Introduction

It is more than ten years by now that the study of superdeformed (SD) shapes in nuclei constitute one of the main venues in nuclear spectroscopy. Today it is well understood that an increased stability of strongly elongated nuclei results from the quantum (shell) effects that manifest themselves, among others, through a lowering of the nucleonic level densities at certain nucleon numbers. Within the anisotropic harmonic oscillator (HO) model, such shell effects arise when the ratios among the three principal frequencies are equal to the ratios of small integer numbers. The strongest shell closures correspond to axially deformed nuclei with the semi-axis ratios 2:1:1 and 3:2:2, the axis ratios being simply related to the oscillator frequencies [1].

The HO model is of course only poor an approximation for the
majority of nuclei, for which the spin-orbit interactions play a determining
role. Yet as it happens, the nuclear mean field obeys approximately a
specific SU(2) symmetry, usually referred to as a pseudo-spin.
(For an early formulation of the pseudo-spin symmetry
see Refs. [2,3]; the contemporary formulation of the
problem is based on the Dirac formalism according to the scheme proposed
in Ref. [4] and further developed recently in
[5,6].)
Taking into account this symmetry allows at the
same time to take care of the strong spin-orbit coupling, and profit from the
simplicity of the HO model.
Indeed, basing on the pseudo-spin symmetry, and
employing a realistic deformed mean field Hamiltonian,
it was possible to predict [7] (after the initial discovery
of the SD band in ^{152}Dy [8] but several years before the
experiments in other regions have been done),
the existence of the whole groups of SD nuclei.
Moreover, the predictions gave also the fact that the
deformations of strongly elongated shapes
may considerably deviate from
the 2:1:1 HO rule; these deviations are now confirmed through
numerous experiments. The abundance scheme for the nuclear SD states
at high angular momenta is well established today in the so-called
*A*190, *A*150, *A*130, and *A*80 regions, see
Refs. [9,10,11,12] for reviews; it includes also
the recently discovered SD states in the *A*60 region
[13,14,15,16,17,18,19,20],
as well as a region of fission isomers in *A*240 nuclei, known
already for a long time but at relatively low angular momenta.

Numerous cluster structures in light nuclei
can also be interpreted as SD states, see
Refs. [21,22] for more details and a more exhaustive
reference list. In particular, the 2:1:1
deformed HO model predicts the SD shell-closures
[1] at particle numbers 2, 4, 10, 16 and 28,
a sequence characterized by an increased stability
at large deformations, and also
compatible with the -cluster structures.
This gives, for example, the -
cluster ground
state of ^{8}Be, or the ^{16}O-
cluster state in
^{20}Ne. Prolonging the same sequence, one may expect stable SD
structures in ^{26}Ne and ^{32}S. Next, doubly-magic
SD states should appear at *N*=*Z*=28 (not to be confused with the
spherical shell closures at the same nucleon numbers). However, because
of the increasing role of the spin-orbit interaction when the nucleonic numbers
increase, these values are slightly modified.
This gives the doubly-magic SD nucleus ^{60}Zn at the center
of the experimentally known SD *A*60 region.

One can see that the SD states in ^{32}S
(although experimentally not discovered to date) constitute a missing
link between the known cluster SD states in very light nuclei,
and the known SD states in the *A*60 region. On the one hand,
the first indications that the cluster SD states in ^{32}S may
exist are provided by the measurements in the ^{16}O-^{16}O
breakup channel [23], and by the ^{16}O-^{16}O
molecular resonances, as quoted in Ref. [24]. On
the other hand, several mean-field calculations, both
non-self-consistent [25] and self-consistent
[26,27,28,29], as well as the -cluster
calculations [30,24], predict in ^{32}S
an existence of the 2:1:1 deformed structures.
It is not clear at the moment, what exactly is the relationship between
the molecular states (a pair of touching ^{16}O nuclei), and the SD
states (a compact matter distribution), although
both classes may correspond to the same axis ratios and deformations.
Such strongly deformed states
should coexist with numerous low-deformation states already
known in this nucleus [31]. In fact, the latter ones are very well
described by the *sd*-shell model [32].

One may expect a number of interesting physical phenomena that can be studied
in the hypothetical SD configurations of ^{32}S, such as
the shape coexistence, competition between various decay
channels, proton neutron pairing and its deformation dependence,
effects related to the time-odd components of nuclear mean
fields, as well as nuclear-molecular and nuclear-cluster
structures. Detailed properties may be significantly influenced
by the presence of intruder states originating from the *N*_{0}=3,
and even *N*_{0}=4 HO shells. With a total number of nucleons
strongly restrained (only 16 per one kind of particles) one
should expect a pronounced variation of shapes from one
single-particle (particle-hole) configuration to another.

In the present paper we aim at theoretical description of the
SD states in ^{32}S and in four neighboring nuclei:
^{33}S, ^{31}S, ^{33}Cl, and ^{31}P. We present predictions
pertaining to detailed spectroscopic information on excitation
energies, spins, moments of inertia, and quadrupole moments
of the SD rotational bands. All these observables may, in a very near
future, become available within the discrete-spectroscopy
measurements using large detector arrays; these observables
have already been obtained experimentally in the other groups of SD nuclei.

The paper is organized as follows. After briefly presenting
in Sec. 2 the theoretical methods we use in this study,
in Sec. 3 we discuss the deformed-shell gaps and
Coulomb effects in ^{32}S, we present a classification of the SD bands,
and describe the level crossings. Results of calculations
for the SD bands in ^{32}S are presented in Sec. 4,
and those for ^{33}S, ^{31}S, ^{33}Cl, and ^{31}P in Sec. 5. Finally, in Sec. 6
we briefly discuss the question of the stability of SD bands, and
Sec. 7 presents our conclusions.