The method to search the solutions
Last modification: 1996/3/25
We determine the ground-state solution of each nucleus by, first,
searching the spherical, a prolate, and an oblate solutions and,
second, comparing the energies of thus obtained solutions. Our
strategy to search for these three solutions for each nucleus is as
follows. The spherical solution is obtained by constraining the mass
quadrupole moments to be zero. The prolate (oblate) solution is
searched in two steps. First, we exert an external potential
proportional to Qz on the initial wavefunction until its
quadrupole deformation parameter satisfies delta > 0.1 ( < -0.1).
Second, we switch off the external potential, let the wavefunction
evolve by itself (or with the acceleration method described in a
different paper), and see if it converges to a deformed local minimum.
If the nuclear shape becomes very close to the sphericity in the
course of evolution, i.e. delta < 0.02 ( > -0.02), we conclude
that the normal-deformation prolate (oblate) solution does not exist
in this nucleus.
For some nuclei with 28 < Z, N < 50, the FRDM [MNM94] predicts
very large deformations delta ~ 0.4. In order not to miss such
large-deformation solutions, we have done additional searches for all
the nuclei in this region, in which we continue to exert the
quadrupole potential until delta becomes > 0.4 ( < -0.3) before
starting the free evolution for the prolate (oblate) solution.
(The initial wavefunction must be located within the potential well
around a solution to obtain the solution after a free evolution.)
These additional searches indeed produced large-deformation solutions.
However, none of them are the ground states unlike in the results of
the FRDM.
In shape-transitional nuclei, the PES often has more than three
normal-deformation minima. However, they are usually very shallow and
it is doubtful that each of them corresponds to a distinct eigenstate
notwithstanding the quantum fluctuation in shape. Therefore, we do
not manage to find out all of these shallow minima.