PHYSICAL REVIEW C, VOLUME 62, 064313

Shell model description of normal parity bands in even-even heavy deformed nuclei

G. Popa,1,* J. G. Hirsch,2,† and J. P. Draayer1,‡

1Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-40012Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de Me´xico, Apartado Postal 70-543, Me´xico 04510 DF, Me´xico

~Received 7 August 2000; published 20 November 2000!

The pseudo-SU~3! model is used to describe the low-energy spectra and electromagnetic transition strengthsin 156Gd, 158Gd, and 160Gd. The Hamiltonian includes spherical single-particle energies, the quadrupole-quadrupole interaction, proton and neutron pairing interactions, plus four rotorlike terms. The quadrupole-quadrupole and pairing interaction strengths are assigned the valuesx523A25/3 and Gp521/A, Gn517/A,respectively. The single-particle energies were taken from experiment but scaled to yield an overall best fit. Forthe other four rotorlike terms, which do not mix SU~3! representations and induce only small changes in thespectra, a consistent set of parameters is given. The basis states are built as linear combinations of SU~3! stateswhich are the direct product of SU~3! proton and neutron states with pseudospin zero. The results are in goodagreement with experimental data, demonstrating the suitability of the model to describe heavy deformednuclei.

PACS number~s!: 21.60.Fw, 21.60.Cs, 23.20.Js, 27.70.1q

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I. INTRODUCTION

Recent improvements in nuclear spectroscopy have mpossible the measurements of lifetimes of highly excitlow-spin states that were previously inaccessible. Ongoexperiments are searching for new energy levels. Forample, in the last few years new levels have been identiin well-deformed rotational nuclei of the lanthanide regio168Er @1,2#, 154Gd @3#, 164Dy @4#, 166Er @5#, 158Gd @6#, 162Dy@7#.

In this paper we report on a study of the156,158,160Gdisotopes that was carried out within the framework of tpseudo-SU~3! model. Four low-lying bands are identifiewith angular momentumJ<8, along with theM1 transitionprobability distribution from the 01 ground state to calculated 11 levels. A comparison of theoretical and experimetal B(E2) transition strengths is also given.

Successful applications of the SU~3! shell model@8# tolight deformed nuclei have led physicists to explore simiconcepts in heavy deformed systems. One of the first clenges encountered when developing a shell-model thefor heavy nuclei is that the splitting of the single-particlevels generated by the spin-orbit interaction is comparain magnitude to the major shell separation of the harmooscillator and thus renders the usual SU~3! symmetry use-less, and with it the logic underlying an SU~3!-based trunca-tion scheme that proved to be so valuable in light deformnuclear systems.

Fortunately, another symmetry appears as a result oflarge spin-orbit splitting; namely, the so-called pseudo-SU~3!scheme which can be appreciated most easily by considethe near degeneracy of the orbital pairs@( l22) j 5( l 22)11/2,l j 5 l 21/2# which together define apseudoshell with one quantum less than the original~parent!

*Electronic address: gabriela@phys.lsu.edu†Electronic address: hirsch@nuclecu.unam.mx‡Electronic address: draayer@lsu.edu

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configuration,h5h21 whereh is the principal quantumnumber of the parent shell@9–11#. The pseudospin doublet

with quantum numbersj 5 j and l 5 l 21 define the subshelstructure of the pseudoshell, which is just the original shless its highestj 5h11/2 level. The physical underpinningof the pseudospin symmetry, and by extension the pseuSU~3! model, have been explored recently in terms of a retivistic ~Dirac! formulation of mean-field results for heavnuclei @12–14#.

A second challenge that is encountered in developinshell-model theory for heavy nuclei is the dimensionalitythe model space which increases very sharply as one mto higher shells. This growth in the dimensionality of thmodel space can only be managed by truncating the mspace to a small, carefully selected subset of the full spSince in light deformed nuclei the quadrupole-quadrupinteraction is dominant so the ground state can be resented by a few irreducible representations~irreps! of SU~3!@15–18#, it is natural to assume one can similarly truncatemodel space in heavy nuclei to those representations, incase of pseudo-SU~3!, that correspond to the largest~pseudo!intrinsic deformation. Indeed, most calculations carriedto date have truncated the space to a single, or at most twthree irreps with a very simple mechanism for generatingsplitting and, in the latter case, mixing of pseudo-SU~3! rep-resentations.

The development of a computer code that is able to cculate reduced matrix elements for any type of physicalerator between different SU~3! irreps @19# has made it pos-sible to include realistic SU~3! symmetry breaking termslike the pairing interaction, in SU~3!-model Hamiltonians.Indeed, recent results using this code show that the paiinteraction is closely tied to the development of triaxialitystrongly deformed systems@20,21#. Furthermore, completemodel-space calculations in thepf shell @18,22# show that avery good description of the low-energy spectra can betained when the Hilbert space is truncated, albeit not soverely, following the same logic as used in thesd shell,

©2000 The American Physical Society13-1

G. POPA, J. G. HIRSCH, AND J. P. DRAAYER PHYSICAL REVIEW C62 064313

TABLE I. The 18 pseudo-SU~3! irreps used in the description of158Gd low-lying bands.

(lp ,mp) (ln ,mn) Total (l,m)

~10,4! ~18,4! ~28,8! ~29,6! ~30,4! ~31,2! ~32,0! ~26,9! ~27,7!~10,4! ~20,0! ~30,4!~10,4! ~18,4! ~32,0!~10,4! ~16,5! ~26,9! ~27,7!~10,4! ~17,3! ~27,7!~10,4! ~13,8! ~23,12!~12,0! ~18,4! ~30,4!~12,0! ~20,0! ~32,0!~8,5! ~18,4! ~26,9! ~27,7!~9,3! ~18,4! ~27,7!~5,8! ~18,4! ~23,12!

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namely, a dominance of the quadrupole-quadrupole intetion. These same calculations also showed that the painteraction is critical for a correct description of momentsinertia.

As a result of these developments, a very powerful shmodel theory for the description of normal parity statesheavy deformed nuclei has emerged. For example, theenergy spectra of Gd and Dy isotopes, theirB(E2) andB(M1) transitions strengths, including both the scissorstwist modes@23# and their fragmentation, have been succefully described using a realistic Hamiltonian@24#.

One difficulty of previous studies has been an apparlack of consistency in choices for Hamiltonian parametersthe theory. In the present work we find a consistent separameters for terms in a realistic pseudo-SU~3! modelHamiltonian. The quadrupole-quadrupole and monoppairing interaction strengths are taken from systematGood agreement with experimental data on the lowest fbands in156,158,160Gd is achieved. The theory also gives corect values for the sumrule forM1 transitions from theground state of these three nuclei, the correct positions o11 energies, and a reasonable reproduction of the fragmtation of theM1 strength.

II. PSEUDO-SU„3… MODEL

The many-particle states ofna active nucleons in a givennormal parity shellha , a5n ~neutrons! or p ~protons!, canbe classified by the following group chain:

$1naN% $ f a% $ f a% ga~la ,ma!

U~VaN!.U~Va

N/2!3U~2! . SU~3!

Sa Ka La JaN

3SU~2!.SO~3!3SU~2!.SUJ~2!, ~1!

where above each group the quantum numbers that chaterize its irreps are given andga and Ka are multiplicitylabels of the indicated reductions. The most important cfigurations are those with highest spatial symmetry@25,18#.

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This implies that only configurations with pseudospin eqto zero need to be considered when considering an enumber of nucleons.

As an example,158Gd will be described. It has 14 protonand 12 neutrons in the 50–82 and 82–126 shells, restively. The number of nucleons in normal~N! and abnormal~A! parity orbitals were determined by filling the Nilssolevels in order of increasing energy with pairs of particlesdeformationb;0.3. This yieldsnp

N58, npA56, nn

N58, andnn

A54 which in turn uniquely determine the highest symmtries in the a5n and a5p chains, U(Va

N).U(VaN/2)

3U(2). The 18pseudo-SU~3! irreps with largest values fothe second order Casimir operator,Q•Q54C223L2, thatwere used in this calculation are listed in Table I.

A. Hamiltonian

The Hamiltonian includes spherical single-particle terfor both protons and neutronsHsp

p[n] , an isoscalarquadrupole-quadrupole interactionQ•Q, neutron and protonpairing termsHP

p[n] , and four smaller ‘‘rotorlike’’ terms thatpreserve the pseudo-SU~3! symmetry:

H5Hspp 1Hsp

n 21

2xQ•Q2GpHP

p2GnHPn

1aJ21bKJ21a3C31asymC2 . ~2!

~Note that tildes, commonly used to denote pseudoquantiare suppressed throughout the article to simplify the notion.! The term proportional toJ2 represents a small correction to the moment of inertia, theKJ

2 breaks the degeneracof the differentK bands within a pseudo-SU~3! irrep @26#,the third term, which is the cubic Casimir invariant opseudo-SU~3!, serves to fix the position of the 01 energiesrelative to one another, and the last one, which distinguisbetweenA-type andBa-type (a51,2,3) internal symmetriesof the rotor, pushes the 11 energies up as they are all banheads ofBa-type structures. The spherical single-particHamiltonian has the form

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SHELL MODEL DESCRIPTION OF NORMAL PARITY . . . PHYSICAL REVIEW C62 064313

Hsps 5(

i s~Csl i s•si s

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wheres stands for protons (p) or neutrons (n). Since onlypseudospin zero states are considered in the present aption of the theory, matrix elements of the spin-orbit partthis interaction vanish identically.

B. Parameters

The pairing and quadrupole-quadrupole interactstrengths were taken from systematics@27,28#. Specifically,since the pairing interaction is invariant under tpseudospin transformation,Gp521/A andGn517/A. On theother hand, it is necessary to rescale the expression foquadrupole-quadrupole interaction strength (x523A25/3) toreflect the effect of the pseudospin transformation onquadrupole operator. To determine this scale factor, csider, for the moment, that a tilde is used to denote pseuquantities. Then the necessary factor can be determinerequiring thatxQ•Q5xQ•Q. A numerical value can be determined by applying this result to the leading ground-stconfiguration:x5(^C2&max/^C2&max)x. For the nuclei underconsideration, C2&max/^C2&max'1.5 so the strength factomultiplying Q•Q in Eq. ~2! needs to be 35A25/351.5323A25/3.

The single-particle strengths that were used are lower tthe standard values@27# by a factor of 4. This means thDp[n] parameters are negative for both protons and neutand of about the same magnitude as those used in prevcalculations@24,31#. This reduction in the strength of thsingle-particle orbit-orbit interaction can be shown to be tto the truncation of the model space. In particular, increasthe single-particle orbit-orbit interaction strength enhanthe mixing of SU~3! irreps in the ground state, and with tha sizable fraction of theM1 strength is shifted to 11 statesthat fall outside the model space. The effect of enlargingmodel space is under further investigation. Thea, b, a2, andasym parameters, which have an overall small effect onspectra, were optimized to yield best fits to experiment.

The strength of the single-particle interaction togethwith the strength of theC3 interaction determines the exctation energy of the excited 01 states. All of the interactionsaffect the various moments of inertia and correct relat

TABLE II. Parameters used in the Hamiltonian~2!.

Parameter 156Gd 158Gd 160Gd

x 0.0077 0.0076 0.0074Dp 20.075 20.075 20.075Dn 20.054 20.052 20.052Gp 0.135 0.133 0.131Gn 0.109 0.108 0.106a31023 22.0 23.0 23.0b 0.14 0.23 0.19asym31023 1.50 1.50 1.62a331024 5.05 7.50 10.00

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spacings between the bands. The parameters of the theorlisted in Table II. It is important to note that they are constent with one another and with those used in a descriptionthe neighboring odd-mass nuclei@29#.

III. RESULTS

The upper part of Figs. 1–3 shows the calculated aexperimental@30# ground state,K52, and two excitedK50 energy bands in(156,158,160)Gd. The theoretical energylevels are in excellent agreement with the experimental dThe model predicts more energy levels in theK52 and inthe two excitedK50 bands.

From Table II we observe thata remains almost constanfor the three isotopes,156Gd, 158Gd, and160Gd. This param-eter is responsible for fine tuning of the effective momentsinertia, and is in agreement with the value used in the neiboring odd-mass study. Theb parameter changes slightlfrom case to case. This is understandable since it fixesK52 band relative to the ground state and the placementhese bandheads is different in the three cases. In this regit is important to note that the second experimental 21 statesare not always the bandhead of aK52 band. For158Gd and160Gd the second 21 energy is theK52 bandhead, but for156Gd it is a member of the first excitedK50 band. Thisdetail is well reproduced by the model.

FIG. 1. Energy spectra of156Gd obtained using Hamiltonian~2!with parameters from Table II. ‘‘Exp.’’ represents the experimenresults and ‘‘Th.’’ the calculated ones. The lower plot gives ttheoretical and experimentalM1 transition strengths from the 01

ground state to the various 11 levels.

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G. POPA, J. G. HIRSCH, AND J. P. DRAAYER PHYSICAL REVIEW C62 064313

The asym parameter, which shiftsBa-type (a51,2,3) in-trinsic SU~3! configurations@24# relative to theA-type, hasan almost constant value for these nuclei. This parameteused to position the 11 states, which are bandheadsBa-type internal configurations, relative to the ground st01, which has anA-type internal symmetry. Thea3 param-eter, which multiplies the third-order Casimir invariantC3 ofSU~3! and which has an eigenvalue that is proportional toirrep’s intrinsic asymmetry (l2m), increases slightly in go-ing from 156Gd to 160Gd. This is consistent with the fact thathe second 01 state moves up in energy in going from156Gdto 160Gd. Note that sincex, the coefficient multiplyingQ•Q and henceC2 is fixed, theC3 term in the Hamiltonianis the only one that directly affects the relative position of t01 energies. This term, which is small relative to the othin the Hamiltonian, was necessary to obtain the detailedproduction of the energy spectra shown in Figs. 1–3, asfound in a previous study@31#. A more detailed analysis othis matter will be the subject of another investigation@32#.

A. E2 Transition strengths

Theoretical and experimental@30# B(E2) transitionstrengths between low-lying states in158Gd are shown inTables III and IV. The quadrupole operator was expresse@25#

Qm5epQp1enQn'ep

hp11

hpQp1en

hn11

hnQn , ~4!

FIG. 2. The same as Fig. 1 but for158Gd.

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with effective chargesep52.25, en51.25. These values arvery similar to those used in earlier pseudo-SU~3! descrip-tions of even-even nuclei@25,24#. @The (h11)/h factors inthis expression have the same origin as the^C2&max/^C2&maxrenormalization ofx multiplying Q•Q in the Hamiltonian.#They are larger than those used in standard calculationB(E2) strengths@27# due to the passive role assigned to tnucleons in unique parity orbitals, whose contribution to tquadrupole moments is parameterized in this way.

In Table III, B(E2) strengths are given for transitionbetween members of the four low-lying bands. The callated results are in good agreement with the known expmental strengths. Electromagnetic transitions from the 2g

1 tothe 01, 21, and 41 states of the ground state band ashown in Table IV. While they are about an order of magtude larger than the measured values, the latter aresmall, about 1023 that of typical transitions between members of the same band, so it is difficult to attach any rsignificance to these differences. This strong~interband! andweak ~intraband! structure of theB(E2) strengths under-scores the significance of the assignment of the levels tKbands.

B. M1 transition strengths

TheM1 strength distributions derived from the calculateigenvectors are shown, along with the corresponding

FIG. 3. The same as Fig. 1 but for160Gd. Note that in this casethe strongestM1 transition (1.12mn

2) to the 11 state at 2.80 MeVoverlaps the experimental transition (0.75mn

2) to the 2.80-MeV 11

state.

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SHELL MODEL DESCRIPTION OF NORMAL PARITY . . . PHYSICAL REVIEW C62 064313

perimental results, in the lower plots that are a part of F1–3. Key features of these distributions are easy to unstand within the framework of the pseudo-SU~3! model. Thebasic structure of the strength distribution is determinedthe SU~3!-symmetry preserving part of the Hamiltoniawhich embodies strongM1 selection rules@31#. In this limitof the theory there is no coupling between different SU~3!irreps and there are at most four nonzeroM1 transitionsbetween the 01 ground state and the various 11 states@24,31#.

The fragmentation of theM1 strength that is predicte~and observed! is a result of symmetry breaking. This breaing of the symmetry is generated by the single-particle apairing interactions that are an integral part of the Hamtonian ~2!. For the nuclei considered, the centroids of t

TABLE III. Theoretical and experimentalB(E2) transitionstrengths for158Gd. The subindices of the angular momentum alabeled with ‘‘g.s’’ for ground state, ‘‘g’’ for the gamma band (K52 band!, ‘‘ a’’ for the first excitedK50 band, and ‘‘b’’ for thesecond excitedK50 band.

J1→(J12)1 Th. (e2b2) Exp. (e2b2)

0g.s.1 →2g.s.

1 5.06 5.0260.152g.s.

1 →4g.s.1 2.59 2.6460.05

4g.s.1 →6g.s.

1 2.276g.s.

1 →8g.s.1 2.12 2.1260.20

2g1→4g

1 1.044g

1→6g1 1.56

6g1→8g

1 1.660a

1→2a1 5.08

2a1→4a

1 2.574a

1→6a1 2.19

6a1→8a

1 2.040b

1→2b1 5.03

2b1→4b

1 2.524b

1→6b1 2.14

6b1→8b

1 2.00

J1→(J11)1 Th. (e2b2) Exp. (e2b2)

2g1→3g

1 2.534g

1→5g1 1.19

6g1→7g

1 0.61

TABLE IV. Theoretical and experimentalB(E2) transitionstrengths between energy levels belonging to different band158Gd. The subindices of the angular momentum follow the saconvention as in Table III.

Ja1→Jb

1 Th. (31022e2b2) Exp. (31022e2b2)

2g1→4g.s.

1 0.60 0.1460.022g

1→2g.s.1 8.3 3.060.4

2g1→0g.s.

1 23.9 1.860.22a

1→4g.s.1 0.13 0.7160.08

2a1→0g.s.

1 0.02 0.1660.02

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experimental and theoreticalM1 strength distributions lie aabout the same excitation energy, and overall there is reaably good agreement between theory and experiment.

In 156Gd the strongest calculatedM1 strength (1.86mn2) to

the 11 state at 3.014 MeV is only 0.065 MeV from thstrongest (1.20mn

2) experimental transition at 3.070 MeVAnd in 158Gd the strongest predictedM1 transition (1.02mn

2

for the 11 state at 3.18 MeV! almost overlaps the strongeexperimentalM1 transition (0.77mn

2 to the 11 energy at 3.20MeV!. The best prediction forM1 strengths is in160Gdwhere the strongestM1 transition (1.12mn

2) to the 11 state at2.80 MeV overlaps the experimental transition (0.75mn

2) tothe 11 state at 2.80 MeV.

While one might like for the theory to give a slightlbetter reproduction of details of theM1 strength distribu-tions, the fact that it does as well as it does is quite remaable since in contrast with the quadrupole-quadrupole inaction, which is part-and-parcel of the SU~3! model, aconsideration ofM1 strengths is not an integral part of thSU~3! theory. As can be seen in Table V, the totalM1strength, which for the full Hamiltonian is between 10 a25 % lower than for its pure SU~3! limit, also shows reasonable agreement with the experimental results. This reducin the strength can be traced to the symmetry mixing coupwith the fact that the model space is strongly truncated. Athe set of parameters used in the present calculation allone to use a larger value forasym which pushes the centroidup in energy, closer to the experimental value.

IV. CONCLUSION

The results of this study show that the normal parbands in the strongly deformed, even-even156,158,160Gd nu-clei can be described within the framework of the pseuSU~3! model. Pseudospin zero neutron and proton confirations with a relatively few pseudo-SU~3! irreps with largestC2 values proved to be sufficient to obtain good agreemwith known experimental results. The Hamiltonian that wused included Nilsson single-particle energies, a commquadrupole-quadrupole interaction, neutron and proton ping interactions with strengths fixed by systematics, and fsmaller rotorlike terms with strengths that were variedoptimize agreement with experiment. The parameter setsresulted from an independent analysis of the three Gdtopes were found to be consistent with one another. Tagreement between theory and experiment extended beenergies, including inter- as well as intrabandB(E2) values

ine

TABLE V. Theoretical and experimental sum rule strengthsM1 transitions from the ground states of156,158,160Gd to their re-spective 11 states.

Nucleus Experiment (B(M1)@mN2 # Theory mix

Pure SU~3!

156Gd 3.40 3.52 2.63158Gd 4.32 4.24 3.08160Gd 4.21 4.24 3.60

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G. POPA, J. G. HIRSCH, AND J. P. DRAAYER PHYSICAL REVIEW C62 064313

and theM1 strength distribution of the ground state.A general conclusion that follows from these results

that the pseudo-SU~3! shell model, with a realistic Hamil-tonian, can be used to describe low-lying normal parity staof strongly deformed rare-earth and actinide nuclei. It sfices to include a relatively small symmetry-adapted, trucated model space. Single-particle and pairing terms inHamiltonian are important to achieving the SU~3! irrep mix-ing that is required to reproduce the observed fragmentaof the ground stateM1 strength. The results suggest thmore detailed microscopic descriptions of other propertie

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heavy deformed nuclei, such asg factors and beta decaysmay finally be within reach of a bona fide microscopic shemodel theory.

ACKNOWLEDGMENTS

This work was supported in part by Conacyt~Mexico!and the U.S. National Science Foundation~PHY-9603006,PHY-9970769, and EPS-9720652 with matching againstlatter from the Louisiana Board of Regents Support Fund!.

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