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\begin{document}
\title{SURFACE CONTOURS AND SHAPES OF SUPERHEAVY ELEMENTS}
\author{S.Niranjani, Department of Information Technology, \\Mohamed Sathak A.J.College of Engineering, Egatur, Chennai-603 103, India.\\ S.Sudhakar and G.Shanmugam,\\ SK Institute of Higher Studies \thanks{www.skhigherstudies.org}, S-2, Lotus colony, Nandanam, Chennai-600 035, India.}
\maketitle
\begin{abstract}
The enormity of data obtained in scientific experiments often
necessitates a suitable graphical representation for analysis.
Surface contour is one such graphical representation which renders
a pictorial view that aids in easy data interpretation. It is
essentially a two-dimensional visualization of a three-dimensional
surface plot. Very recently, it has been shown that Super Heavy
Elements can exist in a variety of shapes - spherical, spheroidal
and ellipsoidal with or without shape co-existence. The shapes of
such nuclei as predicted by us by diagonalizing the triaxial
Nilsson Hamiltonian in cylindrical representation and using the
Strutinsky-BCS corrections are graphically displayed by surface
contours with Origin software. The obtained results are highly
useful in the analysis of the stability of the Super Heavy
Elements. Further, they yield a surprising result that the doubly
magic spherical nucleus after lead (Z=82 and N=126) is SHE (Z=126
and N=184) in the macroscopic-microscopic method itself.
\end{abstract}
\section{INTRODUCTION}
Even though the predicted doubly magic superheavy nucleus having a
lifetime comparable to the age of the earth should have a
spherical shape, there could be other superheavy nuclei having
ellipsoidal shapes (axial and non-axial) with predictable longer
lifetimes. Such a possibility is considered theoretically in this
work by the macroscopic-microscopic method namely the triaxial
Nilsson-Strutinsky-BCS approach. Ellipsoidal shapes for alpha
decaying and fissioning heavy nuclei are not new but well-known
since the time of Hill and Wheeler. After reproducing some well
established experimental quantities for heavy nuclei like alpha
decay Q values, our calculations were extended to the fertile
region of the transitional superheavy nuclei recently synthesized
to determine their shapes by surface contours. The present
macroscopic-microscopic calculations and the resulting contour
diagrams further lead to the fact that the spherical doubly magic
nucleus is $^{310}$126$_{184}$ instead of $^{298}$114$_{184}$
\section{Super heavy element}
A superheavy nucleus is one which is doubly magic beyond
$^{208}_{82}$Pb and has a spherical shape, according to
theoretical predictions. While the predicted magic number for
proton beyond 82 is 114, 120, 124 or 126 according to
non-self-consistent \cite{one}, self-consistent \cite{two} and
relativistic theories \cite{three}, the neutron magic number after
126 is to be 172 or 184 according to those theories. Further, the
size of the superheavy region, whether narrow or broad, is another
interesting problem to tackle. After the advent of recent
experiments by Oganessian \emph{et al}. \cite{four} at Dubna, a
remarkable finding is emerging, according to which there could be
a broad region of long-lived metastable superheavy nuclei having
prolate, oblate or ellipsoidal shapes with very small
deformations. Such a possibility of metastable superheavy
elements directly accessible in heavy ion reactions has already
been predicted by Bengtsson \emph{et al}. \cite{five} as early as
1975 with the triaxial Nilsson-Strutinsky method. However, at that
time, the main focus was the on the stability of such superheavy
elements against spontaneous fission decay. But presently, it has
become clear that the predominant mode of decay of superheavy
nuclei is mainly by a chain of alpha decays which terminate only
at the end by spontaneous asymmetric fission. It seems now that
Z=114 and N=172 are not the final destination but the starting
points for a fertile land of transitional superheavy nuclei
closing on sphericity around Z=126 and N=184 having all possible
ellipsoidal shapes, axial as well as non-axial with small
deformations. This is a dramatic turning point for the quest for
superheavy nuclei.
\section{The Method} Our macroscopic-microscopic method
\cite{six,seven,eight} essentially has two components for the
evaluation of the potential energy of nuclei, namely a smooth
macroscopic liquid-drop model energy part(E$_{LDM}$) and a
fluctuating microscopic part consisting of shell and pairing
corrections ($\Delta E_{shell} \ and \ \Delta E_{pair}$). Thus
the total energy E is given by,
\begin{equation}\label{1}\begin{split}E(Z,N,\varepsilon,\gamma)=
E_{LDM}(Z,N,\varepsilon,\gamma)+\Delta
E_{shell}(Z,N,\varepsilon,\gamma)\\+\Delta
E_{pair}(Z,N,\varepsilon,\gamma)\end{split}\end{equation}
$\varepsilon$ and $\gamma$ being the elongational and the
non-axial parameters. The traditional liquid drop model energy is
\cite{nine},
\begin{equation}\label{2}\begin{split}E_{LDM}=17.9439\left(1-1.7826\left(\frac{N-Z}{A}\right)^{2}\right)A^{\frac{2}{3}}((B_{S}-1)\\+2x(B_{C}-1))\end{split}\end{equation}
where $B_{S}$ and $B_{C}$ denote, respectively, the surface and
coulomb potential energy in units of the spherical values assuming
a homogeneous charge distribution bounded by a sharp ellipsoidal
surface and the fissility parameter \cite{ten}
\begin{displaymath}x=\frac{(Z^{2}/A)}{50.88 \left(1-1.7826\left(\frac{N-Z}{A}\right)^{2}\right)}\end{displaymath}
The single particle energies are obtained by diagonalizing the
triaxial Nilsson-Newton Hamiltonian
\begin{equation}\label{3}\begin{split}h^{0}=-\frac{\hslash^{2}}{2m}\triangledown^{'^{2}}+\frac{m}{2}(\omega_{x}x^{'^{2}}
+\omega_{y}y^{'^{2}}+\omega_{z}z^{'^{2}})\\+C\textbf{$\ell$.s}+D(\ell^{'^{2}}
- \frac{1}{2} N(N+3))\end{split}\end{equation} in cylindrical
representation \cite{eleven, twelve} in the basis states
\begin{equation}\label{4}\vert N_{\rho}N_{z}\Lambda\Sigma >=
\vert N_{\rho}\Lambda> \vert\Lambda> \vert N_{z} > \vert
\Sigma>,\end{equation}where $\vert \Sigma >$ is the spin eigen
function and
\begin{equation}\label{5}\begin{split}\vert \Lambda >=(2\pi)^{-\frac{1}{2}} e^{i \Lambda \phi}\\
\vert N_{z}>=(\sqrt{\pi}
N_{z}!2^{N_{z}})^{-\frac{1}{2}}e^{-z^{2}/2} H_{N_{z}}(z)\\and\
\vert N_{\rho} \Lambda> = \left(\frac{2\Gamma(n+1)}{\Gamma(\vert
\Lambda \vert +n+1)^{3}}\right)^{\frac{1}{2}} e ^{-\rho^{2}/2}
\rho^{\vert \Lambda \vert} L_{n}^{\vert
\Lambda\vert}(\rho^{2}).\end{split}\end{equation} Here we have
substituted $n=\frac{1}{2}(N_{\rho}-\vert \Lambda \vert).$ The
functions $L_{n}^{\vert \Lambda \vert}$ and $H_{N_{z}}(z)$ are the
Laguerre and Hermite polynomials. The $\kappa$ and $\mu$ values
used for the Nilsson Hamiltonian are those given by Bengtsson
\emph{et al}. \cite{thirteen}. The shell and pairing corrections
are then evaluated using the Strutinsky and the BCS methods
\cite{fourteen}.
The shell correction is given by,
\begin{equation}\label{6}\begin{split}\Delta
E_{shell}(N)=E_{N}-\tilde{E}(N)\\ \textrm{where}\
E_{N}=\sum_{n=1}^{N}\epsilon_{n}\\
\textrm{or}\ \Delta E_{shell}(N)=\sum_{n=1}^{N} \epsilon_{N} -
\int_{0}^{N} \tilde{\epsilon}(n) dn\end{split}\end{equation}
Because neither the smoothing range nor the order $p$ represent
physical quantities, the shell correction should be approximately
independent of them as long as they are chosen suitably. The
value of $\gamma$ should be large enough to average over the
levels between major shells, but must not be too large, because
then levels far from the fermi surface would strongly affect the
shell correction. The value $\gamma=1.2\hslash\omega^{0}_{0}$
where $\hslash\omega^{0}_{0}=\frac{45.30}{(A^{1/3}+0.77)}$ MeV in
a sixth-order correction ($p=6$) is near the optimum choice.
The pairing correction for either neutrons or protons is given by
\begin{displaymath}\Delta E_{pair}=E_{pc}-\tilde{E}_{pc}\end{displaymath}
The pairing strength G for neutrons and protons usually are
adjusted to reproduce the average mass differences of neighboring
nuclei that differ by one neutron or one proton. This average
odd-even mass difference is given by the semi-empirical result
\begin{equation}\label{7}\tilde{\Delta}=\frac{12}{\sqrt{A}}\ \textrm{MeV} \end{equation}
In the above formulation of the pairing correction, the value of
G for either neutrons or protons is taken from Dudek et. al
\cite{fifteen}. When parametrizing the pairing constants (average
matrix elements) \cite{sixteen,seventeen} they took as a starting
point the pairing force strength parameters from
\begin{equation}\label{8}\begin{split}G_{n}=\frac{1}{A}[18.95 -0.078(N - Z)] \\
G_{p} =\frac{1}{A} [17.90 + 0.176(N - Z)]\end{split}\end{equation}
where they have been adjusted to experimental data on normal
nuclei by applying the pairing-self-consistent Bogolyubov
formalism without the proton-neutron pairing. When used within the
particle projection formalism, these values have to be decreased,
by about 15\%. In the present application we use the reduction
factor of 0.85 for both the neutrons and the protons, i.e., we
apply 0.85$G_{n}$ and 0.85$G_{p}$ in the monopole pairing
Hamiltonian.
Table I. Q-values for alpha decay of some very heavy and
superheavy nuclei \begin{tabular}[t]{cccccc}\hline\hline
A$_{p}$ & Z$_{p}$ & A$_{d}$ & Z$_{d}$ & Q$_{\alpha}$ (MeV) & Q$_{\alpha}$ (MeV) \\
& & & & Calculated & Experimental \cite{four}
\\\hline
271 & 106 & 267 & 104 & 8.37 & 8.65
\\275 & 108 & 271 & 106 & 9.50 & 9.44
\\279 & 110 & 275 & 108 & 9.75 & 9.84
\\283 & 112 & 279 & 110 & 9.42 & 9.67
\\286 & 114 & 282 & 112 & 10.21 & 10.35
\\290 & 116 & 286 & 114 & 10.63 & 11.00
\\ 294 & 118 & 290 & 116 & 11.42 & 11.81
\\\hline\hline
\end{tabular}
\\We then apply the present model to reproduce the recently
measured alpha decay Q values for nuclei in the very heavy and
superheavy region. Our results are shown above in table I and it
is seen that the experimentally measured Q values for the very
heavy and superheavy nuclei are reproduced within a deviation of
0.4 MeV. Considering the fact that the present model is a simple
one, it is gratifying to note its success in closely reproducing
the alpha decay Q values for the very heavy and superheavy nuclei
considered.
\section{Surface contours}
\begin{itemize}
\item A contour plot is a two-dimensional version of a
three-dimensional surface plot. \item Given a function v=f(x, y),
a surface contour consists of all the curves that connect all the
(x, y) points for a constant v. \item In this work, we use Origin
software to generate the contour plots given the values x, y and
f(x,y). \end{itemize}
The advantages of contour plotting are:
\begin{itemize}
\item Easy detection of areas of rapid change and areas of
constant value. \item Well suited to monochrome and gray scale
reproduction and easy to annotate within the field. \item Widely
used, generally understood by others and can easily be transformed
and projected
\end{itemize}
Hence, results of our calculations are given in the form of
Potential Energy Surface Contours (normalized to
spherical-liquid-drop energy) in the (Q$_{20}$, Q$_{22}$) plane,
where Q$_{20}$ and Q$_{22}$ are the quadrupole moments which can
be related to the Hill - Wheeler polar deformation parameters
Q$_{0}$ and $\gamma$ through the usual relations: $Q_{20}=Q_{0}
cos \gamma$ and $Q_{22}=\frac{Q_{0}}{\sqrt{2}} sin \gamma$.
Q$_{0}$ is proportional to the quadrupole deformation $\beta$ for
weakly deformed systems. The non-axial parameter $\gamma$ gives
the degree of triaxiality. While $\gamma=0^{\circ}$ denotes
prolate shapes, $\gamma=60^{\circ}$ denotes oblate shapes.
Intermediate values of $\gamma$ correspond to triaxial shapes.
We first apply our triaxial Nilsson-Strutinsky calculations to the
doubly magic spherical nucleus $_{82}^{208}$Pb, which produce a
spherical shape with a shell energy of -12 MeV as is well-known.
Then, we consider the long-lived superheavy nucleus
$_{112}^{285}$X which was found to be the starting point of the
transitional region with a deformation $\beta < 0.1$ and prone to
shape coexistence and triaxiality \cite{eighteen} thereby
inhibiting the alpha decay and enhancing its lifetime
\cite{nineteen,twenty}. The superheavy nucleus $_{114}^{286}$X
which was expected to be closing on sphericity turns out to be
transitional and is non-spherical. The nucleus $_{124}^{300}$X is
oblate with $\gamma=60^{\circ}$, $\beta=0.2$ and a fission barrier
of 2.5 MeV. This result is exactly the same as that obtained by
Bengtsson \emph{et al}. \cite{five} which is used as a benchmark
for testing the precision of our calculations. To track down the
doubly magic nucleus after lead, we next consider
$_{126}^{310}$X$_{184}$. It is seen from the figure 1 that the
nucleus $_{126}^{310}$X$_{184}$ is spherical showing that it is
doubly magic. This result is in accordance with the predictions of
the density functional method. However, it is to be noted that
while the doubly magic nucleus $_{82}^{208}$Pb is robust, the
doubly magic superheavy nucleus $_{126}^{310}$X$_{184}$ is fragile
having a shell energy of about -7 MeV and a fission barrier of
about 3 MeV. Between this nucleus and the nucleus $_{112}^{285}$X
we have the transitional region which is very rich in structure
and is important since it may yield long-lived species of
superheavy nuclei. The even Z nuclei with Z=114 - 126 and N=184
are spherical but all of them except Z=126 become non-spherical
when N=172. This leads to the neutron magicity of N=184 and proton
magicity of Z=126.
To summarize, the macroscopic-microscopic calculation namely the
triaxial Nilsson-Strutinsky-BCS approach and surface contour
plotting have been revisited. These closely reproduce the
experimental alpha decay Q-values of very heavy and superheavy
nuclei recently synthesized. Further, they lead to the fact that
the doubly magic spherical nucleus after $_{82}^{208}$Pb$_{126}$
is $_{126}^{310}$X$_{184}$ instead of $_{114}^{298}$X$_{184}$. It
is felt that this result is due to the role of triaxial degree of
freedom in weakly deformed alpha decaying superheavy nuclei.
Undoubtedly, we have reached an exciting phase in our quest for
superheavy elements.
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\begin{figure*}[t]
\centering
\includegraphics*{plot.eps}
\caption{Potential Energy Surface Contour(normalized to
spherical-liquid-drop energy) for $_{126}^{310}$X}
\end{figure*}
\end{document}