VOLUME 77, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 AUGUST 1996

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Shell Model Monte Carlo Studies ofg-Soft Nuclei

Y. Alhassid,1 G. F. Bertsch,2 D. J. Dean,3,4 and S. E. Koonin31Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut

2Department of Physics and Institute for Nuclear Theory, University of Washington, Seattle, Washington3W. K. Kellogg Radiation Laboratory, 106-38, California Institute of Technology, Pasadena, California 9

4Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831(Received 10 August 1995)

We present shell model Monte Carlo calculations for nuclei in the full major shell 50–82 foboth protons and neutrons. For the interaction we use a pairing plus quadrupole derived fromsurface-peaked separable force. The methods are illustrated for124Sn, 128Te, and124Xe. We calculateshape distributions, moments of inertia, and pairing correlations as functions of temperature aangular velocity. Our calculations are the first microscopic evidence ofg-softness of nuclei in thisregion. [S0031-9007(96)00894-0]

PACS numbers: 21.60.Ka, 21.10.Gv, 21.60.Cs, 27.60.+j

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Nuclei with mass number100 # A # 140 are believedto have large shape fluctuations in their ground states.sociated with this softness are spectra with an approximO(5) symmetry and bands with energy spacings interdiate between rotational and vibrational. The geometrmodel describes these nuclei by potential energy surfwith a minimum atb fi 0, but independent ofg [1]. Someof these nuclei have been described in terms of a qufive-dimensional oscillator [2]. In the interacting bosmodel (IBM), they are described by an O(6) dynamisymmetry [3].

Nuclei with 100 # A # 140 fill the major shell be-tween 50 and 82 for both protons and neutrons,conventional shell model calculations in the full spaare impossible for many of these nuclei. However, wthe introduction of shell model Monte Carlo techniqu(SMMC) [4], it has become possible to do exact calcutions (up to a statistical error) in much larger model spa[5] at zero and finite temperatures. This Letter presethe first fully microscopic calculations for soft nuclei wi100 # A # 140 and compares them with the resultsmore phenomenological models.

An important problem is the choice of the interactioIt was recently shown that the realistic residual nuclforce is dominated by a pairing plus quadrupole intertion [6]. We have used such an interaction, wherepairing contains both monopole and quadrupole [7] tewhose strengths are determined by odd-even massferences. The quadrupole interaction is derived fromsurface-peaked separable force [8]. Such an interactiexpected to be a reasonable way of describing defotion and pairing phenomena. A major advantage ofinteraction is that it has a “good” Monte Carlo sign, aaccurate calculations are feasible without using thetrapolation techniques developed to circumvent the “sproblem” [9].

The (isoscalar) surface-peaked interaction is assuto be of the formysr, r0d ­ 2xsdVydrd sdVydr0ddsr 2

0031-9007y96y77(8)y1444(4)$10.00

s-te-l

es

ic

l

d

-s

ts

r-esif-ais

a-s

-n

ed

r 0d, whereV(r) is the mean-field potential. The anguldelta function is expanded in multipoles, and onlyquadrupole component is retained. Thus,

H2 ­ 2Xlm

pgl

2l 1 1P

ylmPlm

212

x :Xm

s2dmQmQ2m : , (1)

where :: denotes normal ordering andPylm, Qm are pair

and quadrupole operators given by

Pylm ­

Xab

s2d,b s jak Yk jbd fayja

3 ayjb

glm ,

Qm ­ 21

p5

Xac

µja

ÜdVdr

Y2m

Üjc

∂fay

ja3 ajc

g2m .

(2)

In (2) a ; n,j denotes a single particle orbit andajm ­s2dj1maj2m. The strength of the quadrupole interactiis determined by self-consistency. A change in tmean-field potential is related to a change in the obody densityr(r ) throughdVsrd ­

Rdr0ysr, r0ddrsr0d.

Using the invariance of the one-body potential undedisplacement of the nucleus, and the separable formthe two-body interaction, we obtain

x21 ­Z `

0dr r2 dV

drdr

dr. (3)

The spherical nuclear density in (3) is calculated frorsrd ­ s4pd21

Pa faR2

asrdyr2, wherefa andRa are theoccupation number and the radial wave function of ora, respectively, and the sum goes over both the cand valence shells. In general we find thatx ~ A21y3

[6]. For 124Xe, Eq. (3) givesx ­ 0.018 MeV21 fm2, butthis value must be renormalized since the interact(1) is taken only in the valence shell. We find thatrenormalization factor of,3 is required to reproducecorrectly the variation of the excitation energy of the fi

© 1996 The American Physical Society

VOLUME 77, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 AUGUST 1996

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21 state in this region. We note that the value of trenormalization factor is expected to be larger thanstandard value of 2 [10] since there areDN ­ 0 matrixelements that are not included in the model spaceaddition to the usualDN ­ 2 elements [11].

The single-particle energies are determined fromWoods-Saxon plus spin-orbit potential using tparametrization of Ref. [12]. For 124Xe, the re-sulting single-particle energies of 0g7y2, 1d5y2,0h11y2, 2s1y2, and 1d3y2 are 23.05, 23.35, 21.07,21.04, and 20.615 MeV for protons and211.79, 212.08, 29.53, 210.21, and2 9.94 MeV forneutrons. The Coulomb potential has the effect of placthe h11y2 proton orbit below thed3y2 and s1y2 orbits.When the central Woods-Saxon potential is used forV(r)in (2), we find that the corresponding matrix elementsthe quadrupole interaction in the proton single-partibasis differ by only a few percent from those in tneutron single-particle basis. We can therefore choeither set.

For the pairing interaction we include only monoposl ­ 0d and quadrupolesl ­ 2d terms with g0 ­ g2.To determine g0 we first extract the pairing gapDfrom the experimental masses of neighboring nuclei [1We then use a particle-projected BCS calculationthe Hamiltonian (1) to find the value ofg0 that willreproduce the experimental gap for a spherical nucwith the same mass numberA. For 124Xe we findg0 ­ 0.15 MeV. The inclusion of quadrupole pairinis important in order to lower the excitation energy of t21

1 state in the tin isotopes to about 1.3 MeV (which2D , 2 MeV when only monopole pairing is includedSince both the monopole pairing and the quadrupquadrupole interaction are attractive, they satisfy the srule in the density decomposition and have a good MoCarlo sign. The quadrupole pairing has componentsviolate the sign rule, but they are all very smallcomparison with the good sign components. Settingthe bad components to zero has no more than a 5% eon the spectrum. In identifying the bad componentsthe interaction, one should use the modified sign rulesince orbits of both parities are present in the 50–82 sh

We begin discussion of our results with the proability distribution of the quadrupole momentQm ;P

r2Y2m. This probability distribution is defined aPsqd ; k

Qm dsQm 2 qmdl (whereq ­ hqmj) and is non-

vanishing forq fi 0 even if the ground state hasJ ­ 0and thuskQml ­ 0. The expectation value of an obserableV in the SMMC is calculated by a weighted integrtion over its values for noninteracting nucleons movinga fluctuating auxiliary field:kVls ; TrsVUsdyTrsUsd,where the single-particle evolution operator associawith an auxiliary fields is Us . In particular,Psqd ~P

s kQ

m dsQm 2 qmdls , but this is difficult to calculatesince it requires all momentskQn

mls. The method usedin a recent SMMC study of deformed nuclei [5] follow

n

f

e

.

s

-net

llctf]l.

d

the prescriptionPsqd ~P

s

Qm dskQmls 2 qmd, which

is valid provided kQnmls ­ kQmln

s. We improve uponthis prescription by explicitly forcing the quadratic moment to be correct. To that end, we calculatesD2

sdmn ­kQmQnls 2 kQmls kQnls for each sample and assumthat higher cumulants vanish. Then (in matrix notation

Psqd ~Xs

1detDs

exp

∑2

12

skQls 2 qdT

31

D2s

skQls 2 qd∏

. (4)

In order for (4) to define a shape distribution, wmust associate a deformationa2m with each quadrupolemomentkQmls . Starting from a deformed nucleus,R ­R0s1 1

Pm amY

p2md, we calculate its quadrupole mome

by expanding its densityrsrd ; r0sr 2 Rd (wherer0 isthe spherical density) to first order in deformation. Wthen find

kQmls ­ 4R0

µZ `

0r3r0srd dr

∂a2m , (5)

wherer0 is calculated as below Eq. (3), but only withthe valence shell. Thea2m are then transformed by rotation to the intrinsic frame wherea20 ­ bcosg anda22 ­ a222 ­ b singy

p2, and (4) is used in the intrin

sic frame to findPsb, gd.Typical shapes and their standard deviation are show

the inset of Fig. 1. Rather than showing directly the shadistributionPsb, gd, we convert it to a free energy surfacthroughFsb, g; Td ; 2T lnfPsb, gdyb3j sin3gjg, wherethe unitary metric

Qm da2m ­ b4j sin3gjdbdgdV has

been assumed [14]. Such free energy surfaces are shin Fig. 1 for 128Te and 124Xe at different temperaturesThese nuclei are clearlyg-soft, with energy minima atb0 , 0.06 andb0 , 0.15, respectively. Energy surfacecalculated with Strutinsky-BCS using deformed WoodSaxon potential [15] also indicatesg-softness with valuesof b0 comparable to the SMMC values. These calculatiopredict for124Xe a prolate minimum withb0 ø 0.20 whichis lower than the spherical configuration by 1.7 MeV, bis only 0.3 MeV below the oblate saddle point, and f128Te a shallow oblate minimum withb0 ø 0.03. Theseg-soft surfaces are similar to those obtained in the Osymmetry of the IBM, or more generally, in cases whethe Hamiltonian has mixed U(5) and O(6) symmetriesa common O(5) symmetry. In the Bohr Hamiltonian,O(5) symmetry occurs when the collective potential enedepends only onb [1]. Our results are consistent withpotential energyV(b) that has a quartic anharmonicity [2but a negative quadratic term that leads to a minimumfinite b0.

We have also estimated totalE2 strengths fromkQ2lwhere Q ­ epQp 1 enQn is the electric quadrupoleoperator with effective charges ofep ­ 1.5e and en ­0.7e, and extractedBsE2; 0 ! 21

1 d, assuming that mos

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VOLUME 77, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 AUGUST 1996

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FIG. 1. Free energy surfaces for128Te and 124Xe in theb-g plane at several temperatures. The contour linesseparated by 0.3 MeV and the lighter shades corresponlower energies. Notice theg softness of the surfaces. Thsurfaces are deduced from microscopically calculated shdistributions (see text). The inset illustrates a few typishapes sampled in the SMMC and their standard deviation124Xe at T ­ 1y3 MeV).

of the strength is in the211 state. We findBsE2; 0 ! 21

1 dvalues of1314 6 10, 2856 6 15, and7248 6 36 e2 fm4

to be compared with the experimental values [16]1660, 3830, and14 900 e2 fm4 for 124Sn, 128Te, and124Xe, respectively. The effective charges used aretheoretical estimates of [11] and are larger than thstandard values because of a low-lying0hv peak in theisoscalar strength function which is outside our mospace. The SMMC calculations reproduce the qualitatrend; the quantitative discrepancy would be reduhad we used a renormalization factor similar topotential field renormalization discussed above. A bequantitative agreement might be achieved by includadditional subshells [17] and an isovector quadrupinteraction.

Information on excited states in SMMC can be obtainfrom strength functions. The energy centroid is givby E ­ S1yS0, whereSn is the nth moment of the relevant strength function, and can be calculated fromfirst logarithmic derivative of the imaginary time responfunction. We calculated the21

1 excitation energy thisway from theE2 response function. The values found1.12 6 0.02, 0.96 6 0.02, and0.52 6 0.01 MeV shouldbe compared with the experimental values of 1.131, 0.7

1446

eto

elr

and 0.354 MeV for124Sn, 128Te, and124Xe, respectively.The discrepancy for128Te and 124Xe is due to contribu-tions from the excited states at the finite temperature uin the calculationssT ­ 0.2 MeVd.

Another signature of softness is the response ofnucleus to rotations. We add a cranking fieldvJz tothe Hamiltonian and examine the moment of inertia afunction of the cranking frequencyv. For a soft nucleuswe expect a behavior intermediate between a defornucleus, where the inertia is independent of the crankfrequency, and the harmonic oscillator, where the inebecomes singular. This is confirmed in Fig. 2 whishows the moment of inertiaI2 ­ dkJzlydv ­ skJ2

z l 2

kJzl2dyT for 124Xe and 128Te as a function ofv, andindicates that128Te has a more harmonic character. Tmoment of inertia forv ­ 0 in both nuclei is significantlylower than the rigid body value (ø43 h2yMeV for A ­124) as a result of pairing correlations.

Also shown in Fig. 2 arekQ2l where Q is the massquadrupole, the BCS-like pairing correlationkDyDl forthe protons, andkJzl. Notice that the increase inI2 asa function of v is strongly correlated with the rapidecrease of pairing correlations, and that the peaks iI2are associated with the onset of a decrease in collect(as measured bykQ2l). This suggests band crossing alothe yrast line associated with pair breaking and alignmof the quasiparticle spins atv ø 0.2 MeV skJzl ø 7hdfor 128Te andv ø 0.3 MeV skJzl ø 11hd for 124Xe. Ourresults are consistent with experimental evidence of bcrossing in the yrast sequence of124Xe around a spin

f

er

led

r

e

e

,

FIG. 2. Observables for124Xe and 128Te as a function ofcranking frequencyv and for two temperatures.I2 is themoment of inertia,Q is the mass quadrupole moment,D isthe J ­ 0 pairing operator, andJz is the angular momentumalong the cranking axis.

VOLUME 77, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 AUGUST 1996

inhngoth

ola

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onscleor

e

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)

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e-R-5-

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E.

nd

t

E.

n-

on,

hys.

of 10h [18]. The alignment effect is clearly seenthe behavior ofkJzl at the lower temperature, whicshows a rapid increase after an initial moderate chaDeformation and pairing decrease as a function of btemperature andv.

We have analyzed the number of correlated pairsthese nuclei in their ground state. For a given angumomentumJ, we define the pair operatorsA

yJMsabd ­

1yp

1 1 dab fayja

3 ayjb

gJM . These operators are bosolike in the sense that they satisfy the expected commutarelations in the limit where the number of valence nucleis small compared with the total number of single-partistates in the shell. In the SMMC we compute the pair c

relation matrix in the ground stateP

M kAyJMsabdAJM scddl,

which is a Hermitian and positive-definite matrix in thspace of ordered orbital pairs (ab) (with a # b). This ma-trix can be diagonalized to find the eigenbosonsB

yaJM ­P

ab caJ sabdAyJM sabd, wherea labels the various boson

with the same angular momentumJ. These eigenbosonsatisfy X

M

kByaJMBgJMl ­ naJdag , (6)

where the positive eigenvaluesnaJ are the number ofJpairs of typea. We have calculatednaJ in the variouspairing channels for124Xe. Since the number of neutronin 124Xe is above 66, neutrons are treated as holes.J ­ 0 and J ­ 2, we can compare the largestnaJ withthe number ofs andd bosons obtained from the O(6) limof the IBM. In the latter we use the exact O(6) formu[3] for the average number of pairs and multiply by trelative fraction of protons and neutrons to find the pcontent for each type of nucleon. For124Xe the SMMC(IBM) results in the proton-proton pairing channel are 0(1.22) s sJ ­ 0d pairs, and 0.76 (0.78)d sJ ­ 2d pairs,while in the neutron-neutron channel we find 1.76 (3.67spairs and 2.14 (2.33)d pairs. For the protons the SMMCd to s pair ratio 0.89 is close to its O(6) value of 0.6However, this ratio for the neutrons, 1.21, is intermedibetween its O(6) value of 0.64 and its SU(3) value of 1.and is consistent with the neutrons filling the middle ofshell. The total numbers ofs and d pairs—1.61 protonpairs and 3.8 neutron (hole) pairs—are below the IBvalues of 2 and 6, respectively. Our fermion-based SMcalculations thus indicate pair correlations forJ . 2.

In conclusion, we have presented the first microscoevidence of softness in nuclei in the 100–140 m

e.

fr

n

-

r

,

cs

region, using the SMMC for the full 50–82 major sheFuture work will include an isovector quadrupole forcHowever, such calculations are more time consumingthis interaction violates the Monte Carlo sign rule.

This work was supported in part by the U.S. Dpartment of Energy, Contracts No. DE-FG-0291-E40608, No. DE-FG0690-ER40561, and No. DE-AC096OR22464, and by the National Science FoundatGrants No. PHY90-13248 and No. PHY91-15574. Coputational cycles were provided on the IBM SP2the Maui High Performance Computing Center. Wacknowledge useful discussions with F. Iachello,Nakada, and A. Ansari. Y. A. acknowledges the hospitity of the Institute for Nuclear Theory at the University oWashington where part of this work was completed.

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[6] M. Dufour and A. Zuker, LANL archives ReporNo. nucl-thy9505010.

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[8] B. Lauritzen and G. F. Bertsch, Phys. Rev. C39, 2412(1989).

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[10] D. R. Bes and R. A. Sorenson, Adv. Nucl. Phys.2, 129(1969).

[11] H. Sagawa, O. Scholten, B. A. Brown, and B. H. Wildethal, Nucl. Phys.A462, 1 (1987).

[12] W. Nazarewicz, J. Dudek, R. Bengtsson, T. Bengtssand I. Ragnarsson, Nucl. Phys.A435, 397 (1985).

[13] A. Bohr and B. Mottelson,Nuclear Structure(Benjamin,New York, 1969), Vol. I.

[14] Y. Alhassid and B. Bush, Nucl. Phys.A509, 461 (1990).[15] R. Wyss and W. Nazarewicz (private communication).[16] S. Ramanet al., Atomic Data Nucl. Data Tables36, 1

(1987).[17] A. P. Zuker, J. Retamosa, A. Poves, and E. Caurier, P

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