% ins96a.tex : report for the proceedings of a work shop at INS
% "Science of short-lived beam '95" (Kakenhi kenkyuu Kai) 
% My talk was given on January 7, 1996.
%
% 18/mar/96 completion, sent to the editor of the proceedings
% 15/mar/96 editing
% 14/mar/96 created by modifying riken96a.tex
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\begin{document}

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\title{
Properties of even-even nuclei with Skyrme-Hartree-Fock method
}

\author{
Naoki Tajima, Satoshi Takahara, and Naoki Onishi\\
Institute of Physics, College of Arts and Sciences,\\
University of Tokyo, Komaba, Meguro-ku, Tokyo, 153, Japan
}

\date{}

\maketitle

We have applied the Hartree-Fock+BCS method with Skyrme SIII force
formulated in a three-dimensional Cartesian-mesh representation$^{1)}$
to even-even nuclei with $2 \leq Z \leq 114$.  In this report, we
summarize the results concerning the atomic masses, the quadrupole
($m$=0, 2) and the hexadecapole ($m$=0, 2, 4) deformations, the skin
thicknesses, and the halo radii. We also discuss the energy difference
between oblate and prolate solutions and the shape difference between
protons and neutrons.

The Skyrme force is an effective interaction widely used in mean-field
calculations. It is a zero-range force modified with the lowest order
momentum dependences to emulate the finite-range effects, a density
dependence to reproduce the saturation, and a spin-orbit coupling
term.  The SIII is one of about twenty parameter sets proposed for the
force and features good single-particle spectra and accurate $N-Z$
dependence of the binding energy.

The advantage of the three-dimensional Cartesian-mesh representation
is that it is suitable to describe nucleon skins, halos, and exotic
shapes as well as properties of ordinary stable nuclei.

The calculation has been done for $N$ ranging from outside the proton
drip line to beyond the experimental frontier in the neutron-rich
side.  We obtained spatially localized solutions for 1029 nuclei,
together with the second minima for 758 nuclei.  We imposed the
D$_{\rm 2h}$ symmetry on the solutions, which means that only even
electric multipole moments do not vanish.  The details of the method
of the calculation are described in Ref.$^{2)}$.

The errors of the calculated ground-state masses from experimental
data$^{3)}$ are negative for most of spherical nuclei, while for
deformed nuclei they are about 3 MeV rather independently of the size
of deformation.  The r.m.s.~value of the errors for 480 even-even
nuclei is 2.2 MeV, which is three to four times as large as the
precision of recent extensively-parameter-fitted mass models like the
TUYY mass formula$^{4)}$ and the finite-range droplet model with a
microscopic shell correction (FRDM)$^{5)}$.  
Nevertheless, the two-proton drip line (in the $N$-$Z$ plane) obtained
by us is almost identical to that of the TUYY and the FRDM.

The calculated electric intrinsic quad\-ru\-pole moments agree well
with those deduced from experimental B(E2)$\uparrow^{6)}$, except for
light nuclei, in which the HF solution often has a spherical shape
when the experimental B(E2)$\uparrow$ is not necessarily very small.
This may be explained by attributing the enhanced B(E2)$\uparrow$ not
only to static deformations but also to collective shape oscillations
around the spherical equilibrium.

The deformation parameters are more useful than the electric multipole
moments for certain purposes.  We define the deformation parameters of
a HF+BCS solution as those of a sharp-surface uniform-density liquid
drop which has the same mass moments as the solution has.  More
specifically, the mass density of the liquid drop is
expressed as

\vspace*{-0.1cm}

\begin{displaymath}
  \rho(\bm{r}) = \rho_0 \; \theta ( R(\hat{\bm{r}})-|\bm{r} |),
  \;\;\;\;\;
  R(\hat{\bm{r}}) = R_0 \; \Bigl( 1 + \sum_{l,m}
  a_{lm} Y_{lm} (\hat{\bm{r}}) \Bigr). \label{eq_radld}
\end{displaymath}

\vspace*{-0.2cm}

\noindent
The necessary and sufficient conditions on $a_{lm}$ to fulfill the
reality of $R(\hat{\bm{r}})$ and the D$_{\rm 2h}$ symmetry are that
$l$ and $m$ are even numbers and $a_{lm}=a_{lm}^{\ast}=a_{l-m}$.  We
set $a_{lm}=0$ for $l \ge 6$ and determine the remaining seven
parameters $\rho_0$, $R_0$, $a_{20}$, $a_{22}$, $a_{40}$, $a_{42}$,
and $a_{44}$ such that the liquid drop has the same particle number,
mean-square mass radius, and mass quad\-ru\-pole ($r^2 Y_{2m}$) and
hexa\-deca\-pole ($r^4 Y_{4m}$) moments as the HF+BCS solution has.

\begin{figure}[tbh]
\vspace*{11.5cm}
\caption{
Difference of the quadrupole deformation parameter $a_{20}$ between
protons and neutrons for the ground states of 1029 even-even nuclei
(denoted by plus symbols) and the first-excited solutions of 758
even-even nuclei for HF+BCS with SIII force.  }
\label{f_a20pn}
\end{figure}

One can find elsewhere$^{2)}$ figures in which thus-de\-fined axial
deformation parameters ($a_{20}$ and $a_{40}$) are graphically
displayed in the $N$-$Z$ plane.  Concerning the non-axial deformation
parameters ($a_{22}$, $a_{42}$, and $a_{44}$), their magnitudes have
turned out very small ($\vert a_{22} \vert$, $\vert a_{42} \vert$ $<
10^{-4}$, $\vert a_{44} \vert$ $\sim 10^{-3}$).

% Difference of shape parameters between protons and neutrons

One of the advantages of mean-field methods over shell-correction
schemes is that the protons and the neutrons do not have to possess
the same radius and deformation.  It is also an advantage of the
Cartesian-mesh representation over the expansion in
harmonic-oscillator bases.  In order to make the best use of this
advantage, we have calculated the liquid-drop shape parameters
separately for protons and neutrons for 1029 ground and 758
first-excited solutions.

Fig.~\ref{f_a20pn} presents a comparison of the values of $a_{20}$ between
protons and neutrons.  The r.m.s.\ difference is 0.012, while the
maximum difference occurs in the ground state of
$^{14}_{\phantom{0}4}$Be$_{10}$
($a_{20}^{\rm neu}=0.36$, $a_{20}^{\rm pro}=0.52$),
which is indicated by letter {\bf A} in the figure.
Symbols indicated by letters from {\bf B} to {\bf F} are the ground
states of
%
$^{16}_{\phantom{0}4}$Be$_{12}$
($a_{20}^{\rm neu}=0.44$, $a_{20}^{\rm pro}=0.59$),
$^{30}_{18}$Ar$_{12}$
($                 0.28$, $                 0.18$),
$^{30}_{12}$Mg$_{18}$
($                 0.18$, $                 0.27$),
$^{36}_{24}$Cr$_{12}$
($                 0.36$, $                 0.28$), and
$^{28}_{18}$Ar$_{10}$
($                 0.18$, $                 0.11$), respectively.
%
Among these nuclei, $^{30}$Ar, $^{36}$Cr, and $^{28}$Ar are outside
the proton drip line.  The absolute value of the difference is smaller
than 0.05 for 98.9 \% of the solutions.

\begin{figure}[tbh]
\vspace*{11.5cm}
\caption{
Same as in Fig.~1
but for the hexa\-deca\-pole deformation parameter
$a_{40}$.
}
\label{f_a40pn}
\end{figure}

Fig.~\ref{f_a40pn} 
is a graph similar to Fig.~\ref{f_a20pn} but for $a_{40}$. The r.m.s.\ 
difference is 0.0063, while the maximum difference is found in
$^{16}_{\phantom{0}4}$Be$_{12}$
($a_{40}^{\rm neu}=-0.127$, $a_{40}^{\rm pro}=-0.031$).
Symbols indicated by letters from {\bf B} to {\bf G}
are the ground state solutions of
%
$^{14}_{\phantom{0}4}$Be$_{10}$
($a_{20}^{\rm neu}=0.065$, $a_{20}^{\rm pro}=0.002$),
$^{18}_{12}$Mg$_{6}$
($                -0.004$, $                -0.065$),
$^{22}_{12}$Mg$_{10}$
($                 0.063$, $                 0.004$),
$^{22}_{10}$Ne$_{12}$
($                 0.005$, $                 0.063$),
$^{22}_{16}$S$_{6}$
($                 0.000$, $                -0.045$), and
$^{18}_{\phantom{0}6}$C$_{12}$
($                -0.048$, $                -0.005$), respectively.
%
Among these nuclei, $^{18}$Mg and $^{22}$S are outside the proton drip
line.

Fig.~\ref{f_R0pn} presents a plot similar to the previous figures but for
the liquid-drop radius $R_0$. The r.m.s.\ difference is 0.18 fm.
Symbols indicated by letters from {\bf A} to {\bf F} are the solutions
of
%
$^{22}_{16}$S$_{6}$
($R_0^{\rm neu}=3.3$ fm, $R_0^{\rm pro}=4.6$ fm),
$^{20}_{14}$Si$_{6}$
($              3.3$, $              4.2$),
$^{16}_{\phantom{0}4}$Be$_{12}$
($              3.9$, $              3.0$),
$^{8}_{6}$C$_{2}$
($              2.5$, $              3.4$),
$^{22}_{\phantom{0}6}$C$_{16}$
($              4.2$, $              3.3$), and
$^{14}_{\phantom{0}4}$Be$_{10}$
($              3.8$, $              2.9$), respectively.
%
Among the nuclei mentioned above, $^{20}$Si is for the excited state
while the others are for the ground states.  $^{22}$S, $^{20}$Si, and
$^8$C are outside the proton drip line.  Hence the maximum difference
for nuclei inside the drip lines occurs in the ground state of
$^{16}$Be.  Incidentally, when we plotted the r.m.s.\ radii
$r_{\rm rms}$ instead of $R_0$, we found that the resulting graph
looked very similar to Fig.~\ref{f_R0pn} except for a scaling relation $R_0$
$\sim$ $(\frac{5}{3})^{1/2}$ $r_{\rm rms}$.

\begin{figure}[tbh]
\vspace*{11.5cm}
\caption{
Same as in Fig.~1 but for the radius $R_0$ (fm) of the corresponding
liquid drop.
}
\label{f_R0pn}
\end{figure}

To summarize, the shapes of proton and neutron density distributions
are not remarkably different from each other according to the results
of our calculations with the Skyrme SIII force. The largest
differences occur in the lightest nuclei.

As for the difference of radial density distribution between protons
and neutrons, one recently pays attention not only to liquid-drop or
r.m.s.\ radius but also to the difference at much lower densities,
i.e., the thicknesses of nucleon skins and the radii of nucleon halos.
We adopt a definition of the skin similar to that proposed in
Ref.$^{7)}$, while the halo radius is defined as the radius at which
angle-averaged mass density is $1.6 \times 10^{-5}$ fm$^{-3}$.
According to the results of our calculation, the nucleon skin grows
monotonously and regularly as nucleons are added to the nucleus.  On
the other hand, the thickness of the nucleon halo (the halo radius
subtracted by $1.2 \times A^{1/3}$ fm) grows very slowly except near
the drip lines, where it changes the behavior completely and expands
very rapidly.  It is also shown that the neutron skin tends to make
the density distribution more spherical.

We have also investigated the systematics of the excitation energy
between the oblate and the prolate solutions and found a clear
difference between below and above the $N=50$ shell magic: For nuclei
with $N < 50$, the oblate solutions often have lower energies than
prolate ones. For nuclei with $N > 50$, oblate ground states are very
rare and found only in nuclei very close to shell magics. This abrupt
change at $N=50$ may suggest that the dominance of prolate
deformations in rare-earth and actinide nuclei should be attributed to
the nature of the Mayer-Jensen major shell compared with that of the
harmonic-oscillator shell

% publicizing publication schedule and anonymous ftp service

All the results of the extensive HF+BCS calculations mentioned in this
report are available electronically as an anonymous ftp service on
the internet: $\mbox{\bf nt1.c.u-tokyo.ac.jp}$.  See a file {\bf
read.me} in the home directory for instructions.  The available
quantities are the binding energies, the Fermi levels, the pairing
gaps, proton and neutron moments ($r^2$, $r^2 Y_{2,0}$, $r^2 Y_{2,2}$
$r^4 Y_{4,0}$, $r^4 Y_{4,2}$, $r^4 Y_{4,4}$), the skin thicknesses,
the halo radii, and the proton/neutron/mass deformation parameters
($a_{20}$, $a_{22}$, $a_{40}$, $a_{42}$, $a_{44}$, $R_0$, $\rho_0$) of
1029 ground states and 758 first-excited local minima.
Single-particle spectra and density distributions of protons and
neutrons are also obtainable for each solution.  Additional
post\-script figures and some FORTRAN source codes to analyze the
results are also provided.

% Acknowledgements

The authors thank Dr.~P.~Bonche, Dr.~H.~Flocard, and Dr.~P.-H.~Heenen
for providing the Cartesian-mesh HF+BCS code {\em EV8}.  They are also
grateful to Dr.~T.~Tachibana and Dr.~K.~Oyamatsu for the TUYY mass
formula code, to Dr.~S.~Raman for supplying the computer file of the
B(E2)$\uparrow$ table, and to Dr.~P.~M{\"o}ller, Dr.~Dobaczewski, and
Dr.~Nazarewicz for discussions.

\vspace{\baselineskip}

\noindent {\bf References}

\vspace{\halflineskip}

\noindent
1) P.\ Bonche, H.\ Flocard, P.-H.\ Heenen, Krieger, and
Weiss, Nucl.\ Phys.\ {\bf A443} (1985) 39.

\noindent
2) N.\ Tajima, S.\ Takahara, and N.\ Onishi,
submitted to Nucl.\ Phys.\ {\bf A}.

\noindent
3) G.\ Audi and A.H.\ Wapstra,
Nucl.\ Phys.\ {\bf A565} (1993) 1.

\noindent
4) T.\ Tachibana, Uno, Yamada, and Yamada,
Atom.\ Dat.\ Nucl.\ Dat.\ Tab.\ {\bf 39} (1988) 251.

\noindent
5) P.\ M{\"o}ller, Nix, Myers, and Swiatecki,
Atom.\ Dat.\ Nucl.\ Dat.\ Tab.\ {\bf 59} (1995) 185.

\noindent
6) S.\ Raman, Malarkey, Milner, Nestor, and Stelson,
Atom.Dat.Nucl.Dat.Tab.\ {\bf 36} (1987) 1.

\noindent 
7) N.\ Fukunishi, T.\ Otsuka, and I.\ Tanihata,
Phys.\ Rev.\ {\bf C48} (1993) 1648.

\end{document}
