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{\LARGE \sf

Skyrme Hartree-Fock-Bogoliubov on 3D-mesh

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\vspace{\baselineskip}

Naoki Tajima\footnote{
e-mail address: tajima at c.u-tokyo.ac.jp} % modified on 04/11/25 

Department of Basic Science,
University of Tokyo, Komaba, Meguro, Tokyo, 153, Japan.

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\vspace{\baselineskip}

% HF with Skyrme force

The Hartree-Fock (HF) method with the Skyrme effective interaction has
been a very successful model to describe the ground-state masses,
structures, and low-energy excitation modes of atomic nuclei.

% mesh representation

As a representation of the HF single-particle wavefunctions, Bonche et
al.\refer{BFH85} introduced a three-dimensional (3D) cubic mesh
representation.  It is more suitable than the oscillator-basis
expansion for the descriptions of low-density tails and
exotic shapes.

% a systematic HF+BCS calculation

Employing this representation, we have done a systematic
Hartree-Fock+BCS calculation with the Skyrme SIII force for the ground
states of 1029 even-even nuclei with $2 \leq Z \leq 114$\refer{TTO96}.
We have also searched systematically for large-deformation solutions
\refer{TTO97}.  From the analysis of the results, we discussed the trends
of the nuclear properties like the mass, moments, and the
skin thicknesses.

This calculation covered the neutron number from outside the proton
drip line to beyond the experimental neutron-rich frontier.  We plan
to extend the calculation to the neutron drip line.  For this purpose,
however, the BCS scheme may not be very adequate: As the Fermi level
approaches to zero, the contribution of positive-energy HF
single-particle orbitals to the pairing correlation may be enhanced so
strongly that its effect may move the neutron drip line by several
nuclides in the nuclear chart.

% HFB in the coordinate space

To include the positive energy states in the pairing correlation, one
needs to switch from HF+BCS to the Hartree-Fock-Bogoliubov (HFB)
method in the coordinate representation.  It is easy to solve with
present computers for spherical case but is still difficult for
deformed case. Up to now, only a method of solution was presented by
Terasaki et al\refer{THF96}. It is a two-step method,
in which the HFB hamiltonian is diagonalized using the HF orbitals as
the basis.

Employing the two-step method, we study the precision of the
low-density tail at large radius as a function of the number of HF
orbitals included in the diagonalization.  In order to activate a
larger space for pairing than Terasaki et al.\ considered, one must
treat a HFB matrix of intractably large dimension.  We propose a
method to eliminate the HF orbitals only negligibly contributing to
the density matrices from the HFB basis to reduce the dimension
substantially.

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\item \label{bib:BFH85}  % Self consistent study of triaxial deformations
         P.~Bonche, H.~Flocard, P.-H.~Heenen, S.J.~Krieger, et al.,
         Nucl.\ Phys.\ {\bf A443} (1985) 39.
\item \label{bib:TTO96} % Extensive HF+BCS calculation with Skyrme SIII force
         N.~Tajima, S.~Takahara, and N.~Onishi,
         Nucl.\ Phys.\ {\bf A603} (1996) 23. The results are obtainable from
         ftp://nt1.c.u-tokyo.ac.jp/hfs3/
\item \label{bib:TTO97} % systematic study of super deformation
         S.~Takahara, N.~Tajima, and N.~Onishi,
         in preparation.
\item \label{bib:THF96} % 3D solution to HFB for drip line nuclei
         J.~Terasaki, P.-H.~Heenen, H.~Flocard, and P.~Bonche,
         Nucl.\ Phys.\ {\bf A600} (1996) 371.
\end{list}

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