Revista Mexicana de Física 43, Suplemento 1 (1997) 58-68

Shell rnodel Monte Carlo studies of nuc1ei

D.J. DEANPhysics Division, Oak Ridge National Laborat07'y

Oak Ridge, TN 37831-6373 USA

ABSTRACT. The pair content and structure of nuclei Jlear N = Z are described in the frameworkof shell-model Monte CarIo (SMMC) calcnlations. Results include the enhancement of J = O,T = 1 proton-ncutroll pairing at N = Z nuclei, ano the lIlarkcd diffcrcIlcc of therlllal propertiesbetween cven-even and odd-odd N = Z nuclei. Additionally 1 present a study of the rotationalproperties of the T = 1 (ground stateL and T = abano mixing seell in 74R.b.

RESUMEN. El contenido de pares y la estructura de los nüclcos cerca de N = Z son descritos enel contexto de cálculos en el modelo de capas Monte Cario (SMMC). Los resultados incluyen elaumento del apareamiento protón-neutrón J = O, T = 1 para mícleos con N = Z, y la marcadadiferencia cn las propiedades térmicas entre los Ilücleos con N = Z par-par e impar-impar.Presento además un estudio de las propiedades rol acionales de la banda base T = 1 Y de lamezcla COIl la banda T = Oencontradas en 74 Rb.

rACS: 21.10.-k; 21.60.Cs; 21.60.Ka

l. INTRODUCTION AND FORMALISM

As ncw radioactivc beam facilities beco me opcrational a wealth of informatioIl 011 nucleiuear the N = Z line has emerged. These experimeutal facilities will coutinue to inereaseom uuderstaudiug of the iu medium isoscalar (T = O) aud isovec!.or (T = 1) protou-neutro u (1"') iuteractiou. Several physical featmes iu these lIuclei are of illteres!. Iu theirgrouIld statcs1 eVCIl-cvell nuclei are domillated by isovedor pair corrclatiolls. In the ab-SCllce of isospill-brcaking forces likc the Coulomb interactioll1 isospill syulIlletry enforcesidentical proton-proton (pp), neutron-nentron (""), and 1m correlations in self-coujugatelluclei. With increasing neutron (or proton) excess, the isovector pH correlations decreasesharply amI the gronud state of even-evell nuclei near ¡1-stability is dominated by isovec-tor.J = O pairiug amoug like nucleons [1,2]. This is the reiLson why 1m correlations arerather difficnlt to explore in stable medium-mass nuclei which have a moderate neutronexccss.

011 the other hand, 1m pairing plays a particularly important role in odd-odd N = Znuclei. With the exception of 34Cl, all N = Z nuclei with JI :s 40 are dominated byisoscalar pairilll!; involving ]Jn pairs in idelltical orhitals. alld so hav(~ground states withT = 01 ,/ > O. lIowcver, the expcrimental ground state spins and isospins of virtuaHy aHodd-odd N = Z nuclei with JI > 40 are T = I and .J = O (lhe only known exception is;,RCU)l in<1icating lhe dominance of isovector pH pairing in these 11tIelei.

FlIrther experimental informalion, inclllding tlw idelltification of the spin. parity, andisospill of tlle grotlnd and excited states of nllelci I"rolll{¡t; As throllgh 98111, \....ill .shcd light

SHELL MODEL MONTE CARLO STUDIES OF NUCLEI 59

on the isospin structure of pairing for A > 60. At the present, 74Rb is the only odd-oddN = Z nucleus in that region that has been studied with any spectroscopic detail [3].Interestingly, there is a change in the dOlllinance between isovector and isoscalar pnpairing with rotational frequency. While the ground state band is identified as isovector(proposed to be the isobaric analogs of states in 74Kr), T = O states becollle energeticallyfavored at about 1.0 MeV of excitation energy.

In this contribution, I will attempt to describe these interesting properties using thetools of shell-lllodel Monte Cario (SMMC) [4,5]. SMMC offers an alternative way tocalculate nuclear structure properties, and is cOlllplementary to direct diagonalization.SMMC cannot find, nor is it designed to find, every energy eigenvalue of the Hamiltonian.It is designed to give thermal or ground-state expectation values for various one- andtwo-body operators. Indeed, for larger nuclei, SMMC may be the only way to obtaininformation on the thermal properties of the system from a shell-model perspective.We are interested in finding the partition function of the imaginary-time many-bodypropagator U = exp( -¡3H) where 13= lIT and T is the temperature of the system inMeV. We calculate expectation values of any observable 11 with

(11) = n- 0.-:1 .n-u (1)

Since H contains many terms that do not commute, one must discretize 13= N, /:;.13.Finally, two-body terms in H are linearized through the Hubbard-Stratonovich transfor-lllation, which introduces auxiliary fields over which one must integrate to obtain physicalanswcrs. The Inethod can be sUlnmarizcd as

Z = n-O = n-exp(-¡3H) --+ n- [exp(-/:;.¡3H)('

N,

--+ J D[a]G(a)n- rr exp [/:;.¡3¡,(anl] ,n:::;l

(2)

where an are the auxiliary fields (there is one a-field for each two-body matrix-elementin H when the two-body terms are recast in qnadratic form), D[a] is the measure ofthe integrand, G(a) is a Gaussian in a, ami h is a one-body Hamiltonian. Thus, theshell-model problem has been transformed frolll the diagonalization of a large matrix toone of large dimensional quadrature. Dimensions of the integral can reach up to 105

for rare-earth systellls, and it is thus natural to use Metropolis randolll walk lllethodsto sample the space. Such integration can 1ll0Stefficiently be perfortned on massivelyparallel computers. Further details are discussed in [5].

Realistic interactions often have a Monte Cario sign problem, that is they have com-plex actions. We have found that a certain class of Hamiltonians has good Monte CarIosign properties, and one performs calculations with these Hamiltonians. Fortunately innuclear physics good Hamiltonians, such as pairing plus quadrupole, are not too far re-moved frOln realistic Hamiltonians so that thc extrapolation is a gcntlc function of thecoupling constant, !J [6].

60 O.J.OEAN

The residual nuelear interaction builds up pairing correlations in a nueleus. Intro-ducing nuelcon creation operators al, these correlations can be studied by defining paircreation operators

(3)

for proton-proton or neutron-neutron pairs, and

(4)

for proton-neutron pairs where "+( -)" is for T = O (Tdefinitions, 1 construct a pair matrix

1) ¡Jn-paJrlng. With these

M~",= L(A~M(ja,jb)AJM(jc,jd)) ,M

(5)

where a = {ja,jb} and a' = (jc,jú and the expectation value is in the ground state orcano nical ensemble at a prescribed temperature. The pairing strength for a given J isthen given by

P(J) = L M~,,,"02:01

(6)

With this definition, Eq. (4), the pairing strength is non-negative, and indeed positive,at the mean-field level. The mean-field pairing strength, PMF(J), can be defined as inEqs. (3) and (4), but replacing tbe expeetation valnes of tbe two-body matrix elementsin the definition of M J by

(7)

wbere "k = (UkUk) is the oeeupation number of tbe orbital k. Tbis mean-field valueprovides a baseline against whieh tme pair correlations can be judged. Genuine paireorrc1ations can then be defined as those in excess of tbe mean-field values

Pcorc(J) = P(J) - PMF(J) . (8)

Note that this is a differcnt dcfinition of thc correlatioIls tha.n that uscd in Ref. 2; however,it leads to the same qualitative results.

In the following l will describe ground state pairing properties in nuelei in tbe massA = 40-80 range. [will then present SMMC results of thermal pairing properties in50MIl alld 5:¿Fc. alld discus.s rolational propcrties of 74Rb. 1 will cOllcludc with a fnlureolltlook fOl'lar¡o;eea!clllations within the SMMC framewOl'k.

SllELLMODELMONTECARLOSTUDIESOFNUCLEI 61

Fea) ~, Zn • -. Me, o Expt, ", ,

Ni,

~ ,Cr ,,, 4 ~~, " - , ,

, '- .~, ,Tie ..'a -&~ -\ - - , N=Z, .-~- - - -t. 'é:. ~.U- - '9c -. . .

-o - *,~ ~..'0 - :~." -

b) ¡ • •- - - - - -'.-. :f - - - - - - - - - - -~- - - ..- - - - - .

.' ' .• •'. .;

•

- - - - - - -'\\:/j' /: ......l/A

---- --- .• ,:'----.1(----- ---- -

•• •Ti •• • e, Me Error• • F•

• • Ni• • Zn •• • N",Z

1.0

;;-Q>

~ 0.0::;.2

-1.0

-20.0

;;-Q>

~ -40.0::;..,

-60.0

-2.044 46 48 50 52 54 56 58 60 62 64 66

A

FIGURE 1. Upper panel (a): Camparisan af the mass excesses !1M as calculated within theSMMC approach with data. Lawer panel (b): Discrepancy between the SMMC results far themass excesses and the data, ó!1M. The salid line shaws the average discrepancy, 450 keV, whilethe dashed lines shaw the rms variatian abaut this value (fram ReL 7).

0.0

.80.0

2. INTERACTIONS AND MODEL SPACES

SMMC calculatians af pi shell nuclei were perfarmed ta study graund-state [7) andthermal [8,9] praperties af nuclei using the Kua-Brown [10] interactian madified in themanapale terms [11]. (The interactian has sllbsequently been dllbbed KB3.) Excellentagreement between theary and experirnent was abserved, indicating the usefulness afthis interaction for pi shell nuclei up to A = 60. The KB3 interaction also successfullyreproduces many of the ground-state properties and spectra of nuc!ei in the lower pishell [7,12]. As an exarnple, Fig. 1 displays the SMMC calclllations of the mass excessesin this rnodel space, relative to the 40Ca core (top panel), and the deviation betweentheory and experiment (bottom panel). 1nvestigations of B(E2)'s, B(M1)'s, and Gamow-Teller operators were also carried out, and gaod agreement in comparison to experimentaldata has been demonstrated [5,7].

62 D.J. DEAN

FIGURE2. Calculated mass excessesare compared with experiment (top panel), and the differencebetween experiment and theory is shown (bottom panel).

In order to investigate heavier systems, the Og9/2 was ineluded in the pf modelspace [10]. However, we found that the coupling between the 017/2 and Og9/2 causes fairlysignificant center-of-mass contamination to the ground state, and we therefore elose the017/2' The model space is thus 0f5/2IpOg9/2' This appears to be a good approxima-tion in systems where both the neutrons and protons cornpletely fill the 17/2 leve!. Themonopole terms of this new interaction were rnodified [131 to give a good descriptionof the spectra of nuelei in the Ni isotopes. Since 56Ni is the core of this model space,the single-partiele energies were deterrnined from the 57Ni spectrum. Mass excesses (toppanel) and differences between experiment and theory (bottom panel) are shown in Fig. 2for this new interaction.

3. PAIR CORRELATlONS NEAR N = Z

It has long been anticipated that J = 0+ proton-neutron correlations play an importantrole in the ground states of N = Z nuelei. These correlations were explored with SMMCfor N = Z nuelei in the mass region A = 48-58 in the pf-shell, and A = 64-74 in the0f5/2 lP09Y2 space. As the even-even N = Z nuelei have isospin T = O, the expectationvalues of A A are identical in all three isovector 0+ pairing channels. This symmetry doesnot hold for the odd-odd N = Z nuelei in this mass region, which usually have T = 1ground states, and (Al A) can be different for proton-neutron pairs than for like-nueleonpairs. (The expectation values for proton and neutron pairs are identica!.)

SIIELLMODELMONTECARLOSTUDIESOF NUCLEI 63

2.5 2.5

N=Z nuclei J=O, T=1 pn2.0 2.0

J=O, T=1 pp/1 1.5 1.5 r«

+« 1.0

1.111 1.0

1v

0.5 0.5

0.0 0.046 48 50 52 54 56 58 60 46 48 50 52 54 56 58 60

A A

FICURE 3. Largest eigenvalues for the J = 0, T = 1 proton-proton (left) and proton-nentron(right) pairing matrix a."a functioll oC mass number.

We find the proton-neutron pairing strength significantly larger for odd-odd N = Znuclei than in even-even nuclei, while the 0+ proton and neutron pairing shows theopposite hehavior, in hoth cases leading to a noticeable odd-even staggering, as displayedin Fig. 3, for the 1'/ shell. Due to the strong pairing of the 17/2, al! three isovector 0+channels of the pairing matrix essential!y exhibit only one large eigenvalne which is usedas a convenient measure of the pairing strength in Fig. 3. This s~ag.l>ering is causedby a constrnctive interference of the isotensor and isoscalar parts of AtA in the odd-oddN = Z nuclei, while they interfere destructively in the even-even nuclei. The isoscalarpart is related to the pairing energy and is found to be about constant for the nncleistudied here.

Figure 4 shows the correlated pairs for N = Z nuclei in the JI = 64 - 74 region of the0/5/2 11'099/2 space. Note that 64Ge exhibits little J = ° pairing, indicating that the 1'3/2subshel! is relatively c10sed for this system. The correlated pairs exhibit a strong .J = O,T = 1 like-particle staggering for the even-even and odd-odd /1- = Z systems, while thennmber of correlated proton-neutron pairs is much larger than the like-particle pairing forthe odd-odd systems. The correlated pairing behavior of N = Z, N = Z + 2, N = Z +4nuclei is ShOWIl in Fig. 4, whcrc DIle clcarly sees the dcercase in T = 1 proton-Ilcutronpairing a.saIle IIl0VCSaway frolIl N = Z.

Another striking feature of proton-neutron pairing can be found through studyingthe therlnal properties of these systems [5,8, 9J. In the case of 52Fe .J = O, proton-nentron and like-particle pairing decrease '1uickly to fermi-gas values at a temperatureof approximately 1.1 MeV (corresponding to an excitation energy of approximately 4MeV) (see Fig. 5). This deerease in pairing behaves like P(T) = Po[1 + exp(T~/¡,)rl,where 7(, '" 1.1 lvIeV, and n = 0.25 MeV, and is the same for both like-particle andprotoll-nclltroIl T = 1 pairing. Far thc case uf 50Mn (shoWJl in Fig. G), the proton-Ilelltroll pairing decrcascs IIlllch more quickly, and is almost zcro at approximately atemperatnre ofO.75 MeV, while the like-particle pairing remains nntil1.1 MeV. Ono al so

64 O.J.OEAN

1.5• --e J=O ppo o __ ¡.--.J=Opn ¡1.0

t T .I • t1

\1• 1-- ;a. 0.5'a., • -.1:\ :¡:

1---1-- r T -10.0 1:1:

Ge Se K,

-0.562 64 66 68 68 70 72 72 74 76 78

A

FIGURE 4. Correlated J = O, T = 1 pairs in the proton-proton and proton-neutron channel forN = Z nuelei in the OJ5/2 lpOg9/2 model space.

•••....-• •

••IP

I III

I !II

••• •• ....

1.00.0

IT .i

-.Il

• •'.• •--- ._-.~. - JI'------_

• • --11::.- •~ _____ .J.______.---

--

1IIh • • •

-35:;- -40.,

-45~w -50

-55:;- 15.,.~ 10~ 5

O8

>=' 7!2.al

65

j 30

::;; 20iD

100.0 1.0

T(MeV)2.0

2

O1.5

~ • 1.0a. 0.5

0.0

1.5~ • 1.0a. 0.5

0.0

2.0:¡ •a. 1.5

1.0T (MeV)

2.0

FIGURE 5. Thermal properties of "Fe. The SMMC results are shown with error bars, while theHnes indicate the mean-field values for the respective pair correlations.

SHELLMODELMONTECARLOSTUDIESOF NUCLEI 65

, ,1

•..-.• .-...'1~'tJ•••• •I •••~

1po;'£tI}

•

:tI

,, •t T .t1,,'.T-

h:!: { •

•••• :11 ••

:t::! "! .i!fj:,.!~---- •.' L

.:.•. - Ji

- ..:.t-t._ --T ~-D.' •.T T.

T, lTj1£~J:•H l~ . -11'.

-25

> -30m -35~UJ -40

-45> 10m::¡.- O~- -lO

8¡::- 7~ 6al

54•. 30:1

:¡¡ 20iií

100.0 0.5 1.0

T (MeV)1.5 2.0

32

A1-v

O-10.8

~ 0.60.' 0.4

0.20.0

2.0

1.0

0.0

-1.01.6

_ '.4

~5. 1.21.00.8

0.0 0.5 1.0T(MeV)

1.5 2.0

FIGURE6. Thermal prorerties oC 50Mn. The SMMC results are shown with error bars, while thelines indicate the mean-field values for the respective pair correlations.

sees a dramatic drop in the dependence of the ¡sospin (T2) = [T(T + 1)] on temperature.Since 50Mn has a T = 1 ground-state, (T2) = 2 at very low temperatures, but decreasestowards zero as the system is heated, indicating a thermal mixture of T = 1 and T = Ostates.

As required by general thermodynamic principIes, the internal energy increases stead-By with temperature. The heat capacity C(T) = dEjdT is usually associated with thelevel density parameter a by C(T) = 2a(T)T. As is typica! for even-even nuclei [81 a(T)increases from a = O at T = O to a roughly constant value at temperatures above thephase transition. We find a(T) '" 5.3:1: 1.2 Mey-I at T ~ 1 MeY, in agreement with theelllpirical value of 6.5 Mey-I [14] for 52Fe. At higher temperatures, a(T) must decreasedue to the finite model space of our calculation. The present temperature grid is not fineenough to determine whether a(T) exhibits a maximum related to the phase transition,as suggested in [8].

The temperature dependence of the energy E = (H) in 50Mn is significantly differentthan that in the even-even nuclei. As can be seen in Fig. 6, E increases approximatelylinearly with temperature, corresponding to a constant heat capacity C(T) '" 5.4 :1: 1Mey-l; the level density parameter decreases like a(T) ~ T- I in the temperature intervalbetween 0.4 MeY and 1.5 MeY. We note that the same linear increase of the energy withtemperature is observed in SMMC studies of odd-odd N = Z nuclei performed with apairing+quadrupole hamiltonian [15] and thus appears to be generic for self-conjugateodd-odd N = Z nuclei.

66 O.J.OEAN

4.0

3.0

--,~---~---- , __ o

74Rb cranked

0.00.0 5.0 10,0

<J,>15.0 20.0

FIGURE 7. < 7'2> as a fUBCtiOIl of J, fOl' ¡"nb.

4. PAIR CORRELATIONS IN 74Rn

Tite band mixillg seen in 74Rb can he meaBured witbill SMMC by adding a crankingtenn to tite sltell-model Hamiltoniall [16]. Tltns Il f- H + w.l,. Note tltat sinee J,is a time-odd operator, tite sigll problem is reintroduced, ami good statistical samplingre<¡uires tite cranking to be carried ont only to finite w. For tite calcnlations sltown illFig. 7, tite largest eranking fre<¡nency was w = 0.4. A signature of tite band mixing is acltange in tite isospin (1'2) as a fnnction of (.1,), wlticlt is sltown in tite figure. We see tltatat about a spin of J, = M" (7'2) moves away from its initial valne of2. A mixing of1' = Oand l' = 1 states forces tite expectation value to decrease. Tltns in a full sltell-modclcalcnlation, we predict tite onset of tltis mixing at approximatcly J, = 4h, consistentwitlt tite spectrum found in Ref. 3.

To understand tite apparent crossillg of tite l' = 1 and l' = Obands, we Itave stndiedthe various pair eorrelations as a fnnction of rotational fre<¡uency. We find tltat titeisovector .1 = O correlations and tite aligned isosealar .1 = 9¡m correlations are mostimportant in this transition. Tltese correlations are plotted in Fig. 7 as a function of (J,),wltere we Itave defined pair corrclations by E<¡.(8). Strikingly tite largest pair correlationsare found in the isovector J = O and isoscalar .1 = 9 proton-neutron citan neis at lowand Itigh fre<¡uencies, respectivcly. Furtltermore, tite variation of isospin witlt inereasingfrcqucncy rcftccts the relative strengths of t.hesc two pn correlations. Qur ealculationelearly confirms that proton-nentron correlations determine tite beltavior of tite odd-oddN = Z nueleus 74Rb, as already supposed in [3].

With increasing fre<¡nency tite isovector .1 = O¡m correlations decrease rapidly to aconstant at (J,) '" 3. Tltis beltavior is accompanied by an increase of tite pn correlationsin tite maximally aligned cltallnel, .1 = 9 T = O wlticlt dominates 7.IRb al. rotational

SUELLMODELMONTECARLa STUDIESOF NUCLEI 67

'OE1.5 Qj• J=Opp

1.0 o J=OpnS

0.5 ~<l. f-o0.0 .0.

-0.5

'J=8T=10.7 o J=9 T=O

S 0.5 ~<l. 0.3 ?< ~

~0.1 _ -.'1 -, -----j r-.I •-0.1 -o

1.8

1.6 •• fc-¡='~ I--i--f8'Z 1.4 -.---<

1.2 T.i,1

1.0 o 5 10 15<J,>

FIGURE8. Seleeted pair eorrelatians and the protan 99/2 aeeupatian number (battam panel) asa fllnetian af (J,). The tal' panel shaws the isaveetar .J = O pp and pn correlatians, while theisoscalar J = 9 and isovector J = 8 pn correlations are ShOWIl in the middle panel.

frcqucneics where the J = O pn eorrelations beeome small. Furthermore, although J = 8,T = 1 pairs exist, they do not exhibit eorrelations beyond the mcan field, as shown inFig.8.

Sinee J = 9pn pairs can be formed only in the 99/2 orbitals of our model spaee,the pairing results suggest inereasing oecupation of this orbital with increasing w. Todcmonstrate this inerease, Fig. 8 shows the 99/2 proton occupation number (equal to thencntron oecupation number) as a funetion of (J,). Whilc the ground state has roughlyonc pro ton in the 99/2 orbital, this number inereases by some 50% at the highest (J,)values we study and tracks the J = Opn pairing correlations. 1t should be noted thatalthough the increased 99/2 occupation number increases the mean-ficld value of the paircorrclations [see Eq. (8)], the J = 9pn correlations as shown in Fig. 8, reflect genuinepairing beyond the meau field values.

5. CONCLUSIONS AND FUTURE PROSPECTS

In these proceedings 1 have inc!icated some af the studies of pairing that are accessibleusing SMMC techniques. As the above results indicate, strong J = O, T = 1 proton-neutron correlations exist for odd-odd N = Z lI\1elei, and die off 'l\lickly as one adds

68 D.J. DEAN

neutrons. Thermal studies of N = Z even-even and odd-odd systems indicate thatproton-neutron isovector pairs break morc rcadily with increasing temperature than dothe like-partiele pairs. Finally, an odd-odd N = Z systcm such a.., 74Rb should exhibita T = I to T = O band mixing as one cranks the system. 80th studics indicate thatisoscalar pairing is more resistant to both thermal and rotational excitation of thc nueleus.

In this procedings I have cndeavored to demonstrate by conccntrating on a particularproblem, which demonstrates that with the SMMC approach a microscopic method isnow at hand which allows dctailed studies of many correlations in nuelei in thc mass rangeA = 60-100. These nuelei will be the focus of futurc expcrimcntal interest at radioactiveion-beam facilities. Spacc does not permit mc to discuss other exciting applications ofSMMC to nuelear systems. 1 will simply list thcm here. We have made a dctailed studyof the ground state properties of pf-shellnuelci ineluding thc quenching of the Gamow-Teller strcngth [7]. We have calculated {3{3-decayIllatrix elements for 76Ge [17]. We havestudied the ,-softness of 124Xe [18]. We havc also begun progralllsin Illlllti-Illajor shellcalculations, and in calculations for rare carth nuelei.

ACKNOWLEDGMENTS

The collaborative elforts of S.E. Koonin, K. Langanke, W. Nazarewicz, F. Nowacki, A.Poves, P.B. Radha, and T. Ressell arc gratcfully acknowledged. Oak Ridge NationalLaboratory is Illanaged by Lockheed Martin Energy Research Corp. for thc U.S. Depart-ment of Energy under contract nUlllber OE-AC05-960R22464. Computational cyeleswere provided by the Center for COlllputational Scienccs at ORNL. OJO acknowledgesan E.P. Wigner Fellowship from ORNL.

REFERENCES

1. J. Engel, K. Langanke, and P. Vogel, Phy .•. Letl. B (1997), in press.2. K. Langanke, D.J. Dean, S.E. Koonin, and P.Il. Radha, Nuel. Phy .•. A (1997), in press.3. D. Rudolph el al., Phy .•. Rev. Letl. 76 (1996) 376.4. G. Lang, C.W. Johnson, S.E. Koonin, and W.E. Ormand, Phy .•. Rev. e 48 (1993) 1518.5. S.E. Koonin, D.J. Dean, and K. Langanke, Phy .•. Rep., in press (1996).6. Y. Alhassid el al., Phy .•. Rev. Letl. 72 (1994) 613.7. K. Langanke el al., Phy .•. Rev. e 52 (1995) 718.8. D.J. Dean el al., Phy .•. Rev. Letl. 74 (1995) 2909.9. K. Langanke, D.J. Dean, P.Il. Radha, and S.E. Koonin, Nuel. Phy .•. A602 (1996) 244.10. T.T.S. Kuo and G.E. Brown, Nuel. Phy .•. 85 (1966) 40.11. A. Poves and A.P. Zuker, Phy .•. Rep. 70 (1981) 235.12. E. Caurier, A.P. Zuker, A. Poves, and G. Martinez-Pinedo, Phy .•. Rev. e 50 (1994) 225.13. A. Nowacki, Ph.D. Thesis, Madrid, Spain, 1995.14. J.J. Cowan, F.-K. Thielemann, and J.W. Truran, Phy .•. Rep. 208 (1991) 267.15. D.-C. Zheng, (unpublished).16. D.J. Dean, S.E. Koonin, K. Langanke, and P.Il. Radha, submitted to Phy .•. Letl. B (1996).17. P.B. Radha el al., Phy .•. Rev. Letl. 76 (1996) 2646.18. Y. Alhassid, G. Bertsch, D.J. Dean, and S.E. Koonin, Phy .•. Rev. Lett. 77 (1996) 1,14t