ReviAta Mexicana de Fúica 39, Suplemento £ {1993} 190-£00

Nuclear detection of galaetic dark matter*

STUART PITTEL, JONATHAN ENGELBartol Research Institute, University of Delaware

Newark, DE 19716, USA

AND

PETR VOGELDepartment of Physies, California lnsitute of Teehnology

Pasadena, CA 91125, USA

ABSTRACT. The elastic scattering of weak1y interacting dark matter particles (assumed to be"neutralinos" -neutral fermions predicted by supersymmetry theories) from nuclei is described.The cross sections associated with this process will govern count rates in laboratory experiments.Particular empbasis is given to a proper description of the structure of the detector nuclei. A briefdiscussion of the prospects for detecting neutralinos in the next few years is given.

RESUMEN. Se describe la dispersión elástica en núcleos, de partículas que interaccionan débilmenteen materia obscura (bautizados Ilneutralinos" -fermiones neutros predichos por theorías desupersimetrías). Las secciones asociadas con este proceso governarán las tazas de conteo enlos experimentos de laboratorio. Se hace énfasis particular en una descripción apropiada de laestructura de los núcleos detectores. Se da una breve discusión de las posibilidades para detectarneutralinos en los próximos años.

PACS: 11.30.Pb; 14.80.Hv; 95.30.Cq

1. INTRODUCTION

Dark matter is the invisible mal erial in galaxies, clusters of galaxies, and on even largerscales. A growing body of evidence [1] suggests that dark matter is significantly moreabundant than the familiar luminous material in slars, elouds of dust, ele. Much of thissubject is still shrouded in mystery, however, and eontemporary physics is ehal!enged bysuch basie questions as: what is the dark matter made of, how much is there, and is itthe same on al! scales?There is currently a majar experimental elfort lo build ultra-sensitive terrestrial de-

tectors aimed at "seeing" dark matter. The basic idea of these experiments is that darkmatter should scatter elastical!y from nuelei in the detectors, albeit with ineredibly lowcount rates. Detectors are currently under development with 73Ge (Ref. [2]), 29Si (Ref. [3]),93Nb (Ref. [4]), 131Xe (Ref. [5]), and 1271+ 23Na (Ref. [6)) as important elements. Clearly,if we wish to interpret the data that come out of these experiments and perhaps even aidthe experimentalists in choosing the optimum detector material, it is essential that we

°Talk presented by S. Pittel at the XVI Symposium on Nuclear Physies, 5-8 January 1993, Oax-tepec, Mexico.

NUCLEAR DETECTION OF GALACTIC DARK MATTER 191

know with reasonable confidence how dark matter candidates interact with nuclei. Therehas been much theoretical elfort on this front in recent years; in this paper, we briellyreview the progress that has been made. A more detailed description, including a morecomplete list of references, can be found in [7].The paper is structured as follows: Section 2 touches brielly on the evidence for dark

matter within galaxies, and summarizes basic properties of "neutralinos" -neutral fer-mions predicted by supersymmetry- which we will assume constitute most of the darkmatter. Section 3 describes the steps required to determine how neutralinos scatter e-lastically from nuclei and presents the relevant formalismo Sections 4 and 5 describecalculations of the spin content of light and heavy nuclei, respectively, which is thecrucial nuclear physics input to neutralino-nucleus cross-sections. Finally, in Section 6,we comment briefly on the prospects for detecting neutralinos within the next few years.

2. DARK MATTER AND NEUTRALINOS

2.1. Evidence lor ga/actic dark matter

Observations suggest that dark matter is present on all scales, constituting up to 99% oflhe lolal mass in lhe universe [1]. Here, however, we concentrale only on the dark matterwithin galaxies, since this is the material thal may register in lerreslrial delectors. Indeed,there now exists parlicularly compelling evidence lhal galaxies are merely the centers ofmuch larger halos filled with dark matler. Analyses of red shifts within galaxies lead to"rotation curves" -plOlS of velocily versus dislance from lhe galactic center for luminousobjects. Beyond the edges of the galaxies, where only a few scattered luminous objectsexist, these curves would fall olf roughly like V1fr if all the maller in the galaxies werevisible. In faet, the curves show no sign of such a drop, implying mass distributions thalextend well beyond the light distributions. It follows thal al least 70-90% of the matterwilhin galaxies is dark.

2.2. Neutm/inos

Primordial-nucleosynthesis calculations imply that al leasl some of the dark matler inthe universe is not in the form of baryons, allhough within galactic halos it may well be.Baryonic dark maller, however, would have to take unusual forms, e.g. very large planets,blaek holes, etc. Since it is not clear why the spatial distribution of lhese objects shouldbe so dilferent from that of ordinary matter, galactic dark matter is often assumed lolake lhe form of stable Weakly-Inleracting Massive Particles (WIMPs). Unforlunately,no such objecls are known lo exisl, and the leading candidate is therefore a hypolheticalparticle predicted in supersymmelric extensions of lhe slandard model [8).Supersymmetry (SUSY) supplies each of lhe known parlicles with a partner of opposite

stalistics. The boson parlners of the known fermions are referred to by lhe same namebut with an s added at lhe fronl, e.g. the squark (a). The fermion partners of the knownbosans are likewise referred lo by lhe same name, but now wilh an ino atlaehed al lheend, e.g. the higgsino (il).

(1)

192 STUART PITTEL ET AL.

The partners oC neutral, colorless particles are obviously the only viable dark-mattercandidates. Qne oC these, the sneutrino (the partner oC the neutrino) has already been ruledout by underground observations. The remaining possibility is the "lightest neutralino"::-a linear combination oC the photino (::Y), the zino (Z), and two higgsinos (H¡ andH2). (The reason there are two higgsinos is that supersymmetric theories must have atleast two neutral Higgs bosons.) The stability oC the lightest combination is ensured bya discrete symmetry ca1led R-parity, which Corbidsthe decay oC an "-ino" purely intoordinary particles. There is very little in the way oC constraints on the mass oC the lightestneutralino. LEp data [91 suggest that it is heavier than about 45 GeV. Furthermore, iCsupersymmetry is to make sense in the context oC particle theory, the maximum massshould most likely be less than a CewTeV. From this point on, we will assume that theseparticles do exist in this mass range, and discuss the physics required to predict theircount rates in detectors on earth.

3. NEUTRALINO INTERACTIONS

Three steps are required to model neutralino-nucleus elastic scattering:

• we must first model the interaction oC neutralinos with quarks;

• we must then Cold in the quark structure oC the nucleon to determine how neutralinosinteract with nucleons;

• fina1ly,we must Cold in the structure oC the target nucleus.

We will only brie!ly review the essential results oC the first two (particle-physics) stepsoC the analysis and then turn to the central Cocus oC this presentation, how to properlytreat the structure oC the target nucleus.Given a SUS Y theory, the effectiveneutralino-quark Lagrangian can be readily obtained

Croma Cewsimple Feynman diagrams. The end result is a Lagrangian containing twoterms,

LN = 2~a,J d4x [X¡I'¡5X :r;(x) +XXS(x)],

where :r;(x) = L Aq~q(x)¡I'/51/Jq(x), and S(x) = L Sq(mq/MW)~q(x)"'q(x) are theaxial-vector and sc~lar hadronic currents, respectively,qX is the neutralino field, and qdenotes the quark !lavor (u, d or s). The first term arises CromZ and squark exchangeand the second primarily CromHiggs exchange. 1'he coeflicients that enter the effectiveLagrangian depend on the masses oC the exchanged particles and the parameters oC theSUSY theory, about which virtua1ly nothing is known though sorne bounds have been setoThe neutralino-nucleon interaction is determined by evaluating the matrix elements oC

the two hadronic currents, :r;(x) and S(x), between one-nucleon states. Both require in-Cormationon the quark structure oC the nucleon, which can to a reasonable approximationbe extracted [7]Cromexperimental data.

NUCLEAR DETECTION OF GALACTIC DARK MATTER 193

Accepting this, we now turn to the question of how neutralinos interact with nuclei.In impulse approximation, the elastic scattering cross section for momentum transferq is determined by two structure functions, one for axial scattering and one for scalarscattering,

(2)

where SA(q) and Ss(q) are defined in [7].Note that the two contributions enter incoher-ently.The maximum momentum that can be transferred from the neutralino to the target

nucleus is qmax= 2¡.tv, where ¡.t is the reduced mass of the neutralino-nucleus system and vis the neutralino velocity. Earlier wegave limits on the neutralino mass. Its average velocityshould be characteristic of our galactic halo, roughly 10-3 c. Using this information, wecan readily show the following:1. For light enough targets, say A ::;30, qmaxis much less than the inverse size of thetarget nucleus. In such cases, the target nucleus appears as a point particle to theprojectile and )Veonly need consider the q = Obehavior.

2. For heavy enough targets, the maximum momentum transfer can be of the same orderas the inverse size of the nucleus. In such cases, we must include finite-q form-factorelfects.With these points in mind, it is useful to isolate the q = Obehavior of the two structure

functions from their finite-q behavior. At q = O,the axial structure function reduces to avery simple form,

1 l' . 128A(O)= 471" (ao + a¡)(JII8pIlJ) + (ao - aI)(JII8nIlJ) , (3)

which depends only on the total spin carried by the neutrons and the protons of thetarget nucleus. Since individual nucleon spins tend to cancel pairwise, the total spinsdepend on the properties of a few nucleons near the Fermi surface only. For finite-q, theaxial structure function describes how the neutron and proton spins are distributed overthe nucleus.The scalar structure function at q = Olikewiseassumes a very simple form,

8s(O) = 2J + 1 ~A2 .471"

(4)

The coefficient ca is much smaller than the corresponding axial coefficients no and al.Nevertheless, since the scalar structure function depends coherently on all of the nucleonsin the nucleus, scalar scattering can be important for heavy-enough nuclear targets. Atfinite-q, the q = O scalar structure function is modulated by the square of the Fouriertransform of the ground state density distribution of the target, which is relatively easyto model for any nucleus. Thus, scalar (or spin-independent) scattering can be treatedwithout any complicated nuclear structure analysis.

194 STUARTPITTEL ET AL.

In the remainder oC this presentation, we will discuss methods that have been devel-oped to treat axial (or spin-dependent) scattering. Here, nuclear structure does play animportant role through the spin structure oC the target nucleus.

4. THE SPIN CONTENT OF LIGHT NUCLEI

To describe spin-dependent elastic scattering by light nuclei, we only require inCormationon the total spins carried by the neutrons and the protons. The earliest calculations [lO] oCthese quantities relied on an extreme single-particle model (SPM) description oC the targetnucleus. Here, the basic assumption is that nucleons pair off to zero angular momentum(and zero spin), so that in an odd-mass nucleus the total nuclear spin arises Crom thesingle remaining unpaired particle.

UnCortunately, the SPM is not appropriate to describe the spin properties oC mostnuclei. For the related magnetic dipole operator, the SPM gives rise to the Schmidtmoments. Even Cornuclei with only a single particle or hole outside a doubly-magic core,the experimental magnetic moments do not lie on the Schmidt lines, and the discrepanciesare even more pronounced away Crom closed shells [11] Many effects are now knownto modiCy the Schmidt predictions [11]. In nuclei in which the closed core is not spinsaturated, MI polarization oC the core by the odd particle plays a significant role. AwayCrom closed shells, core-polarization effects are supplemented by configuration mixingwithin the valence space. Finally, at some level meson exchange currents also contribute.

There is a simple way to improve on the SPM spin estimates without having to resortto detailed nuclear structure calculations. The method goes under the name oC the OddCroup Model (OCM) [12), and was first applied to q = O neutralino elastic scatteringby Engel and Vogel [131. The basic assumption oC this model is that in an odd-massnucleus, containing an even number oC one type oC nucleon and an odd number oC theother, the even group does not contribute to either the orbital or spin angular momenta.The structure oC the odd group, which does contribute, is not defined in detail; rather, itsproperties are fit to known data.

To see how this works, consider the magnetic moment operator,

ji. = ji.p + ji.n = :L (g~Lp + g;Sp),p=p,n

which can be rewritten in terms oC operators Corthe odd (o) and even (e) groups as

(5)

(6)

Evaluating the expectation value oC ji. in the state with M = J and neglecting the contri-bution oC the even group, we obtain Corthe z-component oC the odd-group spin

(7)

NUCLEARDETECTIONOFGALACTICDARKMATTER 195

TABLE1. The spin content of severa!candi-date detector nueleibased on the OddGroupModel. In parentheses are given the corre-sponding results of the extreme Single-Par-tiele Model.

Sp SnI.p 0.46 (0.50) O (O)"CI -0.15 (-0.30) O (O).3Nb 0.36 (0.50) O (O)2.Si O (O) 0.15 (0.50)73Ge O (O) 0.23 (0.50)

TABLEn. The spin content of severa!stablemirror nuelei of potentia! relevance to darkmatter detection, eva!uated using the EOG-M.

Sp Snl.p 0.415 -0.047"Cl -0.094 0.0142.Si 0.054 0.204

Thus, knowing the magnetic moment of the system, its total angular momentum, andthe g-factors, we can estimate the spin content of the odd group and thus of the entirenueleus.In Table 1, we compare the results obtained in the OGM and the SPM for sorne odd-

mass nuelei, most of which are of potential importance to dark-matter detection. Clear1ythere are substantial differences. In 29Si, the difference is more than a factor of three,which translates into roughly an order of magnitude difference in the elastic scatteringcross section. These results illustrate the importance of correlation effects in the nuelear-spin matrix elements; knowledge of such large factors is essential in attempts to interpretdark-matter experiments.In sorne nuelei, it is possible, because of the existence of additional data, to eliminate

the assumption that the even group does not contribute. This can be done wheneverthe target nueleus of interest is part of a mirror pair (related to one another solely bythe interchange of protons and neutrons) with Z = N :i: 1. For each .such pair, threerelevant pieces of data often existo magnetic moments in each of the two nuelei, and theGamow-Teller beta decay from one to the other. The Jt-value for the GT decay is relatedto the difference in the odd and even group spins by

n2(s -S )2= [6170 _ ] ~o e Jt 1 J + 1' (8)

where R = gA/gV is the ratio of the axial-vector to the vector weak-interaction couplingcoefficients and is generally thought to be about 1.0 in nueleL The isoscalar magneticmoment is likewise related to the spin of the even and odd groups by

JlIS == JlZ,N + JlN,Z = J + 0.76(So + Se) + Jlx, (9)

where JlX is a small mesan exchange contribution that can be reliably estimated theoret-ically. Eqs. (8) and (9) can be solved for both the odd and even group spins. We refer tothis method as the Extended Odd Group Model (EOGM) and show sorne representativeresults in Table n. In all the cases considered, the odd group indeed carries most of thespin and the renormalization of the SPM results is larger for the odd group than for the

196 STUART PITTEL ET AL.

even grOUp. Comparing Tables 1 and n, we see that in general the OGM and EOGMprescriptions give similar results.The EOGM results, when available, should be quite reliable. When they are not

available, we can still use the OGM to get reasonable estimates. Alternatively, we canuse detailed microscopic calculations, based for example on the Shell Model. Two suchcalculations have been reported [14] that are of relevance to axial neutralino scatteringby light nuelei. Overall, they lead to results in qualitative accord with those obtainedphenomenologically.

5. SPIN DISTRIBUTIONS OF HEAVY NUCLEI

To describe axial scattering by heavy nuelei, we require information not only on the totalspins carried by the neutrons and protons in the nueleus but also on their distributionsover the nueleus. We cannot extract this information directly from experiment and thusmust resort to detailed microscopic calculations.We will describe two sets of calculations for the spin response of heavy candidate

detector nuelei, one for a l3lXe target nueleus [15] and the other for a 93Nb target [16].We will also comment on calculations that are currently underway for the two otherimportant heavy detector candidates, 73Ge and 1271.

5.1. The Nucleus 13l X e

The nueleus 131Xe has 54 protons and 77 neutrons and is to a good approximation aspherical system dominated by pairing correlations. A natural way to describe pairing isthrough the BCS approximation. In such a treatment, the odd-A ground state is repre-sented as a single quasiparticle built on a fully-paired zero-quasipartiele (Oqp) even-evencoreo In 131Xe, the relevant Iqp state can be expressed as vd

t 10), where vJ creates3/2 3/2a neutron quasipartiele in the dominant 2d3/2 orbit and 10) represents the quasiparticlevacuum of the even-even coreoTo the extent that additional correlations contribute to the ground state they should

do so predominantly in the form of 3qp admixtures, of which there are two possibilities:

[ ]3/2

vt ["t "t]K 10).d3/2 k l

The QTDA permits mixing of the Iqp and 3qp configurations through the quasipartieleHamiltonian.The calculations of Ref. [151 used a model space and effective Hamiltonian that had

been developed earlier to treat the response of 1271to solar neutrinos [17). The amplitudesof admixed 3qp components in the ground state wave function turned out to be verysmall (typically less than about 0.05), but nonetheless they had a significant effect onthe magnetic moment and spin. In particular, they quenched the magnetic moment from1.15 nm to 0.70 nm, in almost perfect agreement with the experimental value of 0.69 nm.

NUCLEARDETECTIONOF GALACTICDARKMATTER 197

1

.8

.•...... .6c''-'-< .4tIl

.2

OO .02 .04 .06

2 (Gey2 /C2)q

FIGURE1. Sample results for the axiaJ structure function SA versus q2 for neutraJino scatteringon 131Xe. The dashed line is the 1qp prediction; the solid line is the full resulto

In Fig. 1, we show a sample axial structure function SA(q) (for a specific choice of theneutraJino structure), ineluding for comparison the pure 1qp results. The large differenceat small values of q reflects the coherent role of 3qp admixtures in quenching the totalspin. However, by a momentum transfer of q2 "" 0.01 GeV2, the effect of correlations iswashed. out and the full results are virtually indistinguishable from the 1qp estimates.

5.2. The Nucleus 93Nb

The nueleus 93Nb has Z = 41 and N = 52 and thus is only a few partieles from doubly-magic 88Sr. As such, it should not exhibit any pronounced collective behavior and shouldbe amenable to a relatively complete shell-model treatment [161.The dominant components of the J~= 9/2+ ground state involve configurations with

the 3 valence protons in the 2PI/2 and 199/2 orbits and the two valence neutrons in the2ds/2 orbit. This small space involves 20 states.As in 131 Xe, small admixtures of configurations involving 1p-1h excitations built on

the dominant configurations can cause a coherent quenching of spin and MI propertiesand must be ineluded. \Ve thus also considered a large space in which one-proton orone-neutron excitations (either from one of the valence orbits or from the 199/2 neutroncore orbit) into any orbit of the sdg shell were ineluded. The resulting space consisted ofroughly 2050 sta tes with J~ = ~+. Here too we assumed an effective Hamiltonian thathad been shown to be appropriate in earJier studies [18].The 1p-1h excitations ineluded in the large-space calculation contributed small ad.

mixtures to the ground state but nevertheless had a significant effect on the magneticmomento The SPM result was 6.79 nm, the small-space result 6.36 nm and the fulllarge-space result 5.88 nm. \Vhile the full result is somewhat smaller than the experimentalvalue of 6.17 nm, a discrepancy in that direction is not too bothersome. It is known [11)

198 STUAIIT PITTEL ET AL.

that meson-exchange currents can renormalize orbital proton g-factors upwards by about10%, raising the calculated magnetic moments without affecting the spin. The fulI wavefunctions therefore appear reasonable for describing the spin properties of 93Nb.The calculated axial structure functions [161 exhibit sorne features that are similar

to those found in 131Xe. In particular, the large-spaee results falI off rapidly with q andapproach the smalI-space results asymptoticalIy. However, the fulI axial structure functionalways differs from the extreme single-particle version, even at large q. It is only spinpolarization of the core that dies off at large q. Valence-space correlations do not, sincethey are nonperturbative.

5.3.3 Still at large: 73Ge and 1271

The most important remaining calculation is the fulI response of 73Ge, which is the activeelement in the detector furthest along in development. Purely microscopic treatments of73Ge are exceedingly difficult. There is good evidence for coexisten ce between sphericaland 'Y-soft deformed shapes in this nucleus, a situation that is too complicated for mostmicroscopic methods. A large-space shelI-model calculation [19] has recently been reportedfor this nucleus, but it does not reproduce the observed magnetic momentoWe are currently pursuing an extension of the Interacting Boson Fermion Model [201

which in principIe can describe the colIective dynamics of 73Ge, including the quenchingof its magnetic moment through spin polarization. The basic idea is to extend the usualIBFM description of odd-mass nudei (whereby a collective odd-mass nudeus is describedas an even-even core of N bosons plus a single quasifermion) to indude configurationsinvolving N - 1 bosons and three quasifermions. The three-quasifermion configurationsincorporate the physics of spin polarization, induding its radial distribution. Similar workhas been reported recently in the context of high-spin phenomena [211. Sorne modificationsare required for our analysis, however. In particular, the interaction that mixes the one-and three-quasipartide configurations must be modified somewhat to properly reflect thephysics of spin polarization. These calculations are currently underway in colIaborationwith Dario Vretenar and should be ready to produce results soon.The detector candidate 1271is also awaiting an adequate treatment. Here, however,

prospects are much better; this nudeus was considered as a neutrino detector in' [171and the methods used th, re -wit;; sorne conceptualIy straightforward modifications toincorporate low-Iying quan.-upole phonons- should be sufficient.

6. PROSPECTS FOR DETECTION

The quantity of interest for detecting neutralinos in terrestrial experiments is the expectedcount rate, usualIy quoted as the number of counts per kilogram of detector material andper month of detecting time. To obtain the count rate from the elastic cross section, wemust fold in information on the neutralino density alld velocity distribution and also on thedetector threshold. A recent discussion of detector technologies, thresholds, backgrounds,etc. appears in [22].

NUCLEAR DETECTlON OF GALACTIC DARK MATTER 199

The event rates obviously depend on the supersymmetry parameters, a discussion ofwhich can be found elsewhere [8]. Here we simply assert that for a large range of theseparameters, count rates in sorne of the new detectors (Nb for example) are expected to beon the order of a few per month per kilogram of material, enough to be detected accordingto the techno-experts. In other regions (particularly for very heavy neutralinos) the ratescan fall to .01 per month per kg or less. If these are the parameters of the real world,terrestrial experiments will not see dark matter in the near future.As such, it is not certain that the new detectors will be able to see galactic neutralinos,

even if they do existo In spite of this, the searches will no doubt go on. And the reasonis clear; if they are successful the payoff for contemporary physics will be enormous.Detection of galactic neutralinos will not only solve the dark-matter problem, but at thesame time by demonstrating supersymmetry will help remove sorne of the arbitrarinessthat currently exists in the standard mode!.

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under grant #PHY.9108011and by the U.S. Department of Energy under contract #DE-F603-88ER-40397.

REFERENCES

1. S. Tremanine, Phys. Today 45, No. 2 (1992) 28.2. B. Sadoulet, in: Proceedings o/ the 4th International Con/erence on Low Temperatures, Dark

Matter, and Neutrinos, ed: N. Booth, Oxford, UK (September 1991).3. B.A. Young et al., Nucl. Inst. Meth. A311 ( 1992) 195.4. R.J. Gaitskell et al., Physica B169 (1991) 445.5. P. Belli et al., Il Nuovo Cimento AI03 (1990) 767.6. BRS collaboration, C. Baeci et al., in: Proceedings o/ the Second International Workshop on

Theoretical and Phenomenological Aspeds o/ Underground Physics, Toledo, Spain (September1991).

7. J. Engel, S. Pittel and P. Vogel, Int. J. Mod. Phys. El (1992) 18. H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 75.9. L3 collaboration, B. Adeva et al., Phys. Lett.B236 (1990) 109; ALEPH collaboration, D.

Decamp et al., Phys. Lett.B236 (1990) 86.10. M.W. Goodman and E. Witten, Phys. Rev.D31 (1985) 3059; J. Ellis and R. Flores, Nucl.

Phys.B307 (1988) 883.11. B. Castel and I. Towner, Modern Theories o/ Nuclear Moments, Oxford, Clarendon Press

(1990).12. A. de Shalit, Phys. Rev.91 (1953) 83; B. Buck and S.M. Perez Phys. Rev. Lett.50 (1983) 1975.13. J. Engel and P. Vogel, Phys. Rev.D40 (1989) 3132.14. A.F. Paeheco and D.D. Strottman, Phys. Rev.D40 (1989) 2131; B.A. Brown, Nucl. Phys.A522

(1991) 221c.15. J. Engel, Phys. Lett.B264 (1991) 114.16. J. Engel, S. Pittel, E. Ormand and P. Vogel. Phys. Lett.B275 (1992) 119.17. J. Engel, S. Pittel and P. Vogel, Phys. Rev. Lett.67 (1991) 426.18. P. Federman and S. Pittel, Phys. Lett.B77 (1978) 29.

200 STUART PITTEL ET AL.

19. K.T. Ressell, in: Proceedings o/ the First Symposium on Nuclear Physics in the Universe, OakRidge, TN, USA (September 1992).

20. F. Iachello and P. van IsacKer, The Interacting Boson-Fermion Model, Cambridge: CambridgeUniversity Press (1991).

21. F. laehello and D. Vretenar, Phys. Rev.C43 (1991) 945.22. P.F. Smith and J.D. Lewin, Phys. Rep. 187 (1990) 203.