Superdeformed bands in one-particle and one-hole neighbors of

In order to analyze polarization effects induced by individual
particle or hole orbitals in ^{32}S, we have also performed the
HF calculations for the four neighboring nuclei: ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P. Among
them, there are two pairs of mirror nuclei, ^{33}S - ^{33}Cl
and ^{31}S - ^{31}P. For each of these nuclei we have
calculated four bands, corresponding to either the four lowest
available particle states (in *A*=33), or the four highest
available hole states (in *A*=31). In other words, in ^{33}S
or ^{33}Cl the neutron or proton is added to the magic 3^{2}3^{232}S configuration, in the [321]3/2(*r*=)
and [202]5/2(*r*=)
orbitals,
which gives the neutron or proton configurations: 3^{2}_{+},
3^{2}_{-}, 3^{3}, and 3^{3*}. Here, by an asterisk we denote the
configuration in which a particle is added not to the lowest
available intruder state, but to the next-to-lowest available
intruder state. Similarly, in ^{31}S and ^{31}P the neutron or proton
is removed from the magic 3^{2}3^{2} ^{32}S configuration, from
the [330]1/2(*r*=)
and [211]1/2(*r*=)
orbitals, which gives the neutron
or proton configurations: 3^{2}_{+}, 3^{2}_{-}, 3^{1}, and
3^{1*}. Again, by an asterisk we denote the configuration in which
a particle remains not in the lowest available intruder state,
but in the next-to-lowest available intruder state.

In Figs. 11 and 12 we show energies of the
calculated HF bands in ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P. One can observe that the mirror nuclei
have extremely similar SD spectra. Bands in the *A*=33
nuclei form pairs of degenerate signature partners, while those
corresponding to the signature partners in *A*=31 are strongly
split, in accordance with the characteristic features of the
corresponding single-particle routhians, Figs. 1 and
2. Note that the ground state bands in the *A*=33
nuclei correspond to non-intruder particle states, and similarly, those
in the *A*=31 nuclei correspond to holes in non-intruder orbitals.

The HF calculations give the energies of rotational bands
on the absolute scale. Therefore, in order to estimate the
available *Q*-value windows for particle emissions, one may
simply compare (at a given value of the angular momentum) the
energies shown in Figs. 6, 11, and
12. Since the rigid-rotor reference energies are the
same at fixed spins, one can directly compare the values given
in the figures. For example, for ^{31}S the yrast energy at *I*=12
is about -244MeV, which shows that none of the ^{32}S bands
shown in Fig. 6, except the 3^{4}3^{4} and 3^{0}3^{4}configurations, can emit a zero-angular-momentum neutron to the SD
states in ^{31}S. Similarly, for ^{31}P the corresponding
yrast energy is -249.5MeV, which opens up the proton emission
channel from several other bands, but not those from the
near-yrast bands shown in Fig. 7.

Let us emphasize
that the angular momentum, ,
carried away by an emitted
particle, dramatically influences the considered *Q*-values in nuclei
around ^{32}S, especially at high spins.
Since after subtracting the rigid-rotor reference, the energies of
bands are fairly flat (Figs. 6, 11, and
12), one can very simply estimate
the *Q*-values at given *I* and
values to be by an amount of
++1)]0.05MeV larger than those at =0.
For instance, at *I*=20,
and with the angular-momentum transfer =2 (or 3),
the additional energies in a daughter nucleus
are 4.3 (or 6.6)MeV. Consequently, the protons emitted
through the high angular-momentum (e.g., *N*_{0}=3) orbitals are among
the most likely candidates for the band-to-band emission mechanism.
From the results presented in the figures one may precisely
estimate the *Q*-value windows for particles carrying out any given amount
of the angular momentum from any given band.

The illustrations of the dynamical moments in ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P, shown in
Figs. 13 and 14, indicate an extreme similarity
of the results in mirror nuclei.
This suggests that several among the SD bands in the mirror
nuclei around ^{32}S might manifest the ``identical band'' phenomenon.
Comparing these results with those in the magic SD band in ^{32}S,
Fig. 9, one sees that particles in the intruder
[321]3/2 orbitals and extruder [202]5/2 orbitals, respectively
add and subtract 1/MeV(at high spin) with respect to the
magic core. Variations of
,
that correspond to the
intruder and non-intruder hole states, are of the similar
order.

By calculating differences between one-body observables, like the
angular momentum or quadrupole moment, determined in ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P, and
in ^{32}S, one can identify basic single-particle properties of
all important orbitals around the SD ^{32}S magic-core
configuration. These differences correspond not only to the
bare average values of the observables, calculated for given
orbitals, but also include complete polarization effects. It
is known that in the SD *A*150 nuclei, the
single-particle alignments [42], and charge
quadrupole moments [43,44], constitute additive
quantities with respect to adding and subtracting particles
from the magic SD configurations of ^{152}Dy. An analogous
observation is also confirmed by calculations in the SD
*A*60 nuclei [29,45,46]. In the
present paper we have verified the additivity of alignments and
quadrupole moments between the SD bands in ^{32}S, and in ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P.
Tests of this principle in other nuclei around ^{32}S are left
for a future publication.

In Figs. 15 and 16 we present the obtained
relative alignments
and proton quadrupole moments
,
respectively. Since the relative alignments
pertain to the total angular momentum, the effects of neutron
and proton orbitals, obtained in the *N*=16 and *Z*=16 nuclei,
respectively, are almost identical. For
relative proton quadrupole moments, the effects of neutrons and
protons are different, because neutrons contribute only through
the polarization effects, while for protons one also has the
bare direct contribution. In Figs. 15 and 16
we also indicate by which particle- or hole-orbital differ the
bands in ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P form the magic SD band in ^{32}S.

One can see that the relative alignments generated by various
orbitals differ considerably. Therefore, the relative alignments may
serve as distinct fingerprints of orbitals in SD nuclei around
^{32}S. In particular, the second intruder, hole-orbital
[330]1/2(*r*=-*i*), gives rather large negative relative alignment, while
the positive-parity, hole-orbital [211]1/2(*r*=+*i*), gives a rather
constant alignment of about -1,
and hence may be at
the origin of hypothetical yet another class of
identical bands in this region.

The relative proton quadrupole moments of orbitals around
the magic *N*=*Z*=16 SD gap are fairly constant in function
of the rotational frequency. Values corresponding to
intruder orbitals are usually much larger than those
corresponding to positive-parity orbitals.
Hence, one can easily understand the origin of groups of ^{32}S bands
having significantly different quadrupole moments, see Sec. 4.2.
As far
as the polarization effects alone are concerned, the
extruder particle orbitals [202]5/2(*r*=)
carry almost the same
effect as the intruder hole-orbitals [330]1/2(*r*=). Needless
to say, these are the main orbitals which are at the origin
of the SD shapes in ^{32}S.