Revúta Mexicana de Fí3ica 40, Suplemento 1 (199.•) -17-62

Pseudo SU(3) model and abnormal parity states*

JORGE G. HIRSCHDepartamento de Física

Centro de Investigación y Estudios Avanzados del IPNApartado postal 14-740, 07000 México, D.F., México

OCTAVIO CASTAÑOS AND PETER O. HESSInstituto de Ciencias Nucleares, Universidad Nacional Autónoma de México

Apartado postal 70-543, 04510 México, D.F., MéxicoReceived 6 January 1994; accepted 11 March 1994

ABSTRACT.The most important features of the pseudo SU(3) model are reviewed. This descriptionof heavy deformed nuclei is based on pseudo spin symmetry, and is able to correctly predict thecollective spectra and their transition amplitudes, using appropriate etrective charges. It also givesa very good description of the exotic double beta decay, a hard test of any nuclear model. Thesimplest pseudo SU(3) assumptions describe nucleons in the abnormal parity states as havingseniority zero. This idea is very useful as a mathematical simplification, and is also able to givea nice description of the backbending phenomena, but has been challenged in a recent studyabout etrective charges and BE2(Ot --+ 2t) systematics. We carefully checked this statement andconclude that the seniority zero approach is a formal way of assign abnormal parity nucleons apassive role. The contribution of the abnormal parity nucleons to the nuclear quadrupole momentsis finite but only a fraction of their asymptotic Nilsson values.

RESUMEN.Se revisan los aspectos más importantes del modelo pseudo SU(3). Esta descripciónde los núcleos pesados deformados está basada en la simetría de pseudo-espín y es capaz de pre-decir correctamente los espectros colectivos y sus amplitudes de transición usando cargas efectivasapropiadas. También provee una muy buena descripción del exótico decaimiento beta doble, unadifícil prueba para cualquier modelo nuclear. El modelo pseudo SU(3) más sencillo describe a losnucleones en los estados de paridad anormal teniendo antigüedad cero. Esta es una simplificaciónmatemática muy útil y puede dar una aceptable descripción del "backbending", pero recientes es-tudios sobra cargas efectivas e intensidades BE2(Ot --+ 2iJ la han cuestionado. Nosotros revisamoseste postulado y concluimos que la aproximación de antigüedad cero es una manera formal de asig-nar a los nucleones con paridad anormal un rol pasivo. La contribución de estos nucleones a losmomentos cuadrupolares nucleares es finita, pero solo una fracción de sus valores asintóticos.

PACS: 21.60.Fw; 21.60.Cs

1. INTRODUCTION

The pseudo SU(3) model has been developed during the last two decades [1-41. Its mainaim is to extend the successful application of the SU(3) symmetry from light to heavy

• Work supported in part by Conacyt under project 1570-E9208.

47

48 JORGE G. HIRSCH ET AL.

deformed nuelei. This possibility is based in the existence of the pseudo spin symme-try [5]. Based on this concept, it was possible to construct a transformation from thereal deformed single partiele states, where the spin-orbit interaction completely destroysthe three dimensional harmonic oscillator structure of the single partiele spectra presentin light nuelei, to the pseudo SU(3) states, where the "pseudo" spin orbit interaction isnegligible, and the underlying SU(3) symmetry is recovered for normal parity states.

Having this new SU(3) symmetry, it has been found that many important features ofheavy deformed nuelei can be reproduced. For example, the spectra and E2, MI, M3transition amplitudes for sorne rare earth and actinides were correctly predicted usingappropriate effective charges [4,61. AIso the exotic double beta decay half lives of sornenuelei were predicted with notably accuracy [7j.

These large capabilities of the pseudo SU(3) model lead us to the conelusion that thepseudo SU(3) wave functions have a significant overlap with the exact wave functionsof heavy deformed nuelei, or at least are the most important part in the phenomenadescribed with the mode!. Core polarisation, as in shell model calculations, is simulatedusing effective charges.

Single partiele wave functions, both in deformed Nilsson or Woods Saxon potentials,exhibit the intruder states of abnormal parity as nearly unmixed for normal deformations(f3 ::; 0.3) [8,9]. In this limited single-j subspace pairing strongly competes with thequadrupole-quadrupole interaction. These unique parity nueleons were assumed to haveseniority zero, providing a framework in which the whole wave function is of the SU(3)type, an assumption that greatly simplifies calculations.

A comparison of effective charges needed in the pseudo SU(3) and the Single ShellAsymptotic Nilsson Model (SSANM) exhibits the limitations of the seniority zero ap-proach [lOj. It shows that effective charges must be renormalized by nearly 25% to com-pensate for the lack of quadrupole moment associated with the nueleons in the abnormalparity states.

In this paper we examine again sorne relevant characteristics and predictions of thepseudo SU(3) mode!. In particular, a possible description of the backbending phenom-ena based on the seniority zero approach is reviewed [21. We present a discussion of thequadrupole moments and BE2j == BE2(Oi --+ 2i) intensities in the pseudo SU(3) andSSANM models. We confirm the results found in [10], that the quadrupole moments inthe normal parity sector are the same in both models. We also discuss several predictionsfor the quadrupole moments of the abnormal parity nueleons.

2. THE PSEUDOSU(3) MODEL

A tractable shell model theory of heavy deformed nuelei requires asevere truncation ofthe spherical model basis. Oí course the goal of truncation, to reproduce the essentialphysics found in low-lying sta tes of a larger space in a smaller one, can only be achievedif the basis selection is made relative to those parts of the interaction that dominate thelow-energy structure [4].

The organization of the basis states relies on uncovering an SU(3) symmetry in the struc-ture of higher major shells, where SU(3) is the symmetry group of the three-dimensional

PSEUDO SU(3) MODEL AND ABNORMAL PARITY STATES 49

harmonic oscillator (HO). Since for lower ds-shell nuclei the nuclear shell structure is notmuch different from that of the HO, SU(3) was proposed by Elliott [11] as a reasonableds-shell symmetry, a one that could be used to truncate the full space(103-S states) downto tractable size (101-Z states). Since Q. Q is dominant and Q. Q = 4Cz - 3L. L, whereCz is the second order Casimir invariant of SU(3), irreducible representations (irreps) ofSU(3) which have the largest values of Cz should dominate the structure of low-Iyingstates. In nuclear physics SU(3) irreps are labeled hy (.\,1') [4] and

(CZ) = (.\+ Id 3)(.\ + 1') - .\1' . (1)

Basis sta tes belonging to this "leading" irrep of SU(3) are those which have the largestintrinsic qlladrupole deformation, (Qo) "" (Q . Q) 1/2 Asevere truncation scheme wouldrestrict hasis sta tes to the leading irrep. Full space ds-shell calculations have confirmedthat the leading irreps do indeed comprise 60-80% of the yrast (lowest state of a givenspin) eigenstates.For the higher major shells the spin orbit and centrifugal streching perturbations com-

pletely destroy the HO shell structure. In all cases, however, there are new major shells.These new major shells are comprised of all the j subshells of the corresponding HO majorshell except for the largest j subshell which is pushed into the next lower major shell. Theremaining j subshells are grouped together and called the normal (N) parity orbitals ofthe new shel!' In addition to these, there is the highest j subshell from the next higher HOshell. This "intruder" level has the parity opposite to the other levels in the major shelland is called the abnormal (A) (or unique) parity leve!. As an example, consider the r¡= 5major shell of the HO. It has the degenerate subshells hll/Z, h9/Z, h/z, 1s/z, P3/Z' PI/Z'

The corresponding nuclear major shell, however, has the subshells h9/Z, h/z, 1s/z, P3/Z'

PI/Z and the abnormal parity ¡13/Z level with the splittings shown in Fig. 1 (deformationf = O) for protons. This breakdown of the HO structure means that the "real" SU(3)symmetry is not expected to be good even though Q . Q remains a dominant interaction.Now consider the normal parity set of orbits; namely, the h9/Z, h/z, 1s/z, P3/Z and PI/Z

that come from the r¡ = 5 HO shell. Note that the j values (ignore 1 for the moment)are exactly the j values which appear in the '7 = 4 HO shell. This suggest that one canmap the 1 and s (preserve j) for these levels onto i and S, Le, j = 1+ s - i+ s = jsuch that h9/Z, h/z, 1s/z, pj/z, PI/Z - 99/Z,ih/z,ds/z,d3/Z,sl/z, This mapping schemegrew out of the studies of pseudo spin-orbit doublets [5]. lt has been used by Raju,Draayer and Hecht [11 in the discussion of the ground state magnetic moments of odd-A deformed nuclei and by Strottman [121 in a study of Ni-Cu-Zn isotopes. Hecht [13]has used the scheme to investigate decoupled negative parity spectra of the Au nucleiwhile Braunschweig and Hecht [141 used it in an analysis of the core deformation fornuclei in the proton rich Xe-Nd region. The scheme has also been used in attempts atproviding a microscopic shell-model interpretation of high-spin phenomena in Ge and Banuclei [3]. Using an effective interaction comprised of operators which form an integritybasis for the SU(3) - R(3) algebra was shown to be sufficient to reproduce almost exactly,within a single leading irreducible representation of SU(3), the ground and gamma bandrotational structure of eight rare earth and four actinide nuelei, reproducing accuratelythe interband and intraband E2 strength [41. A rigorous test of the theory was provided

50 JORGE G. HIRSCH ET AL.

PROTONS

>-oa::wzw

o., 0.4

FIGURE 1. Nilsson level scheme for the TI = S(;¡= 4) proton shell. At deformation f = O the levelsare labeled with spherical shell model qnantum numbers, while at f # O the asymptotic pseudoquantum numbers labels ñ[;¡ñ,A) are given.

by the prediction, in essentially the same nuclei, of a number of 1+ states with strong MItransitions to ground state, as well as E2 and M3 transition strengths, using the real MI,E2 and M3 operators [6J. lt was also possible to describe the two neutrino mode of thedouble beta decay in 150Nd and 238U in good agreement with the experimental values [7J.This latter prediction is far from trivial, given this exotic decay has challenged nuclearstructure descriptions for more than a decade.

Recently an analytic expression for the transformation that take us from the normalparity orbitals to the pseudo-space was introduced [ISI. By applying this transformationto the spherical Nilsson Hamiltonian it can be shown explicitly that the strength of thepseudo spin-orbit interaction is almost zero and the orbitals j = i:l: 1/2 are nearly degen-erate doublets [4,16,171. The relabeled levels form a major shell for the pseudo oscillatorpotentia1. The symmetry of this oscillator is, of course, pseudo SU(3) (the "pseudo" de-notes the pseudo-shell realization but the abstract algebra is just SU(3).) This "real"to "pseudo" shell mapping is a (restricted) unitary transformation and does not changeanything as far as exact full space results are concerned. A tensor decomposition of thereal Q. Q interaction into its pseudo SU(3) components shows that it is predominantlyQ . Q, Le. the quadrupole-quadrupole interaction in the pseudo shell. Also, although itis the spin-orbit interaction that destroys the HO shell structure, note the behavior ofthe Nilsson levels labeled by their ñ[;¡ñ,AI quantum numbers in Fig. 1. For deformationsappropriate to well-deformed nuclei, f "" 0.3 { wit_h f = 0.9SI1), one sees that the levelsare grouped into pseudo spin-orbit doublets (fl = A :l: 1/2). This implies that the pseudospin-orbit splitting is small, supporting the assumptions of the model, and the formalresults for the transformation [IS]. Similarly, the equivalent of the centrifugal streching

PSEUDO 5U(3) MODEL ANO ABNORMAL PARITY STATES 51

interaction which spreads the ivalues is less strong here than in the HO picture. Also notethat adding nueleons to the elosed shell tends to create an intrinsic state with the largestpossible total ñ, (for fixed deformation in prolate deformed nuelei, ñ, decreases as energyincreases) and hence the largest quadrupole momentum,

(2)

An additional complication for heavy nuelei is that the valence protons (1f) and neutrons(v) are filling different majors shells. Thus for a given nueleus there are two open shells,one for protons and one for neutrons, each comprised of a set of normal parity levels andthe associated abnormal parity leve!. Since the nuelear interaction is assumed to conserveparityas well as the numbers of protons and neutrons, one might be led to consider a basisbuilt by weak coupling configurations of the four separate spaces. However, in a previousstudy [161 it was found that if the residual interaction contains a significant Q •. Qv part,then coupling the leading proton pseudo SU(3) configuration to the leading neutron pseudoSU(3) configuration leads to an SU(3) scheme whose leading irreps dominate the low-lyingstructure. To this strong coupled space we couple weakly the remaining abnormal parityspaces.

3. TIIE ABNORMAL PAIUTY STATES

In the abnormal parity parts of the neutron and proton shells it was assumed in previouswork [4,6) that low seniority configurations were the most important ones. Thus, onlyseniority zero configurations were taken into account.

In order to understand this seniority zero postulate, we will review in some detailthe microscopical description of the backbending phenomenon. It has been explained byStephens and Simon [8,181 as an alignment of two neutrons in the intruder i13/2 shell. Ifthese two neutrons, instead of rotating around the 3-axis, align along the rotational axisof the nueleus, this adds and additional 13/2 + 11/2 = 12 units of angular momentum.Therefore, the nueleus can decrease its collective rotation while increasing its total angularmomentum through the addition of single-partiele angular momentum.

These are the basic ideas which support a microscopic look at backbending, in a workdone by Ratna Raju, Hecht, Chang allC1Draayer [21. The rotational core is naturallyprovided by the normal protons and neutrons strongly coupled in the pseudo SU(3) basis.The abnormal parity hll/2 orbital provides the seniority zero and two states, and theobserved backbending in 1263a is reproduced. In Fig. 2 their results are displayed. Theyselected the irrep ()" ji) = (24, O) for the 1,p pseudo-shell for both protons and neutrons.Its spectrum is shown in the right hand side of Fig. 2, and having been obtained for aHamiltonian with a surface delta interaction, it resembles very much a rigid rotor limit(pure Q . Q interaction). The first column of the far right hand side shows the spectrumobtained for the partieles in the pure subspace of the abnormal hll/2 level, with oneseniority zera state and five seniority two statcs, couplcd to 1 = 2,4,6,8,10, which are notdegenerated beca use of the presence of a Hamiltonian with a surface delta interaction in theabnormal sector. The Hilbert space is constructed as the direct product of the normal and

52 JORGE G. HIRSCH ET AL.

10

'~:8otO9

8

7

--"6

:;G --",.

5>-" 'oa:..,

4 ",Z.., --",, __ 'o

2 --'

----'.----'

O__ o

"

-

",

••_0-.'~, .',-

•,•'--

EXPERIMENT 1ST EXCT. YRAST 2ND EXCT. (f'p)18 (hll/Z)8CALCULATION (:i:¡¡)o(24,O) To4

T'3

FIGURE 2. Energy spectra for 126Ba. On the right hand side are spectra obtained diagonalizingseparately the interaction in the ji> (~,¡¡)= (24,0) and the hll/2, V = 0,2 spaces. In the centeris the composite result which indudes the effect of the interaction between the normal-parity andthe h" /2 nudeons.

abnormal subspaces, coupled to good angular momentum, and good isospin. An interactionHamiltonian able to mix the normal and abnormal spaces is introduced. The resultingspectrum of the camposite product space is shown in the middle, and in the left hand sideare the experimental results. The backbending of the yrast band between spins 1= 10-14is obvious to the trained eye. The relative intensities of the three interaction channels(normal, abnormal and mixed) were selected lo reproduce the experimental information.The mixing channel must be low enough to allow the backbending elfect to occur. lf themixing is too strong, the rotational and seniority two states would became mixed suchthat no sudden change in the ground band, i.e. backbending, can be seen.

This microscopical description of the backbending phenomenon provides a plausiblepicture of the role played by the particles in the abnormal parity states. But it mnstbe stressed that this is not the most usual description of the backbending phenomenon.Although the main ideas are the same as in other models, there exist a crucial dilference:the pairs of abnormal parity nucleons are described as seniority zero ones, i.e. they arecoupled to angular momentum zero in the spherical basis. The most common approach is to

PSEUDO 5U(3) MODEL ANO ABNORMAL PARITY STATES 53

fill Nilsson levels with pairs of nucleons with angular momentum projeclion zero, and thento project these states in order to obtain states with good total angular momentum [8,181.These pairs in the deformed basis could have partial occupation, being then describedas 'luasiparticles. 13ut even in this case the deformed zero 'luasiparticle state does notcoincide with the seniority zero one.

4. QUADRUPOLE MOMENTS

Now we give a more detailed look to the quadrupole transition strength. The reducedelectric 'luadrupole transition intensity BE2T is given in terms of the electric quadrupolemoment Q 1101 by

(3)

with

and

Q" = (2+ Mlq~IO+O)

= J161r/5 ¿(2+ Mlr~(i) Y2M{f,,(i))10+0), (}= 1r, V

(4)

(5)

where Q" represents the 'luadrupole transition amplitude, and in this case is essentiallythe same as the intrinsic 'luadrupole momento

In the aboye expressions the real 'luadrupole operators appear. In order to evaluatethem between pseudo 5U(3) states, they must be expanded in the pseudo 5U(3) basis,as explained in [6). In Table II of this reference the explicit expansion of the quadrupoleoperator in its pseudo 5U(3) components is shown, and it becomes evident that Q isproportional to (j, given the (A, /l) = (1,1), J( = 1, L = 2, S = O component has largelythe greatest component. 13ut there exist a numerical factor, associated with the fact thatthe pseudo shell has one phonon less than the real one, and, as can be checked from thistable Il, it can be expressed approximately by

Q _ TI" + 1Q- _ íi" + 2 Q-o •....... 0-_ o"

'1" '1"+ 1(6)

It is important to notice that the 10+) and 12+) states are strongly coupled states in thepseudo 5U(3) basis. They are labeled by their total (A,I'), their orbital angular momentumL with projection M, their spin S = O (which gives J = L), and an additional label J(distinguishillg tlle diffcrcllt rotatiollai bands. The final cxpression for Qo becomes

(7)

54 JORGE G. HIRSCH ET AL.

where, for the leading strong eoupled irrep

(8)

The final expressions to be evaluated are [4,61

(9)

and

t)(10)

The aboye expressions seem sligthly involved, but they are easy to evaluate given the{... }(9 - A/1 eoeffieient) and the (... , ... 11 ... ) (SU(3) redueed Clebseh-Gordan eoeffieient)are wen known quantities, and there are available eomputer eodes [191 whieh make theiruse nearly as easy as the more kn{lwn R(3) Clebseh-Gordan eoeffieients. The explieitexpression for C2 is given in Eq. (1). In Table 1 we present, for twelve heavy deformednuelei, their nnmber of protons and neutrons in the normal parity states (11~ and 11[:),their irreps (A., 1'.) and (Av,I'v), the quadrupole transition amplitudes Q. and Qv and,in the last eolumn, their approximate intrinsie values (in the deeoupled basis)

Q;nt 1)0 + 1(2' )o = -- Aa + /10 1 O' = 7T, V.1)0

(11)

From Table 1 it is elear that the intrinsie Q values represent a very good approximationto the exaet ones.

5. THE SINGLE SHELL ASYMPTOTIC NILSSON MonEL (SSANM)

This model was proposed in Re£. [201 and used extensively in [101. Its aim is to give asimple way to ealculate intrinsie quadrupole moments for nuelei in different shens. Itsansatz is that "a nuelens is as deformed as it can be in a single shen".

As an attempt to give some mieroseopieal support to the model, in Refs. [10,20J itis suggested that the behavior of the Nilsson energy leve!s as a funetion of deformation

PSEUDO SU(3) MODEL AND ABNORMAL PARITY STATES 55

TABLE l. Protan and neutron occupation numbers in normal spaces n': I n:;, dominant irreps(..\11"7 J.LlI")' (>'V! ILtJ), quadrupole moments Q1I"I Qv and intrinsic quadrupole moments Q~nt1 Q~nt fOIsame rare earth and actinide nuclei.

Nuclei nN nN ("•. 1'.) ("v,!,v) Q. Q~nt Qv Q~nt• v

154Sm 6 6 (12, O) (18,0) 31.5 30.0 45.3 43.2156Gd 8 6 (10,4) (18, O) 32.3 30.0 44.5 43.2160Dy 10 8 (10,4) (18,4) 31.9 30.0 44.5 43.2164Dy 10 10 (10,4) (20,4) 31.9 30.0 54.5 52.8166Yb 12 8 (12,0) (18,4) 30.9 30.0 50.7 48.016BYb 12 10 (12,0) (20,4) 30.9 30.0 55.5 52.8174Yb 12 14 (12, O) (20,6) 30.4 30.0 58.3 55.2232Th 4 10 (12,2) (30,4) 32.3 31.2 77.1 74.7234U 6 10 (18,0) (30,4) 44.2 43.2 77.2 74.723BU 6 12 (18,0) (36, O) 44.4 43.2 86.3 84.8240U 6 14 (18,0) (34,6) 43.9 43.2 89.1 86.3

shows that, beyond the deformation of f3 '" 0.15 (or, equivalently, , '" 0.14), the slopesof the energy curves for different Nilsson levels beco me almost constant for most of thelevels. The slope of the Nilsson energy curve is proportional to the quadrupole momentof the state. Hence, the constancy of this slope would imply that the quadrupole mo-ments of the Nilsson orbits change very little beyond f3 '" 0.15. Because most deformednuclei have larger deformations, it can be assumed that the quadrupole moments of thevaríous occupied Nilsson orbits would be approximately constant for different nuclei. Thechange in the intrinsic quadrupole moment from one nucleus to another is therefore dueto the different number of valence nucleons occupying the available orbits with the largestquadrupole moments. The spectrum of the quadrupole moments of the Nilsson states withlarge deformation is obtained by consideriug the single-particle sta tes in a majar shell tobe degenerate and then diagonalizing the qo operator in that space. Their eigenvalues arejust the mass quadrupole moments of these deformed single-particle states.For the normal parity states, we decided, instead of diagonalizing the aboye mentioned

matrix, to use the pseudo SU(3) formalism described aboye. We construct the states in thepseudo shell i¡ = '1 - 1, with asymptotic quantum numbers (i¡,ñ" A)O = A:!: 1/2 and theparity 7T = (-1)", and their intrinsic quadrupole moments are given by (2) and (6). Theresults are exhibited in Table 2. In the last column the q¡¡ values of Table 2 of Re£. [101are reproduced, and it is evident that they are very similar.We have demonstrated the assumption that the SSANM without the abnormal par-

ity states must give results analogs to the pseudo SU(3) to a very good approximation.In [10] it is explained as sorne kind of "redefinition" of the quadrupole operator, a misun-derstanding of what is simply a mathematical transformation, but the conclusion remainsvalidoIt remains to be discussed the role of the abnormal parity states. The pseudo SU(3) and

the SSANM have exactly the same assumption (the nucleus is as deformed as it can be)

56 JORGE G. HIRSCH ET AL.

TABLE2. The first column exhibits the Nilsson labels (Ñ,ñ"A)O". Quadrupole moments in thenormal space evaluated in the pseudo SU(3) and the SSANM are shown in the last two columns.

(Ñ,ñ"A)WSU(3) q~SANMqn

(3,3,O)! + 7.5 7.4

(3,2,I)r 3.7 3.8

(3,2,1)r 3.7 3.5

(3,1,2)~+ 0.0 0.2

(3,1,2)~+ 0.0 -0.2

(3,I,O)r 0.0 -0.2

1- 96 9.6(4,4,0)2(4,3,1)! - 6.0 6.2

3- 6.0 5.9(4,3,1)2

(4,2,2)r 2.4 2.7

(4,2,2)r 2.4 2.2

(4, 2,O)!- 2.4 2.2

and the same results in the normal parity sector. Is it correct to give a similar deformationto the nudeons in the abnormal parity part?

6. EFFECTIVE CHARGES

The distortion of the dosed shells by an added partide can be simply understood as a con-sequence of the nonspherical field generated by the extra partide. The order of magnitudeof the e/fect can be estimated by observing that the eccentricity of the density distribu-tion is of order A -1, and hence, the potential shoule! acquire a similar shape. The orbit ofeach proton in the dosed shell is thus slightly distorted and acquires an extra quadrupolemoment of the order A-1 Q,p ane! of the same sign as the mass quadrupole moment of thepolarizing partide. The total induced quadrupole moment is of the order [211

From this expression it is quite natural to infer

Zepal = A e

with

(12)

( 13)

.tI +e'll" = e epOl1 ( 14)

The effective charge eeff was introduced to re!lect the polarization e/fect of the coreo

PSEUDO 5U(3) MODEL AND ABNORMAL PARITY STATES 57

This is the approach selected in [10]' with and additional parameter ~ > 1 multiplingepol in the neutron case, to reproduce the accepted trend that the polarization charge issomewhat larger for a neutron than for a proton [21]. Their expressions are

Zeeff = ce-v 'A (15)

This parametrization of the effeCtive charges is not unique. For example, in their clas-sical study of spherical even-even illlclei using the pairing plus quadrupole hamiltonian,Kisslinger and 50rensen [231 adopted the parametrization ee: = 2.0 e and e~ff = 1.0 e, aswell as Ring and 5chuck did in their book ( [8]' pages 65 and 389), where a microscopicdescription of the effective charges is provided. In sorne cases, different sets of eeff areassociated with the same nuclei near closed shells (see Re£. [221 for nuclei near 208Pb).

In the spirit of trying to compare different models, in order to obtain a deep under-standing of the underlying physics, we will retain the philosophy postulated in [10], butpro pose a different parametrization, one which will not disqualify the pseudo 5U(3) ap-proach from the beginning, and will allow us to perform a very detailed analysis of boththe 55ANM and the pseudo 5U(3) mode!. This proposal is

e~ = e(1 + ~Z/A), (16)

where the para meter ~ can vary from O to 4, describing the situation from the bare protoncharge to one strongly correlated with the coreo In this parametrization we loose theslightly increase of e~ol over e~ol, but we could well say that it is at least as valid as theother one.

Figure 3 shows the behavior of

cmodel = Q( exp)Q(model)

(17)

for the 55ANM and the pseudo 5U(3). In Fig. 3a the parametrization (15) is used, whilein Fig. 3b the alternative (16) is employed, in the case of 168Er. Figure 3a exhibits thebehavior of Fig. 4 of [10]' and for ~ = 2.1 CSSANM = 1 and CP,SU(3) "" 1.5, and is the basicsupport for their claim of a 50% underestimation of Q in the pseudo 5U(3). In Fig. 3a itis clear, also, that the limit Cp,SU(3) = 1 is never achieved, for the allowed values of ~.Figure 3b exhibit the behavior of the same quantities, using the parametrization (16). Inthis case, it is clear that both models are able to reproduce the experimental data, witheffective charges

55ANM: e~ff = 1.7; e~ = 0.7

pseudo 5U(3): e~ff = 2.2; e~ff = 1.2(18)

In summary, there are at least two different parametrizations of the effective charges whichcan reproduce the experimental data. This result weakens in sOllle way the conclusionsin [101. But it is indisputable that the quadrupole lIIoments in the normal parity sector are

58 JORGE G. HIRSCH ET AL.

2.5a)

b)

2.0

-¡¡."E 1.5Ü

1.0

0.51.0

2.5

--

1.5

---

2.0, 2.5

SU(3)SSANM

------3.0

0.51.0 1.5 2.0 2.5 3.0 3.5 4.0,

2.0

-¡¡."~ 1.5Ü

1.0 --- ---

SU (3)SSANM

--- ---

FIGURE 3. Quolienl belween lheorelical and experimenlal Q values for IOSEr, for lhe pseudoSU(3) (solid line) and lhe SSANM (dashed line). In Fig. 3a the paramelrizalion (15) was used,while Fig. 3b refers lo (16).

the same in the pseudo SU(3) and the SSANM description. The seniority zero assumptionin the abnormal parity levels reduces the total quadrupole moments predicted by thepseudo SU(3) as compared with the SSANM ones, reslllting in greater effective chargesin any parametrization. 'Ve interpret this reslllts saying that abnormal parity valencenudeons adiabatically follows the behavior of the normal parity ones.

7. HIGHER SENIORITY STATES

The single-j case of the SSANM was discllssed by Eisenberg and Creiner [24] as theextreme single-partide mode!. Civen the quadrupole operator qo does not mix states withdifferent parity, the abnormal parity state remain decollpled, and its matrix elementsare [8,251

(19)

PSEUDO SU(3) MODEL AND AUNORMAL PARITY STATES 59

16a) 14.... ---'"

/ ,/ / ,,12 /

I ,10 I ,

I \o I \8 I \o-

I \6 I \I

I \4 I 2qp \

I \2 1. SSANM \

\

00 2 4 6 8 10nA

b) 20 ~,,\\

15 \\,

o \o- 10 \\\\

5 -- 2qp \

- - - - SSANM \\\

2 4 6 8 lO 12 14

nAFIGURE 4. The total qlladrllpole deformation in the abnormal parity space plotted against thenumbcr nA oí particlcs in the abllormal space. Thc solid Hne refcrs to the two quasiparticle esti-mation, while the dashed lined refers to the SSANM description. Fig. 4a (4b) was performed to asingle shell with j = 11/2 (j = 13/2).

where HO wave functions were used. This numbers are exactly those given in Table IVof [10] for the abnormal parity states.As a way to investigate the spherical quasiparticle content of the abnormal sector, let

us analyze the contribution of two ,!uasiparticle states, coupled to angular momentum 2,to the <¡uadrupole moment. \Ve use the I3ogoliubov- Valatin transformation [8]

with

tajm

(ljm (20)

- -( l)i+m(ljm - - aj-m (21 )

Given we are restricted to a state with nA particles in a single shell with angular momen-tum j, the u and v coefficients are

v = {~}1/22j + 1 {

2j + 1 - nA } 1/21l= -----

2j + 1(22)

60 JORGE G. HIRSCH ET AL.

The t\Vo quasipartide state with angular momentum 2 is

where l...11M denotes angular momentum coupling ami Djm 10)operator is expressed in second quantization as

qM= L (j¡m¡lqMlhm2)a}¡m¡aj,m2

Jimd2rn2

and, for a single j and in terms of quasipartide operators, becomes

(23)

O. The quadrupole

(24)

The quadrupole moment is

--12 J2j : 1 (jllqllj) uv

{l(l + 2)(21 + 3) nA (21+ 2 - TIA) }1/2

5(1+ 1)(2l + 1)

(26)

(27)

for nA partides in a levcl with j = l + 1/2.In Fig. 4 we bave made a plot of qA(j) VS nA for the hll/2 ami ;13/2 levels. In the same

figure we plotted the quadrupole moments obtained for the abnormal states in the SSANM,filling the single-j level with pairs having the maximal possible quadrupole moments. Itis apparent that for two partides both results are quite similar (with two partides onecan have only states with seniority zero or two) but for any other number of partidesthe SSANM quadrupole moment is twice the two-quasipartide one, exhibiting a higherseniority conten\. Giving such dominance of higher seniority states in the ground state ofeven-even nudei imply the use of the seniority zero description of the backbending is anoversi mpIifica t ion.

It is also interesting to analyze the quadrupole moment assigned in di!ferent models toone partide in the lowest energy state in the abnormal sector. This is the state with lowestprojection of the total angular momentum m = n = 1/2. In the single shell approach, thisstate has j = '/+ 1/2, and using (19) its quadrupole moment is q~;ngl.j = '/('/+ 2)/(21]+ 1).On the other side, in the asymptotic limit, i.e. allowing maximum mixing bet\Veen all themembers of the 1]shell with the same orbital angular momentum 1, the state (1/1]0)1/2 has,using (2), a quadrupole moment q;;"ym = 2ry. This asymptotic value is 3-4 times greaterthan the single-j one. The fact that in both deformed Nilssoll and Woods-Saxon potentialthe intruder states remains essentially unmixed for deformations f :::: 0.2-0.3 18,91 supportsthe single-j assignment. It also implies that the n = 1/2 state has not a constant slope,¡.c. it ha.s not rcached its maximal quadrupolc morncnt at this dcformatioll, as can beconfirmed in Fig. l.

PSEUDOSU(3) MODELANDABNOHMALPAHlTYSTATES 61

In conelusion, we can say that filling each single partiele state with nueleons with theirmaximum quadrupole moment is meaningful for those occupying normal parity states.1I0wever, doing the same in the case of unique parity states contradicts microscopic Nilssonami \Voods-Saxon calculations. It is reasonable to assume that nueleons in the uniqueparity sector share the deformation of the others, and then they cannot be described aspure seniority zera states. The effective charges used in the pseudo SU(3) description ofnuelei ineludes the renortnalization of '" 25% required for the fact the abnormal statesrcma.incd "'frozcn" in the Il1odel.

8. CONCLUSIONS

Some general features of the pseudo SU(3) model were presented. Its predictive power formany properties of heavy deformed nuelei was exhibited and associated with its successfuldescription of valence nueleons. Core polarisation effects are ineluded via the effectivecharges.In the particular case of BE2¡ transitions, the main ideas praposed in [ID) were ana-

lyzed. Their assumptions about the quadrupole moments in the pseudo SU(3) formalismwere rigorously checked, and found correct as a first appraximation. In the normal paritysector SU(3) states with largest deformation are lowest in energy, ami therefore result inthe same states of maximal deformation as in the SSANM.In the abnormal parity sector the two and four quasiparticle content was shown as a

necessary interpretation of the contribution to quadrupole moments of nueleons in theseorbitals. These quadrupole moments are non-zero, but only a fraction of their asymptoticlimiL The seniority zera appraach is exhibited as a very useful Illathelllatical appraach,which in general is assigning nucleons in unique parity orbitals a passive rale, but needsgreater effective charges to compensate for this assumption.

ACI{NOWLEDGMENTS

The authors want to thank the kind hospitality received from Pral. Dr. \V. Scheid duringtheir visit to the Institut fur Theoretische Physik, JLU-Giessen, where Illuch of this workwas done. Interesting comments fram Dr. J.P. Draayer, Dr. S. Ralllan and Dr. K.II. Bhattare also acknowledged.

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