arXiv:nucl-th/0305021v1 8 May 2003
Beyond–mean–field–model analysis of low-spin normal-deformed and superdeformed
collective states of32S,36Ar,38Ar and40Ca
M. Bender,1H. Flocard,2and P.-H. Heenen1
1Service de Physique Nucl´ eaire Th´ eorique, Universit´ e Libre de Bruxelles, C.P. 229, B-1050 Bruxelles, Belgium
2CSNSM, Bt.104, F-91405 Orsay Campus, France
(Dated: May 8, 2003)
We investigate the coexistence of spherical, deformed and superdeformed states at low spin in
32S,36Ar,38Ar and40Ca. The microscopic states are constructed by configuration mixing of BCS
states projected on good particle number and angular momentum. The BCS states are themselves
obtained from Hartree-Fock BCS calculations using the Skyrme interaction SLy6 for the particle-
hole channel, and a density-dependent contact force in the pairing channel. The same interaction is
used within the Generator Coordinate Method to determine the configuration mixing and calculate
the properties of even-spin states with positive parity. Our calculations underestimate moments of
inertia. Nevertheless, for the four nuclei, the global structural properties of the states of normal
deformation as well as the recently discovered superdeformed bands up to spin 6 are correctly
reproduced with regard to both the energies and the transition rates.
PACS numbers: 21.10.Ky, 21.10.Re, 21.30.Fe, 21.60.Jz, 27.30.+t
The existence of deformed bands in the spectrum of nu-
clei whose ground state is spherical has been established
since the sixties for16O, and since the early 1970s for
40Ca . With the modern large multi-detector γ arrays
like Euroball and Gammasphere, many more normal-
deformed and even superdeformed rotational bands have
been uncovered in these systems, such as those explored
up to very high spin in doubly-magic40Ca [2, 3], and
the adjacent transitional nuclei36Ar [4, 5] and38Ar .
The occurence of well-deformed prolate structures in such
magic or close-to-magic nuclei is understood as resulting
from a drastic reorganization of the Fermi sea in which
the oblate deformation driving last level of the shells are
emptied while orbitals originating from the f7/2shell are
On the theoretical side, the new structures have been
explored with the cranked mean-field (MF) method 
in which such a reorganization is naturally taken into
account. On the other hand, thanks to conceptual and
numerical progress that took place over the last decade,
the shell model method is now also in position to analyze
spectra in which spherical and well-deformed configura-
tions coexist. However, in the latter case, the complex-
ity of the calculation often prevents full scale diagonal-
izations in complete shells and specific choices must be
made for the extension of the basis in order to keep com-
putations within reach of present technology. In some
sense, prior to the diagonalization, the expected physics
is introduced in the Hamiltonian to use. This choice is
vindicated by the outcome the of calculations and su-
perdeformed bands built on particle excitations to the
pf shells do come out. The cranked MF method does
not have to make such an a priori choice; if a reorganiza-
tion of the Fermi sea is required, it will occur naturally as
a consequence of energy optimization. The present work
illustrates a class of methods attempting to bridge the
gap betwen these two approaches while remaining close
to the MF in spirit. Indeed, the introduction of a di-
agonalization within a class of MF collective states and
the restoration of symmetries (particle number, angular
momentum) broken at the mean-field level transfers the
physical description from the intrinsic to the laboratory
frame where the shell model naturally operates.
In the next section, we provide a quick overview of the
qualitative features of our method, which is described in
more detail in . The necessary formalism and notations
are also introduced. In the third section, we present our
results for the four nuclei40Ca,38Ar,36Ar and32S; a
selection which keeps in touch with recent experimental
progress. Moreover, taking into account an earlier work
on16O , this choice allows us to test our method on a
set of nuclei illustrating most of the spectroscopic diver-
sity of the sd shell region of the nuclear chart. In this sec-
tion, our results are compared with data and with those
provided by shell model, cranked MF, or other extended-
II. THE METHOD
A. Effective Hamiltonian and Collective Hilbert
The N-body physical states analyzed in this work
are contained in the Hilbert space spanned by solu-
tions |β2? of Constrained Hartree-Fock-BCS (CHFBCS)
equations . In those equations, the constraint is im-
posed on the axial mass quadrupole moment operator
ˆQ20. The notation β2is a label standing for any quan-
tity in one to one correspondance with the expectation
value Q2= ?β2|ˆQ20|β2?. The single-particle wave func-
tions from which BCS states are constructed are dis-
cretized on a three-dimensional mesh. As explained in
Ref. , this technique provides accurate solutions of
the mean-field equations. Note that our calculation does
not assume the existence of an inert core.
As two-body interaction in the particle-hole chan-
nel of the Hamiltonian ˆH, we have chosen the SLy6
parametrization  of the Skyrme force.
our previous studies were performed with the SLy4
parametrization. Both sets have been fitted on the same
set of observables, but differ by the way the center-of-
mass motion is treated. In SLy4, only the diagonal
part of the cm energy is substracted from the total en-
ergy, while in SLy6, the full cm energy is extracted self-
consistently. This difference makes the calculations with
SLy6 more time consuming.
also the surface tension of the Skyrme interaction and,
in this respect, the SLy6 parametrization seems to be
more satisfactory . This better surface tension is the
motivation for the choice of SLy6 in the present study.
The pairing force is a zero-range, density-dependent force
acting predominantly at the surface of the nucleus in or-
der to describe the pairing effects in the particle-particle
T = 1, Tz= ±1 channel. The parameters of the latter
force  are identical for neutrons and protons and taken
without readjustment from Refs. [7, 8].
calculations, for each nucleon species, the active pairing
space is limited to an interval of 10 MeV centered at the
Fermi surface. The present study does not therefore in-
volve the definition of a new set of forces. It relies on
well established interactions tested within the mean-field
approach over a wide range of nuclei and phenomena cov-
ering the nuclear chart. This work is thus part of a pro-
gram whose aim it is to perform an additional evaluation
of this Hamiltonian by taking into account the effects of
For the sake of an easier comparison with the litera-
ture on quadrupole collective spectroscopy, we adopt the
sharp edge liquid drop relation to relate the β2deforma-
tion parameter and the axial quadrupole moment Q2
The cm operator affects
As in earlier
where the nuclear radius R in fm at zero deformation is
related to its mass A according to the standard formula
R = 1.2A1/3. In this paper, only axial prolate and oblate
deformations are considered. As we are mostly concerned
with low-energy collective spectroscopy, we consider a
range of values of β2covering all deformations such that
the excitation energy of the constrained BCS states with
respect to that of the spherical configuration |β2= 0? is
at most 20 MeV.
In the CHFBCS equations, we also introduce the cor-
rection terms associated with the Lipkin-Nogami pre-
scription (see Ref.  for a description and for further ref-
erences). Indeed, within schematic models, these terms
have been shown to make the BCS solutions closer to
those which would result from a full variation after pro-
jection on the particle number. Thus, they should be
appropriate for calculations such as ours where a projec-
tion on N and Z is anyhow carried out at a later stage.
Another benefit of the LN method is that it suppresses
the collapse of pairing correlations which may otherwise
occurs when the density of single-paricle states around
the Fermi level is small. It therefore ensures a smooth
behavior as a function of the quadrupole moment of the
overlap and energy kernels which intervene in the Gen-
erator Coordinate Method (GCM).
Several fundamental symmetries are broken in the BCS
states |β2?, which are eigenstates neither of the parti-
cle numbers nor of the angular momentum.1We restore
these symmetries by means of a triple projection: first
on the proton and neutron numbers Z and N, then on
the total spin J. For a given nucleus, the BCS state |β2?
has been determined with the usual constraints ensuring
that the expectation values of the proton and neutron
numbers have the correct Z and N values. In the fol-
lowing, we only select that component of the BCS state
|β2? which is an eigenstate of the particle number opera-
tors for the same values, discarding the Z ± 2,4,... and
N ± 2,4,... components. For this reason, we do not in-
troduce particle number labels and use the notation |β2?
for the particle-number projected state.
By contrast, the CHFBCS equations leading to the
states |β2? do not include any constraint on the angular
momentum expectation value. Starting from the N and
Z projected states, we will consider separately all the col-
lective Hilbert spaces spanned by the components |J β2?
resulting from a projection of the |β2?’s on the subspace
of the even total angular momentum J. Hereafter, the
notation |J,β2? stands for any of the states associated
with the 2J + 1 values of the the third component of
the angular momentum. Note that, because the initial
CHFBCS equations do not break the reflection or the
time-reversal symmetry, the model provides no informa-
tion on odd-spin states.
Finally, we diagonalize the HamiltonianˆH within each
of the collective subspaces of the non-orthogonal bases
|J,β2? by means of the Generator Coordinate Method
(GCM) [9, 14, 15]. This leads to a set of orthogonal
collective states |J,k? where k is a discrete index which
labels spin J states according to increasing energy. As for
|J,β2? the notation |J,k? stands for any state of the spin
multiplet. A byproduct of the GCM are the collective
wave functions gJ,k(β2) describing the distribution of the
states |J,k? over the family |J,β2?. All collective prop-
erties discussed hereafter are directly computed from the
N-body physical states |J,k?.
We stress that the correlations introduced by the dif-
ferent configuration mixings of the initial mean-field wave
functions |β2? achieve several goals. First, the particle-
number projection corrects a deficiency of the BCS de-
scription of pairing in finite systems. Second, the an-
1Among others that we do not discuss are translational, and
gular momentum projection separates the dynamics ac-
cording to spin and allows a direct calculation of elec-
tromagnetic moments and transition probabilities in the
laboratory rather than the intrinsic frame. Finally, per-
forming a configuration mixing over the coordinate β2
by the GCM, we construct a set of orthonormal states
|J,k? in which the large-amplitude quadrupole collective
correlations are taken into account .
B. Charge multipole and transition moments
The angular-momentum projection performs a trans-
formation to the laboratory frame of reference and, there-
fore, an intrinsic deformation cannot be unambiguously
assigned to the projected states |J,k?. For instance, all
states |0,k? have a quadrupole moment equal to zero.
On the other hand, any multipole operator can be calcu-
lated in a straightforward, although sometimes tedious,
manner directly from matrix elements involving the BCS
It is, however useful to extract quantities analogous
to intrinsic frame quantities from physically well-defined
observables such as spectroscopic or transition moments,
in order to achieve contact with standard mean-field ap-
proaches. For instance, when transition probabilities sug-
gest that the two states |Ji,k? and |Jf,k? can be inter-
preted as forming a rotational band, one can define a
charge intrinsic moment Q(t)
c2(J,k) (c stands for “charge”
and t for “transition”) from the B(E2,Ji→ Jf) accord-
ing to the standard formulas . Our model relies on
the mixing of purely axial quadrupole deformed configu-
rations and describes K = 0 positive parity bands only.
Moreover, since the CHFBCS equations do not include
a spin cranking term, the quality of the description of
|J,k? states will deteriorate with spin J. In the follow-
ing, we will consider only spins up to 10. A first possible
definition of the intrinsic quadrupole moment Q(t)
B(E2,J → J − 2)
?J 020|J − 20?2e2,(2)
whith the notation of Ref.  for Clebsch-Gordan coeffi-
cients. Within the rigid rotor model, one can also define
an intrinsic quadrupole moment Q(s)
for ”spectroscopic”) of a state with spin J (J ?= 0) related
with the spectroscopic quadrupole moment
c2(J,k) (s stands here
Qc(J,k) = ?J,k|ˆQc22|J,k?, (3)
in the laboratory frame, via the relation
c2(J,k) = −2J + 3
In Eq. (3), it is understood that the bra and kets both
correspond to the M = J component of the 2J + 1 mul-
tiplet. The values of Q(s)
c2(J,k) and Q(t)
c2(J,k) are equal
mass deformation (β2 and Q2). The thin solid curve gives
?β2|ˆ H|β2? (MF), while the thick solid, dashed, dotted and
dash-dotted curves correspond to ?J,β2|ˆ H|J,β2? (PMF) for
the values J = 0, 2, 4 and 6 respectively. The energy origin
is taken at ?β2 = 0|ˆ H|β2 = 0?.
40Ca; Projected energy curves versus the
when the rigid rotor assumption is strictly fulfilled. The
differences between the values of these two quantities will
tell us how well this assumption is satisfied by the calcu-
lated physical states. Instead of Q(s)
we will equivalently consider the dimensionless quantities
c2(J,k) calculated according to a relation
similar to Eq. (1) in which the mass A is replaced by the
number of protons Z since we are dealing with the charge
Technical details on the evaluation of Qc(J,k) and
B(E2) will be given in an appendix below.
c2(J,k) and Q(t)
c2(J,k) and β(t)
The ground state of the N = Z doubly-magic nucleus
40Ca is spherical. As in several magic nuclei, the lowest
excited state is a 3−state at 3.74 MeV. This state as
well as other negative parity states is not included in
the Hilbert space of our calculation. Its study requires
parity breaking mean-field calculation involving octupole
deformations as done in a previous study performed for
Pb isotopes .
The lowest states of several even parity rotational
bands are known since the early 70’s . More recently,
several normal- and also superdeformed bands have been
identified up to very high spin thanks in particular, to a
recent Gammasphere+Microball experiment .
In Fig. 1, we have plotted the particle projected
mean-field deformation energy curve (denoted below
gram for neutrons. Except for an overall Coulomb shift, the
proton spectrum is almost identical. The dashed curve gives
the Fermi energy.
Nucleus40Ca; Self-Consistent HFBCS Nilsson dia-
as MF) ?β2|ˆH|β2? and the particle number and spin-
projected mean-field curves ?J,β2|ˆH|J,β2? (denoted be-
low as PMF) for low spin values. In addition to the ex-
pected well marked spherical minimum, the MF curve
displays a superdeformed secondary minimum around
β2= 0.5. The J = 0 PMF curve is almost flat over the
range −0.2 ≤ β2≤ 0.2. It also presents a secondary min-
imum around β2= 0.5. The minima of the higher spin
PMF curves correspond to both oblate and prolate defor-
mations at β2≈ ±0.2 and to the prolate superdeforma-
tion β2= 0.5. This latter minimum becomes the lowest
one from spin 4 upward. These features of the mean-
field curves are consistent with the HFBCS self-consistent
single-particle diagram shown in Fig. 2, where a small gap
is visible at β2= 0.4. At β2= 0.6, a prolate deformation-
driving orbital, originating from the f7/2shell, is occu-
pied, creating a deformed shell gap. Note that the de-
formed gap at β2= 0.9 shows up as a softening of the
MF and PMF curves.
In Fig. 3, along with the MF and PMF curves, we show
the excitation energies
of the GCM states for the spins J = 0, 2, 4 and 6. From
the GCM collective wave-function gJkof each state |J,k?,
a mean deformation can be calculated
This quantity does not necessarily coincide with the de-
c2(J,k) calculated from Eqs. (3) of (4), but
provides a convenient indicator of the weight of deformed
mean-field states in the projected state. It can be dif-
ferent from zero for J = 0 for which β(s)
In particular, it allows one to detect the 0+band head
FIG. 3: Nucleus40Ca; MF ?β2|ˆ H|β2? (thin solid) and PMF
?J,β2|ˆ H|J,β2? (thick solid) deformation energy curves. The
ordinates of short horizontal segments give the energy EJ,k
of the GCM states (Eq. 5). The abscissa of the black points
indicate the mean deformation (¯β2) of the corresponding col-
lective wave-function gJ,k (Eq. 6). The energy origin is taken
states of40Ca. The ground-state 0+wave function is drawn
with a thick solid line. The wave functions of the ND and SD
bands are drawn with dashed and dotted lines respectively.
Collective GCM wave-functions gJ,k for low-spin
of a deformed band and to perform a band classifica-
tion of the states pending confirmation from the analysis
of the in-band transitions. The magnitude of the axial
quadrupole collective correlations is given by the differ-
ence between the energies of the spherical configuration
and of the GCM ground state, E(0,1). It is equal to
TABLE I: Energy, spectroscopic moment and transition amplitudes of some of the ND (upper part) and SD (lower part) bands
of40Ca. The k labeling of the state refers to our calculation. The columns five to eight give the intrinsic moments, see Sect.
IIB. The last column gives the excitation energy of the most likely corresponding experimental state .
(MeV) (e fm2) (e2fm4) (e fm2)(e fm2)(MeV)
0.526 MeV, a value consistent with estimates in other nu-
clei . Among the many J+
at large deformation,¯β2≈ 0.5, a subset (03, 22, 41, 61)
whose members can plausibly be assigned to a superde-
formed band (SD). This band is interpreted within the
shell model as a 8p-8h band. The other excited states
have smaller¯β2 values. At lower excitation energy, we
will analyze whether the states (02, 21, 42, 62) can be in-
terpreted as the observed normal deformed (ND) band.
In Fig. 4, we have plotted the collective wave func-
tions gJ,k. The position of their nodes can only be un-
derstood if one keeps in mind the fact that, because
the exchanges of intrinsic axes are included in a three-
dimensional rotation, the quadrupole dynamics is not
undimensional. It takes place in the full quadrupole β2,
γ plane along the six lines associated with the gamma
angles γj= 2j π/6, j = 0, ..., 5 . The underlying
group structure also imposes that all non-zero-J collec-
tive wave-functions vanish at β2= 0.
kGCM states, one observes
(right) excitation spectrum of even spin and positive parity
Comparison of experimental (left) and calculated
Fig. 5 compares the calculated and experimental exci-
tation energies of the low spin states. The GCM states
(right-hand side) can be assigned to ND or SD bands
according to their¯β2 value. On the left side, we have
drawn the experimental states to which these bands most
likely correspond. In addition, at the extreme left side
of the picture, we have plotted two other states observed
within the same range of spins and excitation energies
(2+at 5.249 MeV and 4+at 6.509 MeV). However, we
feel entitled to eliminate them from our discussion of the
GCM results because, accordingto some theoretical mod-
els , they correspond to a K = 2 band which, cannot
be described within the present purely axial model.
The upper part of Table I collects data for the states
of the ND band (labelled “band 2” in ). They have
already been analyzed by Nathan et al.  in the early
70s in terms of a 4p-4h structure.
The GCM bandheads are at about the right excitation
energy. However, the overall spectrum is too spread out.
Since this is a general trend of this calculation, we defer
a discussion of possible causes to the conclusions section.
On the other hand, the B(E2) values, or equivalently
c2(J,k) moments, of the E2 transitions originat-
ing from the 2+
2are in nice agreement with
the experimental value Q(t)
our calculation, the transition down from the 4+
duced due to a coupling with the 4+
band. This coupling is probably a spurious consequence
of the too large spreading of the GCM ND band which
pushes the ND 4+level in the vicinity of the SD 4+state.
Another consequence of the ND band spreading is that
the SD 4+and 6+states are the lowest ones, while ex-
periment suggests that this band is not yrast below the
highest observed spin (J = 16). Our calculated value for
the monopole transition matrix element |M(E0)| = 1.3 e
fm2is smaller, but of the same order of magnitude as the
1and the 6+
c2(J,k) = 74 ± 14 e fm2. In
1state of the SD
experimental value (2.6 ± 0.1) e fm2. Note that no
effective charge is used in our calculation.
In the MF curve, the barrier between normal deforma-
tions and the SD minimum does not exceed 0.5 MeV as
seen in Fig. 1. The excitation energy of this SD mini-
mum is 10.1 MeV, a value much larger than the value
of 5.213 MeV observed for the SD bandhead (band 1 in
). Angular-momentum projection reduces this excita-
tion energy to 6.1 MeV. Finally, the quadrupole corre-
lations due to configuration-mixing push the excitation
energy of the SD bandhead to 6.52 MeV. As seen in
Fig. 3, this small increase is related to the contribution
of quadrupole correlations to the energy, which is larger
for the 0+
although the GCM result shows a significant improve-
ment with respect to MF ones, the final outcome is not
as satisfactory in40Ca as it is in16O , where a better
agreement with experiment had been found.
The GCM states (0+
with an average deformation¯β2= 0.5. Their properties
are given in the lower part of Table II. As compared
with experiment, see Fig. 3, the bandhead is slightly too
high in excitation energy and the moment of inertia too
small. Still, the agreement is much better than for the
ND band. One notes an irregularity for the 4+state,
due to the already mentioned mixing whith a nearby ND
state. The calculated transition quadrupole moments for
the various transitions in the SD band are in good agree-
ment with the experimental value Q(t)
given in Ref. , especially if one takes into account that
in our calculation the 6+
rates are decreased by a mixing of the 4+
less deformed 4+
1state. The proximity of the values of
c2shows also that the SD band satisfies the
A more recent analysis of transition quadrupole mo-
ments within the SD band presented in Ref.  suggests
spin to values around 120 e fm2for the 6+→ 4+tran-
sition. Our calculation is in agreement with this finding,
although it predicts the change a few units of angular
momentum too low.
1ground state than for the 0+
1) form a SD band
2level with the
c2drops from values around 180 e fm2at high
Particle-hole analysis of the GCM states
The ND and SD bands are often interpreted in terms
of 4p-4h and 8p-8h configurations respectively. This is
for instance the case in RMF calculations without pair-
ing performed with the NL3 parametrization by the Lund
group . Similarly, a shell model calculation by Caurier
et al.  in a restricted model space assigns the ND and
SD bands to 4p-4h and 8p-8h configurations, respectively.
It also finds that calculated transition probabilities re-
lated to quadrupole moments cluster around Q(t)
e fm2, and are slightly decreasing with angular momen-
tum. It, thus, seemed of interest to perform a similar
TABLE II: Properties of the self-consistent np-nh HF states
and overlap with low-lying 0+GCM states. The second and
tird columns give the excitation energy with respect to that
of the (0p-0h) HF ground state of the HF states (Enp-nh) and
of its J = 0 component (EJ=0np-nh). The next two columns
give the the HF Intrinsic quadrupole moment Q2 and the
associated deformation β2. The last three columns give the
squared ovelaps |?J = 0k|np-nh?|2with the lowest |J = 0,k?
GCM states. The normalization of the particle-hole states
|np-nh? is chosen such that the norm of its J = 0 component
is equal to 1.
(MeV) (MeV) (fm2)
2p-2h(p) 11.151 8.896
2p-2h(n) 11.462 9.071
6p-6h(p) 15.586 11.846
6p-6h(n) 15.791 12.002
12p-12h 31.625 26.565
0.0020.0000 0.000 0.6163 0.2170 0.0363
0.143 0.1189 0.0413 0.0330
0.143 0.1083 0.0421 0.0317
0.414 0.0107 0.1131 0.0938
0.582 0.0000 0.0124 0.1626
0.584 0.0000 0.0110 0.1495
0.777 0.0000 0.0043 0.1378
1.308 0.0000 0.0000 0.0000
analysis of our results.
The definition of a particle-hole excitation is far from
unambiguous. For instance, in the shell model, it is de-
fined with respect to the spherical oscillator basis used
in the calculation. However carefully this basis is cho-
sen, it remains a technical rather than a physical refer-
ence. In keeping with the spirit of a mean-field method,
we have defined the np-nh states, |np-nh?, by requiring
that, for each particle-hole configuration, they minimize
FIG. 6: Square of the overlap ?J = 0,β2|np-nh? between the
HF |np-nh? states (n = 0, 2, 4, 6, 8) and the particle and
spin J = 0 projected mean-field states |0,β2?. Because the
two 2p-2h and the 6p-6h overlaps are very similar, only one
for each is shown. The normalization of a particle-hole state
|np-nh? is chosen so that the norm of its J = 0 component is
equal to 1.
the Hartree-Fock energy with the same Skyrme Hamil-
tonian used in the projected GCM study. A deformed
state |np-nh?, thus, differs from a particle-hole configura-
tion built directly by a simple redefinition of the orbital
occupations in the spherical HF ground state |0p-0h?.
Since the definition of |np-nh? does not involve the pair-
ing channel, all such states are Slater determinants and
have the right particle number.
struct one state for 0p-0h, 4p-4h and 8p-8h configura-
tions and two states for 2p-2h and 6p-6h ones. Because
of the self-consistent redefinition of the orbitals, these
states are not orthogonal to each other. Although their
structure is very different, it turns out that the energy
of the spin J = 0 component extracted from the 2p-2h,
4p-4h, and 8p-8h states are almost degenerate, as can
be seen in Table II. They are also very excited with re-
spect to both the experimental and the GCM bandheads.
There is no obvious explanation for this near-degeneracy,
which was already noticed by Zheng et al.  in a much
more schematic mean-field model.
The overlap between the particle-hole states and the
J = 0 PMF states |J = 0,β2? is plotted in Fig. 6 as
a function of β2.The overlap for the |0p-0h? is con-
stant over the range −0.2 ≤ β2 ≤ 0.2. This range cor-
responds to a nearly constant PMF energy, as seen in
Fig. 1. The 2p-2h overlap is small and vanishes outside
this range. The 4p-4h overlap is gaussian shaped and
peaked at β2= 0.5. The 8p-8h curve is peaked at an
even larger deformation of β2 = 0.9 and its maximum
reaches a rather large value (≈ 0.8) indicating that the
J = 0 component of the 8p-8h state is very close to to the
|J = 0,β2= 0.9? PMF solutions. All these different de-
formations are consistent with the locations of deformed
shell gaps observed in Fig. 2. On the other hand, the
overlap of the 8p-8h state is almost zero at the deforma-
tion β2≈ 0.5 of the GCM SD band.
These results are already an indication that an analysis
in terms of pure particle-hole excitations can only provide
a crude approximation. Another indication is that the
energies of the |np-nh? states are several MeV larger than
those of the |J = 0,β2? states (Table I). This is confirmed
by the overlaps given in the last three columns of Table II.
It is seen that, with the exception of the GCM ground
1, which, as expected, has a large overlap with
the spherical |0p-0h?, neither the 4p-4h nor the 8p-8h
HF solutions can be convincingly assigned to the ND or
SD GCM bandheads. This is in line with findings of an
earlier study of the light doubly-magic nucleus16O .
In this way, we con-
With the same techniques, we analyze three other nu-
clei, penetrating each time somewhat deeper into the sd
shell. As for40Ca, a recent experiment  has found
high-spin states of38Arwhich can be interpreted in terms
of bands, one of them corresponding to large deforma-
tion.In both nuclei, the 0+bandheads have not yet
FIG. 7: Nucleus38Ar; Deformation energy curves ?β2|ˆ H|β2?
(thin solid line) and ?J,β2|ˆ H|J,β2? with J = 0, 2, 4 and 6
corresponding to thick solid, dashed, dotted and dash-dotted
lines, respectively. The ordinates of the short horizontal seg-
ments give the energy EJ,k (Eq. 5) of the GCM states. The
abscissa of the black points indicates the mean deformation
¯β2 (Eq. 6) of the GCM collective wave-function gJ,k. The
energy origin is taken at E0,1.
FIG. 8: Same as Fig. 5 for the nucleus38Ar
been seen. A shell-model analysis has also been done in
Ref.  within a restricted Hilbert space, as a full s-d,
f-p shell model calculation is presently beyond compu-
tational limits. Nevertheless, the overall quality of the
agreement constitutes strong evidence of the importance
of f-p excitations. Since the40Ca subsection has shown
that the real underlying structure is likely to be more
complex than a single pure mp-nh excitation, it is worth
noting that our model has the ability to describe coherent
multiple excitation to higher shells.
Keeping in mind the limitations of the present calcu-
lation, we concentrate our analysis on the lower part of
collective, even parity bands which, in Ref. , are la-
beled 1 and 2, respectively. For instance, our collective
space does not include two-quasiparticle configurations
and misses the lowest 2+state which is well understood
band(lower part) GCM states of38Ar. The k labeling of the state refers to our calculation. The columns five to eight give the
intrinsic moments (see Sect. IIB). The last column gives the experimental excitation energy of the most likely state assigment
Energy, spectroscopic moment and transitions amplitudes of some of the ND band (upper part) and the SD
(MeV) (e fm2) (e2fm4) (e fm2)(e fm2) (MeV)
spin states of38Ar. The ground state wave function is drawn
with a thick solid line. The wave functions of the ND and SD
bands are by dashed and dotted lines, respectively.
Collective GCM wave functions gJ,k for the lowest
in terms of a π[d3/2]−2
signement is compatible with the pairing gap ∆pat the
proton Fermi surface that we obtain. Indeed, the ex-
citation energy of this two-quasiparticle configuration is
approximated in perturbation by 2∆p, which is equal to
1.96 MeV, rather close to the observed 2.17 MeV.
The MF and PMF energy curves of38Ar presented in
Fig. 7 are very reminiscent of40Ca. The MF curve ex-
hibits a well-defined spherical minimum. It is consider-
ably softened by angular momentum projection, as shown
by the J = 0 PMF curve. Moreover, there is a clear in-
J=2configuration [24, 25]. This as-
TABLE IV: Out-of-band transitions and branching ratios for
38Ar. The k labeling of the state refers to our calculation.
Data are taken from .
k→ (J − 2)+
k′ B(E2;J → J − 2) Branching Ratio
dication of a shell structure around β2= 0.5. At this de-
formation, the angular momentum projection on J = 0
lowers the energy by approximately 5 MeV, reducing the
excitation energy to slightly more than 6 MeV. The GCM
results presented in Figs. 8 and 9 are also similar to those
obtained for40Ca: i) an isolated spherical ground state,
ii) a set of states which can be grouped into a moderately
deformed band (ND band) starting at 3.4 MeV and iii)
a SD band with a 6.22 MeV bandhead. Keeping in mind
that the present calculation predicts a too small moment
of inertia, we estimate that the ND band starts at the
correct energy while the SD band is located too high by
about 1.8 MeV. These results are comparable in quality
to those reported of Ref. , which however, in contrast
to us, predict slightly too large moments of inertia.
The calculated in-band transition probabilities – which
are not yet measured –, the deformation and spectro-
scopic moments are given in Table III. They confirm
the status of the SD band as a rotational band. The
FIG. 10: Same as Fig.7, but for the nucleus36Ar.
only existing electromagnetic transition data to be com-
pared with our results concern branching ratios for sev-
eral states of the two bands. Our results, are given in
The low-energy spectrum of the N = Z nucleus36Ar
is that of a transitional nucleus. In contrast to40Ca and
38Ar, a ND band is built directly on the ground state
. Recently, a SD band has been observed up to a
16+state [4, 5]. The energy of the 0+bandhead has
been proposed to be at 4.3 MeV. Supporting data can be
found in Ref. . In the former references, the data are
analyzed with the help of both shell model and cranked
Nilsson-Strutinsky calculations. The shell model repro-
duces the ND band energies very well. Both methods
explain the SD band in terms of the promotion of four
particles to the p-f shell. According to the diagram of
Fig. 2, this excitation is associated with the deformed
shell gap at β2≈ 0.6. The calculations indicate that this
SD band behaves as a good rotor. They also predict the
position of the bandhead to better than 1 MeV. Both the-
oretical methods reach rather good agreement on B(E2)
values. Results of similar quality have been obtained by
more schematic projected shell model calculations how-
ever with appropriately adjusted parameters .
As can be seen in Fig. 10, the MF curve still presents
a spherical minimum. It displays neither a secondary
minimum nor even a plateau for a deformation close to
β2= 0.6. The projection on spin markedly changes the
picture. The ground state becomes oblate. At spin 0,
the SD shell effect still does not show up as a minimum
while higher spin PMF curves present a well-defined min-
imum which gradually shifts from β2= 0.45 to 0.65. The
properties of the GCM states, shown in Figs. 11 and 12,
qualitatively reproduce the observed features with one
FIG. 11: Same as Fig. 5 for the nucleus36Ar.
ND and one SD band. Note that the 4.4 MeV, 2+level
involves a two quasi-particle excitation and is, therefore,
not included in our variational space. On the other hand,
the SD bandhead energy is predicted too high by about
1.4 MeV and, as for the nuclei studied above, the mo-
ments of inertia are too small. In Fig. 12, one sees that
the collective wave functions of the SD band are peaked
around β2≈ 0.55. This value is larger than that calcu-
lated in Ref. [4, 5].
Table V confirms the oblate nature of the ND band,
as can be seen from the g0,1wave function and the rotor
behavior of the SD band. This feature is not present in
the data. From Tables V and VI, one sees that the ob-
served in-band B(E2) values are slightly overestimated,
while the out-of-band transition rates are correctly repro-
duced. Our results show that the inclusion of collective
states of36Ar. The ground-state band and the SD band are
drawn with solid and dotted lines respectively.
Collective GCM wave-functions gJ,k for low-spin
TABLE V: Energy, spectroscopic moment and transition amplitudes of ground band (upper part) and SD band (lower part)
GCM states of36Ar. The k labeling of the state refers to our calculation. The columns five to eight give the intrinsic moments
(see Sect. IIB). The last two columns give the experimental excitation energy and B(E2) ↓exp corresponding to the most likely
state assigment [4, 5, 29].
(MeV) (e fm2) (e2fm4) (e fm2)(e fm2)(MeV)(e2fm4)
60 ± 6
372 ± 59
454 ± 67
TABLE VI: Out-of-band transitions in36Ar. The k labeling
of the states refers to our calculation. Data are taken from
Refs. [4, 5].
k→ (J − 2)+
k′ B(E2) ↓ (e2fm4)
quadrupole correlations supports the results obtained by
more schematic models such as the cranked Nilsson cal-
culations presented in Refs. [4, 5]
The MF energy curve for the mid-shell, N = Z, nu-
cleus32S, presented in Fig. 13, diplays a flat spherical
minimum. As can be expected from the single-particle
spectrum diagram of Fig. 2, it does not show any struc-
ture around β2≈ 0.5. Note that it presents an inflex-
ion point at very large deformation, β2≈ 1. In contrast,
the J = 0 angular momentum projected curve exhibits
two well-defined minima at small deformation, the oblate
one being the lowest. One also notes a stabilization of
the SD minimum which, however, never becomes yrast.
These results are similar to those of other mean-field ap-
proaches [30, 31, 32, 33]. Note that the authors of 
suggest that the SD structure may be stabilized at high
spin by non-axial octupole deformations.
The GCM calculation leads to the results displayed in
Figs. 14 and 15, and the moments and transition rates
given in Tables VII and VIII. In agreement with experi-
FIG. 13: Same as Fig. 7, but for32S.
FIG. 14: Same as Fig. 5 for the nucleus32S.
ment, the levels and energies are those of an anharmonic
vibrator with a characteristic 0+
1and a (4+
TABLE VII: Data for the low-lying states in32S which can be interpreted as anharmonic vibrator states. The k labeling of
the states refers to our calculations. Experimental data are taken from  and .
(e fm2)(e2fm4) (e fm2) (e fm2) (MeV) (e2fm4)
FIG. 15: Collective GCM wave-functions gJ,k for the lowest
spin states of
drawn with a thick solid line. The dashed line corresponds
to the 0+, 2+and 4+members of the two-phonon triplet,
while the SD rotational states are drawn with a dotted lines
32S. The first 0+(ground state) and 2+are
TABLE VIII: Energy, spectroscopic quadrupole moment and
E2 transition amplitudes of the SD band states of32S. The
k labeling of the state refers to our calculation. The columns
five to eight give the intrinsic moments (see Sect. IIB).
(MeV) (e fm2) (e2fm4) (e fm2)
sequence. The vibrational pattern is slightly more promi-
nent in the calculated ratios of B(E2) values than in the
observed ones. This may be due to an innacuracy in
the description of the 2+
1state which is almost spher-
ical in our calculation while the data suggest a strong
prolate deformation. In comparison with similar calcula-
tions with the Gogny force , the SLy6 force appears
to be less performant on this specific point. On the other
hand, the overall pattern and, in particular, the triplet of
two-phonon states is better described by our calculation.
The GCM predicts a SD band with a large deformation
β2≈ 0.9, starting at the high energy of 12 MeV. However,
this energy must probably be scaled down, presumably
by the same factor than the one that can be deduced
from a comparison of experimental and GCM spectra on
IV.DISCUSSION AND CONCLUSIONS
This article belongs to a series exploring a model start-
ing from a mean-field description of nuclear structure and
an effective Hamiltonian valid for the entire nuclear chart.
The model attempts a global description of the nuclear
ground states and low-energy spectra. Here, the SLy6
Skyrme interaction and a zero-range density dependent
pairing force have been used. The building blocks are
self-consistent N-body states, based on the HF+BCS
method.Extensions in progress involve the Hartree-
Fock-Bogoliubov (HFB) method and the cranked HFB
method. No assumption is made on the existence of a
core and of valence orbitals. The collective basis is gen-
erated by a constraint on the intrinsic axial quadrupole
moment. This limits the model to even spin and parity
collective structures. The symmetries broken at the level
of the mean field are restored by means of particle num-
ber and angular momentum projection. This transfers
the physical description from the intrinsic to the labo-
ratory frame while yielding moments and transitions to
be compared directly to experimental data. Finally, the
dynamical effects associated with the selected constraint
are incorporated by means of a diagonalization of the
initial Hamiltonian within the collective basis (GCM).
Other ways to generate collective bases have already
been explored by our group (see for instance Ref. ),
by contraints on monopole, triaxial quadrupole, and oc-
tupole deformations or by the construction of a two-
quasiparticle excited basis. Essentially, these extensions
do not entail a modification of the numerical tools but re-
quire more computing time and they have only included
the restoration of the particle-number symmetry.
A first test study on the magic nucleus16O presented in
Ref.  yielded results demonstrating that the extension
of a MF method through restoration of broken symme-
tries results in a significantly improved description of the
level structure and other spectroscopic properties of this
nucleus. The present work studies four light nuclides –
32S,36Ar,38Ar,40Ca – to test the model on systems il-
lustrative of a variety of properties encountered in the s-d
shell. Another reason to select these nuclei is the recent
discovery of superdeformed structures in the three heavi-
est. Such structures are generally interpreted as resulting
from the interaction between s-d and f-p shells.
The results presented in this work are encouraging.
The general trends seen for low-lying collective levels are
well described both with regard to the relative energy
positions within the spectrum of a given isotope and to
the evolution from nucleus to nucleus. In particular, as
in data, the magic and thus isolated nature of the ground
state of40Ca and38Ar contrasts with36Ar and32S, where
the ground state is strongly coupled to other compo-
nents of the spectrum.Moreover, at small excitation
energies, the vibrational features of32S are well repro-
duced. Within the same calculational frame, we also find
the recently discovered SD structures. We confirm the
conclusions drawn from the cranking model and the shell
model that the existence of SD bands is connected to a
partial occupation of deformation-driving orbitals origi-
nating from the f7/2shell. However, in the specific case
of40Ca, a detailed analysis has shown that the associa-
tion of SD bands with 4p-4h configurations must not be
taken too literally. Quadrupole collectivity generating a
coherent multi-excitation of p-h configurations of various
complexities is probably more representative of the na-
ture of the SD bandhead wave functions. Finally, the E2
transition matrix elements of these SD bands agree rather
well with the data, indicating that our method correctly
reproduces the average quadrupole deformation. We re-
call that no effective charge is involved in our calculation.
Our calculations display a systematic deficiency in that
they predict level densities which are too small. This fea-
ture manifests itself through the high excitation energy
of the bandheads and the small values of the moments
of inertia. At this point, we can only surmise the causes
of this problem, as it may be traced either to the model,
or to the Hamiltonian, or to both. However, two results
seem to exonerate the Hamiltonian, at least as the dom-
inant source of discrepancy. First, the results for the
magic nucleus16O were rather good and, just as in this
work, angular momentum projection was shown to lead
to large energy gains by amounts of the order of 5 MeV,
reducing by almost an order of magnitude the MF excita-
tion energies of the deformed configurations (see Figs. 3,
7, 10 and 13). Second, similar calculations with another
well-tested effective interaction, i.e. the Gogny force, lead
to the same phenomenon . In this and earlier refer-
ences from the same authors, a scaling factor (≈ 0.7) is
introduced to compensate for the too low level density. It
has been justified on the basis of a comparison between
solutions of a cranked mean-field method and a projected
method on non-cranked states. In other words, the de-
ficiency does not originate either from the Hamiltonian,
nor from the model, but from a too strict limitation of the
MF basis. The best description of the (N0,Z0) nuclide is
obtained by projecting out the (N0,Z0) component of the
MF solution constrained to have the same mean values
of theˆ N andˆZ operators. In the same way, the opti-
mal description of a state of spin J should result from
the projection of a cranked MF state constrained to have
?ˆJx? = J. These points are also discussed in . One
underlying dynamical explanation has to do with the an-
tipairing cranking effect, which increases the moment of
inertia as needed to improve our results. In the present
work, by projecting out a J = 6 component out of a non
angular momentum MF solution, we miss the pairing re-
duction which affects a ?ˆJx? = 6 cranked MF state.
This discussion also delineates possible lines of future
developments. On the one hand, in selected cases sug-
gested by experimental data, it may become necessary to
enlarge the collective space by taking into account other
deformations (as suggested, for instance, by Yamagami
and Matsuyanagi ). Finally, a promising extension
of the present work would involve the projection of an-
gular momentum cranked solutions. Work is presently
underway along these three directions.
This research was supported in part by the PAI-P5-
07 of the Belgian Office for Scientific Policy. We thank
R. V. F. Janssens and R. Wyss for fruitful and inspir-
ing discussions. M. B. acknowledges support through a
European Community Marie Curie Fellowship.
THE QUADRUPOLE MOMENT IN THE LAB
In this appendix, we derive the formulas which per-
mits to calculate quadrupole transition matrix elements
between symmetry restored mean-field states. Some of
the equations given in Ref.  contain misprints that are
For sake of simple notation, we omit particle-number
projection throughout this appendix. It does not alter
any of the formulas given here concerning angular mo-
mentum. Different from above, we will explicitely in-
clude the angular momentum projection M in the no-
tation. Starting from time-reversal, parity, and axially
symmetric (with the z axis as symmetry axis) mean-field
states |q? obtained with a constraint on the quadrupole
moment labelled as q, projected states |JMq? with total
angular momentum J and angular momentum projection
where NJq = ?q|ˆPJ
obtained applying the angular-momentum projector 
00|q?1/2is a normalization factor, are
MK=2J + 1
whereˆR(Ω) = e−iαˆ Jze−iβˆ Jye−iγˆ Jzis the rotation op-
erator and DJ
function, which depend both on the Euler angles Ω =
(α,β,γ), see  for details.
The projected GCM states |JMk? are given by
where the non-orthonormalset of weight functions fJ,k(q)
is determined variationally by solving the Hill-Wheeler
equation. The collective wave functions gJ,k(q) given in
MK(Ω) = e−iMαdJ
|JMk? =dq fJ,k(q)|JMq?, (9)
figures 4, 9, 12 and 15 above are obtained from the fJ,k(q)
by a transformation involving the hermitean square root
of the norm operator, see  and references therein for
The spectroscopic quadrupole moment of the GCM
state |JMk? is defined as
×?J,M = J,q|ˆQ20|J,M = J,q′?. (10)
?J,M = J,k|ˆQ20|J,M = J,k?
For J and J′integer and even – as assumed here – the
matrix element of the projected mean-field states enter-
ing Eq. (10) can be expressed in terms of the reduced
matrix element ?Jq||ˆQ20||Jq′? 
?J,M = J,q|ˆQ20|J,M = J,q′?
√2J + 1
The reduced matrix element involving different values
of J appears in the expression of the reduced transition
probability between the GCM states k′and k with angu-
lar momentum J′and J, respectively
k′ → Jk) =
The reduced matrix element is evaluated by the com-
mutation of one of the two projection operators in the
M′K′ with the quadrupole opera-
tor. After a lengthy, but straightforward application of
angular momentum algebra and taking advantage of the
symmetries of the mean-field states, one obtains
√2J + 1(2J′+ 1)
1 + (−)J
0µ(β)?q|e−iβˆ Jy ˆQ2µ|q′?
where only the “left” states have to be rotated. A similar
expression, where only the right states have to be rotated,
is given in .
 A. M. Nathan and J. J. Kolata, Phys. Rev. C 14, 171
 E. Ideguchi, D. G. Sarantites, W. Reviol, A. V. Afanas-
jev, M. Devlin, C. Baktash, R. V. F. Janssens, D.
Rudolph, A. Axelsson, M. P. Carpenter, A. Galindo-
Uribarri, D. R. LaFosse, T. Lauritsen, F. Lerma, C. J.
Lister, P. Reiter, D. Seweryniak, M. Weiszflog and J. N.
Wilson, Phys. Rev. Lett. 87, 222501 (2001).
 C. J. Chiara, E. Ideguchi, M. Devlin, D. R. LaFosse,
F. Lerma, W. Reviol, S. K. Ryu, D. G. Sarantites, C.
Baktash, A. Galindo-Uribarri, M. P. Carpenter, R. V. F.
Janssens, T. Lauritsen, C. J. Lister, P. Reiter, D. Sew-
eryniak, P. Fallon, A. G¨ orgen, A. O. Macchiavelli, and
D. Rudolph, Phys. Rev. C 67, 041303(R) (2003).
 C. E. Svensson, E. Caurier, A. O. Macchiavelli, A. Juoda-
galvis, A. Poves, I. Ragnarsson, S.˚ Aberg, D. E. Ap-
pelbe, R. A. E. Austin, C. Baktash, G. C. Ball, M. P.
Carpenter, E. Caurier, R. M. Clark, M. Cromaz, M. A.
Deleplanque, R. M. Diamond, P. Fallon, M. Furlotti, A.
Galindo-Uribarri, R. V. F. Janssens, G. J. Lane, I. Y.
Lee, M. Lipoglavsek, F. Nowacki, S. D. Paul, D. C. Rad-
ford, D. G. Sarantites, D. Seweryniak, F. S. Stephens, V.
Tomov, K. Vetter, D. Ward and C. H. Yu, Phys. Rev.
Lett. 85, 2693 (2000).
 C. E. Svensson, A. O. Macchiavelli, A. Juodagalvis, A.
Poves, I. Ragnarsson, S.˚ Aberg, D. E. Appelbe, R. A. E.
Austin, G. C. Ball, M. P. Carpenter, E. Caurier, R. M.
Clark, M. Cromaz, M. A. Deleplanque, R. M. Diamond,
P. Fallon, R. V. F. Janssens, G. J. Lane, I. Y. Lee, F.
Nowacki, D. G. Sarantites, F. S. Stephens, K. Vetter and
D. Ward, Phys. Rev. C 63, 061301(R) (2001).
 D. Rudolph, A. Poves, C. Baktash, R. A. E. Austin, J.
Eberth, D. Haslip, D. R. LaFosse, M. Lipoglavsek, S. D.
Paul, D. G. Sarantites, C. E. Svensson, H. G. Thomas, J.
C. Waddington, W. Weintraub and J. N. Wilson, Phys.
Rev. C 65 034305 (2002).
 A. Valor, P.-H. Heenen and P. Bonche, Nucl. Phys.
A671, 145 (2000) .
 M. Bender and P.-H. Heenen, Nucl. Phys. A713, 39
 M. Bender, P.-H. Heenen and P.-G. Reinhard, Rev. Mod.
Phys. 75, 121 (2003).
 P.-H. Heenen, P. Bonche, J. Dobaczewski, H. Flocard, S.
Krieger, J. Meyer, J. Skalski, N. Tajima and M. S. Weiss,
Proceedings of the International Workshop on “Nuclear
Structure Models”, Oak Ridge, Tennessee, March 16-
25, 1992, R. Bengtsson, J. Draayer and W. Nazarewicz
(edts.), World Scientific, Singapore, New Jersey, London
and Hong Kong, 1992, page 3.
 E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R.
Schaeffer, Nucl. Phys. A635 (1998) 231, Nucl. Phys.
A643 (1998) 441(E).
 M. Bender, K. Rutz, P.-G. Reinhard and J. A. Maruhn,
Eur. Phys. J. A7, 467 (2000).
 J. Terasaki, P.-H. Heenen, H. Flocard and P. Bonche,
Nucl. Phys. A600, 371 (1996).
 D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 (1953).
 P. Bonche, J. Dobaczewski, H. Flocard, P.-H. Heenen and
J. Meyer, Nucl. Phys. A510 (1990) 466.
 K. E. G. L¨ obner, Gamma-ray transition probabilities in
deformed nuclei, in W. D. Hamilton [ed.], The elec-
tromagnetic interaction in nuclear spectroscopy, North–
Holland, Amsterdam, Oxford, 1975, page 141.
 A. Messiah, M´ echanique Quantique, Dunod, Paris, 1960.
 P.-H. Heenen, A. Valor, M. Bender, P. Bonche and H.
Flocard, Eur. Phys. J. A11, 393 (2001).
 P. Bonche, J. Dobaczewski, H. Flocard and P.-H. Heenen,
Nucl. Phys. A530, 149 (1991).
 W. J. Gerace and A. M. Green, Nucl. Phys. A93, 110
(1967); Nucl. Phys. A123, 241 (1969).
 M. Ulrickson, N. Benczer-Koller, J. R. MacDonald and
J. W. Tape, Phys. Rev. C 15, 186 (1977).
 E. Caurier,F. Nowacki,
The superdeformed excited band of
 D. C. Zheng, D. Berdichevsky and L. Zamick, Phys. Rev.
C 38 (1988) 437.
 B. H. Wildenthal and E. Newman, Nucl. Phys. A118,
 E. R. Flynn, O. Hansen, R. F. Casten, J. D. Garrett and
F. Ajzenberg-Selove, Nucl. Phys. A246 , 117 (1975).
 P. M. Endt, Nucl. Phys. A521, 1 (1990).
 H. R¨ opke, J. Brenneisen and M. Lickert, Eur. Phys. J.
A14, 159 (2002).
 G.-L. Long and Y. Sun. Phys. Rev. C 63, 021305(R)
 S. Raman, C. W. Nestor, Jr. and P. Tikkanen, Atom.
Data Nucl. Data Tables 78 (2001) 1.
 H. Molique, J. Dobaczewski and J. Dudek, Phys. Rev. C
61, 044304 (2000).
 R. R. Rodriguez-Guzm´ an, J. L. Egido and L. M. Robledo,
Phys. Rev. C 62, 054308 (2000).
 M. Yamagami and K. Matsuyanagi, Nucl. Phys. A672,
 T. Tanaka, R. G. Nazmitdinov and K. Iwasawa, Phys.
Rev. C 63, 034309 (2001).
 A. Kangasm¨ aki, P. Tikkanen, J. Keinonen, W. E. Or-
mand, S. Raman, Zs. F¨ ul¨ op,´A. Z. Kiss and E. Somorjai,
Phys. Rev. C 58, 699 (1998).
 M. Babilon, T. Hartmann, P. Mohr, K. Vogt, S. Volz and
A. Zilges, Phys. Rev. C 65, 037303 (2002).
 P. Ring and P. Schuck, The Nuclear Many-Body Prob-
lem, Springer Verlag, New York, Heidelberg, Berlin, 438
 D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii,
Quantum Theory of angular momentum, World Scien-
tific, Singapore, 1988.
 R. Rodriguez-Guzm´ an, J. L. Egido and L. M. Robledo,
Nucl. Phys. A709, 201 (2002).
A. Poves and A. Zuker,