Quadrupole collective correlations and the depopulation of the superdeformed bands in mercury

Nuclear Physics A519 (1990) 509-520 North-Holland

QUADRUPOLE DEPOPULATION

COLLECTIVE

CORRELATIONS

OF THE SUPERDEFORMED

AND THE

BANDS

IN MERCURY

P. BONCHE’, J. DOBACZEWSKI 2,3 II. FLOCARD3, P.H. HEENEN4, S.J. KRIEGER’, J. MEYER6 and MS. WEISS5

1 Service de Physique lMorique, CEN Saclay-91191, Gif sur Yvette Cedex, France ’ institute of Theoreticul Physics, Hoia 69, PL 00681 Wursaw, Poland 3 Division de Physique ~~o~q~e*, ~P~-914~ Orsay Cedex, France 4 Service de Physique ~~cZ~aire brusque, U.L.B.-C.P. 229-1050 Bnrssefs, Belgium ’ Lawrence Livermore National Laboratory, University of Calijbmia, Livermore CA 94550, USA 6 IPNL, IN2P3-eNRS/Uniuersiti C. Bernard Lyon 1, 69622 Villeurbanne Cedex, Fence

Received 2 April 1990 (Revised 3 August 1990) Abstract: Fully self-consistent generator coordinate method calculations have been performed on a basis of quadrupole constrained Hartree-Fock plus BCS wave functions for the five even mercury isotopes ‘PO-198Hg.The GCM results support conclusions drawn from previous HFi BCS calculations. The predicted evolution of superdeformed band head (shape isomer) properties as a function of the neutron number is consistent with the data. Using calculated transition matrix elements, we evaluate in-band versus out-of-band quadrupole decay and explain the sudden termination of the superdeformed band.

1. Introduction

A previously published work based on fully microscopic Hartree-Fock plus BCS (HF+ BCS) calculations with the Skyrme interaction SkM* “) has adumbrated shape isomerism and concomitant superdeformation in the Hg isotopes. Similar inferences have been made in Nilsson-St~tinsky 2-8) and in self-consistent calculations 9-1’), although these can differ from ours in their precise predictions. More recently, the superdeformation phenomenon has been observed, first in 19’Hg [ref. r2)J, then in several isotopes of Hg and Pb [refs. 8*13-17)].In this article, we elaborate on our previous calculations by improving on the mean field des~~ption of nuclear states. Using the generator coordinate method (GCM) 1s,‘9), we diagonalize the microscopic two-body hamiltonian within the basis set of constrained HFC BCS wave functions. The GCM provides values for the energy of the ground and excited states including the shape isomer which take into account the effect of correlations in the collective degree of freedom. The GCM will allow us to discuss the qualitative modifications of the shape isomeric stability, as well as the change of the superdeformed (SD) moment of inertia as a function of the neutron number. With the GCM, one can also evaluate quad~pole transition matrix elements between states. We will use * Unite de recherches des Universitb Paris XI et Paris VI associie au

C.N.R.S.

0375-9474/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

P. Bonche ei al. / Quadruple collective correlations

510

these microscopic results to discuss not only the angular momentum at which the SD band is observed to end, but also the rapidity of its demise.

2. The generator coordinate method The solid lines in fig. la-e show the deformation energy curves obtained from constrained HF+ BCS calculations in coordinate space 20) using the SkM* effective

-40

-20

0

20

40

60

Q lb)

Fig. 1. (a-e) HF+ BCS deformation energy curves (MeV) as function of the mass quadrdpole moment Q (in b). Tbe ordinates of the horizontal bars are the energies of the GCM states. The mid-segment abscissae give the GCM mean values of the quadrupole moment. The solid dots give the same information for the decoupled states (see text). The dashed vertical line points to the HF+ BCS SD minimum and the arrow to the GCM SD state. (f) HF+ BCS lowest minimum (open circle), HFi BCS SD minimum (open square), HF+ BCS barrier top (open triangles) and GCM SD (bars) excitation energies relative to the GCM ground state.

511

f? Bonche et al. / Quadrupole collective correlations

nucleon-nucleon

potential

*‘) and a seniority

tions I). The HF+ BCS energies of the constraining the level of mean shape isomeric tion values

are plotted

axial mass quadrupole field theory,

one would

interaction as functions

of the absolute

pairing

operator:

identify

the experimental

and local minima

correla-

of the expectation

moment

states with the HF+ BCS wave functions

are those

to describe

6 = 22*-y*

value - x2. At

ground

whose quadrupole seen in fig. la-e.

and

expectaA better

approximation, really the next step beyond this theory, is to construct the wave functions as linear combinations of the Slater determinants (Q(Q)), where IO(Q)) denotes the self-consistent HF+ BCS solution labeled according to the mean value Q of the quadrupole operator. These linear combinations are determined by a minimization of the expectation value of the Skyrme plus pairing two-body hamiltonian 6 This is the GCM method. Ek is represented by the integral:

in which the weight

I

function

In it, the kth nuclear

fk satisfies

eigenstate

the Hill-Wheeler

1Pk) with energy

equation:

{(~(a)lBl~(o’))-Ek(~(Q)l~(O’))}fk(O’)d0’=0.

(2)

It is useful to note that the GCM incorporates automatically the effects of ingredients of the Bohr collective hamiltonian 22,23) method such as the zero-point motion energy and the effective collective masses. Moreover, it does not rely on assumptions such as the gaussian overlap approximation nor on the validity of quadratic expansions of the kernels “) which are often used to derive microscopically these ingredients ‘l). We emphasize also that, although we will sometimes use semi-classical reasoning and language to explain our results, we are reporting results of an exact diagonalization and do not use BKW barrier of the shape isomers. In our case, each Slater

penetrability

determinant

arguments

to discuss

I@( Q)) for the Hg isotopes

the stability

is represented

by some ten megabytes of information, which describe the wave functions of all (partially) occupied orbitals. Handling that much information is a challenge in itself. In addition, one must understand questions of numerical precision in the calculations of the kernels appearing in eq. (2). Furthermore, the basis is overcomplete, so that one needs to devise insensitive means of dealing with spurious (i.e. zero) eigenvalues of the normalization matrix (@(~)l@( Q’)). It is also important to determine how fine a grid in Q is necessary to adequately represent the integral GCM solutions eq. (1). Lastly, one must deal with the imprecision in particle number that arises from linear combination of BCS solutions which are individually constrained to have the correct average particle number but are not eigenstates of the number operator 24). All of these questions have been addressed in a technical paper 25). Suffice it to say that we are satisfied that we have solved the Hill-Wheeler equation

512

R 3onc~e et al. j ~a~~~ole

e~lfect~veco~?eZutjons

with a precision of 0.02 MeV. As indicated above, we have confined ourselves to cases of axial deformations. This assumption greatly simplifies the calculation, and although it must only be considered as a first approximation for the lowest states, it is certainly correct for the shape isomer. The gamma degree of freedom will be incorporated in a future work, where we expect it to make only modest corrections to the present results. 3. GCM results The short horizontal bars in fig. la-e show the results of our GCM diagonalization of eq. (2) for the five even Hg isotopes (N = 110-l 18) considered in this work. The ordinate is the total energy of the states (where the GCM ground state is taken as zero) and the abscissa of the center of the bar the average of the quadrupole moment for that particular GCM state. This permits us to test the validity of our previous predictions based upon the simple mean field theory. In fig. lf we have drawn the energy of the Hartree-Fock minimum (open circles) relative to the GCM ground state as the zero of energy. These points indicate the ground state energy in the mean field approximation. The short horizontal bars above are at the excitation energies of the GCM state corresponding to the shape isomer or band head for the SD band. The open squares are the positions of the minima of the second well, i.e. the mean field predictions for the energies of the shape isomer. What is pertinent is that the energy shifts of the two Hartree-Fock minima relative to their corresponding GCM eigenstates do not change and are nearly equal. This means that our previous interpretation of the excitation energy of the shape isomer relative to the nuclear ground state is not altered by the collective correlations induced by configuration mixing. From fig. la-e, one can see that, except for 19’Hg, the HF absolute minimum provides an excellent approximation of the GCM ground state quadrupole moment. This can also be seen qualitatively in the solid-line curves in the left part of fig. 2 which picture the amplitude of the GCM wave functions* for the ground state. These have a simple structure and are approximately peaked at the value of the quadrupole moment of the lowest HF minimum. For reference purposes, we also show the wave function of the first excited state as a dashed line in fig. 2. The evolution of the breadth of the ground state-wave function and of the proximity of the first excited state as a function of neutron number N is consistent with observations of so-called shape coexistence at lower neutron numbers. This is a general trend in these isotopes; the heaviest have a simpler and the lightest a more complex collective quadrupole structure. This manifests itself both in shape coexistence at small values of Q for the lighter isotopes, and in the structure of the GCM eigenstate which we identify with the shape isomer based upon the large value of the expectation value of its quadrupole moment (see fig. 1). The latter *

The relationship between collective wave functions and the fk(Q)‘s is discussed in refs. ‘9V25).

P. Bonche et al. f Quadmpole collective correlations

I,,,

,,,

I

-40

-20

,,,

,,,/,,,

,/,,/,,,,,,,,,,,,,,,,,11

40

0 a(;

60 -40

-20

40

0

60

Q 1;

Fig. 2. GCM collective wave functions for five Hg isotopes as function of the mass quadrupole moment Q (in b). Left-hand side: ground state (solid) and first excited state (dash) wave function. Right-hand side: same jnformation for the SD state (solid), its nearest neighbor in energy (dot) and the SD decoupled state (dashed line, see text).

has more cont~butions at small values of the quad~pole moment (i.e. components of first well HFf BCS wave-functions) in the lighter Hg. This GCM eigenstate is interpreted as an intrinsic state. The mechanism which drives both phenomena is the proximity to shell closure N = 126 which renders the ground state more spherical and produces a less penetrable barrier between the first and second wells in the HF-t- BCS total energy representation. A measure of this is the energy of the top of the barrier between the two HF+BCS wells shown by the open triangles in fig. If.

514

P. Bon&e et al. / ~ad~p~l~

collective c~~relatians

The modifications of the deformation energy curves as a function of N strongly affect the shape isomeric state. In fig. la-e the vertical dotted lines point to the mass quadrupole moment of the second well minima. One notes that the mean value (6) of the best candidate GCM eigenstate (vertical arrow on the Q-axis) is close to that of the HF+BCS minima at large N and slowly moves to lower Q-values as one removes neutrons. The same trend can be observed in the values of proton quadrupole moments listed in table 1 which appear compatible with present experimental data (20* 2e - b in the superdeformed band of “‘Hg, see ref. ““)). This phenomenon is explained by the changes in the thickness and height of the barrier separating the first and second well. The smaller of either of these quantities, the more likely the GCM will mix HF states of similar energy in the first and second well and induce a shift towards lower deformations. As noted above, this is related to the neutron shell closure. Before drawing experimental consequences from this mixing, we will perform one further theoretical step. Degeneracy implies some lack of specificity of the observables used to describe a quantum mechanical state. If one examines the solid and the dotted lines in the right part of fig. 2, which represent, respectively, the GCM wave function for the shape isomer and its nearest energetic neighbor in the first well, one sees that, as a function of Q, they can be very diffuse; especially so for the lightest Hg isotopes. Moreover, they display a similar nodal structure for low values of Q. We can resolve the approximate degeneracy by a further diagonalization of the quadrupole moment operator in the subspace generated by these two GCM states. This additional decoupling is not essential for most of our conclusions but clarifies our interpretation of the role of these states in the observed termination of the SD band. Our reasoning is that a decoupling according to (6) provides the relevant basis for a study of superdeformation. Indeed, a slight mixing existing at zero spin is likely to disappear at larger angular momentum, because states with a larger deformation are energetically favoured by rotation. In the analysis of the de-excitation scheme of the SD band proposed in the next section we will therefore use the decoupled states as band heads. As expected from the HF+ BCS deformation TABLE 1 Proton quadrupole moments {&) of the SD states for five Hg isotopes. The first column gives the mass number A, the three following give the values of (Q) in barn as calculated by the HF+BCS, the GCM and the GCM plus decoupling (see text) methods, respectively. A

HF+ BCS

GCM

GCM + dec.

190 192 194 196 198

17.6 18.5 18.5 18.5 18.6

11.8 16.5 17.5 17.9 18.4

16.0 17.5 18.0 18.2 18.4

515

l? Bonche et al. / Quadrupole collective correlations

energy

curves,

the decoupling

19*Hg. The resulting wave functions

eigenvalues

quadrupole

less effective

are shown

for the decoupled

fig. 2. The third column to proton

becomes isomer

as one goes from

by the solid dots in fig. la, e and the

are the dashed

lines in the right part of

of table 1 shows also that the decoupling moment

values

19’Hg to

close to those

predicted

prescription by the mean

leads field

method.

Note also that although the Q-values of the two GCM states may be strongly affected (as in 19’Hg), their energy does not change by more than 100 keV. Examining the wave functions of the shape isomer and its first well companion, it is clear that even after decoupling, they will mix in r9’Hg, and less so in the more neutron rich isotopes until one reaches 19’Hg where they will hardly mix at all. We

conclude that r9’Hg would have a much better chance of a long-lived shape isomer than r9’Hg. However, in view of the standard techniques used to populate the SD band, this is a somewhat academic point because, as we shall discuss it in the next section, the SD bands empty before reaching the band head which would be the potential shape isomer. The interpretation of the results obtained with or without decoupling procedure is the same: the quadrupole moment of the eigenstate smaller with decreasing neutron number. For purposes we can as a first approximation

assume

in the second well becomes of inferring the SD spectrum

that the moment

of inertia

of the shape

isomer is given by the rigid body value. Although quantum mechanical effects must be present, this value is consistent with those used to interpret the data on the rotational band built upon the shape isomer in 240Pu [refs. 27328)].With this assumption, our calculations reproduce the band head moment of inertia inferred from the recent measurement on 190-‘94Hg. Moreover, our results are compatible with the observation of 192Hg and 194Hg having the same moment of inertia while that measured in 19’Hg is slightly smaller 29). Concomitantly, we predict that the moment of inertia for the SD band in 196Hg and r9’Hg should be the same as observed in 192Hg and 194Hg.

4. Depopulafion

of the superdeformed

hand

The results of the previous section can now be applied to predict de-excitation rates within and out of the SD band. An essential ingredient for such a calculation is a set of quadrupole radiation matrix elements. These have been calculated from our theoretical results making only the standard collective model assumption that the rotational states can be described as a product of a collective rotor times an intrinsic GCM state. In addition, we must introduce phenomenological estimates of the moments of inertia of the bands to permit an extrapolation of our GCM results to non-zero spin. For the SD band, selecting a moment of inertia is simple. As discussed in the previous section, the rigid body moment of inertia is appropriate. The case for the spectrum built upon states in the first well is more heuristic. We assume that our

516

P. Bon&

et al. / Quadrupale

colteetive come&ions

collective GCM states at small values of the quadrupole moment are the band heads for rotational bands which function as doorways at the appropriate values of the angular momentum. Choosing a moment of inertia for these bands permits us to construct a rotational spectrum based on GCM states at small quadrupole moments, i.e. in the first well. We have chosen the following phenomenological relationship between the moment of inertia & of a given band k and the quadrupole moment Qk = ( W, 161 W,) of its band head,

A=~a2+b2(Q~/Qd2.

(3)

The coefficients a = 12.5A2/MeV and b = lOOh’/MeV entering this formula have been selected to reproduce the rigid body moment of inertia at large Q and to have the order of magnitude suggested by experiment 30) at small Q. Finally, the energy Ef of the first state in each band was taken as the energy of the GCM intrinsic state Ek corrected by the rotational energy

-c(j2),

Et+= Ek-29k

where (p) is the HFSBCS expectation value of the total angular momentum for the quadrupole moment value of the band head. From the calculated spectrum and using standard formulae for transition rates 19), we can follow the population of the SD band as a function of angular momentum, assuming it to be unity at say 30R. Our results are shown on the right side of fig. 3 for the five Hg isotopes. One notices that the transition out of the SD band takes place very rapidly: a nearly complete depopulation occurring in two to three transitions, consistent with experiment. Furthermore, we calculate that the depopulation will begin at about 8 to 14A which is also qualitatively consistent with experiments in the Hg isotopes. The left side of fig. 3 displays the spectrum of depopulating gamma rays expected from our calculated transitions. One sees a systematic change in the pattern as a function of neutron number, with the predicted spectrum hardening with increasing N. If this is the correct mechanism for the emptying of the SD rotational band, then perhaps these gamma rays could be observed. It is at this point that the assumption of the first well states acting as doorways comes back to complicate the prediction of the depopulating transitions. Although dilution of the doorway among complicated states is expected to preserve the integrated strength, the spreading may render the observation of these gamma rays more difficult relative to background. While we have made reasonable assumptions on the spectra of the rotational bands built upon our GCM states, the conclusions we have drawn are not very sensitive to them. One recalls that the out-of-band versus in-band branching ratios Rk(f) associated with quadrupole transition is inversely proportional to the square of the quantity Wk = (Y,, / t$i~~~}/( tPs, 16 1!Pk) and depends on the fifth power of the ratio of transition energies: &(l) = (~~t(&)/E~)5/ W”,. The SD band will

P. Bonehe et al. / Quadruple

0

2

(M:“j ET

6

collectiue correlations

0

2

4

8

8

10 12 14 16 18 20 22

I

Fig. 3. Decay of the superdeformed band for five Hg isotopes. Left-hand side; relative intensities of the depopulating E2 gamma rays as a function of their energy. The bar widths have no significance. Right-hand side; evolution of the population of the SD band as a function of spin. The vertical arrow indicates the spin below which the SD band is excited with respect to one (or several) positive-parity band(s).

therefore disappear around a spin value l,, such that one of the Rk(&) = 1. The corresponding de-excitation gamma rays will have an energy approximately equal to W2’5 times that of the last observed SD transition. The calcuiated values of W2” vary from about 10 in 190-‘p2Wgto 40 in 198Hg. This alone explains the hardening of the de-excitation spectrum from 1.5 to 6 MeV with increasing IV. In particular,

P. Bonche et al. / Quadrupole collective correlations

518

this feature

is not directly

related

heavier

isotopes

relevant

factor in the calculated

reasonable

(see fig. If). The large value

assumptions

1, the relevant

ratios

precise

made

choice

to the increasing

sudden

R(I)

vary rapidly

for the moment

of the ratio

depopulation

on the evolution

band-head W*”

excitation

is also the only

of the SD band.

Indeed,

of the ratio Ey”‘/E’,” as a function around

of inertia

I,,. Hence,

in the with

of spin

independently

of the

$&‘s, one finds that the SD band,

once it begins to depopulate, does it in few transitions. It is relevant to be concerned that other than quadrupole

transitions

our results. We believe that because the majority of the observed lies at high excitation energy centered at about 13 MeV, direct transitions will not be important enough to alter our conclusions.

will alter

dipole strength out-of-band El We do know in

specific cases, such as the gamma ray decay of the shape isomer in 236T238U[refs. 31T32)] that the dipole transition can be comparable to that of the quadrupole. However, this happens at sufficiently low energy as to be likely not applicable to our situation. Moreover, we expect that the small amount of dipole strength existing at low excitation will begin only at energies similar to that of the shape isomeric state. Therefore, we think it reasonable that any rotational band built upon this dipole strength will have very small transition energies which will inhibit direct de-excitation of the SD band by dipole transitions. It is not clear that the mechanism we have found germane in say dysprosium. Here, the SD band ends spins (=24h) [ref. “)I and a discussion such as ours, results at zero spin is no longer valid. In particular,

relevant in the Hg isotopes is at higher energies and higher which is based on microscopic the zero spin SkM* HF+ BCS

deformation energy curve of “*Dy does not display any local minimum at large Q. Further calculation will be required to understand the range of applicability of the mechanism

we have elucidated

based

upon

GCM

states used as band

heads.

5. Conclusions We have compared the energy and position in collective space of the GCM eigenstates with those obtained in previous HF+BCS calculations. We conclude from this study that the excitation energy of the SD band head or shape isomer predicted by the HF+ BCS method is unaltered by the introduction of quadrupole collective correlations. With the notable exception of 19’Hg, this is also true for its mean quadrupole moment values, giving us confidence in the accuracy of our former predictions of the quadrupole deformation of the SD band in 19*-19’Hg. Making this assumption built upon experience that the moment of inertia at such large deformations is that of a rigid rotor, leads us to understand why the moments of inertia in 19*Hg and ‘94Hg are the same. Using the decoupling procedure described in sect. 3, we find that the moment of inertia in 19’Hg should be slightly smaller cranked Hat-tree-Fock calculathan in the heavier isotopes ‘92-198Hg. Preliminary tions

on r9’Hg with broken

time reversal

symmetry

indicate

that the odd particle

P. Bonche et al. / Quadrupole

collective correlations

519

does not change the deformation of the SD band. This appears consistent with experiment. Lastly,we have employed our results to analyze the termination in angular momentum of the SD band. Using rotational bands built on each of the GCM eigenstates and microscopically calculated transition matrix elements within and out of the SD band, we find that it terminates at approximately 8 to 14h, in agreement with experiment. We also reproduce the abruptness of the termination and make predictions on the energy of the quadrupole SD depopulating transitions. As a consequence of our results, it appears highly improbable that one can reach the SD band head or shape isomer starting at high angular momentum and that alternative experiments involving a direct feeding of SD states at low spin have to be devised. We acknowledge fruitful discussions with R.R. Chasman, B. Haas, R.V.F. Janssens, M. Meyer and W. Nazarewicz. We wish to thank J.D. Anderson, J.A. Becker and E.A. Henry for their patient illumination of the experimental mysteries. We gratefully acknowledge the Centre de Calcul Vectoriel pour la Recherche for extended computing facilities. This work was supported in part by the NATO grant RG 85/0195, in part by the US Department of Energy under Engineering Contract NW7405ENG-48, in part by SDIO/IST administered by NRL, in part by DOEDOD(OM)MOU and in part by the Polish Ministry of National Education under Contract CPBP 01.09.

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@adrupole

collective correlations

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