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\begin{document}

\def\baselineskipTaj{0.57cm} \baselineskip=\baselineskipTaj
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\title{
Extensive Deformed HF+BCS calculation \\ with Skyrme SIII force
}

\author{
N.~Tajima, S.~Takahara, and N.~Onishi\\
Institute of Physics, College of Arts and Sciences,\\
University of Tokyo, Komaba, Meguro-ku, Tokyo, 153, Japan
}

\date{}

\maketitle

\begin{abstract}
We have applied the Hartree-Fock+BCS method with Skyrme SIII force
formulated in a three-dimensional Cartesian-mesh representation to
even-even nuclei with $2 \leq Z \leq 114$.  While the details will be
published elsewhere, we summarize in this report the results
concerning the atomic masses, the quadrupole ($m$=0, 2) and
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.  We present nine figures which are not included in the
forthcoming paper but illustrate some arguments given in the paper.
Our results can be obtained electronically via computer network.
\end{abstract}

%----------------------------------------------------------------------

%---------------------------------------------------------------------

\noindent {\bf 1. INTRODUCTION}

\vspace{\halflineskip}

Among theoretical attempts which aim at treating all the nuclides in a
single framework, the simplest category seems to be the mass formula.
Hence, let us focus the discuss of this section on the nuclear masses,
although the purpose of our calculation is not restricted to the
masses.  The most familiar mass formula will be that of Bethe and
Weizs{\"a}cker\cite{We35,BB36}, which expresses the nuclear masses as
a function of the number of neutrons ($N$) and that of protons ($Z$)
as,

\vspace*{-0.35cm}

\begin{displaymath}
  E(N,Z) = a_{\rm V}         A      + a_{\rm S}     A^{2/3}
         + a_{\rm I} (N-Z)^2 A^{-1} + a_{\rm C} Z^2 A^{-1/3},
\end{displaymath}

\vspace*{-0.2cm}

\noindent
where $A=N+Z$.  By changing the values of four coefficients ($a_{\rm
V}$, $a_{\rm S}$, $a_{\rm I}$, and $a_{\rm C}$), the r.m.s.\ error
from experimental data\cite{AW93} of even-even nuclei can be decreased
down to 3.5 MeV.  The form of this formula originates in the liquid
drop picture of the atomic nucleus together with a parabolic
approximation to the dependence on the total isospin.  The first step
to decrease the error is to take into account the shell effect. It
should be noticed here that the shell effect is strongly dependent on
deformation.

Taking into account the shell effect, the TUYY mass
formula\cite{TUY88}, for example, achieved an r.m.s.\ mass error of as
small as 538 keV.  The number of fitting parameters are 6 for the gross
part (corresponding to the parameters of the Bethe-Weizs{\"a}cker
formula) while that for the shell part is as many as 269: There is one
parameter for each value of $Z$ in an interval $1 \le Z \le 112$
and one for each value of $N$ in $1 \le N \le 157$.  Generally
speaking, less number of parameters are preferable for
the reliability of the extrapolation to nuclei not synthesized
yet.  One usually switches to less phenomenological models in order to
reduce the number of parameters.

The most elaborate one among such models seems to be the finite-range
droplet model with a microscopic shell correction (FRDM), whose latest
update was done by M{\"o}ller et al.\ \cite{MNM95}.  Another
extensive calculation was carried out by Aboussir et al.\ \cite{APD92}
in the extended Thomas-Fermi plus Strutinsky integral method (ETFSI).

The former as well as the latter methods can be regarded as
approximations to the Hartree-Fock (HF) equation.  The
straight-forward solutions of the equation including deformation
require long computation time for global calculations even with
present computers.  Such global results are not yet available to the
public as far as we know.  This is the reason why we have embarked on
an extensive HF+BCS calculation.

Incidentally, as far as the nuclear mass concerns, a novel model based
on the shell-model configuration mixing was proposed
recently\cite{DZ95}, which achieved an r.m.s.\ error of 375 keV with
28 parameters.

%---------------------------------------------------------------------

\vspace{\baselineskip}

\noindent {\bf 2. SET UP OF THE CALCULATION}

\vspace{\halflineskip}

The Skyrme force is an effective interaction widely used in mean-field
calculations. It is essentially a zero-range force but 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\cite{BFG75} is one of the many
parameter sets proposed for the force and features good
single-particle spectra and accurate $N-Z$ dependence of the binding
energy.

The advantage to express single-particle wavefunctions on a
three-dimensional Cartesian mesh 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 Refs.~\cite{TOT94,TTO96a}.

\vspace{\baselineskip}

\noindent {\bf 3. RESULTS}

\vspace{\halflineskip}

\noindent {\bf 3.1. nuclear mass}

%-----------------------------------------------------------------------
% table of atomic mass differences

\begin{table}  \label{t_massdev}
\begin{tabular}{lrrrrrr}
\hline
 & \multicolumn{1}{c}{AW'93} & \multicolumn{1}{c}{TUYY}
 & \multicolumn{1}{c}{FRDM}  & \multicolumn{1}{c}{ETFSI1}
 & \multicolumn{1}{c}{SIII}  & \multicolumn{1}{c}{macro} \\
\hline
TUYY     &0.52 (480)&               &               &              & & \\
FRDM     &0.68 (462)& 4.31 (1521)   &               &              & & \\
ETFSI1   &0.74 (430)& 4.27 (1472)   & 2.74 (1742)   &              & & \\
SIII     &2.22 (480)& 2.59 (\pz 977)& 2.50 (\pz 958)&2.26 (\pz 940)& & \\
macro    &3.55 (480)&17.25 (2228)   &16.07 (2246)   &8.29 (1895)   &
          5.71 (1029)& \\
sph-SIII &6.21 (480)& 6.79 (\pz 977)& 7.00 (\pz 958)&6.75 (\pz 940)&
          4.99 (1029)&8.60 (1029)\\
\hline
\end{tabular}
\caption{
The r.m.s.\ differences of nuclear masses between experiment and
theoretical models in MeV.  In parentheses are the number of even-even
nuclei to take the mean value. See text for explanation.
}
\end{table}

In Table~1, we show the r.m.s.\ differences of the masses of even-even
nuclei between theoretical models as well as between the experiments
and the models.  In the table, {\bf AW'93} represents the experimental
atomic mass table by Audi and Wapstra\cite{AW93}, {\bf TUYY} the mass
formula of Tachibana et al.~\cite{TUY88}, {\bf FRDM} the finite-range
droplet model\cite{MNM95}, {\bf ETFSI1} the extended Thomas-Fermi
Strutinsky integral method with the SkSC4 force\cite{APD92}, {\bf
SIII} the HF+BCS results using the Skyrme SIII force, {\bf macro} the
Bethe-Weizs{\"a}cker type function fitted to AW'93, and {\bf sph-SIII}
the HF+BCS with SIII solutions obtained with a spherical code (the
{\sl SKHAFO}\cite{Re91}).  The r.m.s.\ deviation from AW'93 is 2.2 MeV
for SIII, which is 3-4 times as large as that of 0.52 MeV for TUYY,
0.68 MeV for FRDM, and 0.74 MeV for ETFSI1.  It should be noticed,
however, that the parameters of FRDM, TUYY, and ETFSI were fitted to
all the available recent experimental mass data while the parameters
of the SIII force were determined by fitting to the masses and charge
radii of only seven spherical nuclei.  In addition, the number of the
fitting parameters is 275 in TUYY, 19 in FRDM, and 8 in ETFSI, while
it is only 6 in the SIII force.  The result of the spherical code is
worse than that of the Bethe-Weizs{\"a}cker formula, which vindicates
the importance of deformation in the evaluation of the shell effect.

%-----------------------------------------------------------------------
% Fig.1: {f_drip} : Two-nucleon drip lines

We show two-nucleon drip lines calculated with the TUYY mass formula,
the FRDM, and the HF+BCS with SIII in Fig.~\ref{f_drip}.  The
two-proton (-neutron) drip line passes through two even-even nuclei
whose two-proton (-neutron) separation energies have different signs.
As one can see, the two-proton drip line of SIII is almost identical
to that of FRDM and TUYY.  As for the two-neutron drip line, too, the
distance between the lines for SIII and FRDM looks of the same size as
that between the lines for FRDM and TUYY.

%-----------------------------------------------------------------------
% Fig.2 : {f_merr} : Mass error versus Bsurf

In Fig.~\ref{f_merr}, the error of the nuclear masses calculated with
HF+BCS with SIII are plotted versus $B_{\rm surf} -1$, where $B_{\rm
surf}$ is the ratio of the surface area to that of the spherical
shape.  The surface area is calculated using liquid-drop shape
parameters defined in section 3.5. The error from experiment is
negative for most of spherical nuclei, while for deformed nuclei it is
$\sim$ 3 MeV rather independently of the size of deformation.

%-----------------------------------------------------------------------
% Fig.3 : {f_gap}

\vspace{\halflineskip}

\noindent {\bf 3.2. pairing gap}

We compare the experimental and the calculated pairing gaps in
Fig.~\ref{f_gap}.  The former are calculated with the 5-point formula,
while the latter are obtained directly from the BCS part of the HF+BCS
scheme.  The figure contains 252 (1029) proton and 299 (1029) neutron
experimental (theoretical) gaps.
Plus and x symbols indicate proton and neutron gaps, respectively.
In the left-hand side, the gaps are
plotted as functions of the mass number $A$. The empirical formula
$\Delta = 12 A^{-1/2}$ MeV is also drawn with solid curve.  The
experimental points fall close to the empirical curve (left-top
portion).  The calculated points also fall near the curve for heavy
nuclei but they are raised for $50 < A < 100$ (left-bottom portion).
In lighter nuclei ($A < 50$), the calculated gaps are located below
the curve, which is due to our setting the maximum pairing force
strength to 0.6 MeV.  In the right-hand side, the agreements between
the experimental and theoretical gaps are displayed as a scatter plot.
The diagonal line is drawn to facilitate the comparison. One can see a
trend that the difference is small for small pairing gaps (i.e., for
heavy nuclei) but large for large gaps (i.e., for light nuclei).

A likely origin of the overestimation for $50 < A < 100$ is
that the pairing active space above the Fermi level (we use $\le 5$
MeV) is smaller than the wavelength of the shell oscillation ($\hbar
\omega_{\rm osc}$ $\sim$ $41 A^{-1/3}$ MeV) for small $A$; such a
situation does not fit to the continuous spectrum approximation.  The
problem is that it is not always possible to enlarge the pairing
active space because positive energy states, whose wavefunctions are
not spatially localized, should have zero occupation probability.  In
future calculations for $A<100$, it is desirable to improve the method
to determine the pairing force strengths.


\vspace{\halflineskip}

\noindent {\bf 3.3. moments}

The calculated electric intrinsic quad\-ru\-pole moments agree well
with those deduced from experimental B(E2)$\uparrow$\cite{RMM87},
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.  It
may also be related to the complicated landscapes of the potential
energy curves of many shape-coexisting nuclei in mass region
$A$=50-100. See Ref.~\cite{TTO96a} for details.

\vspace{\halflineskip}

\noindent {\bf 3.4. shapes of nuclei in {\boldmath $4 \le Z, N \le 6$}}

%-----------------------------------------------------------------------
% Fig.6 : {f_becpes}

% Beryllium and Carbon PES

Among the 1029 even-even nuclei which we calculated,
$^{8}_{4}$Be$_{4}$ has the largest deformation ($a_{20}$=0.62).
This situation coincides with the two-alpha-cluster picture. The
potential energy curve is given in Fig.~\ref{f_becpes}.
The abscissa is $\delta$ defined as $\delta$ $\equiv$ 3 $\langle Q_z
\rangle$ / 4$\langle r^2
\rangle$ with $Q_z$ $\equiv$ $2 z^2 - x^2 - y^2$.

Concerning $^{10}_{\phantom{0}6}$C$_{4}$ and
$^{10}_{\phantom{0}4}$Be$_{6}$, the experimental B(E2)$\uparrow$
values indicate large deformations: $\vert a_{20} \vert$=0.82 for
$^{10}$C and 1.1 for $^{10}$Be.  However, the potential energy curves
in Fig.~\ref{f_becpes} have only the spherical minimum.  The quantum
fluctuation in shape may be able to account for the large
B(E2)$\uparrow$ since the curves are very soft toward prolate
deformations.

The largest deviation from the experimental $a_{20}$ deduced from
B(E2)$\uparrow$ occurs in $^{12}$C.  For this nucleus the experimental
B(E2)$\uparrow$ is very large (corresponding to $\vert a_{20} \vert
$=0.59) and an oblate intrinsic deformation with a triangular
three-alpha-cluster configuration has been suggested. However, the
HF+BCS calculation with the SIII force, as well as other widely-used
Skyrme forces of the SkM$^{\ast}$ and the SGII, gives a potential
energy curve which has only one minimum at the spherical shape. An old
Skyrme force SII gives an oblate minimum with
$\delta=-0.27$\cite{Va73}.

Let us briefly report on the three-alpha-cluster linear-chain state of
$^{12}$C. By extending the potential energy curve in
Fig.~\ref{f_becpes} to larger $\delta$, we have found the first
excited minimum at $\delta \sim 1.0$, in good agreement with the
result of the Nilsson model for the linear-chain state
($\delta$=1.1)\cite{ZZC91}.  The excitation energy from our
calculation is 21 MeV. Though it is much larger than the experimental
value of 7.654 MeV, the overestimation will be improved by the angular
momentum projection.


\vspace{\halflineskip}

\noindent {\bf 3.5. shape parameters}

Since the deformation parameters are more useful than the electric
multipole moments for certain purposes, let us also discuss the
deformation parameters.  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.  The mass
density of the liquid drop is expressed as

\vspace*{-0.4cm}

\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}

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.

%-----------------------------------------------------------------------
% Fig.4 : {f_rmsrr0}

Let us check the feasibility of thus-defined shape parameters by
comparing them with model-independent quantities.

As one can see from Fig.~\ref{f_rmsrr0}, the liquid drop radius $R_0$
agrees well with the r.m.s.\ radius $r_{\rm rms}$ multiplied by
$(\frac{5}{3})^{1/2}$ (the diagonal line).
Owing to deformation and surface diffuseness,
the former tends to be slightly smaller than the latter.
For 5361 points plotted in the figure (mass, proton, and neutron
moments of 1029 ground and 758 first-excited solutions), the maximum
and the r.m.s.\ deviations of $R_0$ from
$(\frac{5}{3})^{1/2} r_{\rm rms}$ are 0.3 fm and 0.06 fm,
respectively.

%-----------------------------------------------------------------------
% Fig.5 : {f_dela20}

As for the quadrupole deformation parameter, Fig.~\ref{f_dela20} shows
the relation between $a_{20}$ and $\delta$ defined in section 3.4.
The solid line
represents $a_{20} = (\frac{16}{45}\pi)^{1/2} \delta$, which is the
leading-order expression for thin-surface density distributions.  If
we fit a polynomial through order three with the leading order
coefficient fixed to the above value, the resulting function is
$a_{20}$ = $(\frac{16}{45}\pi)^{1/2} \delta$ $-0.47 \delta^2$ $+0.78
\delta^3$.  The maximum and the r.m.s.\ deviations from this function
are 0.05 and 0.007, respectively.

One can find elsewhere\cite{TTO96a} figures in which thus-de\-fined axial
deformation parameters ($a_{20}$ and $a_{40}$) are graphically
displayed in the $N$-$Z$ plane.
Comparisons with the results of FRDM are made there, too.
Concerning the non-axial deformation
parameters ($a_{22}$, $a_{42}$, and $a_{44}$), their magnitudes have
turned out very small ($\vert a_{22} \vert$, $\vert a_{42} \vert$ $<
10^{-4}$, $\vert a_{44} \vert$ $\sim 10^{-3}$).

\vspace{\halflineskip}

\noindent {\bf 3.6. shape difference between protons and neutrons}

%-----------------------------------------------------------------------
% Fig.7 {f_a20pn} : difference of a_{20} 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.  Difference of the quadrupole
deformation parameter $a_{20}$ between protons and neutrons are shown
for the ground states of 1029 even-even nuclei (denoted by plus
symbols) and the first-excited solution of 758 even-even nuclei for
HF+BCS with SIII force.
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.

%-----------------------------------------------------------------------
% Fig. 8 : {f_a40pn} : Difference of hexadecapole deformation
%                      between protons and neutrons

Fig.~\ref{f_a40pn} is 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$_{12}$
($a_{40}^{\rm neu}=-0.127$, $a_{40}^{\rm pro}=-0.031$), which
is indicated by letter {\bf A}.
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. 9 : {f_R0pn} : Difference of liquid-drop radius
%                     between protons and neutrons

Fig.~\ref{f_R0pn} presents a plot similar to the previous two 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 quite similar to Fig.~\ref{f_R0pn} except for a scaling relation
$R_0$ $\sim$ $(\frac{5}{3})^{1/2}$ $r_{\rm rms}$.

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.
It should be reminded here that our calculation does not reach the
neutron drip line except in light mass region.

\vspace{\halflineskip}

\noindent {\bf 3.7. energy difference between prolate and oblate states}

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 major shells of not the harmonic-oscillator shell
but the Mayer-Jensen shell. See Ref.~\cite{TTO96a} for a figure on
which the above argument is based.


\vspace{\halflineskip}

\noindent {\bf 3.8. nucleon skins and halos}

As for the difference of radial density distribution between protons
and neutrons, we study not only the liquid-drop or r.m.s.\ radius but
also 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.~\cite{FOT93}, 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 found that the neutron skin tends to make the density
distribution more spherical. See Ref.~\cite{TTO96a} for the details.

%-----------------------------------------------------------------------
% acknowledgements

\vspace{\baselineskip}

The authors thank Dr.~P.~Bonche, Dr.~H.~Flocard, and Dr.~P.-H.~Heenen
for providing a Cartesian-mesh HF+BCS code {\sl EV8}\cite{BFH85}.
They are also grateful to Dr.~T.~Tachibana and Dr.~K.~Oyamatsu for the
TUYY mass formula code, to Dr.~S.~Raman for the computer file of the
B(E2)$\uparrow$ table, and to Dr.~P.~M{\"o}ller, Dr.~Dobaczewski, and
Dr.~Nazarewicz for discussions.

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\bibitem{TTO96a} % HFS3 (ev8 calc. with SIII)
         N.~Tajima, S.~Takahara, N.~Onishi, submitted to
         Nucl. Phys. {\bf A}.
\bibitem{Re91} % spherical HF+BCS code with Skyrme force
         P.-G.~Reinhard, in ``Computational Nuclear Physics 1'',
         ed. K.~Langanke, J.A.~Maruhn, and S.E.~Koonin,
         (Springer-Verlag, Berlin, 1991) 28.
\bibitem{RMM87} % B(E2) 0 -> 2 tabulation
         S.\ Raman, C.H.\ Malarkey, W.T.\ Milner, C.W. Nestor, JR.,
         and P.H.\ Stelson,
         Atom.\ Data Nucl.\ Data Tab.\ {\bf 36} (1987) 1.
\bibitem{Va73} % Separation between HF and BCS, oblate 12C with SII
         D.~Vautherin, Phys.Rev.{\bf C7} (1973) 296.
\bibitem{ZZC91} %Skyrme HF, three-alpha linear chain state of 12C
         L.~Zamick, D.C.~Zheng, J.A.~Caballero, E. Moya de Guerra,
         Ann.\ Phys.\ {\bf 212} (1991) 402.
\bibitem{FOT93} % Definition of skin
         N.~Fukunishi, T.~Otsuka, and I.~Tanihata,
         Phys.\ Rev.\ {\bf C48} (1993) 1648.
\bibitem{BFH85}  % Self consistent study of triaxial deformations
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\end{thebibliography}

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\newpage

\noindent {\bf FIGURE CAPTIONS}

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\item \label{f_drip}
Two-nucleon drip lines from three\\
mass models.
\item \label{f_merr}
Error of the atomic masses from\\
HF+BCS with SIII.
\item \label{f_gap}
Experimental and calculated pairing gaps.
\item \label{f_becpes}
Potential energy curves of even-even nuclei in $4 \le Z, N \le 6$.
\item \label{f_rmsrr0}
Relation between liquid-drop radius\\
and r.m.s.\ radius.
\item \label{f_dela20}
Relation between two kinds of quad- \\
rupole deformation parameters.
\item \label{f_a20pn}
Difference of quadrupole deforma- \\
tion between protons and neutrons.
\item \label{f_a40pn}
Difference of hexadecapole deforma- \\
tion between protons and neutrons.
\item \label{f_R0pn}
Difference of liquid-drop radius between protons and neutrons.
\end{list}

\end{document}
