Page 1

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

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; 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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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 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

Page 6

6

experimental value (2.6 ± 0.1) e fm2[21]. 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

[2]). 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 [8], 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. [2], 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

Q(t)

c2shows also that the SD band satisfies the

rotor criterion.

A more recent analysis of transition quadrupole mo-

ments within the SD band presented in Ref. [3] suggests

that Q(t)

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+

3state. Thus,

3, 2+

2, 4+

1, 6+

1) form a SD band

c2= 180+39

−29e fm2

1→ 4+

1and 4+

1→ 2+

2level with the

2transition

c2and Q(s)

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 [4]. Similarly, a shell model calculation by Caurier

et al. [22] 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

c2= 170

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.

state

EHF

(MeV) (MeV) (fm2)

EJHF

Q2

β2

|?Jk|JHF?|2

0+

2

0+

1

0+

3

0p-0h

2p-2h(p) 11.151 8.896

2p-2h(n) 11.462 9.071

4p-4h

11.767 8.729

6p-6h(p) 15.586 11.846

6p-6h(n) 15.791 12.002

8p-8h

13.247 8.732

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

73

73

211

297

298

396

667

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.

Page 7

7

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. [23] 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

state 0+

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 [8].

In this way, we con-

B.

38Ar

With the same techniques, we analyze three other nu-

clei, penetrating each time somewhat deeper into the sd

shell. As for40Ca, a recent experiment [6] 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. [6] 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. [6], 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

Page 8

8

TABLE III:

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

[6].

Energy, spectroscopic moment and transitions amplitudes of some of the ND band (upper part) and the SD

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+

1

6+

2

3.44

3.61

7.32

12.41

0.0

6.9

0.7

8.3

—

104

130

47

—

-24

-2

-21

——

72

68

39

—

0.33

0.31

0.18

-0.11

0.01

0.09

3.94

5.35

7.29

0+

3

2+

2

4+

2

6+

1

6.22

6.61

8.11

10.34

0.0

-36

-52

-59

—

508

611

611

—

124

142

146

–—

160

147

140

—

0.72

0.62

0.63

0.56

0.64

0.66

4.57

6.05

7.49

FIG. 9:

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 [6].

J+

k→ (J − 2)+

k′ B(E2;J → J − 2) Branching Ratio

(e2fm4) Calc.Exp.

2+

4+

6+

1→ 0+

1→ 2+

2→ 4+

1

15

22

126

2

14

73

5 (3)

5 (1)

2

2+

2+

4+

4+

6+

2→ 0+

2→ 0+

2→ 2+

2→ 2+

1→ 4+

1

1

1

3

34

110

2

1

0.540 (13)

3

1

1531 (6)

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. [6], 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

Page 9

9

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

Table IV.

C.

36Ar

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

[26]. 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. [27]. 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 [28].

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

FIG. 12:

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

Page 10

10

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].

GCM

State

J+

k

EJ,k

Qc

B(E2) ↓

Q(s)

c2

β(s)

c2

Q(t)

c2

β(t)

c2

Eexp

B(E2) ↓exp

(MeV) (e fm2) (e2fm4) (e fm2)(e fm2)(MeV)(e2fm4)

0+

1

2+

1

4+

1

6+

2

0.00

2.80

7.43

13.65

0

13

12

-1.3

—

44

103

93

—

-45

-34

3.3

——

47

60

55

—

0.22

0.28

0.26

0.00

1.97

4.41

9.18

-0.21

-0.16

0.02

60 ± 6

0+

2

2+

2

4+

2

6+

1

5.90

6.67

8.87

11.13

0—

366

536

715

—

112

133

171

—

0.52

0.62

0.80

—

136

137

151

—

0.63

0.64

0.71

4.33

4.95

7.76

9.92

-32

-48

-68

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].

J+

k→ (J − 2)+

k′ B(E2) ↓ (e2fm4)

Calc.Exp.

2+

2+

4+

6+

2→ 0+

2→ 0+

2→ 2+

1→ 4+

1

0.3

178

3.3

57

4.6±2.3

3

1

2.5±0.4

5.3±0.8

1

quadrupole correlations supports the results obtained by

more schematic models such as the cranked Nilsson cal-

culations presented in Refs. [4, 5]

D.

32S

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 [33]

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+

1, 2+

1and a (4+

1,2+

2,0+

2)

Page 11

11

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 [34] and [35].

J+

kE (MeV)

(MeV)

Qs

Transition B(E2)

Q(s)

c2

Q(t)

c2

Eexp

B(E2)exp

(e fm2)(e2fm4) (e fm2) (e fm2) (MeV) (e2fm4)

0+

1

2+

1

0+

2

2+

2

0.00

3.22

6.32

7.04

0.0

2.3

—

-0.7

—

38

144

157

0.02

2.8

94

—

2.3

—

-0.7

—

43

12

74

0.8

12

58

0.00

2.23

3.78

4.28

2+

0+

2+

2+

2+

4+

1→ 0+

2→ 2+

2→ 2+

2→ 0+

2→ 0+

1→ 2+

1

61 ±3

72

54 ±3

11 ±2

1

1

1

2

4+

1

7.35 11.7

1

124.4672

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

respectively.

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).

J+

k

EJ,k

(MeV) (e fm2) (e2fm4) (e fm2)

Qc

B(E2) ↓

Q(s)

c2

β(s)

c2

Q(t)

c2

(e fm2)

β(t)

c2

0+

5

2+

5

4+

4

6+

3

12.3

12.7

13.6

14.8

0—

719

767

1057

—

136

176

206

—

0.77

1.00

1.17

—

190

164

184

—

1.08

0.94

1.05

-39

-64

-82

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 [31], 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

40Ca,38Ar,36Ar and32S.

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

Page 12

12

initial Hamiltonian within the collective basis (GCM).

Other ways to generate collective bases have already

been explored by our group (see for instance Ref. [18]),

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. [8] 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 [31]. 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 [36]. 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 [32]). 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.

Acknowledgments

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

FRAME

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. [7] contain misprints that are

corrected here.

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-

Page 13

13

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

M

|JMq? =

1

NJq

ˆPJ

M0|q?,(7)

where NJq = ?q|ˆPJ

obtained applying the angular-momentum projector [36]

00|q?1/2is a normalization factor, are

ˆPJ

MK=2J + 1

8π2

?

dΩ DJ∗

MK(Ω)ˆR(Ω),(8)

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 [37] 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

MK(β)e−iKγa Wigner

|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 [9] and references therein for

more details.

The spectroscopic quadrupole moment of the GCM

state |JMk? is defined as

Qc(J,k) =

?

?16π

×?J,M = J,q|ˆQ20|J,M = J,q′?. (10)

16π

5

?J,M = J,k|ˆQ20|J,M = J,k?

?

=

5

dq

?

dq′f∗

J,k(q)fJ,k(q′)

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′? [37]

?J,M = J,q|ˆQ20|J,M = J,q′?

=?JJ20|JJ?

√2J + 1

?Jq||ˆQ20||Jq′?.(11)

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

B(E2,J′

k′ → Jk) =

e2

2J′+ 1

+J

?

?

M=−J

+J′

?

M′=−J′

+2

?

µ=−2

|?JMk|ˆQ2µ|J′M′k′?|2

=

e2

2J′+ 1

????

dq

?

dq′f∗

J,k(q)fJ′,k′(q′)?Jq||ˆQ2||J′q′?

????

2

.(12)

The reduced matrix element is evaluated by the com-

mutation of one of the two projection operators in the

expressionˆPJ

M′K′ with the quadrupole opera-

KMˆQ2µˆPJ′

tor. After a lengthy, but straightforward application of

angular momentum algebra and taking advantage of the

symmetries of the mean-field states, one obtains

?Jq||ˆQ2||J′q′? =

√2J + 1(2J′+ 1)

NJqNJ′q′

+2

?

µ=−2

1 + (−)J

2

?J′02µ|Jµ′?

π/2

?

0

dβ sin(β)dJ

0µ(β)?q|e−iβˆ Jy ˆQ2µ|q′?

(13)

where only the “left” states have to be rotated. A similar

expression, where only the right states have to be rotated,

is given in [38].

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40Ca,preprint