Beyondmeanfieldmodel analysis of lowspin normaldeformed and superdeformed collective states of^{32} S,^{36} Ar,^{38} Ar, and^{40} Ca
ABSTRACT We investigate the coexistence of spherical, deformed, and superdeformed states at low spin in 32S, 36Ar, 38Ar, and 40Ca. 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 HartreeFock BCS calculations using the Skyrme interaction SLy6 for the particlehole channel and a densitydependent 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 evenspin 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.

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ABSTRACT: Modern nuclear structure theory is rapidly evolving towards regions of exotic shortlived nuclei far from stability, nuclear astrophysics applications, and bridging the gap between lowenergy QCD and the phenomenology of finite nuclei. The principal objective is to build a consistent microscopic theoretical framework that will provide a unified description of bulk properties, nuclear excitations and reactions. Stringent constraints on the microscopic approach to nuclear dynamics, effective nuclear interactions, and nuclear energy density functionals, are obtained from studies of the structure and stability of exotic nuclei with extreme isospin values, as well as extended asymmetric nucleonic matter. Recent theoretical advances in the description of structure phenomena in exotic nuclei far from stability are reviewed. Comment: 18 pp, plenary talk, International Nuclear Physics Conference (INPC 2004), Goeteborg, Sweden, June 27  July 2, 2004Nuclear Physics A 09/2004; · 1.53 Impact Factor  SourceAvailable from: Peter Ring
Article: Configuration mixing of angularmomentumprojected triaxial relativistic meanfield wave functions
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ABSTRACT: The framework of relativistic energydensity functionals is extended to include correlations related to the restoration of broken symmetries and to fluctuations of collective variables. The generator coordinate method is used to perform configuration mixing of angularmomentumprojected wave functions, generated by constrained selfconsistent relativistic meanfield calculations for triaxial shapes. The effects of triaxial deformation and of K mixing is illustrated in a study of spectroscopic properties of lowspin states in 24 Mg.Physical Review C 04/2010; 81:044311. · 3.72 Impact Factor  SourceAvailable from: Alexey P. Severyukhin[Show abstract] [Hide abstract]
ABSTRACT: By means of HFB calculations with independent constraints on axial neutron and proton quadrupole moments Q{sub n} and Q{sub p}, we investigate the large amplitude isoscalar and isovector deformation properties of the neutronrich isotope ²°O. Using the particlenumber and angularmomentum projected generator coordinate method, we analyze the collective dynamics in the {l_brace},{r_brace} plane. The parametrization SLy4 of the Skyrme interaction is used for all calculations in connection with a densitydependent zerorange pairing interaction. Our results show that already for this moderately neutronrich nucleus the transition moments are modified when independent neutron and proton collective dynamics are allowed.Physical Review C 01/2007; 75(6):064303064303. · 3.72 Impact Factor
Page 1
arXiv:nuclth/0305021v1 8 May 2003
Beyond–mean–field–model analysis of lowspin normaldeformed 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, B1050 Bruxelles, Belgium
2CSNSM, Bt.104, F91405 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 HartreeFock BCS calculations using the Skyrme interaction SLy6 for the particle
hole channel, and a densitydependent 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 evenspin 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
I.INTRODUCTION
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 [1]. With the modern large multidetector 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 doublymagic40Ca [2, 3], and
the adjacent transitional nuclei36Ar [4, 5] and38Ar [6].
The occurence of welldeformed prolate structures in such
magic or closetomagic 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
filled.
On the theoretical side, the new structures have been
explored with the cranked meanfield (MF) method [2]
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 welldeformed 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 meanfield 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 [7]. 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 [8], 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
MF calculations.
II.THE METHOD
A.Effective Hamiltonian and Collective Hilbert
Space
The Nbody physical states analyzed in this work
are contained in the Hilbert space spanned by solu
tions β2? of Constrained HartreeFockBCS (CHFBCS)
equations [9]. 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= ?β2ˆQ20β2?. The singleparticle wave func
tions from which BCS states are constructed are dis
cretized on a threedimensional mesh. As explained in
Page 2
2
Ref. [10], this technique provides accurate solutions of
the meanfield equations. Note that our calculation does
not assume the existence of an inert core.
As twobody interaction in the particlehole chan
nel of the Hamiltonian ˆH, we have chosen the SLy6
parametrization [11] 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 centerof
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 [12]. This better surface tension is the
motivation for the choice of SLy6 in the present study.
The pairing force is a zerorange, densitydependent force
acting predominantly at the surface of the nucleus in or
der to describe the pairing effects in the particleparticle
T = 1, Tz= ±1 channel. The parameters of the latter
force [13] 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 meanfield
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
quadrupole correlations.
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
Most of
The cm operator affects
As in earlier
β2=
?
5
16π
4πQ2
3R2A
,(1)
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 lowenergy 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 LipkinNogami pre
scription (see Ref. [9] 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 singleparicle 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 particlenumber 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
timereversal symmetry, the model provides no informa
tion on oddspin states.
Finally, we diagonalize the HamiltonianˆH within each
of the collective subspaces of the nonorthogonal 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
Nbody physical states J,k?.
We stress that the correlations introduced by the dif
ferent configuration mixings of the initial meanfield 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
isospin invariance.
Page 3
3
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 largeamplitude quadrupole collective
correlations are taken into account .
B. Charge multipole and transition moments
The angularmomentum 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
states β2?
It is, however useful to extract quantities analogous
to intrinsic frame quantities from physically welldefined
observables such as spectroscopic or transition moments,
in order to achieve contact with standard meanfield 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 [16]. 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)
is thus:
c2(J,k)
Q(t)
c2(J,k) =
?
16π
5
B(E2,J → J − 2)
?J 020J − 20?2e2,(2)
whith the notation of Ref. [17] for ClebschGordan 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,kˆQc22J,k?,(3)
in the laboratory frame, via the relation
Q(s)
c2(J,k) = −2J + 3
J
Qc(J,k). (4)
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
FIG. 1:
mass deformation (β2 and Q2). The thin solid curve gives
?β2ˆ Hβ2? (MF), while the thick solid, dashed, dotted and
dashdotted curves correspond to ?J,β2ˆ HJ,β2? (PMF) for
the values J = 0, 2, 4 and 6 respectively. The energy origin
is taken at ?β2 = 0ˆ Hβ2 = 0?.
Nucleus
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
β(s)
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
quadrupole moment.
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),
c2(J,k) and β(t)
III. RESULTS
A.
40Ca
The ground state of the N = Z doublymagic 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 meanfield calculation involving octupole
deformations as done in a previous study performed for
Pb isotopes [18].
The lowest states of several even parity rotational
bands are known since the early 70’s [1]. 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 [2].
In Fig. 1, we have plotted the particle projected
meanfield deformation energy curve (denoted below
Page 4
4
FIG. 2:
gram for neutrons. Except for an overall Coulomb shift, the
proton spectrum is almost identical. The dashed curve gives
the Fermi energy.
Nucleus40Ca; SelfConsistent HFBCS Nilsson dia
as MF) ?β2ˆHβ2? and the particle number and spin
projected meanfield curves ?J,β2ˆHJ,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 selfconsistent
singleparticle 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
EJ,k= ?J,kˆHJ,k?
(5)
of the GCM states for the spins J = 0, 2, 4 and 6. From
the GCM collective wavefunction gJkof each state J,k?,
a mean deformation can be calculated
¯β2=
?
β2gJ,k(β2)2dβ2. (6)
This quantity does not necessarily coincide with the de
formation β(s)
c2(J,k) calculated from Eqs. (3) of (4), but
provides a convenient indicator of the weight of deformed
meanfield 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
c2(J,k) vanishes.
FIG. 3: Nucleus40Ca; MF ?β2ˆ Hβ2? (thin solid) and PMF
?J,β2ˆ HJ,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 wavefunction gJ,k (Eq. 6). The energy origin is taken
at E0,1.
FIG. 4:
states of40Ca. The groundstate 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 wavefunctions gJ,k for lowspin
of a deformed band and to perform a band classifica
tion of the states pending confirmation from the analysis
of the inband 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
Page 5
5
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 [2].
GCM
State
J+
k
EJ,k
Qc
B(E2) ↓
Q(s)
c2
β(s)
c2
Q(t)
c2
β(t)
c2
Eexp
(MeV) (e fm2) (e2fm4) (e fm2)(e fm2)(MeV)
0+
2
2+
1
4+
2
6+
2
3.99
5.40
9.28
13.39
0.0
2.2
18.7
34.3
—
112
16
187
—
7.8
51.3
85.8
——
75.2
23.9
77.4
—
0.30
0.09
0.26
3.35
3.91
5.28
6.93
0.031
0.201
0.336
0+
3
2+
2
4+
1
6+
1
8+
1
10+
6.52
7.30
8.15
10.53
13.07
16.18
0.0
38.2
35.5
64.1
66.5
70.1
—
447
373
557
882
962
——
0.525
0.383
0.628
0.619
0.63
—
150
115
133
163
169
—
0.60
0.44
0.51
0.64 9.856
0.66 12.338
5.21
5.63
6.54
7.97
133.9
97.6
160.2
157.9
161.2
1
0.526 MeV, a value consistent with estimates in other nu
clei [18]. 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 8p8h 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 [19]. The underlying
group structure also imposes that all nonzeroJ collec
tive wavefunctions vanish at β2= 0.
kGCM states, one observes
FIG. 5:
(right) excitation spectrum of even spin and positive parity
states in40Ca.
Comparison of experimental (left) and calculated
Fig. 5 compares the calculated and experimental exci
tation energies of the low spin states. The GCM states
(righthand 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 [20], 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 [2]). They have
already been analyzed by Nathan et al. [1] in the early
70s in terms of a 4p4h 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
the Q(t)
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[2]. In
2is re
1state of the SD