Classification of SD bands

For the conserved parity, =,
and signature, *r*=,
quantum numbers, the space of single particle states is
separated into four parity/signature blocks,
,-*ir*]=(++,+-,-+,-).
By constructing a particle-hole
excitation we necessarily arrive at a rearrangement
among the four blocks of levels - one class of rearrangements always leading
to the occupation of all the lowest levels *within* each block.
It turns out that such states form a majority among the low-lying
bands studied here.
Supposing that the lowest states in each of the blocks are occupied,
one may describe in the standard way the
many-particle configurations by giving the numbers of states
occupied in each block. In this notation, the SD ^{32}S magic
configuration is given by the (4,4,4,4) occupation numbers,
both for neutrons and protons, while the ground-state
configuration reads (5,5,3,3).

All configurations that are examined below are built by exciting
particles from the four levels below, to the four levels above the
neutron and/or proton Fermi energies at the SD shape. The
remaining orbitals below the Fermi levels will always be occupied.
Therefore, the single-particle neutron or proton active spaces are
composed of 8 orbitals (4 intruders and 4 non-intruders) that
contain 4 particles. This leads to *C*^{8}_{4}=70 possible many-body
SD configurations for neutrons and *C*^{8}_{4}=70 SD configurations for
protons. The fact that among the bands studied in this article
always the lowest states in each parity/signature block are
occupied (other cases, where necessary, will be explicitly
mentioned) reduces these numbers from 70 to 19 neutron or proton
configurations necessary to control the low-energy rotational bands
constructed within the discussed active spaces.

Further, we use the observation that for both intruder states,
[330]1/2 and [321]3/2, the *r*=+*i* signature partners are always
below the *r*=-*i* partners (for all deformations and rotational
frequencies). Hence, the intruder orbitals should preferably be
occupied in that order of increasing energy, i.e., when one
particle occupies the negative-parity orbitals, it will occupy the
[330]1/2(*r*=+*i*) orbital, when two - they will occupy the [330]1/2(*r*=+*i*) and [330]1/2(*r*=-*i*)
orbitals, and when three - they will occupy the [330]1/2(*r*=+*i*), [330]1/2(*r*=-*i*),
and [321]3/2(*r*=+*i*) orbitals, etc. This rule reduces the number of available
configurations from 19 to 9. Finally, we reject two more
configurations, as described below, and we are left with 7
configurations to be considered for neutrons and for protons.
Although such a preselection of configurations may appear to be
quite arbitrary, it is in fact based on the requirement that one
wants to end up with a restricted set of low-energy configurations
only.

Figure 3 shows schematically the single-particle
orbitals (top), as well as all the considered here particle-hole
configurations (bottom). The same diagram is valid both for
neutrons and protons. The four intruder-states that are close to
the Fermi energies are all characterized by the principal HO
quantum number *N*_{0}=3. Following the well-established notation,
we denote the neutron or proton intruder occupations by the symbol
3^{n/p}, where *n* or *p* are the numbers of the occupied neutron
and proton intruder states, respectively.

As illustrated in the figure, for the 3^{0} or 3^{4} configurations,
there are four or none particles, respectively, in the positive-parity
states, and hence, in our predefined phase space, these configurations
are unique. For the 3^{2} configuration, two particles occupy
the positive-parity states, and we restrict our considerations
to only one (out of three) occupation variant, namely, we require both particles
to occupy the two signatures of the lower orbital [211]1/2.
Hence, in the following, symbol 3^{2} pertains to this particular
configuration. Finally, if one or three intruder states are occupied,
i.e., in the 3^{1} or 3^{3} configurations, there are
accordingly, three particles or one particle in the positive-parity
orbitals, and here an additional label is necessary. We
distinguish the corresponding configurations by introducing subscripts
+ or -, i.e., by using symbols 3^{1}_{+}, 3^{1}_{-}, 3^{3}_{+}, 3^{3}_{-}.
The subscripts correspond, (i) in the 3^{3} configurations, to the
signature of the occupied [211]1/2(*r*=)
orbital, and (i)
in the 3^{1} configurations, to the
signature of the occupied [202]5/2(*r*=)
orbital. Whenever symbols
3^{1} or 3^{3} without the subscripts are used, they pertain to both
such configurations.

After having preselected the 7 neutron and proton configurations,
we have at our disposal 49 configurations of the whole nucleus,
which we denote by 3^{n}3^{p}, and when necessary supplement by the signature
subscripts + or -, as described above.
For example, the magic SD configuration of
^{32}S is denoted by 3^{2}3^{2}, and the ground-state
configuration reads 3^{0}3^{0}.

A manifest symmetry between neutrons and protons implies a manifest symmetry
between the corresponding rotational bands. We have verified that those bands
which are mirror images obtained from one another by replacing the neutrons
by the protons and *vice versa* lead to almost identical results.
However, because of the larger spatial extensions of intruder
orbitals as compared to positive-parity orbitals, the (very small) Coulomb
shifts will always slightly bring down the *p*>*n* configurations
below those with *p*<*n*. Consequently, in the following we consider only
the 3^{n}3^{p} configurations for *p**n*.
Introducing these last arguments into our selection
scheme, we end up with 30 ^{32}S
configurations to be considered in the further analysis.