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
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,
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