PHYSICAL REVIEW C VOLUME 29, NUMBER 4

Intraband y transitions between ' C+' C molecular states

APRIL 1984

K. Langanke and O. S. van Roosmalen8'. K. Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125

(Received 12 September 1983)

Following the ideas of the double resonance mechanism we argue that recent measurements of thegamma/particle decay branching ratio in ' C+' C might indicate the existence of molecular reso-nances in inelastic channels. To detect molecular states in the elastic channel we suggest that theexperiments be repeated at lower energies. To this end, we have calculated the y-decay properties of' C+ ' C molecular states in a microscopically founded potential model.

I. INTRODUCTION AND DISCUSSIONOF RECENT EXPERIMENTAL DATA

The pronounced structures found in elastic and inelastic' C+ ' C cross sections' are frequently interpreted interms of a series of rotating dinuclear complexes (molecu-lar bands). To test this hypothesis, there have been at-tempts' to measure the gamma transition between twomembers of a proposed molecular band in ' C+. ' C. Theentrance channel energy in these experiments was chosento coincide with resonant structures found in the elasticand mutual 2+ cross sections near E, =25.5 MeV,where particularly the inelastic cross section exhibits apronounced structure with a width of a few hundred keV.Since these structures are interpreted as arising from aJ=14 resonance, an intraband y transition should occurto the J=12 member of the molecular band, which isidentified with the structure in the cross sections atE, =20 MeV. The experimental data ' exclude amolecular interpretation of the J=14 resonance in theelastic channel, but they are compatible with the assump-tion of fragmentation of a shape resonance in the elasticchannel at this energy, as we discuss below.

Based on the results of microscopic studies of heavy-ioncollisions, it was recently suggested that the structures inthe ' C+' C cross sections are due to the coexistence ofboth molecular and shape resonance bands. It was ar-gued that molecular states (restricted to partial waves withl & 12 due to the influence of the Pauli principle) give riseto the intermediate resonant structure at lower energies viaa doorway mechanism, while at higher energies theresonant structure in the elastic and inelastic cross sec-tions is caused by a double resonance mechanism inwhich shape resonances in the elastic channel couple tomolecular states in inelastic channels. In the model ofRef. 8, the resonant structure found in the elastic and mu-tual 2+ cross sections at E, =25.5 MeV is explained bya coupling of the J=14 shape resonance in the elasticchannel to a molecular state in the double 2+ channel.This picture is compatible with the gamma/particle decaybranching ratios for the elastic channel resonance as foundin Refs. 5 and 6. As we discuss at the end of this section,it might even be supported by the experimental data ofRef. 5, which show a few events which can be interpretedas an intraband decay of the proposed molecular state in

the inelastic channel.A more reliable test of the coexisting band hypothesis,

and of the existence of ' C+' C molecular states in gen-eral, is expected from measurements of thegamma/particle decay branching ratios for the proposedmolecular resonances at lower energies. However, due tothe small y energies of an intraband transition at lowerenergies (say E, & 15 MeV), it is not clear whether the ywidths of the molecular states are large enough to makesuch measurements feasible with present experimentaltechniques. To illuminate this point, we have calculatedthe y widths of members of a ' C+' C molecular bandemerging from a potential model. The basic requirementsof a microscopic theory are partly satisfied by the intro-duction of a Pauli projector, which ensures that the wavefunction of relative motion does not give rise to many-body states which violate the Pauli principle. The poten-tial parameters are adjusted to reproduce the experimentalenergies.

If the resonant structure in the cross section atE, =25.5 MeV is indeed caused by the couphng of ashape resonance in the elastic channel to a molecular statein the mutual 2+ channel, the measurements of Refs. 5and 6 should contain events related to an intraband decayof the inelastic channel molecular resonance as well asthose arising from the decay of the shape resonance in theelastic channel. It is therefore essential to know the y en-ergy of the intraband transition, which is given by the or-bital angular momenta of the final and initial states andby the rotational constant of the band. Recent measure-ments of the spin alignments clearly indicate that thepronounced resonant structure in the double 2+ channel atE, =25.5 MeV is dominated by a configuration inwhich both the intrinsic spins I; and the orbital angularmomentum l point into the same direction (aligned chan-nel), and so suggest a classification of the resonance asI~ ——I2 ——2 and I=10, where II z are the fragment spins.Since a decay within the molecular band is expected to beconsiderably faster than a decay involving a deexcitationof one of the ' C nuclei, the molecular resonance shoulddecay to a final state with II I2 ——2 and 1=8. U——nder theassumptions that the band structure in the inelastic chan-nel is similar to that in the elastic channel' and that therotational constant of the molecular band in ' C+' C isslightly smaller than 100 keV (in agreement with an

29 1358 1984 The American Physical Society

INTRABAND y TRANSITIONS BETWEEN ' C+' C MOLECULAR. . . 1359

analysis of the experimental data" and a theoretical pre-diction' ), the y energy of the decay should be Ez -3.5MeV. The resonant structure found in the mutual 2+channel at E, =22 MeV is thus a candidate for the finalstate of the transition within the inelastic band.

Reference 5 presents a diagram in which the observedevents are plotted as a function of the reaction Q valueand the y energy. The data of this experiment can beanalyzed for events arising from the suggested intrabanddecay. Allowing for some uncertainty in the rotationalconstant of the molecular band and for the width of theresonance, we assume a y energy of E& ——3.5+0.3 MeVand a reaction Q value Q= —8.86 MeV Er —for eventsstemming from a decay of the molecular resonance intothe mutual 2+ channe1. In fact, the experimental data ofRef. 5 show four events which are compatible with theseconstraints on Er and Q. The number of these events isconsiderably greater than the estimate of random coin-cidences (0.3—0.6) given in Ref. 5.

The data of Ref. 6 also show events with aQ = —12.4+0.3 MeV, but, in the representation in Ref. 6,these events cannot be distinguished from y decays arisingfrom the (02+,2+) inelastic channel, which has Q = —12. 1

MeV.

II. POTENTIAL MODEL DESCRIPTIONOF INTRABAND y TRANSITION

As suggested above, direct evidence for the existence of' C+'2C molecular states can be gained from a measure-ment of the gamma/particle decay branching ratio atlower energies, where the elastic cross section exhibits pro-nounced resonant structure. To obtain an estimate of the

y widths of these states, we have calculated the E2 transi-tion in a molecular ' C+ ' C band arising in a microscopi-cally founded potential model for the elastic channel. Inour study, we assume that the ' C+ ' C molecular statesin the elastic channel can be described by an antisym-metrized product

Pnl &C1 C2 unlI

~Ic'I @2 unl) (2)

I g &unl I gl & A[el 42 l(xn)]n Pnl

where 412 are fixed internal wave functions describingthe ' C fragments in their ground state with internal spinI=0. They are approximated as the harmonic oscillatorshell model ground states in an SU(3)@SU(4) representa-tion (1-s coupling scheme) with a width parameter b = 1.7fm. The wave functions unl(x) are the radial harmonicoscillator wave functions of width P =b /p, where p isthe reduced mass. The normalization factors pnl are de-

fined as

and ensure that the many-body wave functions Vl areproperly normalized. The 24-body antisymmetrizer is A.The unknown wave functions gl(x) are determined bysolving the equation of relative motion,

III'd l(l+ 1)Iri, +V{x)+, —E gl(x) =0,

2tu dx 2PX(3)

where Al projects off the Pauli-forbidden states of relativemotion. ' ' The potential in Eq. (3) was fitted to repro-duce a series of resonances at the experimentally knownenergies. We adopted a Gaussian form for the nucleus-nucleus potential,

V(x ) = Vo expX2

(4)

and varied the depth parameter Vo subject to the follow-ing conditions:

(1) The potential should exhibit a rotational band ofnarrow, molecularlike resonances. The energies of thesestates should coincide with the energy ranges in which theelastic cross section is dominated by resonances of thesame spin assignment, allowing an interpretation of thecalculated resonances as doorway states.

(2) In agreement with a previous microscopic study, '

the potential should exhibit two lower lying series of reso-nances in order to ensure the correct number of nodes inthe relative wave function.

Both conditions can be fulfilled by choosing Vo ———190MeV. The width parameter of the Gaussian potential(a=3.3 fm) was adopted from Ref. 15. The Coulombpart of the potential was assumed to be that of a homo-geneously charged sphere with radius 3.8 fm. The ener-gies of those resonances which we interpret as the' C+' C molecular band are given in Table I; each ofthese lies within the range of experimentally observedresonant structures with the same spins. They also agreefairly well with the molecular resonances predicted in thecalculation of Ref. 12.

Note that we assume a different internal structure forthe ' C nuclei than that in Ref. 15; the numbers of Pauli-forbidden states in our calculation are different, nl 1 for-—l = 10 and nl ——0 for l = 12. Our potential is l independentand more attractive than that of Ref. 15, which might berelated to the fact that the spin-orbit potential in Ref. 15was too repulsive.

In the bound state approximation' the E2-transitionmatrix element between two states with angular momen-tum l and I' is given by

Tf, = (e,I Q I

e', .) = g 1

n, n +Pn!Pn'I''(gllul)(u I Igl)(4, 4 ulIAQA I@I 42 u I), (5)

where Q is the quadrupole operator. Following arguments similar to those in Refs. 17 and 18 for the a+ ' C system, the24-particle matrix element in (5) can be rewritten as

& C I='@&='u.

l I~Q~

IC I ='C'2='u'I

& =»~ & 4 I='

I QI I@I='&+S xi & u.l I Q-I I u'I & (6)

1360 K. LANGANKE AND O. S. van ROOSMALEN 29

TABLE I. Resonance energies in MeV, internal quadrupolemoments Qo in b, B(E2) values in e b for intrabaud transitions,and corresponding y widths in eV for the ' C molecular bands.

02468

10

E,—5 ~ 5—4.8—3.2—0.8

2.66.8

Qo

1.251.251.231.181.09

B(E2,l —+I —2)

0.03120.04750.05190.05210.0470

7.7X10—'6.1X10-41.3X10 '5.0X 101.3 X10—'

0

46810

4.85.36.48.2

10.814.2

2.502.712.912.913.06

0.11870.19200.24170.27150.2973

2 9X102.5 X10—'6.2X10—'2.6X 108.0X 10-'

where U~ is given by

vM (@I @2 &niI

~I

@& c'2 &n I'with M=2n+1=2n'+1' and p~L defined as in Eq. (2)with 2N+L = min(2n+1, 2n'+1'). The operators Q& andQ,d in (6) are the isoscalar parts of the quadrupole opera-tor which act only on the coordinates of the first ' C frag-ment and the relative motion, respectively. 4]= de-scribes the first ' C nucleus excited to its first 2+ level inthe SU(3)SU(4) representation. The matrix element(4~=

I Qt I

@~= ) is evaluated from the experimentallyknown lifetime of the 2+ level at 4.43 MeV in ' C. Itssign (negative) is adopted consistent with the theoreticalintrinsic quadrupole moment of the ' C ground state andthe experimental quadrupole moment of the 2+ excitedstate in ' C (Ref. 19). The kernels vM and p„t are calculat-ed from the normalization kernels in an SU(3) labeledbasis (obtained using the formalism described in Ref. 20)performing SU(3)/R(3) recoupling transformations. ' Tokeep the numerical effort within reasonable hmits this cal-culation is restricted to 2n+I =M (22. For higher oscil-lator excitations M) 24 the values of p„t (which for agiven 1 approach to p„t~1 with increasing n ) are extrapo-lated assuming a similar n dependence as for the u+ Caand ' 0+ ' 0 systems where the kernels are known analyt-ically. The accuracy of the kernels thus obtained is es-timated to be better than 3%. The accuracy in Tf; is ex-pected to be even better, since, in the expansion of the rel-

I

B(E2,1'~l)=,I Tf, I2l'+1 (8)

while the corresponding y width is

ative wave function in terms of oscillator functions, theexpansion coefficients decrease with increasing n and thequantities Qp„I /p„ I which occur in the matrix elements(5) [after substitution of (6)] approach 1 asymptotically,and reach the value -0.9 already at 2n+1=22. The ker-nels U~ are set to zero for M )24, as by then these asymp-totically vanishing kernels are smaller than 0.07 and thesecond term in (6) is much larger than the first.

In addition to a band of bound states corresponding torotational excitations of the Mg ground state, the po-tential adopted in our study exhibits two bands of molecu-lar states. The lower of these has a bandhead at E,= —5.5 MeV and an internal quadrupole moment [calcu-lated in a manner similar to Eq. (5)] of Qo ——1.25 b (seeTable I). The excited molecular band, which we adopt todescribe the experimentally observed ' C+' C resonances,starts at E, =4.8 MeV, in agreement with othertheoretical approaches. ' The internal quadrupole mo-ments of these excited states increase from Qo ——2.5 b forthe 1=2 molecular resonance to Qo ——3.05 b for the 1=10state (Table I). The reason for this is that the molecularstates stretch with increasing angular momentum: Theirhigher energies are closer to the potential barrier, and con-sequently their internal quadrupole moment increasesslightly with l. Note that the lower molecular band agreesrather well with results of other theoretical ap-proaches, ' which, however, did not predict excitedmolecular states. This is of interest for thegamma/particle decay branching ratio, as our calculationsuggests that the experimentally observed resonances cor-respond to the excited molecular band with Qo=2. 85 brather than to the lower molecular band with Qo ——1.25 b.In the analysis of Refs. 5 and 6 an internal quadrupolemoment of Qo

——1.8 and 1.6 b was assumed, extractedfrom the study of Ref. 24.

The molecular band structure exhibited by our ' C+ ' Cpotential is very similar to that found in microscopic stud-ies of the ' 0+' 0 system, where also two bands ofhighly deformed states are predicted. In the ' 0+' 0system, there might be some experimental evidence for theexistence of both types of molecular bands.

The B(E2) value for a transition from a molecular statewith orbital angular momentum l' to a state with angularmomentum I is given by

I r(eV) = 1.22&(10' A'(eV s)[E&(MeV)] B(E2,1'~l)(e b ) .

The results for the B(E2) values and the y widths as, cal-culated for the two molecular bands, are listed in Table I.Our calculated B(E2) values agree nicely with the well-known rigid rotor formula '

151' 1'B(E'2,1'~l ) = ergo . (10)

32m(41' —1)The y widths for states in the energy range of interest

I

(E, (14 MeV) are smaller than 1 eV. If we adopt areasonable total width of the molecular resonance (includ-ing spreading and escape widths) of less than =300 keV,then I r /I „,)9X 10 for the 1=8, 10 molecular reso-nances as entrance channel states. This ratio is within thesensitivity of modern crystal ball techniques. Because ofthe higher y energy expected, one of the pronounced1=10 resonances at E, =13.5 MeV is preferred over

INTRABAND y TRANSITIONS BET%BEN ' C+' C MOLECULAR. . .

one of the 1=8 states at E, =10.5 MeV as a possible in-itial state in a measurement of the gamma/particle decaybranching ratio. On the other hand, the disadvantage ofadopting an 1=10 initial state is that this state seems tocouple resonantly to the 2+ channel, while for energiesE, m &12 MeV coupling to inelastic channels becomesnegligible. Our study therefore suggests using an 1=8state as an initial state. On experimental grounds, theI =8 resonance at E, = 10.25 MeV has Iecently been in-terpreted as a molecular state and might therefore be

used to measure the gamma/particle decay branching ra-tio.

ACKN0%'LED GMENTS

The authors are greatly indebted to Charles A. Barnesand Steven E. Koonin for Inany fruitful discussions andcomments. This work was supported in part by the Na-tional Science Foundation (Grants No. PHY82-15500 andPHY82-07332) and by the Deutsche Forschungsgemein-schaft.

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