Revista Mexicana de Física 42, Suplemento 1 (1996) 227-242

The projected shell rnodel and its newest applicationsto high-spin spectroscopy

YANG SUN1,2,3., JING-YE ZIIANG2, MIKE GUIDRy2,3, AND DA HSUAN FENG4

1 Joint Institute /or Heavy Ion ResearehOak Ridye National Labomtory, Oak Ridye, Tennessee 37831, U.S.A.

2Department o/ Physies, University o/ Tennessee, Knoxville, Tennessee 37YY6, U,S.A.3Physies Division, Oak Ridye National Labomtory, Oak Ridye, Tennessee 37831, U.S.A.

4Department o/ Physies and Atmospherie Seienee, Drexel UniversityPhiladelphia, Pennsylvania 1Y104, U.S.A.

ABSTRACT. The Projeeted Shell Model is a shell model developed for medium to heavy nuclearsystems. It was primarily designed to study nuclear structure of high-spin states in normallydeformed systems, and has recently been generalized to describe superdeformed bands. The modelis expected to be a powerful tool for studying the physics related to the new "(-ray detectors, suchas Gammasphere, Eurogam, and GASP. A general introduction to the rnodel is given and threetopics are discussed: 1) A systernatic ca1culation and a statistical comparison with the crankedshell model in treating rotationalnuclei; 2) Application to superdeformed hands; 3) The poten tia!existen ce of tH = 4 bifurcation in the PSM.

RESUMEN.El Modelo de Capas Proyectado (MC?) es un modelo de capas desarrollado parasistemas nucleares de medianos a pesados. Fue diseñado primeramente para estudiar estructuranuclear de estados de alto espín en capas con deformación normal, y ha sido recientemente gen-eralizado para describir bandas superdeformadas. Se espera que este modelo sea una herramientapoderosa para estudiar la física relacionada con los nuevos detectores de rayos 1, como Gamma-sphere, Eurogam y GAS? Se da una introducción general al modelo y se discuten tres tópicos:1) Un cálculo sistemático y una comparación estadística con el modelo de capas de manivela para eltratamiento de núcleos rotacionales; 2) Aplicación a bandas supercleformes; la existencia potencialde una bifurcación !::J.I = 4 en el MCP.

PACS: 21.60.Cs; 21.10.Re

l. INTRODUCTIOl'

\Vith the advent of a new generation of high precision ,,¡-ray detector arrays (e.y., Gamma-sphere of L13L, Eurogam of Strasbourg and GAS? of Legnaro), nuclear structure physicsal h¡gh spins is again a very active field. In recent years, the number of the high spin mea-surements has been growing and the range of spins has moved higher and higher, thanksto rapidly developing experimental techniqnes. Furthermore, an incredibly sharp energyresolution in the ,,¡-ray measurement makes it postiible to detect extremcly delicate andfine structure such as the e£fect of identica! bands, and of 6.I = 4 bifurcation .

• Invited Talk presented by YS at the XIX Symposium on Nuclear Physics, Oaxtepec, México,January 3~6, 1996.

227

228 YANG SUN ET AL.

The "standard" model for nuclear structure is the spherical shell model, which assumesan independent single-particle picture where the valence nucleons move in a spherical plusspin-orbit potentia!. It is now commonly accepted that nearly all nuclear properties shouldcome from the diagonalization of the many-body Hamiltonian matrix, usually within oneor a few major shells, with elfective residual interactions between these valence nucleons.However, except for several narrow windows in the N (neutron number) and Z (protonnumber) landscape, a straightforward application of the shell model is unattainable. Thesituation is especially acute for heavier systems. If one insists on calculating such nucleiby a brute force shell model, then one must be ready to diagonalize Hamiltonian matriceswhich are astronomical in dimensionality. Even if one could do such a large scale calcula-tion, one can probably not interpret the outcome as simply as one do es with a deformedfield.The Projected Shell Model (PSM) [1] has been developen in order to meet the stan-

dards imposed by quality of measurements from modern experimental techniques, and toovercome the known difficulties of the spherical shell mode!. Although many high spinphenomena can be studied by the mean-field plus cranking descriptions, it has been ourstrong belief that a proper shell model approach will be able to explain quantitatively thefinest features of observed data, and could provide us with a unified understanding of everaccumulating high-spin data. The material presented in this paper seems to indicate thatthis is the case.The paper is organized as follows: In Sect. 2, we briefly introduce the PSM. A detailed

description of the model has recently been published as a review article II]. In Sect. 3,a systematic calculation for normally deformed even-even rare earth nuclei is presenten.To test the theory's ability to describe the general features of nuclear systems, we makea statistical comparison of our results with those from another popular high spin model-the cranking mode!. Generalization and first application of the PSM to a superdeformedsystem is given in Sect. 4. Our examples show that this model might be a powerÍ'l1 toolfor analyzing data coming from the large 'Y-ray detectors. A potential existence of 6./ = 4bifurcation in the PSM is discussed in Sect. 5, which provides an alternative interpretationfor recently observed bifurcation obsen'ed in some superdcformed bands, and predicts thesame elfect in normally neformed systems. Finally, a short summary is given in Sect. 6.

2. TIIE PROJEeTED SIIELL MODEL

The many body Hilbert space in the PSM begins with the deformed (Nilsson-type) singleparticle basis. Pairing correlation is treaten by successive BeS calculations for the Nilssonstates. \Vhile this shell model basi, viola tes rotational symmetry, it can be restoren bystandard angular momentum projection techniques [2). Thus, the shell model truncationis carried out within the quasiparticle states with the vacuum 1<1».In construction of the PSM Hilbert space, we shall iutroduce the angular momentum

projected wave functiou:

( 1 )

(2)

TIIE PROJEeTED 51lELL MODEL AND 11'5 NEWE5T. . . 229

where 1< labels the basis states. Acting on an intrinsic state I'I'K)' the operator PJ.¡ K willgenerate states of good angular momentum. Here, we shall only consider a well-deformedsystem and the intrinsic states have axial symmetry; thus J( is a good quantum numberfor I'I'K)' The basis states I'I'K) are spanned by the set, for example,

{<>tl4», a¡;<>¡jatl4»},{ata¡j 14»} ,

{14», ata~j 14»,a¡,a¡,I4», ata~jat a¡,I4» } ,for odd neutron, odd-odd, and even-even nuelei, respectively. The quasipartiele vacuumis 14» and a," (a;") is the quasipartiele annihilation (creation) operator for this vacuum;the index ni (p;) runs over selected neutron (proton) quasipartiele states and 1< runs overthe configuration. The single partiele configuration contains three majar shells for eachkind of nueleons, which appears to be adequate for the applications discussed here. Fornormally deformed rare- earths, we use N = 4, 5 and 6 for neutrons and N = 3, 4 and 5 forprotons. The size of the basis, which should iuelude the most important configurations, isdetermined by qp energy truncation, typically 2 MeV aboye and below the Fermi energy.The PS~l rotationally invariant Hamiltonian is taken as

(3)

(4)

In the aboye equation, j¡o is the spherical siugle-partiele shell model Hamiltonian. Thesecond term is the quadrupole-quadrupole inleraction and the lasl lwo terms are themonopole and quadrupole pairing inleractions, respectively. The interaction strengths aredelermined as follows: the quadrupole iuleraction strength X is adjusted so that the knownquadrupole deformation '2 from the Ilartree-Fock-Bogoliubov self-consistenl procedure [3]is obtained. The monopole pairing strength GM takes lhe form

r N - Z] 1GM = lGI :¡:G2 A .A-,

where the minus (plus) sigu is for neutrons (protons), and GI and G2 are adjusted to theknown energy gaps. For normally deformed rare earths, GI = 20.12 and G2 = 13.13. Thequadrupole pairing strength GQ is assumed lo be proportional lo GM and the proportion-alalily conslant GQ/GM is approximately 0.2. Finally, the "coefficients of expansion" fKin Eq. (1) are delermined by diagonaliziug the rotationally invariant Hamiltouian in thisbasis.

3. SYSTEMATIC DESCRll'TION AND STATISTICAL ANALYSlS OF NOltMALLY DEFORMED

NUCLEl

Slatistical analysis of lhe properlies of measured momenls of inerlia in superdcformed andnormally deformed nuelei has received cousiderable atlenlion recentl)". There have been a

230 YANG SUN ET AL.

few theoretical calculations addressing sorne specific cases, but before the work describedhere there was no published attempt to reproduce the global features of the experimentalstatistical distribution. Such an endeavor provides a strong test of a theory's ability todescribe the general features of nuclear structure. In this section we apply the PSM andthe Cranking Model with particle number projection to such an analysis for normallydeformed nuclei [41.

The experimental sample used in the present statistical analysis consists of the yrastbands at normal deformation in even-even nuclei with Z = 66-78 and N = 86-116, atotal of 2145 band pairs. \Ve include in the data set all known spin states 15] up to thefirst band crossing, or up to 1 = 10+, whichever comes first. For each possible pair, weevaluate the average absolute difference in the kinematic moment of inertia

(5)

where the transition energy is E~ = El - EI-2, with (1) or (2) specifying the nucleusI and 2, respectively, and the average absolule difference in lhe dynamical momenl ofinertia

5:1(2) = I(Imax/2 - 1)

Im•x 15E~(l) - 5E~(2)1{; (5EN) + 5E~(2))/2'

(6)

where 5E~ = E~ - E~-2.The PSM and lhe normal Cranking Model (Nilsson + Strutinski + BCS approach) wilh

particle number projection (see, for instance, Re£. [6]) have been used to calculale thesesame quantilies for comparison wilh the available dala. In all calculations, we have chosenfor lhe two models minima! seis of paramelers lhal are expecled to describe low-spinspectroscopy. Furlhermore, these parameter seis were selecled according lOcommonly usedprescriplions for each model, and were fixed before the slatislical analysis was performed.Thus, no paramelers have been adjusted lo improve the agreemenl wilh the statislicaIdala for either mode!.

Figures I and 2 compare dala with calcu!alions for the slatistical distributions. One seesfrom these resulls lhat lhe PSM gives very good overall agreemenl wilh lhe observations,except for an overprediction al extremely low values ane! a general underprediclion alvery large values of lhe e!iscrepancy (where slalislics are poor) that will be discussee! fur-ther below. Conversely, lhe cranking resulls are in poor agreement with lhe observations,exhibiting a broad and ralher Aal e!islribution that is in sharp conlrasl lo the empiricaldislribulion lhat peaks strongly near a value of aboul 20.

AIlhough lhe PSM accounls well for lhe global momenl of inerlia slalistics, close ex-amination reveals SDIllC systematic and instructivc discrepancies. Prom Figs. 1 and 2 ,lhe PSM produces higher values lhan observee! in the region of \'ery small devialions ane!lower values than observed in the regio n of large deviations. This is presumably relaledlo the simplicity of lhe PSM approach, which uses a basis lruncaled al a fixed deforma-lion. Although the lwo-body correlalions incorpomled lhrough lhe Hamiltonian (3) in the

THE PROJECTEDSHELLMODELANDITSNEWEST... 231

300

250

200rJ)-e 150:JO -Ü -. ,.1 ••100 ,

50

oo 20 40 60

• Exp.PSMCSM

,-,, ,-' ,,,,,,..- ...-- ,

••• - •• ---1_ •• •

80 100 120 140 1608 J(1)

FIGURE l. Statistical distributions for the kinematic moment of inertia in 2145 pairs of bands. Thenumber of times each value of bJ(l) occurs is plotted vs. bJ(l). FiIled cirdes are calculated fromEq. (1) using the experimental data; the solid line from PSM and the dashed line from CrankingModel calculations.

subsequent diagonalization account for sorne amount of deformation fiuctuation, we maygene rally expect the model to work better for well-deformed than for deformation-soft nu-elei. Figure 3 shows a systematic high spin calculation for 3(1) in Er, Yb, and Hf isopotes,with states up to spin 34 n. One sees that for the low spin stales discussed aboye the theoryworks rather well for those well-deformed nuelei with neutron number larger than 92. Thetheory fails to describe the more rapid increase of the moment of inertia at low spins forlighter rare earths, which is a characteristic feature of deformation-soft nuelei. Therefore,the PSM produces more very similar bands than are actually observed (the overshoot ofthe theory at the far left side of Figs. 1 and 2), and fewer highly non-identical bands thanobserved (the undershoot of the theory at the far right of these figures).The good overall agreemeni between the data and the PSM results (and the correspond-

ing poor agreement between the data and the cranking results) for the 63(1) and 63(2)distributions suggests that the shell-model configuration mixing and angular momentumprojection implicit in the PS,,¡ are important in reproducing the experimental low-spinmoment of inertia variation from one nueleus to another. The same conelusion has alsobeen drawn in our study of identical bands at normal deformation [71.

4. QUANTITATlVE DESCRII'TION OF SUI'ERDEFORMED llANDS

The topic of sllperdeformation (SD) has been at the forefront of nuelear structure physicssince observation of the SD band in 152Dy [81. The existence of enhanced deformation,diminished pairing, and less frequent band crossings makes phenomena such as identical

232 YANGSUN ET AL.

• Exp.PSMCSM

O'.,..• 1_,-'•• ,

• .'. , ..'•• • • ••• •••• ,40 60 80 100 120 140 160

O J(2)

400

350

300

250lJl •-e 200:JoÜ 150

_1-,100 o'

o,.,50

OO 20

FIGURE2. As in Fig. 1, but for the variation of the dynamical moment of inertia 63(2) calculatedfrom Eq. (2).

bands [9] and t,I = 4 bifurcation [10,11] easier to detect in SO than in normally deformed(NO) systems.

Superdeformation is both theoretically and experimentally a well-established concept,but, except for a few examples reported recently (see, e.g. Ref. (121), the measured SObands are not lirmly tieu to the nuclear gronnu state by complete decay sequences. Thus,the spin assignments for superdeformed statés remain uncertain. This leaves hypothesessuch as the existence of '1uantized alignment [13] as open questions [14). Other impor-tant physical quantities such as the static moment of inertia :f(1) for SO bands cannotbe uniquely determined because one does not know the angular momenta. Mean-lield de-scriptions of the cranking type are of limited utility for these questions because angularmomentum is not a conserved quantity in those theories.

The PSM was initially constructed to treat normally deformed high-spin states [1].Application of this model to superdeformation requires an assignment for the interactionterms of the two-body Hamiltonian. Although there is no reason of principIe to expectthat these interactions will be the same as those found to be appropriate for normaldeformation, we adopt here a minimalist approach: we attempt to calculate the propertiesof SO states using the same Hamiltonian, Eq. (3), as employed for the NO case. TheNilsson parameters useu to construct the basis states of Eq. (2) are well-established forthe normal deformation region and are simply extrapolated to the larger ueformationregion. Our preliminary results of the SO sta tes in the A ~ 130 mass region indicate thatthe quadrupole plus pairing type interaction 12] may also work well for SO systems. Inthis section, we shall report on results obtained for 131,132Ce as a representative exampleof these calculations.

For a SO band uescription, it is known that high j intruder orbitals play an important

...,'"t'l""E;~t'l

Qt'l

"'"'"t'l,..,..:::o"t'l,..>z";:j'"Zt'l:::&lel

172Hf

170n

168Er

170Hf _"

168Yb~¿

166Er

168Hf

166Ybr¿

164Er

166Hf

164Yb

162Er

164Hf

162yb

160Er

100.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 ül-

28160

130

100

70

40

160•

130

100

70

40

FIGURE 3. Syslemalic comparison of calculaled kinemalic momenl of inerlia J(l) vs. w2 wilh lhe dala for Er, Yb, and Hf isolopes.Definilions of lhe quanlilies: 2J(l) = (21 - l)/w, "'" = E,(I)/2, and E,(I) = E(I) - E(I - 2).

'"'"'"

234 YANGSUNET AL.

131Ce

..................••

100 •:> ,•• - -. exp.- - -. exp ,

132CeQ) 90 o-o th, o-o th

:;; ,,N 80 ,:S ,,O 70:;;ro 60.2E 50

'"e 40>.O 30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Rotationai frequency [MeV h-1]

FIGURE 4. Comparison of calculated SD dynamical moment of inertia 3(2) with the data forl32Ce [18Jand 131Ce [24J.Definitions oí the quantities: 3(2) = 4/[Eo(I) - Eo(I - 2)J.

role [15]. Therefore, the configuration space for the SD calculation has to indude suchorbitals. For the SD A ~ 130 mass region, N = 3, 4 and 5 shells were employed for theprotons, and for neutrons, N = 4, 5 and 6. For both nudei I31.I32Ce, the basis states areobtained at the same deformation of <2 = 0.38. The strength of the monopole pairinginteraction in Eq. (4) is critical for a quantitative discussion of the moment of inertia.We employ monopole pairing strengths [16J with Gl = 18.0 and G2 = 14.4. Althoughour present choice of the strengths of Eq. (4) may not be optimal, systematic calculationsshow that they work surprisingly well for this mass regio n [17). Finally, the quadrupolepairing strength GQ in Eq. (3) is assumed to be 16% of the monopole pairing GM, inagreement with the average ratio of quadrul'0le to monopole pairing strengths found tobe appropriate for normally deformed rare-earth nudei [1).Qur results are compared with the data [18] in Fig. 4, where the dynamical moment of

inertia 3(2) is plotted as a function of rotational frequency nw. The large peak appearingat nw ~ 0.36 in 132Ce corresponds to a near-simultaneous crossing of four bands: the SDground-state band (g-band), a band associated with the alignment of a iI3/2 neutron pair,and two bands cOIresponding to the alignment of "11/2 proton pairs (see Fig. 5 fOI theirconfigurations). Because of the strong decoupling effect, this 2-quasineutron state becomosyrast after the crossing with the g-band. The two quasiproton bands rise quickly away fromyrast, and so playa minor role in the structure of the presently-known yrast sequence inthe nudeus 132Ce. However, the bands with proton alignment playa significant role in theyrast structure of the adjacent odd-neutron nudeus I3ICe, where the neutron alignmentis blocked. In fact, the much less pronounced peak seen in 3(2) at Ítw ~ 0.42 in I3ICeof Fig. 4 is due to the smaller crossing angle associatod with the proton alignments (seeFig. 5) and stronger band interactions. This explains the observed differences in dynamicalmoment of inertia between l3ICo and 132Ce at rotational frequencies near nw ~ 0.40.For a better understanding of these results, let us now look at the band diagram of

132Ce shown in Fig. 5. A band diagram is defined [1] as tho diagonal elements of thoHamiltonian in the projected basis of Eq. (2) vorsus spin. \Ve concentrate on two kindsof band crossings: the first crossing between the g-band and the 2-qp bands at spin 22n,

TIIE PROJECTEO SHELL MOOEL ANO ITS NEWEST... 235

4 8 12 16 20 24 28 32 36 40 44 48

Spin (h)

22

20

18

16

- 14>(\)

~ 12->-Ol 10•..(\)eW 8

6

4

2

OO

• yras! bandvi'312 [112,-312]nh11l2 [1/2,-312), [-312,5/2J4-qp bandg-band

- - - - -..

.';" .0'-;" •.........••

:."",,.... ....4

4••••

132Ce

FIGURE 5. Band diagram (bands before confignration mixing) and the yrast band (the lowest bandafter configuration mixing, denoted by dots). Only the lowest-Iying bands in eaeh eonfigurationare shown. Far the 2-qp statcsj [{ -quantum numbers are given in square brackets. The 4-qp statesare combinations of ane 2-quasineutron and ane 2-quasiproton bands shown in this figure.

and the second crossing between the 2-qp and the 4-qp bands at spin 38 h. The gain ofalignment can be read from the plot. In this energy vs. angular momentum plot, rotationalfrequency w = ~fis the slope of the curve. For the 2-quasineutron band, one finds thatw == O until the system begins to rotate at spin 12 h (see the !lat portion of the curve inFig. 5). This is a perfect alignment proeess, sinee 12 h is the maximum angular momentumthat a pair of j = 13/2 particles can contribute. The same perfect alignment can be seenf.or h11/2 protou pairs, which do not rotate until spin lOh. Thus, f.or a 4-qp configuration(2-quasinentrou + 2-quasiproton), the band begins to approximate collective rotationalmotion near 22 f, = 12 ft + 10 ft. \Ve find that bands with alignment of neutron pairs builtupon h9/2 and h11/2 orbitals lie high in energy (more than 1 MeV higher than the 2-qpbands presented here at the crossing spin 22 h) and are unlikely to playa significant rolein the yrast region; this is contrary to the results of Ref. [191.

236 YANG SUN ET AL.

•.- - -. SO yrast band, expO-e SO yrasl band, th

132Ce

.---. SDyraslbanó,expo-e so yraslband,lh

ND yrast band, e.p- NO yrastband, lh

;;-ID 2.0::;;~ 1.6IDai 1.2>-~ro 0.8EE 0.4ro

<.?0.0

O 4 8 12 16 20 24 28 32 36 40 44 O 4 8 12 16 20 24 28 32 36 40 44 48

Spin Spin

FIGURE 6. Comparison of lhe ealculated -¡-ray energy wilh lhe dala for 132Ce (SD laken from [18]and ND from [23)) and l3ICe (SD laken from [24)).

After the lirst band erossing, the observed yrast band of 132Ce is mainly 2-quasineutronin structure, as indicated in Fig. 5, while that of 131Ce is one i13(2 neutro n plus a pair ofh11(2 protons. For 132Ce, lhe second band crossing occurs al spin 38 ñ (see Fig. 5). Thissecond crossing has nol eulered olher lheorelical [19) discussions, probably beca use ilselfecl on lhe observed bands is small. Indeed, even in lhe plOl of 3(2) in Fig. 4, one canhardly see the elfecl. However, such a crossiug implies a rearrangmenl of lhe wavefunctional lhal spin for the yrast stmcture and should have other observable consequences. \Vesuggest that the second band crossing may play an important role in understanding theobserved anomalous behavior of tH = 4 bifurcation (or tl.l = 2 staggering) in 132Ce [20].The good agreement with data encourages us to make a theoretical spin assignment

for the observed SD bando In Fig. 6, we plot the gamma-ray energy E, as a function ofspin l. For comparison, the ND yrast band of 132Ce is also plotted. The ND calculationis carried out in such a way that the shell model space is now lruncated al a deformationof €2 = 0.21 [161- For the SD band of 132Ce, the calculation coincides best with lhe datawhen we place the measured [18) lirsl gamma-ray energy at. spin 24 h. This is a posit.iveshift. of 6 ñ relat.ive t.o t.he spin assignment. given in fief. [21]. For t.he SD band of 131Ce, weplace t.he previously assigned band head 1 = 21/2h [22] t.o spin stat.e l = 33/2ñ. There isagain a 6 h shift, ait.hough from the plot, 8" may also be possible. This is a large shift, butwe note that the same amount of spin assignment error was reported in the neighboringnueleus 133Nd [12] aft.er the complete decay out of t.he SD band had been establishedexperimentally.

f:ll = 4 B1FURCATION

Qne of the most active areas of research in high spin spectroscopy is f:ll = 4 bifurcation(or tl.l = 2 staggering). To date, the phenomenon has been conlirmed in at least in twocases: SD bands of 149Gd [lO] and 194Hg [Il], and has been reported for other cases.The observation is as follows: the rotational sequences in the superdeformed nuelei with

TIIE PROJECTED SIlELL MODEL AND ITS NEWEST... 237

angular momentum di/fering by two (E2 sequenee) can split into two branehes, one havingspins 1,1 + 4,1 + 8... , and the other 1+ 2, 1+ 6,1 + 10.... The staggering amplitude isvery small, of the order of 0.5 keV. Clearly, without the new generation of ,,-ray arrays,this e/feet would not have been observed. The bifureation in the superdeformed bandscan persist for about 5 1.0 7 oseillations. There are suggestions in the literature [10,26]that this may be eaused by a fourfold symmetry (henee the name C4) in nuclear systems.Quite reeently, we investigated this phenomenon with the PSM and o/fered an alternativeexplanation [25]: the observed bifureation is a kind of quantum meehanieal interfereneeeoming from the eonfiguration rotation from intrinsie 1.0 laboratory frame and the shellmodel eonfiguration mixing.In Fig. 7, we present our numerieal results from realistie ealculations. The quantity we

use 1.0 illustrate the staggering is defined in Ref. [U]. As reeently been notieed, the five-point referenee [U] used in extraeting the "staggering" may exaggerate the real e/feet ifnot applied earefully (see, e.g., Ref. [27]). Sinee this phenomenon depends on the variationof E1 values, it should also be direetly ohservable in the variation of the dynamieal momentof inertia, 3(2). In the bottom left part of Fig. 7 we show the dynamical moment of inertia,3(2); a clear step strueture is observed that is eorrelated with the 6.1 = 2 staggering inthe Eh) values. In the ealculations shown in this seetion, we have employed the sameIlamiltonian given in Eq. (3). Thus, no new terms have been added 1.0 the Hamiltonian,nor have any parameters been adjusted, to produce these bifureation e/feets.In the left part of Fig. 7, we observe clear 6.1 = 4 bifureation as a funetion of angular

momentum for both even- and odd-spin sequenees in 166Tm, from a Hamiltonian thateontains neither an explieit interaetion of Y44 form, nor an 14 ter m 1.0 eouple intrinsiestates di/fering in 1< quantum numher by four. On the other hand, the right part of Fig. 7exhibits no obvious bifureations for a similar ealculation in 162Tm and the eorresponding3(2) behaves smoothly. Thus, the ealculations suggest that hifureation e/feets oeeur forsome nuclei and not. for ot.hers.There are several quest.ions one may ask in at.t.empt.ing 1.0 understand t.his physies. First,

what is the inherent. meehanism in the PSM whieh eould give rise t.o a 6.1 = 4 bifureation.Seeond, why should the effeet oee1lr for some nueei, and not for the ot.hers. To answersueh questions, let us write down a mat.rix element. of t.he llami1tonian in the projeetedbasis states of Eq. (2) [28]

whieh is analogous to t.hat. of t.he part.icle-rotor model wit.h symmetrized wavefunetion [29],with t.he phase factor (-1)/- J(' responsible for the signature dependenee assoeiat.ed witht.he usual twofold symmetry of the rot.or.The 6.1 = 4 bifureat.ion e/fect. now follows direct.ly frolIl t.he int.egral (7) and basie

properties of the funetion dl,J(' (¡3), which may be expressed in the form [3D]

(8)

...l<1;;z::>U)

"z>:

" t ' , , , , , , J "1.51- 166Tm 1.5 162Tm

~ 1.0 1- Neg. parity 1.0I Pos. parity>al-'" 0.5 .•. 0.5 •........ ...•.. 4".. ;""""Ol ...•... ... 1.." • ..a - - -O , , •

O O •. ..."" I •••• .•• ~"..'.. O O - - -. - .. ..'" -c. )"" . __"¡;:: •. •. ...• "" .....' •• .." ".. ••. •. \Q) • ....' • 'o. oc- .' \en -0.5 '.' -0.5 \Ol 'ro •éñ -1.0 -1.0 "

-1.5 -1.5

-2.0 ' , '1 I --'- -2.0, " "

~ 60 166Tm 60 162Tm •

~ 55 Neg. parity ,,' ~ _ 55 Pos. parity ,.'~ .......•' .. .-N 50 ,. 50 ,.~.... ••.--- .---.... ,.O 45 " 45 ,.~ "ro 40 - - -o - - -. . 40 ~ .•u ,'E 35 - 35 ..-ro~ 30 - 30r ..-O

25 ' I I I '2512 14 16 1B 20, 22 24 26 12 14 16 1B 20 22 24 26

Spin [h) Spin[h)FIGURE 7. Plot illustrating ó.I = 2 staggering effec!. Left: '66Tm, exhibiting elear staggering in LlE(-¡) for both even (filled cireles)

OC and odd (filled squares) spin sequences, and step structure in 3(2) (for simplicity, only the even spin sequence is shown). Right: '62Tm,~ exhibiting no such structures. Quantity "staggering" is defined as in Re£. I11J.

TIIE PROJECTEO SIIELL MOOEL ANO ITS NEWEST... 239

Integration

4.0e-02K=1, K'=5

2.0e-02

O.Oe+OO

-2.0e-02

-4.0e-02O 4 8 12 16 20 24 28 32 36 40

Spin

FIGURE 8. t'!.! = 4 bifurcation effect directly frotll the geotlletric properties of stllall-d fnnction.

where t'!.~~)'" d~~)(/3= 1) and ,,(x) = Re(iK-K'eix). Upon insertiou of Eq. (8) iu Eq. (7),the oseillatory augular depeudeuee of the iutegraud is seeu to reside primarily iu theproduets ,,(v/3) siu /3 (the other faetors are usually smooth aud uou-oseillatory). Thus, iu-depeudeut of the t'!.J( = J( - J(' value for the two eoufiguratious eoupled by this matrixelemeut, Eq. (7) iuvolves au iutegraud factor siu(/3) eos(v/3) or siu(/3) siu(v/3) with iutegra-tiou limits from O to l' for eaeh term iu the sum over v iu Eq. (8). Thercfore, for valuesof v iu the fuuetiou ,,(v/3), that differ by four, the eoutributious from the " fuuetiou tothe iutegral are iu phasc, while for values differiug by 2 the eoutributious are out of phase.Siuee v ruus over values up to ! iu the sum of Eq. (8), the eoutributious add iu-phase forstates haviug !'J.! = 4. As oue example, we show the results of the d-fuuetiou iutegratiou,

JoI d/3siu /3dk w(/3), iu Fig. 8.Thus, for either ter m iu the integrands of Eq. (7), there are basieally two faetors:

the small-d funetion aud the matrix elemen!. The former is purely geometrie aud thelatter eoutaius the detailed dynamies siuee it iuvolves the Hamiltouiau and two intrinsiestates. It is partieularly iuterestiug that the !'J.! = 4 bifureation derives from the geometrieproperties of small-d fuuetiou, but the dynamies thorough the Hamiltoniau matrix elerneutgoverns the presenee or ahseuee of the effeet iu particular real nuclear systems.

To auswer the seeond question raised aboye, we note that the interplay arnoug thed-funetion aud the matrix elements determines the maguitude of the !'J.! = 2 staggeringeffee!. Iu mauy cases, one expeets this rnatrix element to be approximately Gaussiau iuthe angle /3. A uarrow width of the Gaussiau will suppress the staggeriug amplitude, aud iteau disappear eompletcly if the distributiou is lH'aked too sharply iu (3, or coneentrated attoo low values of (3.Tlms, observatiou of the eff"ct n'quin's that the iuteractiu¡,; states have

240 YANG SUN ET AL.

I I . ,

136pm

1- SO band -•, ', ,J'. , , .,

>-- , , "" , , ,", , ,, , , , , , , .•

'ft , , , , ,• , , , ',••• •

1-

, , , . , • , ,I , , ,

136pm,

1-,,

SO band,,

1- ,,,,1- ,,

- ..•--..- - ..- - - ---.- .•.. - ..-

, , , , ,

:> 2Q)=.Ol

Oe'CQ)OlOlca .2-(/J

-4~:::- 55Q)

:2N 50E.o 45:2Oi 40.~Eca

35e>-O

3024 26 28 30 32 34 36 38 40 42 44

Spin [h]FIGURE9. The same as Fig. i but for the superdeformed band in 136Pm.

4

their overlap distributed over a sufficiently large regio n of (3 when the system is rotatedbetween intrinsic and laboratory frames. This requirement suggests qualitative constraintson the physical situations that are favorable for observation of t.! = 4 bifurcation 1311.

From a variety of such calculations for rare-earth nuelei, and physical reasoning of thesort exemplified aboye, we have arrived at theoretical criteria for optimizing the experi-mental observation of these staggerings if they result f1'Omthe mechanism p1'Oposed aboye.These criteria are discussed in nef. 1311.

To illustrate the more common existence of this e!fect in SO nuelei, we present inFig. 9 a calculation for a superdeformed band in an odd-odd nueleus, 136Pm. The toppart exhibits the t.E(-y) staggering while the bottom part illustra!es the :7(2) step struc-ture. In the calculation <2= 0.38 is again assumed, as used in the 131.132Cecalculations.We learned reccntly that sllch staggering structure may have bccll observed ex pe rimen-tally in this nueleus, and that further measurement is planned to confirm this observa-tion 1321.

TIIE PROJECTEO SIIELL MOOEL ANO ITS NEWEST. .. 241

Finally, we stress that the extreme sensitivity of the proposed mechanism to the mi-croscopic structure of the near-yrast quasiparticle bands implies that a systematic andhigh-resolution investigation of t!.I = 4 bifurcation could provide a sensitive new probeof band structure in the near-yrast regio n of both superdeformed and normally deformednuclei. This is particularly true for situations where bands cross at very small angles andwith strong interactions: situations where little information can be obtained by the usualalignment or moment of inertia analyses.

5. SUMMARY

In this paper, we have presented sorne of the newest applications of the Projected ShellModel to high spm spectroscopy. The statistical comparison with moment of inertia datashows the ability of the PSM to describe the global features of rotational nuclei, withoutfine tuning parameters for individual cases. The successful application to SD bands indi-cates that the model could be a powerful tool for analyzing data from the new large "(-raydetectors. The alternative explanation of t!.I = 4 bifurcation provides us with new under-standing, and suggests that the band interaction nature of the effect could be a sensitivenew probe of band structure.

ACK!':OWLEDGMENTS

The Joint Institute for Heavy Ion Research has as member institutions the University ofTennessee, Vanderbilt University, and the Oak nidge National Laboratory; it is supportedby the member institutions and hy the U. S. Department of Energy through ContractNo. DE-AS05-76ER04936 with the University of Tennessee.Theoretical nuclear physics research at the University of Tennessee is supported by the

U.S. Department of Energy through Contract No. DE-FG05-93ER40770.Oak Ridge National Laboratory is managed by Lockheed Martin Energy Research Corp.

for the U. S. Department of Energy under Contract No. DE-AC05-960R22464.

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242 YANG SUN ET AL.

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