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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

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 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 [6].

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

filled.

On the theoretical side, the new structures have been

explored with the cranked mean-field (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 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 [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 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 [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= ?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

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2

Ref. [10], 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 [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 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 [12]. 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 [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 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

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 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. [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 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

isospin invariance.

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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 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

states |β2?

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 [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 020|J − 20?2e2,(2)

whith the notation of Ref. [17] 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

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

?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?.

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 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 [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

mean-field deformation energy curve (denoted below

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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; 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

EJ,k= ?J,k|ˆH|J,k?

(5)

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

¯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

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

c2(J,k) vanishes.

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

at E0,1.

FIG. 4:

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

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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 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 [19]. The underlying

group structure also imposes that all non-zero-J collec-

tive wave-functions 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

(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 [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 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

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