)

The superheavy nuclei along to the $\beta$-stability line are known to be
$\alpha$-emitter. The $\alpha$-decay half-life of the nucleus showing shell
closure is believed to be comparatively larger than the neighboring nuclei.
Thus, to confirm the magic number corresponding to a particular neutron number
N, it is beneficial to calculate half-life of $\alpha$-decay. The
investigation of $\alpha$-decay of nuclei gives information about their degree
of stability and possibility of existence in nature. Here we take the nucleus
296124 (Z = 124 and N = 172) for the calculation of $\alpha$-decay energy
patr97 .

Figure 4:  (Color online) The $\alpha$-decay ($Q_{\alpha}$-energy) chain from
Z = 124 to Z = 92.

The Qα-energy and half life (Tα) are compared with available experimental data
as shown in Table 1. The Qα-energy is calculated using the following equation:

$Q_{\alpha}(N,Z)=BE(N,Z)-BE(N-2,Z-2)-BE(2,2).$

In the equation, BE(N,Z) is binding energy of the parent nucleus having N
neutrons and Z protons, and BE(N - 2, Z - 2) is the binding energy of daughter
nucleus after emission of an $\alpha$-particle (BE(2,2)). The binding energy
of $\alpha$-particle (4He) is 28.296 MeV. The Qα energy values are in good
agreement with experimental data audi03  as well as FRDM moel97  as shown in
Table. The decay chain is also plotted in Fig. 4 which shows good agreement
with experiments as well as FRDM calculations. The half-life log10 Tα(s)
values are estimated using the phenomenological formula viol66 ;

$log_{10}T_{\alpha}(s)=\frac{aZ-b}{\sqrt{Q_{\alpha}}}-(cZ+d)-h_{l}og,$

where Z is atomic number of parent nucleus, and the other parameters are; a =
1.66175, b = 8.5166, c = 0.20228, and d = 33.9069. The values of the
parameters are used from Sobiczewski et al. sobi89 . The hindrance ($h_{log}$)
caused by odd number of protons and/or neutrons is zero here.

Table 1:  The Qα and Tα calculated using NL3 parameter set in RMF. The results are compared with finite range droplet model (FRDM)moel97  as well as the available experimental dataaudi03 .The binding energy is in MeV and half life is in seconds. | RMF(NL3) | FRDM | Expt.
---|---|---|---
A | Z | BE | Qα | Tα | BE | Qα | Tα | BE | Qα | Tα
296 |  | 124 |  | 2056.01 |  | 14.11 |  | 10-6.41 |  |  |  |  |  |  |  |
292 |  | 122 |  | 2041.83 |  | 12.98 |  | 10-4.68 |  |  |  |  |  |  |  |
288 |  | 120 |  | 2026.51 |  | 14.28 |  | 10-7.66 |  | 2023.06 |  | 13.92 | 10-7.02 |  |  |
284 |  | 118 |  | 2012.49 |  | 13.92 |  | 10-7.50 |  | 2008.69 |  | 13.10 | 10-5.95 |  |  |
280 |  | 116 |  | 1998.12 |  | 13.14 |  | 10-6.54 |  | 1993.49 |  | 12.42 | 10-5.10 |  |  |
276 |  | 114 |  | 1982.96 |  | 11.88 |  | 10.-4.48 |  | 1977.62 |  | 12.33 | 10-5.44 |  |  |
272 |  | 112 |  | 1966.55 |  | 11.68 |  | 10-4.60 |  | 1961.66 |  | 11.61 | 10-4.45 |  |  |
268 |  | 110 |  | 1949.93 |  | 11.33 |  | 10-4.38 |  | 1944.97 |  | 10.94 | 10-3.47 | 1943.53 | 11.7 | 10-5.2
264 |  | 108 |  | 1932.96 |  | 10.56 |  | 10-3.14 |  | 1927.62 |  | 10.57 | 10-3.18 | 1926.67 | 10.59 | 10-3.2
260 |  | 106 |  | 1915.22 |  | 9.66 |  | 10-1.42 |  | 1909.90 |  | 9.93 | 10-2.15 | 1909.06 | 9.90 | 10-2.07
256 |  | 104 |  | 1896.59 |  | 8.12 |  | 102.73 |  | 1891.53 |  | 8.75 | 100.59 | 1890.56 | 8.93 | 100.05
252 |  | 102 |  | 1876.41 |  | 8.33 |  | 101.25 |  | 1871.98 |  | 8.35 | 101.19 | 1871.35 | 8.54 | 100.52
248 |  | 100 |  | 1856.44 |  | 7.07 |  | 10.5.15 |  | 1852.03 |  | 7.64 | 102.91 | 1851.57 | 8.0 | 101.60
244 |  | 98 |  | 1835.21 |  | 7.25 |  | 103.57 |  | 1831.38 |  | 6.90 | 105.01 | 1831.22 | 7.32 | 103.30
240 |  | 96 |  | 1814.17 |  | 5.93 |  | 108.68 |  | 1809.98 |  | 6.52 | 105.81 | 1810.28 | 6.40 | 106.36
236 |  | 94 |  | 1791.81 |  | 4.30 |  | 1018.29 |  | 1788.21 |  | 5.77 | 108.54 | 1788.41 | 5.87 | 108.03
232 |  | 92 |  | 1767.81 |  | 3.41 |  | 1025.66 |  | 1765.695 |  | 5.14 | 1011.18 | 1765.98 | 5.41 | 109.50

###  III.5 Density Distribution

The neutron and proton density distributions for Z = 124 and 120 nuclei are
plotted in Fig. 5. The nuclei with N = 172, 184 and 198 are taken as
representative cases for detailed investigation of the internal structure. The
charge distribution of both Z = 120 and 124 show that the center part of
nuclei have very low density indicating a hollow inside.

Figure 5:  (Color online) The density of selected isotopes of Z = 120, 124
nuclei with NL3 parameter set. Figure 6: Two-dimensional density contours for
nuclei 284,290,292,304,318120 shown using NL3 parameter set. Figure 7:  Same
as Fig. 6 but for nuclei 288,294,296,308,322124.

Figure 8:  The neutrons and protons density distribution for nuclei
296,308,322124.

.

In order to have an insight into the arrangement of nucleons we plot the two
dimensional contours for some selected nuclei. The density contours for
284,290,292,304,318120 and 288,294,296,308,322124 nuclei are shown in Figs. 6,
7, and 8. In general, it is clear from the figures that the central region in
all nuclei except 308124 nucleus have considerably low density. In case of
isotopes of Z = 120, as shown in Fig. 6, N = 170 nucleus is slightly deformed
($\beta_{2}$ = 0.021) and all other(N = 164, 170, 184 and 198) are spherical
in their ground state. The nuclei with Z$\geqslant$120 have large number of
protons and hence considerable Coulomb repulsion among protons. The strong
repulsion changes the entire distribution of nucleons. The doubly magic
nucleus 292120 is largely studied previously bend99 ; dech99 ; pei05  and is
predicted to be semi-bubble. In the present calculations using RMF(NL3), semi-
bubble structure of these nuclei can be clearly seen in Fig. 6. The hollow
region at the center is spread over the radius of 1 - 2 fm. This may suggest
that these nuclei might be a fullerene type structure consisting of 60
$\alpha$-particles and a binding neutron per alpha and/ or few neutron
clusters. The clusters of some heavier nuclei might be possible. The density
distribution of 288,294,296,308,322124 nuclei is shown in Fig. 7. In this case
the density of nucleus N = 184 is more at the central region while all other
nuclei studied here are showing bubble type structure. The low density region
extends up to $\sim$ 2 fm. The nuclei with N = 164, 170, 172 and 198 are near
spherical ($\beta_{2}$ = 0.041, 0.056, 0.034 and 0.023 respectively) whereas N
= 184 is spherical in shape. In order to give further insight into the
arrangement of nucleons, we plot the density distribution of neutron and
proton separately (Fig. 8). It is clear from the figure that both neutrons as
well as protons are shifted from the central region except for N = 184
nucleus.

##  IV Conclusion

In the present work we use RMF(NL3) model to explore the structure of
superheavy nucleus Z = 124. The results of our calculations are compared with
macro-microscopic FRDM prediction. We calculate binding energy, quadrupole
deformation parameter ($\beta_{2}$), two neutron separation energy (S2n), and
decay half-life (T1/2) for the isotopic series of Z = 124 and for the
consistence of our results we calculate the same quantities for Z = 120
nucleus. The quadrupole deformation parameter at heavier side of series show
more deviation from FRDM values. The two neutron separation energy shows the
sudden fall in energy at neutron numbers N = 172, 184, and 198 indicating the
magic structure. The $\alpha$-decay energy and half-life are also calculated
and compared with the experiments and FRDM results which seem to be in good
agreement. The density profile of the selected nuclei shown that the
depression in the density at the central region of the nuclei with the
exception of 308124\. This nucleus is the only candidate which does not show
the depression at the center. Finally, this theoretical investigation of
ground state properties of Z = 124 nuclei may be helpful for an experimental
exploration to locate the “island of stability” which is expected to be
existed in the large Z superheavy region.

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