Z. Phys. A 354, 125-134 (1996)

Rare-earth nuclei and the pseudo-SU(3) model*

D. Troltenier1, C. BahrF, J:.Escher3, J.P. Draayer4

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Received: 8 September 1995Communicated by W. Weise

ZEITSCHRIFT

FOR PHYSIKAco Springer-Verlag 1996

Abstract. An extended version of the pseudo-SU(3) modelis used to describe low-energy positive-parity states ofthree rare-earth nuclei in a full proton-neutron formalism.Along with other residual interactions, the Hamiltonianincludes the pairing which induces mixing between differ­ent representations of SU(3). Excitation energies, B(E2)values, B(Ml) values, quadrupole moments, and gR factorsof 136Xe, 138Ba, and 204Hg are calculated in a truncationfree environment and the results are compared with theexperimental data and other theories.

PACS: 21.60.Cs; 21.60.Fw

1 Introduction

More than two decades have passed since the pseudo­SU(3) model was first introduced [1]' During that timeperiod, and especially over the last few years, the pseudo­SU(3) model has been used to describe a wide variety ofvery different nuclear phenomena [2, 3, 4]. Despite thislong history,' however, the nucleon-nucleon interactionsused in almost all the pseudo-SU(3) model investigationshave been highly schematic in nature, a feature that is dueprimarily to technical difficulties related to the calculationof SU(3) matrix elements of more general interactions.Recently a code was released [5] that lifts this limitationand led to the introduction of interactions in pseudo-

* Work supported in part by a gram rrom the U.S. National ScienceFoundation.

1 E-mail: trolteni@rouge.phys.lsu.edu~Present address: Department or Physics, University or Toronto,60 St. George St., Toronto, Ontario M5S IA7, C:lnada. E-mail:bahri@physics.utoronto.ca.J E-mail: phesch@lsumvs.sncc.lsu.edu~E-mail: phdryr@lsumvs.sncc.lsu.edu

SU(3) model calculations that independent shell-modelstudies indicate are very important for an adequate de­scription of experimental data. Among these is the pairinginteraction, which is known to playa prominent role inmany interesting physical phenomena. In two recent stud­ies the effects of pairing correlations on collective excita­tion spectra, B(E2) values, quadrupoLe moments, momentsof inertia, and gR values were investigated [6, 7].These systematic studies, which were carried out todeepen our understanding of the effects of pairing in anuntruncated pseudo-SU(3) model configuration space,serve as a backdrop for the calculations presented in thiscontribution.

The objective of the present study is to describe withinthe framework of the pseudo-SU(3) model properties ofthree special rare-earth nuclei and compare the results toother theories (semi-realistic and conventional shell­model calculations, particle-plus-vibrational-core model)and the experimental data. Specifically, excitation ener­gies, B(E2) values, quadrupole moments, and gR values of136Xe, 138Ba, and 20ol.Hgwill be conside~ed. (Similar re­sults for lol.°Ce were reported in an earlier publication [7].)These particular nuclei have been selected because theycan be treated in a truncation free environment: 136Xe and138Ba are N = 82 nuclei that can be represented, respec­tively, by four and six valence protons outside the Z = 50shell while 204Hg has two proton holes in the N ~ = 4 shelland two neutron holes in the Ny = 5 shell.

Recently the pseudo-SU(3) model has been used tostudy double beta (f3{3) decay in heavy deformed nuclei[4]. These investigations focus on questions relating to thepossibility of physics beyond the standard model of theelectroweak interaction. A thorough examination of thechart of nucleids shows that there are only a limitednumber of systems where this decay mode is, at least inprinciple, experimentally detectable [8]. This follows be­cause the more common decay modes, such as consecutivesingle {3-decays or electron capture, are strongly sup­pressed. 136Xe and 20ol.Hgare among the set of f3f3-decaycandidates; indeed, experiments on 136Xe have alreadybeen performed [9]. A study of these nuclei thereforeprovides useful information for the ongoing attempts to

(1)

In contrast with the basis state situation, the structureof the Hamiltonian must be considered in somewhatgreater detail since an understanding of the physical effectof each of its terms is important for a full appreciation of

a rseudo-SU(3) model description of the N = 82 nuclei13 Xe, 138Ba, and of 2°~Hg that is given in Section 3. Inthis regard, note that the N = 82 cases are closed neutronshell systems so matrix elements of operators that act onneutron degrees of freedom vanish. On the other hand204Hg has both proton and neutron valence nucleons s;both particle types playa role in the dynamics as well asinteractions between them. A general form for the pseudo­SU(3) model Hamiltonian, HpSU(3)' that covers both casesis given by:

The first four terms in HpSU(3) can be understood asa many-particle extension of the single-particle NilssonModel Hamiltonian, where H 0 denotes the spherical oscil­lator part which is diagonal in the pseudo-SU(3) modelbasis space [10]. Moreover, since the configuration spaceis restricted to a single shell for the valence protons andlikewise for the neutrons, the Ho term only contributes tothe binding and has no effect on other observables such asthe excitation spectra. The algebraic quadrupole operator,Q", is used to define the quadrupole deformation drivingterm, Q". Q", and the multiplicative constant 1. is thequadrupole-quadrupole interaction strength parameter.To complete the analogy with the Nilsson Model,HpSU(3) includes the sum of the squared single-particleorbital angular momenta for both protons ( -+ L.i)fJ andneutrons ( ....•.'D.lt). Within the single-particle NilssonModel picture these terms are known to effectivelyflatten the harmonic oscillator potential (D" and D.assume negative values for heavy nuclei) for higherangular momentum states which mimics a feature that isautomatically included in potentials of the Woods-Saxontype.

The identical-nucleon pairing interaction is denotedby H'j, and Hp for protons and neutrons, respectively, withGIt and G. standing for the corresponding pairing strengthparameters. (The pairing interaction was only recentlyincorporated into the pseudo-SU(3) model and the sys­tematic studies that have been carried out suggest that ithas a profound influence on low-energy properties ofnuclei [6, 7].) The JZ and K; operators are, respectively,the square of the total angular momentum and its projec­tion on the intrinsic body-fixed z-axis. (Previous attemptsat describing the experimental low-energy spectra ofheavy deformed nuclei indicated the importance of boththe J2 and K; in describing their rotational excitationsand band splitting features [11, 14].) In this contributionthe strength parameters of H PSU(3) are determinedthrough a best-fit procedure aimed at obtaining optimalagreement between theoretical and experimentalexcitation energies, B(E2) values and quadrupolemoments. Before proceeding with a discussion of results inthe next section it is useful to compare the best-fit

126

describe the (J(J-decay mode within the pseudo-SU(3)model [4].

This contribution is organized as follows: T-he grouptheory that underlies the pseudo-SU(3) model is reviewedin the first part of the next section. Regarding the latter,only a very limited number of results are given becausea rather extensive discussion of the correspondence be­tween the quantum numbers, Casimir operators, andpseudo-SU(3) wave functions can be found in previouslypublished articles [3,6,7]. On the other hand, the pseudo­SU(3) Hamiltonian and a choice for the correspondinginteraction strength parameters is considered more care­fully and in greater length because they are of special

importance for a theoretical description of the three nuclei136Xe, 138Ba, and Z04Hg. In the final part of Sect. 2,definitions of experimentally measured quantities are re­viewed. A comparision of pseudo-SU(3) model results toother theories and the available experimental data is pre­sented in Sect. 3. A summary is given in Sect. 4 as weB asan outline of expected developments in algebraic shell­model theories that are based on the pseudo-spin symmetry.

2 The pseudo-SU(3) model

The pseudo-SU(3) model is a many-particle shell-modelbased theory that takes full advantage of pseudo­spin symmetry, which in heavy nuclei is manifest in thenear degeneracy of the orbital pairs [(l- l)j=l-ll'J.,(l + l)j=l+ I/Z]. It is also a theory that takes full account ofthe Pauli Exclusion principle. Like most other shell-modelschemes, the proton and neutron configuration spaces ofthe pseudo-SU(3) model are usually restricted to a singleoscillator shell. (This limitation can be lifted by employinga symplectic extension of the theory, which introduceshigher-shellcorrelations in a very natural way [14].) A re­cent extension of the pseudo-SU(3) model takes the spindegrees of freedom into account in a full proton-neutronformalism [6, 7]. This extended theory was used to inves­tigate the effect of different nuclear interactions on thelow-energy properties of even-even nuclei where the influ­,ence of pairing correlations is known to play an importantrok ,

As an algebraic shell-model theory the pseudo-SU(3)scheme exploits powerful group theoretical methods inthe construction of basis functions and for the calculation

of required matrix elements. Specifically, basis states arelabeled by eigenvalues of Casimir operators of the under­lying symmetry groups and additional indices that arerequired to resolve multiplicities in the group reductions.Since a full discussion of these matters can be found

elsewh.ere, see for example [3, 6, 7J, only the results will bereprod-\iced here. In terms of the space (U(N)+-+[f]),shape (SU(3) +-+(..1., p)), orbital (SO (3) +-+L), spin (S) andtotal angular momentum (J) as well as the various multi­plicities (0: for the U (N) ::> S U (3) reduction, p for theproduct of two SU(3) irreps, and K for the SU(3) ::> SO(3)reduction) the basis states have the form (subscript 1t forprotons and y for neutrons):

I {m" [j,,]0:1t (}'It, p,,), m. [J.Jo:. (..1.., P.)} P (..1., p)

KL{S",S.}S;JM).

HpSU(3) = Ho - ~ Q"'Q" + D,,~.tt + D.i)t - G"H'f.lc h

- G.Hp + aK; + bJ2 • (2)

Table 1. A list of all parameters used in the pseudo-SU(3) calcu­lations for the nuclei 136Xe, 138Ba, and 204Hg. The first seven rowsrefer to the Hamiltonian of Eq. (2) while the last row gives theeffective proton charge used in the transition operator of Eq. (3). (Noeffective neutron charge was used.) In this application of the theoryno parameters enter into the expression for the magnetic dipoleoperator in Eq. (4) since the values g: = 1 and g: = 0 were used andfor the configurations selected the spin part of the operator does notcontribute (see text).

Parameter 136Xe138BalO4Hg

X [MeV]

0.005580.003730.003793D. [MeV]

- 0.088716- 0.104308- 0.05116D. [MeV]

0.00.0- 0.01921G. [MeV]

0.159710.152620.10755G. [MeV]

0.00.00.09559a [MeV]

- 0.060060.053170.05398b [MeV]

0.0129150.007518- 0.010762

e.11 [eJ

1.271.562.34

parameter values which are listed in Table 1 to indepen­dent estimates:

• The quadrupole-quadrupole interaction strength para­meter X is expected to behave roughly like x­242A-s/3MeY [12J or X-119A-s/3MeY [13]. ForA = 138 this yields, respectively, 66 keY and 32 keYwhich are about an order of magnitude larger than theparameter values for 136Xe and 138Ba obtained

through the best-fit calculations (see Table 1). On theother hand, the pseudo-SU(3) model value is very sim­ilar to those obtained in pseudo-symplectic calculations[14J. For A = 204 the corresponding numbers are34 keY and 17 keY which is closer to the value used in

the pseudo-SU(3) calculations presented here.• In the Nilsson model the parameter D" and D. multi­

plying the squared proton and neutron orbital angularmomentum operators are roughly equal to- 0.30 MeY and - 0.19 MeY in the rare earth region

[17J which is not far from the parameter values of thepseudo-$U(3) best-fit calculations.

• The proton and neutron pairing strengths G" andG. are expected to decrease like ciA [15J where theconstants of proportionality c" (c.) are assumed to beapproximately 17 MeY (23 MeV) in reference [16J and27 MeY (22 MeV) in reference [12]. For the A = 138case these estimates yield respectively 0.123 MeY and0.196 MeY for the values of G" so the best-fit values for136Xe and 138Ba are in the expected range. For thel04Hg case the proton (neutron) pairing strength esti­mates are 0.083 MeY (0.113 MeV) and 0.132 MeY(0.108 MeY), respectively, which is very close to thevalues obtained in the best-fit calculation.

Although these comparisons should not be taken too

seriously, for example because the estimates for X assumestrong deformation and neither of the A = 138 cases nor

the A = 204 system can be so characterized, the approx­imate agreement between parameters obtained from gen­eral considerations and those obtained in best-fit pseudo­SU(3) model calculations demonstrates its consistencywith other theories, thereby suggesting that the pseudo­$U(3) model is built on a correct physical picture.

Before discussing comparisons of results for 136Xe,1J8Ba, and 204Hg to experimental data and other theories,it is important to write down definitions for transitionoperators and other quantities. This is done in the nextsection.

2.1 Transition operators and observables

The electric quadrupole transition operator is defined as

(3)

where the notation of references [3, 7J is employed. (Notethat in the applications that follow the effective neutron

charge e. is set to zero.) Likewise, the magnetic dipoleoperator can be shown to be given by

T11l(Ml) == PN(g:L~1l + g~L~1l + g~S~1l+ g~S~Il) (4)

where PN denotes the nuclear magneton, and g: (g~) andLrll (S~Il) are, respectively, the orbital (spin) gyro-magneticfactor and orbital (spin) angular momentum for protons(0" = 1<) and neutrons (0" = v) .

The SO(3) reduced matrix elements of a tensor oper­ator TJM between states of initial (final) angular mo­mentum and projection Ji and Mi (Jf and Mf) is defined by[20J

<ijJfIITJllyJ;)<J)v!i, J MIJfMf) == <YfJfMfITml;'jJ)vI;)

where Yi and Yf represent additional quantum numbersthat are required to uniquely define the initial and finalstates, and <'fJflIT JIIYJi) stands for the reduced matrixelement. It follows from this, see also [7, 19J, that thereduced transition probability for electric quadrupoleradiation is given by

and that for magnetic dipole radiation by

The definition of the electric quadrupole moment is givenby

ru;; J(2J - 1)Q(yJ) == ..)5-5 (J + 1)(2J + 3) (yJIIQ21IyJ)

and that or"the magnetic dipole moment by

from which the definition of the gyro-magnetic factor iseasily determined to be

0.0 0--------°- .. 0_-_----- 0Pseudo-SU(3) Exp. Wildenthal Baldrige

& Larson

,4--Z-- __._._2__.-.-2---.---- 2

6--------6 - ..------i- __. 6__ ._..._4_ .. --.-- 4

, 4

2

1.2

1.6

3.2

~>~6 2.4

Fig. 1. From left to right the figure depicts the excitation spectrumof 136Xe as calculated within the pseudo-SU(3) model, the experi­mental values, results of the calculation by Wildenthal and Larson,and, on the very right, the energies as obtained in the model used byBaldrige. (The thin dashed lines are provided to guide the eye inconnecting the levels.) Note that in order to obtain a better agree­ment to experimental data the spectra of Baldrige (60 keY) andWildenthal and Larson (70 keY)were shifted downwards. In addi­tion, so more details in the upper spectra can be shown, the spacingsbetween the ground and the first excited states are not to scale

pseudo-SU(3) model shows reasonable agreement withthe experimental data, taking into account all the experi­mental levels up to about 3 MeV. Note that in addition tothe already established experimental levels, theory pre­dicts the existence of 0 +, 3 +, 5 +, and 6 + states in theenergy range between about 2.2 Me V and 3 MeV. Thecalculations by Wildenthal and Larson and by Baldrigealso seem to be in reasonable agreement with experi­mental data but only up though about 2.5 MeV. Neitherof these models account for the three 4 + states around

2.6 Me V or for the large number of 2 + states above2.65 MeV. A common feature of all three calculations is

the prediction of at least one low-lying 3 + state and of_a second 5+ state slightly below 3 MeV. Further materialfor an evaluation of the theoretical results is provided inTable 2 which lists experimental B(E2) values togetherwith the results of the pseudo-SU(3) model and the shell­model calculations by Wildenthal and Larson. The config­uration space of both of these models is restricted to singleshells which means that higher shell correlations are notincluded and therefore an effect like core polarization isnot taken into account. This is a common defect of almost

all shell-model theories (with the exception of the pseudo­symplectic shell model [14J ) and is usually compensatedfor by the use of effective charges in the calculation ofB(E2) values and quadrupole moments. In Table 2 theeffective charge of the Wilden thai and Larson calculationwas fixed in order to reproduce exactly the experimentalB(E2; 21 -jo 4d value. (Usually the B(E2; 01 -jo 2tl value is

3.1 136Xe

128

3 Discussion of results

As indicated above, in this contribution the pseudo-SU(3)model is used to investigate properties 9f 136Xe, 138Ba,and Z04Hg with the results compared to experimentalnumbers and other theoretical predictions. The 136Xe and138Ba cases are taken to be closed neutron-shell systems(N = 82 shell) with four and six valence protons, respec­tively, outside a closed proton Z = 50 core. The Z04Hgcase is characterized by two proton and two neutron holesin the N" = 4 and N. = 5 shells, respectively. Becausethese are all relatively simple configurations, it was pos­sible to carry out the pseudo-SU(3) model calculationswithout additional truncahon. Note, however, that theconfiguration space that was used is only composed ofstates with zero proton and neutron spin. This corres­ponds to selecting U(N) irreps with the highest spatial­symmetry (see [7J ) which means that all effects resultingfrom spin excitations are suppressed. This is not too crudean approximation because the spin-flip modes have beendetermined by experiment to lie at significantly higherenergies than most of the states considered in the presentanalysis.

As mentioned in the introduction, the nucleus 136Xe hasrecently received increased attention due to interest in the{J{J-decay problem [9J and in connection with new gR­

factor measurements [21]. Since this nucleus has a parti­cularly simple configuration space (four valence protonsand a closed neutron shell) it has also been the subject ofseveral other theoretical studies [22, 23, 24, 25J and, inparticular, the present pseudo-SU(3) model calculationswill be compared with corresponding results obtained byBaldrige [26J and by Wildenthal and Larson [27].

Baldrige performed a so-called semi-realistic shell­model calculation by employing a realistic Bruckner reac­tion matrix as the lowest order Hamiltonian and then

corrected for neglected effects through the addition ofphenomenological pairing and multipole interactions[26]. The strength parameters of these added terms wereadjusted to fit the energy spectrum of 13~e and withoutfurther adjustments this effective interaction was used forthe calculation of the excitation energies of 136Xe. Theother calculation that the pseudo-SU(3) model results willbe compared with was performed by Wilden thai andLarson who adjusted the strength of a modified surface~ interaction and the single-particle energies of the con­ventional shell model to reproduce the experimental dataof the (A = 136 - 140) nuclei in a truncated Og7/Z - Ids/z

configuration space [27].The experimental excitation energies of 136Xe (below

3.2 MeV) are shown in Fig. 1 in comparison to the resultsobtained by Baldrige (-jo "large space" calculation in[26]), the calculations by Wilden thai and Larson [27J andthe pseudo-SU(3) model. (Note that in order to obtainbetter agreement to experimental data the spectra of Bal­drige and Wilden thai and Larson were shifted downwardby 60 ke V and 70 ke V, respectively (see ground state en­ergy level in Fig. 1).) Except for the first 4 + state, the

Table 2. Comparison of experimental andtheoretical B(E2, II - I f) values of 136Xe inunits [e2fm~]; The first column gives theinitial and final values of the angularmomentum, II and If; the second columnlists the experimental values; the results ofthe pseudo-SU(3) model calculations aredenoted in the third column; and the fourthcolumn contains the theoretical data as

derived from the calculations byWilden thai and Larson (see text). (In orderto get a better agreement to experimentaldata, an effective charge of e:f f = 2.40 e wasused to calculate the results in the fourthcolumn whereas Wilden thai and Larson for

reasons of simplicity originalIy usede:JJ = l.Oe in Reference [27}.)

Ii - If

01 -2121-414~-61"61 - 81

81 -10121-2221 -4221 -4322 -4122 - 42

41-4241-4)41-5141-62

1850 ± 850, 3462 ± 61193.5 ± 4.5 .0.794 ± 0.03

102913427

100236269

2333338190.16

7310560

B(E2)n [e2fmJ.]

352193.513.3

675

91.8

used but here that is not done because its appears there isconsiderable uncertainty in its value: first, the error bar onthe value reported by [39J is rather large, and second,Speidel, et aI. [21J recently reported a value nearly twiceas large as that reported previously.)

The data listed in Table 2 show that the pseudo-SU(3)model and the shell-model calculation of Wilden thaI and

Larson give to within roughly a factor of two the sameresults and both models overestimate the experimentalB(E2; 41 -+ 6d by more than an order of magnitude. Inconsidering these results, it is important to note thatfurther work seems to be required to make a better deter­mination of B(E2; 01 -+ 2d. Further predictions of quad­rupole moments that were calculated within the frame­work of the pseudo-SU(3) model are listed in Table 3.

Finally it is important to consider how the magneticproperties of 136Xe and 138Ba are treated in the pseudo­SU(3) model and for this it is necessary to recall the expres­sion for the magnetic dipole operator in Eq. (4). Since theconfiguration space for the current implementation of thepseudo-SU(3) model is made up of states of maximal spacesymmetry, the proton and neutron configurations havezero spin so that matrix elements of the proton andneutron spin operators in Eq. (4), Sill and S~Il' vanish. Inaddition, since the N = 82 nuclei 136Xe and 138Ba have novalence neutrons, the matrix elements of the neutron or­bital angular momentum operator in Eq. (4), L~",vanishas well. In summary, for 136Xe and 138Ba the total angularmomentum equals the proton orbi'tal angular momentumand, therefore, the magnetic dipole operator is propor­tional to the total angular momentum, and consequently,diagonal in the basis space. As a consequence the pseudo­SU(3) model predicts vanishing Ml-transition probabilit­ies which to a large extend is supported by the availableexperimental data [39]. For 136Xe the B(E2; 6z -+ 61)> 3'10-4 W.u. value seems to be the only available data

point. The 138Ba case will be discussed below. In addition,the current version of the pseudo-SU(3) model predicts thegyro-magnetic factor to be constant and equal to g~ == 1while the available experimental data points [21] givegR(21) = 1.20 ± 0.25 [21] and gR(2d = 0.8 ± 0.15 [39Jwhich are therefore compatible with the theoretically de­termined number. The situation is of course different for

ZO.l.Hg because the proton and neutron orbital angular

Table 3. Prediction of quadrupole moments for 136Xein units of ebas obtained within the pseudo-SU(3) model. The column on the leftgives the angular momentum I, the energy sequence number :x,andthe parity It; and the column on the right lists the results of thepseudo-SU(3) model calculations, Q(I,)PSU(3)' in units of [eb].

Q(I:)PSU(31 [eb]

0.14- 0.17

0.200.06

-0.090.130.47

0.180.11

- 0.420.280.3!

-0.09-0.28

momenta both contribute to the matrix elements of the

magnetic dipole operator even though the spin sector stillyields vanishing matrix elements (see Sect. 3.3).

3.2 138Ba

The nucleus 138Ba has six valence protons in the LV = 4shell and a closed LV = 5 neutron shell. A review of the

experimental results for 138Ba can be found in [31] anda report on theoretical calculations that have been carriedout can be found in [28, 29, 30] and references therein. InFig. 2 the experimental excitation energies (center) belowabout 3 MeV are compared to the pseudo-SU(3) modelresults (left) and the predictions of Larson et aI. (right) asdetermined from a conventional shell-model approach[29J which is very similar to what was briefly described inSect. 3.1.

Both models show reasonable agreement with the ex­perimental data, although a final evaluation for the statesbetween about 2.7 and 3 MeV is still out due to experi­mental uncertainties in the determination of parity and

Fig. 2. The experimental excitation spectrum of 138Ba (center) incomparison to the results of the pseudo-SU(3) model (leJc) and theshell model used by Larson et al. (righc). The dashed lines in betweenthe level bars are provided to guide the eye between correspondinglevels

0.0 0---Pseudo-SU (3)

B(E2)pSU(3) [elfm"']

2415157

0.72482

125232891503

0.51347172

38

0.6810.10.1010.971

17091

0.2630

440

6.1

420 ± 51

> 15

> 7.2

> 0040> 801

40 ± 17

B(E2)e;r. [elfm"]

2415 ± 6421.87 ± 0.839

3.2 ± 0.4

10.0 ± 10.083.2 ± 12

01-2121 -4141-6161 - 81

81 - 101101-12101 -22O2 - 21

O2 - 22

21-2221-2321 - 3121 - 3221 -4221 - 43

22 - 41

31 -4131 - 42

41-4241-5151-6151-71

Ii -1/

Table 5. The B(E2) values of 138Ba ~ calculated within the pseudo.SU(3) model (- B(E2)pSt'13J are lIsted together with the experi­mental d,ata (- B(E2h:.~p)in units of [elfm"]. (See Table 2 for anexplanation of the notatIOn.)

---12 ----0Exp. Larson et al.

'---64 ---4······ 4

2 •• '-"'--- 2 "-"---

["('i';';i); ?:., .

212= ---3 ....•..'...=02650_ ••••••••••J:- '=,34 ====-=. (2.3',4)=455 -'--40' 2)43---~;'''''' ,2 --~.-.·---2 .. 404----:;:0:, •.---40,2)-- ..---12

322 '" •• '4 ---"'_ .---53.:::-::..-::...-:....-_-_-3

o ----.-.';:~. 40_.~.·. 52 ._.-~, 62::::_~240,2,3)6---"--"'---6' ..

3

..-.>Cl.l

6 2.5>,00•..Cl.l

ci3 2c::

.9B

.~ 1.5tI.l

130

Table 4. Comparison of experimental and theoretical excitationenergies for higher angular momentum yrast states in 138Ba. Thefirst row denotes the angular momentum I and the parity It, withexperimental uncertainties in these assignments indicated by paren·theses. The second and third row are respectively the experimentaland pseudo-SU(3) model energies while the last row lists the relativeerrors. (The experimental data are taken from Reference [39]).

r (7t)8t(91)lOtllt(l2t)

Ee;rp [MeV]

3.3603.1834.1583.622-4.690

Epsu(3) [MeV]

3.2413.2174.0763.5985.3254.512

EExp - Epsu(J)0.035

- 0.0110.0200.007 0.038-Epsu(3)

spin assignments. Note that the pseudo-SU(3) model doesnot account for the second excited 6+ state, predicting itto lie about 700 keV higher than the experimental level,while the calculation by Larson et al. describes this statevery nicely. On the other hand, there seems to be experi­mental evidence for a low-lying (1, 2, 3) level at 2.19 MeVwhich cannot be found in the calculation by Larson et al.

but which can be identified as a 2+ level in the pseudo­SU(3) model calculation.

In addition to the data shown in Fig. 2, in Table 4 theenergies of the experimentally known higher angular mo­mentum yrast states are compared with the pseudo-SU(3)model calculation. In spite of the very good agreement, itis necessary to be cautious since at these higher energiesadditional physical effects may occur which are not takeninto account in the present version of the pseudo-SU(3)model. The experimental B(E2) values are compared tothe pseudo-SU(3) model results in Table 5. The pseudo­SU(3) model results were generated with an effectivecharge of ee!! = 1.56 e as this results in exact agreementwith the experimental B(E2; Ot --+- 21) value. Consideringthe fact that the experimental results change by about3 orders of magnitude with the experimental error barsbecoming particularly large for higher excited states, tJ]epseudo-SU(3) model description does reasonably well.

Results for quadrupole moments of 138Baare given inTable 6. Experimental values are only available for thelowest 2+ state. Considering the large deviations in the

Table 6. Pseudo-SU(3) results for quadrupole moments of the 138Ba yrast states are listed in units of [eb] in the second row (- Qpsu(JJ andcompared to experimental data in the third row( - QE;rp)' The first row denotes the angular momentum I, the energy sequence number:l, andthe pari ty It.

IX It2t 3+4+5t6t7+8t9tlOt"11 1

Q(I:)PSU(3) [eb]

- 0.08- 0.12- 0.01- 0.26- 0.31-0.27-0.31- 0.31- 0.35- 0.39

± 0.08 (7) Q(I:)exp [eb]- 0.14 (7)

-0.11 (7)

"-----Z-·... __4_4---

'---4

2--_.__Z_. ". "'---2

fi-=:-i',3"

~z_ _.(z') i?s= , I.Z"3.: ZZ Z --- '---6'?" :JH 4'·'

36 0 """"="",,,,;~~:F· 1~2--;-'n :':':':'-...-",::'---I,ZZ,3' =610___ --=.i_2,3I --- --- Q~l.!'::U,1:., ---2

10..... " .•.-:,...•

o I ••••••••:. ••••~--O---~ "'-:'''===3ZZ--- ", -10,,'"....---0

2.5

)orted data, it is not possible to draw any conclusionsnceming the quality of the pseudo-SU(3) description.

Picking up with the discussion of magnetic propertiesit was' started at the end of Sect. 3.1, recall that the

factors of 13SBa are predicted to be constant. Theperimental data - which were taken from [39, 40J - areXP(21) = + 0.72 ± 0.11, g~.xP(4d = + 0.80 ± 0.14, andXP(61) = + 0.977 ± 0.02 which i! certainly compatibleth the prediction g~SU(3) == 1. In addition, note that theail able experimental data on the magnetic transitionobabilities B(Ml; 1, -+ If), are typically estimated to be) larger than 10-3 Weisskopf units, with one notableception, namely the B(M 1; 2 + -+ 3 +) = 0.3 W.u.:tnsition from the relatively high-lying 2 + state at640 Me V [39]. Again it seems that the experimental data'e comparable with a decription of magnetic properties; predicted by the current pseudo-SU(3) model imple­entation.

o 0------_--------_.-----_0Pseudo-SU(3) Experiment Covello

& Sartoris

here have been a number of experimental efforts aimed: understanding the structure of Z04Hg. More often thatot, these studies were stimulated by an observed changel the low-lying structure of the mercury isotopes at ZOOHg

32, 33, 34, 35J. However, the high level of experimentalttention given Z04Hg does not seem to have stimulatedcorresponding amount of theoretical interest [34, 36,

7]. This is particularly surprising since it is experi­lentally difficult to produce proton-deficient nuclei nearle doubly closed shell nucleus zosPb [37J while the smallumber of proton and/or neutron holes makes this regioneadily accessible to (large-scale) shell-model calculations.ls mentioned in the introduction, there is some hope thathis situation will change in view of the current interest in1f3-decay in this region., The experimental energy spectrum of Z04Hg is shownn the center of Fig. 3. Also shown are the correspondingesults obtained with the pseudo-SU(3) model (left) and inalculations of Covello and Sartoris (right) [36]. Theluthors of the latter publication used a model in whichwo interacting proton holes are coupled to a sphericaliquid-drop type core that is allowed to performquadru­)ole surface vibnitions. Since neutron holes are not ex­

)licitly a part of the configuration space, the model shouldlot be expected to describe excitation energy structureslominated by single-particle features which are supposedo appear aboR about 1.5 MeV. In view of this and thenodel's simplicity it is surprising how well it accounts for:he energies of the low-lying states and the corresponding:{uadrupole moments and B(E2) values (see below). Thislolds not only for Z04Hg but also for the neighboring,sotopes 19S-202Hg (see [34, 36J) which indicates the;trong collective character of the energetically lowest>tates in the neutron rich mercury isotopes [32].

At higher energies the experimental energy spectrum ismore complicated than predicted by the model of Covelloand Sartoris. The pseudo-SU(3) model seems, on the otherhand, to account for virtually all of the experimental levelsand in addition predicts a few new ones that remainundetected: a second excited 0 + state at about 1800 ke V

Fig. 3. The experimental energy spectrum of IO"Hg (center) is com­pared with the results of the pseudo-SU(3) model (lefc) and calcu­lations made by Covello and Sartoris (right). Levels which wereexperimentally identified to possess negative parity are not shownand cases where the available experimental information does notallow any parity assignment are indicated by an asterisk. In all othercases the parity of the indicated experimental level is positive. Theangular momentum assignment of the experimental levels generallyfollows the assignment from [39] with the exception of the level at1851 ke V where the experimental data seem to allow a J = 3 stateThis assumption is supported by the fact that the results of boththeories predict a low-lying J = 3 state of positive parity. Finally,since a number of the experimental spin and parity assignments arenot absolutely sure, the dashed lines between the pseudo-SU(3) levelsand the experimental energies which are given to guide the eyeshould be considered tentative

and a 5 + state at about 2300 keV. The available experi­mental information is insufficient to establish a completepicture above about 1.5 MeV because the parity and/orthe angular momentum assignment for a number of levelsis uncertain. It is for this reason that the dashed linesbetween corresponding levels, which are intended to guidethe eye, are somewhat tentative between the higher energylevels predicted by the pseudo-SU(3) model and the ex-perimental data. -

The experimental yrast B(E2, I -+ 1+ 2) values areshown in Fig. 4 together with corresponding pseudo­SU(3) results and the results obtained by Covello andSartoris and in the shell-model studies carried out byRydstrom et al.

The latter authors fitted empirical interaction matrixelements to three A = 206-nuclei and used this Hamil­

tonian to make shell model predictions for several lighternuclei among which was Z04Hg [37]. Both the pseudo­SU(3) model and the shell-model calculation of Rydstrom,et al. seem to describe the, experimental data very well,although final conclusions can only be drawn once moreexperimental data points become available. The goodagreement of the calculation of Rydstrom et al. and the

132

Yrast

B(E2; I -> I+2)-Values for 204Hg

0.5•

Experiment~

XPseudo-SU (3)0.4

.0-Covello-& Sartoris•.......•

aN aRydstrom et al..0 N <U'--'..-.,

0.3<'l +-/\ i

·1

I 0.2- a<'l CI.l X'-'

a:l 0.1 ~ .0-a

X0

. .n¥

0->2

2->44->66->88->10 10->12Transition Angular Momenta

Table 7. Comparison of selected B(E2, Ii ....• I f) value predictions forthe nucleus 20oloHgin units of e2b2: The first column lists the initial

and final ,values of the angular momentum, Ii and I/> the secondcolumn gives the results of the pseudo-SU(3) model calculationsB(E2)psu(3) and the third column contains the results obtained by th~model of Covello and Sartoris, B(E2)cs (see text).

Ii-+ I, B(E2)psu(3) [e2b2)B(E2)cs [e2b2)

01 -+ 2z

0.074 0.002501 -.2)

0.049

Oz -.21

0.023 0.26Oz-. 22

0.057 0.0721 -+ 2z

0.053 0.02721-2)

0.002

21-42

0.043

22 -41

0.021 0.012z -.42

0.114

41 -.4z0.003

4 Summary and outlook

An extended version of the pseudo-SU(3) model has beenused to describe the positive parity low-energy propertiesof the rare-earth nuclei 136Xe, 138Ba, and 204Hg. Theinteraction that was used is composed of a many-particle

Table 8. Comparison of selected quadrupole moments for the nu­cleus 20oloHgin units of [eb): The first column lists the angularmomentum I, the parity It, and the energy sequence number z; thesecond column denotes the experimental value; the third columncontains the result of the pseudo-SU(3) model calculation; and thefourth column presents the results obtained by the model of Covelloand Sartoris.

+0.26

Q(l.)cs [ebJ

+0.29- 0.21

Q(l,JPSU()l [ebJ

+0.62+ 0.41+0.35+0.32+ 0.01+ 0.74+0.64+ 0.47+0.76+0.63

Q(l,JEXP [ebJ

+ 0.40 ± 0.20

values of the orbital gyro-magnetic factors were set asfollows: g: == 1 and g: == 0 for proton and neutrons, re­spectively. Table 9 lists the pseudo-SU(3) model gR valuesin comparison to the results of the model used by Covelloand Sartoris and one experimental data point which isslightly smaller than both calculated values. In general thegR values of the pseudo-SU(3) model are somewhat small­er in magnitude than the predictions of the model used byCovello and Sartoris.

To complete the analysis of the magnetic properties of204Hg, in Table 10 the pseudo-SU(3) predictions for re­duced magnetic dipole transition are given where in thedefinition of the transition operators the same values forg: and g: as above were employed.

Fig. 4. The experimental B(E2; I -. I + 2) values for yrast bandstates are shown ( -. full diamonds) in comparison to results obtainedwith the pseudo-SU(3) model ( -. crosses), calculations of Covelloand Sartoris ( -+ triangles), and theoretical data published by Ryd­strom et al. (-. squares). The abscissa indicates the yrast stateangular momenta for each transition. In comparison to the results ofRydstrom et al. the pseudo-SU(3) data indicate a decrease of collec­tivity at higher angular momenta while the results of both are inexcellent agreement with the experiment

experimental B(E2) values is somewhat surprising as theircalculated 204Hg excitation spectrum does not seem verycompetitive [37]. The B(E2) results from the work ofCovello and Sartoris in Fig. 4 are difficult to assess sinceonly two numbers are available [36J. Additional pseudo­SU(3) model predictions for B(E2) values can be found inTable 7.

Table 8 lists the pseudo-SU(3) quadrupole momentsfor 204Hg in comparison to the results obtained by themodel of Covello and Sartoris and one experimental valuewith a relatively large error bar.

In general the results of Covello and Sartoris for thelow-lying states are smaller in absolute magnitude thanthe pseudo-SU(3) model predictions. Note that thepseudo-SU(3) model predicts a positive sign, indicatingoblate nuclear deformation, for all the low-lying states.(Predictions for quadrupole moments of states of higherenergies are given for completeness and future reference.)In contrast with this, the results of Covello and Sartorisindicate that the second 2 + state is prolate.

To conclude this section, consider the magnetic prop­erties of 204Hg. The expression for the magnetic dipoleoperator as used within the pseudo-SU(3) model is givenin Eq. (4). As mentioned in Sect. 5, the configuration spaceof the pseudo-SU(3) model version presented in this con­tribution is restricted to S,. = Sy = S = 0 states whichmeans that the contributions of the spin sectQrin T 11'(M1)vanish identically. However, contrary to the situation for136Xeand 138Ba,in this case the neutron orbital angularmomentum contributes to the matrix elements of themagnetic dipole operator. In order to simplify matters andto reduce the number of parameters .in the theory, the

Table 10. Pseudo-SU(3) predictions of B(M1) values for selectedtransitions in the low-lying positive-parity spectrum of 204Hg inunits of the square of the nuclear magneton, ()l1f)2. The left columndenotes the initial and final values of the angular momentum, Ii andII and the right column lists the reduced magnetic dipole transitionprobability, B(M1)I'SUI3), as obtained within the pseudo-SU(3)model.

133

1. R.D. Ratna-Raju, J.P. Draayer, K.T. Hecht: NucI. Phys. A202(1973) 433.

2. J.P. Draayer, K.J. Weeks: Ann. Phys. 156 (1984) 41.3. O. Castaiios, J.P. Draayer, Y. Leschber. Ann. Phys. 180 (1987)

290.

plus one for the effective proton charge. (No effective

neutron charge was used nor were the parameters enteringthe magnetic dipole operator considerd to be adjustable.)

Calculated excitation energies, B(E2) values, quadru­pole moments and ~R factors were .compared to experi­mental data and vanous other theones (semi-realistic andconventional shell-model theories as well as a particle­plus-spherical core model). The quality in the pseudo­SU(3) model description of the experimental data wasfound to be at least competitive with any of the other

theories although improvements for the pseudo-SU(3)model are certainly possible if the number of parameters isincreased and/or if slightly different interaction terms areused.

In view of anticipated future developments of thepseudo-SU(3) model, the results presented in this contri­bution serve several purposes: .• The work presented in this contribution as well as in

very recent pairing correlation studies focussed oneven-even systems and a logical continuation of theseinvestigations is the extension to odd-even and odd­odd nuclei. The mathematical formalism that is used in

this contribution contains all the necessary ingredientsfor such an extension, so the current study can be takenas the S = 0 limit of a theory that is expected to beapplicable to S :;C 0 nuclei as well. And in this regard, itis important to have S = 0 result for the various para­meters, as a starting point for the S :;C 0 cases.

• Although there have already been several attempts todescribe the nuclear [3[3-decay within the pseudo-SU(3)model, the Hamiltonian used in those calculations ishighly schematic since, for example, the pairing interac­tion is neglected [4]. The [3[3-decay investigations couldcertainly be improved by the inclusion of pairing cor­relations which would also imply that more than oneSU(3) irreducible representation is considered for eachnucleus. In this regard, the present studies of the nuclearproperties of the [3[3-decay candidates 136Xe and 204Heshould prove useful for future enhancements to those

. investigations.• The purpose of the ongoing studies of the pseudo-SU(3)

model is to obtain a deeper understanding of the dy­namics of this model so it can be utilized for futureapplications of its extension, the pseudo-symplecticmodel. This model is not restricted to single shells forprotons and neutrons but rather takes higher shellcorrelations into account. Large space calculationswithin the pseudo-symplectic model are computerwisevery costly and it seems expedient to look for possiblenumerical simplifications which in turn necessitatesstudies like those of this contribution which help pro­vide insight into the effects of the different interactionterms on the overall dynamics of nuclear systems.

References

B(M 1)I'SUI3) [()l,y)2]

0.400.800.140.010.0010.440.050.220.050.240.0010.78

I" gR,(IJEXf' CJ.t1f]g ",(IJI'SU(3) CJ.t1f]g",(IJcs CJ.t,v].II

0.5021

0.40 ± 0.1 0.530.6022

0.390.5423

0.9031

0.3741

0.410.5642

0.5651

0.3761

0.3481

0.36

able 9. Comparison of selected gyro-magnetic factors for the nu­cleus 204Hg in units of the nuclear magneton, Il,y: The first columngives the angular momentum I and the energy sequence number 1:;

the second column denotes the experimental value, gR,(IJEX':' thethird column gives the result of the pseudo-SU(3) model calculation,g",(IJI'SU()); and the fourth column gives the resolts obtained usingthe model of Covello and Sartoris, g",(IJcs.

extension of the Nilsson Hamiltonian and residual inter­actions. The latter consists of three terms: the squaredoperators of the total angular momentum, its intrinsic:-axis projection, and a pairing term. While the first twoterms were previously found to be of importance in de­scribing certain band-splitting features in excitationspectra of heavy deformed nuclei, the pairing is a newfeature incorporated into applications of the pseudo­SU(3) model.

The strength parameters of these various terms in theinteraction were determined in a best-fit procedure aimedat an optimal description of experimental excitation ener­gies, B(E2) values and quadrupole moments. The valuesdetermined in this way were found to be compatible withindependent phenomenological parameter estimates. Thenumber of parameters entering the calculation for 136Xeand 138Ba equals six: five for the interaction plus one forthe effective charge. Since the neutron degree of freedomhad to be considered for Z04.Hg, the number of parametersin that case was larger (eight): seven to fix the Hamiltonian

01 .••• IIO2 .••• 11

11 ...• 21

11 ...• 22

I II .••• 23; 21 .••• 22

21 .••• 2321 .••• 31

22 .••• 31

, 31 .••• 4131 .••• 42

,41-42

134

4. O. Castaiios, J.G. Hirsch, O. Civitarese, P.O. Hess: ~ucf. Phys.A571 (1994) 276; J.G. Hirsch, O. Castaiios, P.O. Hess: Nucl.Phys. A in press.

5. C. Bahri, J.P. Draayer: Compo Phys. Comm. 83 (1994) 59.6. D. Troltenier, C. Bahri, J.P. Draayer: Generalized Pseudo-SU(3)

model and Pairing, Nucl. Phys. A586 (1995) 517. D. Troltenier, C. Bahri, J.P. Draayer: Effects of Pairing in the

Pseudo-SU(3) model, Nucl. Phys. A589 (1995) 75.8. T. Tomoda: Rep. Prog. Phys. 54 (1991) 53 and references therein.9. J.-L. Vuilleumier, F. Boehm, J. Busto, K. Gabathuler, H. Hen­

rikson, D. Imel, V.J. Jergens, L.W. Mitchell, M. Treichel, J.-c.Vuilleumier, H. Wong: Nucl. Phys. B31 (Proc. Suppl.) (1993) 80.

10. S.G. Nilsson: Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) No.1 6.11. H.A. Naqvi, J.P. Draayer: Nucl. Phys. A516 (1990) 351; H.A.

Naqvi, J.P. Draayer. !,+ucl. Phys; A536 (1992) 297.12. M. Baranger, K. Kumar: Nucl. Phys. AllO (1968) 490.13. DJ. Rowe: Phys. Rev. 162 (1967) 866.14. D. Troltenier, J.P. Draayer, P.O. Hess, O. Castaiios: Nucl. Phys.

576 (1994) 351.15. D.R. Bes, R.A. Sorensen: Adv. Nucl. Phys. 2 (1969) 129.16. R.F. Casten In: Nuclear Structure from a Simple Perspective

(Oxford University Press, 1990).17. P. Ring, P. Schuck: In: The Nuclear Many-Body Problem

(Springer-Verlag, New York, 1980).18. D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii: In: Quan­

tum Theory of Angular Momentum (World Scientific, Sin­gapore, 1988).

19. J. Eisenberg, W. Greiner: In: Nuclear Theory, Vol. 1, NuclearModels (North-Holland Physics Publishing, 1987).

20. M.E. Rose: In: Elementary Theory of Angular Momentum(Wiley, New York, 1957).

21. K.H. Speidel, H. Busch, S. Kremeyer, U. Knopp, J. Cub, M.Bussas, W. Karle, K. Freitag, U. Grabowy, J. Gerber: Nucl.Pbys. A552 (1993) 140.

22. R.J. Lombard: Nucl. Phys. A1l7 (1968) 365.23. K. Heyde, M. Waroquier, G. Vanden Berghe: Phys. Lell. 35B

(1971) 211.

24. O. Scholten, H. Kruse: Phys. Lett. 125B (1983) 113.25. F .. Monti, G. Bonsignori, M. Savoia, Y.K. Gambhir. Nuovo

Clmento Phys. Vol. 104.-\ (1991) 3126. W.J. Baldrige: Phys. Rev. C18 (1978) 530.

27. B.H. WildenthaI, D. Larson: Phys. Lell. 37B (1971) 266.28. M. Waroquier, K. Heyde: Nucl. Phys. A164 (1971) 113.

29. D. Larson, S.M. Austin. B.I:!. Wildent~al: Phys. Rev. C9 (1974)1574; D. Larson, S.M. AuStlh, B.H. Wtldenthal: Phys. Rev. Cll(1975) 1638.

30. G.M. Ewart, N. de Takacsy: Phys. Rev. C17 (1978) 303.31. 1. Dioszegi, A. Veres, W. Enghardt, H. Prade: J. Phys. GI0 (1984)

969.

32. Y.K. Agarwal, C. Gunther, K. Hardt, P. SchUler, J. Stachel, H.J.Wollersheim, H. Emling, E. Grosse, R. Kulessa, W. Spreng: Z.Phys. A320 (1985) 295.

33. A. Bokisch, K. Bharuth-Ram, A.M. Kleinfeld, K.P. Lieb:Z. Phys. A289 (1979) 231; M.T. Esat, M.P. Fewell, R.H. Spear,T.H. Zabel, A.M. Baxter, S. Hinds: Nucl. Phys. A362 (1981) 227'D.A. Craig, H.W. Taylor: J. Phys. GI0 (1984) 1133; W.R. Keb{J. Billowes, J. Burde, M.A. Grace, A. Pakou: Nucl. Phys A448(1986) 123;

34. R.A. Gatenby, E.W. Kleppinger, S.W. Yates: Nucl. Phys A492(1989) 45. .

35. M. Vergnes, G. Berrier-Ronsin, G. Rotbard, J. Skalski, W.Nazarewicz: Nucl. Phys A514 (1990) 381.

36. A. Covello, G. Sartoris, Nucl. Phys AI04 (1967) 104; A. Covello. G. Sartoris: Nucl. Phys A149 (1970) 41. '

37, L. Rydstrem, J. Blomqvist, R.J. Liotta, C. Pomar: Nucl. Phys., A512 (1990) 217.

38. Y. Leschber, J.P. Draayer: Phys. Lett. B190 (1987) 1; O. Casta­iios, J.P. Draayer, Y. Leschber: Z. Phys. A329 (1988) 33.

39. Nuclear Data Sheets, ed. by National Nuclear Data Center,Brookhaven National Laboratory, Upton, NY 11973, USA.Updated experimental data were obtained via e-mail atBNLND2.DNE.BNL.GOV.

40. P. Rhagavan: Atomic Data, Nuclear Data Tables 42 (1989)189.