PHYSICAL REVIE% C VOLUME 34, NUMBER 4 OCTOBER 1986

Molecular interpretation of the oscillations of the fusion excitation functionfor the a+ Ca system

F. MichelI'aeulte des Sciences, Universite de l'Etat, B-7000 Mons, Belgium

G. ReidemeisterPhysique Nucleaire Theorique, CP229, Universite Libre de Bruxelles, B-1050Bruxelles, Belgium

S. OhkuboDepartment ofApplied Science, Kochi 8'omen's Uniuersity, Kochi 780, Japan

(Received 27 December 1985}

The oscillations observed in the experimental a+~Ca fusion excitation function between E =10and 27 MeV are described within the frame of the optical model. Use is made of an existing opticalpotential giving a precise description of the elastic scattering data for E )24 MeV on a broad angu-

lar range, which is extrapolated to lower incident energies. A correct description of elastic scatteringexperimental excitation functions between E =12 and 18 MeV, as well as of the fusion excitationfunction, is achieved by using different imaginary geometries for both processes. The oscillations

appearing in the calculated fusion excitation function are due to maxima in the even-I transmission

coefficients, for l ranging from 6 to 12; these maxima correspond to shape resonances in the under-

lying potential„which are associated with states belonging to an excited positive parity band with

X=14 in the a+ Ca system. The same potential predicts an X =12 positive parity band of stateswhose energies are in good agreement with the Ti experimental ground state band. It is shown

that the correct reproduction of the spacing between the fusion oscillations achieved here results

from the use of a deep real potential, which automatically yields a decoupling between positive and

negative parity bands.

I. INTRODUCTIONFusion excitation functions have been measured for a

wide range of heavy-ion systems (for a recent survey ofthe available experimental data, see, e.g., Ref. 1). Theaverage behavior of the data can be satisfactorily ex-plained within the frame of various semiclassical or po-tential models. Of particular interest3 has been the obser-vation, for a limited number of light systems, of a broadoscillatory structure in the measured fusion cross sectionsas a function of the incident energy; in particular oscilla-tions, a few MeV's wide, have been seen for the '2C+'zC(Ref. 4), ' C+ ' C (Ref. 5), ' C+ ' 0 (Refs. 6 and 7),' C + ' 0 (Ref. 8), and ' 0+ ' 0 (Ref. 9) systems, as wellas for systems composed of ' C or ' 0 plus various 4nsd-shell nuclei. ' The finer structure possibly superim-posed on this gross structure is not considered here.

For symmetric systems the observed structures havebeen interpreted as being due to successive shape reso-nances whose angular momenta increase in steps of twounits;" this spacing is peculiar to these systems since evenwaves only are active in this case. For nonsymmetric sys-tems like ' C + ' 0 it has been repeatedly pointed out thatthe shallow potentials currently in use for describing theelastic scattering data predict an odd-even alternation ofthe successive shape resonances, resulting in a smallerspacing and a smoother behavior for the predicted oscilla-tions, in contradiction to experiment. It has been suggest-ed' that the parity dependence of the optical potentialthought to be present for these systems could result in a

staggering of the odd and even partial waves, bringing thespacing of the calculated structures in agreement with thedata.

In this paper we investigate another nonsymmetric sys-tem where broad oscillations were detected some timeago, ' i.e., a+ Ca. Fusion cross sections were measuredat the time between E =10 and 27 MeV in connectionwith the anomalous large-angle scattering (ALAS) ob-served in the elastic channel (see, e.g., Ref. 14 and refer-ences therein); the large difference observed between thea+ Ca and the neighboring system a+ Ca fusion crosssections was shown' to support the optical model inter-pretation put forward to explain the backward enhance-ment phenomenon. ' ' However, the broad oscillatorystructure seen in the a+ Ca fusion data was not studiedin the work of Eberhard et al. ,

' and —to ourknowledge —it has never been investigated since.

Because of the ALAS puzzle, the a+ Ca system hasbeen extensively studied both from an experimental and atheoretical point of view. In particular, the exceptionallylow absorption which was shown to be responsible for thephenomenon has allowed a precise determination of thereal part of the underlying optical potential down to verysmall distances, as compared to most neighboring systemswhere the stronger absorption precludes such an accuratedetermination Therefo. re we feel that the preciseknowledge of the interaction potential extracted for thatparticular system may help to clarify the mechanismunderlying the fusion oscillations.

34 1248 1986 The American Physical Society

34 MOLECULAR INTERPRETATION OF THE OSCILLATIONS OF. . .

II. ANALYSIS

The extensive optical model analysis performed by Del-bar et al. '" of elastic a+~Ca data for energies rangingfrom E =24. 1 to 166.0 MeV resulted in the extraction ofa global optical potential whose real and imaginary parts,of squared Woods-Saxon shapes, have energy-independentgeometrical parameters. The simplest version of this po-tential ("potential A") can be written as

V(r)= Vc(r) Uof—(r,R,a) iso—f (r,Rw, aw),

with

f(r,R,a) =I 1+exp[(r —R)/a] I

(2.1)

(2.2)

and where Vc(r) is the Coulomb potential due to a uni-formly charged sphere of radius 1.3A'~i fm=4.446 fm.The geometrical parameters have the values

R =4.685 fm,

a = 1.290 fm,

Rw=6 000 fm

aw ——1.000 fm .

(2.3)

The real well depth varies linearly with energy accordingto

Uo ——198.6(1—0.00168E~) MeV; (2.4}

the energy behavior of the imaginary strength was shownto be adequately represented by a linear prescription be-tween 24 and 62 MeV:

8'0(E~) =(2.99+0.288E~) MeV, (2.5)

while some saturation is needed at higher energy.The simple potential defined by Eqs. (2.1)—(2.5) was

shown to give a remarkable description of the complicatedevolution of the data on the broad investigated energy andangular ranges, including the backward enhancement seenat low energy and its progressive disappearance aboveabout 50 MeV. Moreover, this potential is unique since itfits high energy data exhibiting rainbow scattering andsatisfying the criteria of Goldberg and Smith. '

Extrapolation of that potential at lower energies is com-plicated by the appearance of Ericson fiuctuations'which prevent a detailed comparison of calculated crosssections with existing high resolution data. Even a com-parison with energy-averaged angular distributions likethose supplied by Bisson et a1. is hindered by the pres-ence below 18 MeV of a sizeable compound elastic contri-bution which culmi. nates at about E =10 MeV. Takingthis contribution into account would require a Hauser-Feshbach calculation, which was not attempted here.Therefore we restricted the study of the low-energy prop-erties of the Delbar et al. potential to a comparison of itspredictions with the excitation functions of Robinsonet a/. ' which span the 12—18 MeV energy range.

Two minor modifications were imposed to the energydependences [(2.4) and (2.5)] of the original potential.First, the linear energy dependence of the imaginary depth[cf. Eq. (2.5)] was multiplied by the cutoff factor

F(E )=[I—exp[ (—E —E' ')/5]I,where E' '=3.7 MeV is the threshold energy and 6 wasfixed rather arbitrarily at the value of 5 MeV. This modi-fication guarantees that absorption cancels at the thresh-old; at the lowest energy investigated here for elasticscattering (i.e., EN=12 MeV), it causes a departure ofsome 20% from the original parametrization. Second, wefixed the real well depth to the value UO=180 MeV (thecorresponding real potential will be referred to as D180 inthe following), which is somewhat lower than that ob-tained from Eq. (2.4) for the energy range considered here.It is to be noted in this respect that prescription (2.4),whose slope parameter was essentially determined by thehigh energy data, does not prove entirely satisfactory inthe 20—30 MeV laboratory energy range, since it predictsvolume integrals per nucleon pair Ji/4A of some 370MeV fm, whereas model-independent analyses of thesame data ' lead to volume integrals of about 350MeV fm; our choice Uo ——180 MeV corresponds toJi /4A =350 MeV fm . Also there exist theoretical indi-cationsi that some saturation of the real a-nucleus po-tential should appear at low energy. An independentempirical indication of the need for some saturation withrespect to prescription (2.4) comes from spectroscopicconsiderations concerning the energy location of themembers of the ground state band of Ti with respect tothe threshold (see below}.

A comparison of the predictions of the potential thusdefined with Robinson et al. low energy excitation func-tions' is presented in Fig. 1; it can be seen that the overallenergy trends of the data are very satisfactorily repro-duced by the calculation. At c.m. angles 8, =148.9'and 176.1', the calculated cross section is some~hat toolarge, which could indicate some underestimation of theabsorptive strength. The only serious discrepancy appearsat 158.1', where the calculated cross section displays abroad minimum around E 12 MeV whereas experimentindicates none; however, examination of the results of Bis-son et al. indicates that the Hauser-Feshbach contribu-tion at this angle is quite sufficient to fill the predictedminimum. In contrast, the other minimum predicted byour calculation at 8, =139.7' for E =15 MeV is nothidden by the compound nucleus contribution, in agree-ment with calculations of Bisson et al. Although notpresented here, the predictions of our potential have alsobeen compared with the lower energy excitation functionsof John et al. ' between E~=5.0 and 12.5 MeV. Goodagreement is obtained with the average experimental crosssections; the largest discrepancy consists in an underes-timation of the experimental data for 8, ~90', which ismaximum for energies where the compound elastic crosssection is the largest.

As can be seen in Fig. 2, the total reaction cross section

og n.A, g (21 + 1)(1———~Si

~)

1

calculated with our potential does not show any conspicu-ous structure, and it is larger at all energies than thefusion cross section of Eberhard et al. ' (It should benoted that the data of Ref. 13 are not corrected for the en-

1250 F. MICHEL, G. REIDEMEISTER, AND S. OHKUBO 34

Ca(a, u) 8, =158.1'

59.0

148.9

139,7

9010-

76.1'

" ~'~iiM Iji12 14 16

E (MeV)

18

FIG. 1. Comparison of the elastic scattering excitation func-tion for the o.+~Ca system, calculated with the D180 potential{dashed line), with the experimental data of Ref. 18 {points),

1500-

40

1000-

15

E~ (MeV)

FIG. 2. Reaction (o.~, solid line) and fusion (o~, dashed line)excitation functions, calculated with the D180 potential, togeth-er with the experimental data of Eberhard et al. (Ref. 13}{points}. The experimental errors, which are of the order of 5%,are not shown for clarity.

ergy loss in the target, which is stated to be about 850 keVat E~=10 MeV. Taking this correction into accountwould bring the calculated reaction cross section in betteragreement with the fusion data at the lowest energies;however, it was not taken into account here since thiscorrection is rather small and energy dependent. ) Al-though slightly larger, our calculated reaction cross sec-tion is in good agreement with that extracted by Bissonet al. from their optical model plus compound nucleusanalysis.

For light heavy-ion systems like ' C+ ' C, it has beenpointed out" that the oscillations observed in the fusiondata could be described qualitatively by empirically reduc-ing the absorption needed for the optical model descrip-tion of the elastic scattering data. For heavier systemssuch as ' 0+ Pb, an empirical reduction of the rangeof the imaginary potential which correctly describes thereaction excitation function and elastic scattering has alsobeen shown to successfully reproduce the fusion excita-tion function. More recently the success of this type ofprescription has been invoked2 to support the neglect ofthe surface part of the imaginary potential in the calcula-tion of proton and alpha-particle fusion cross sections.This recipe has been thoroughly used in the systematicstudy of Hatogai et al. , who were able to consistentlyreproduce the experimental fusion and elastic scatteringdata for heavy-ion systems ranging from ' C+ ' C to

Ca+ Ca, using a common real potential but differentimaginary geometries for the two processes. While theimaginary potential Wz they use for fusion calculations isshort ranged, an additional longer range potential Wz isused for calculating elastic scattering cross sections; thelatter is interpreted as being responsible for the directtransitions taking place preferentially in the peripheral re-

gion and depleting the elastic channel. The same idea thatthe imaginary potential responsible for fusion is substan-tially shorter ranged than that accounting for elasticscattering and reaction data also underlies several otherpotential models of fusion such as the boundary conditionmodel of Afanas'ev and Shilov or the distorted wave cal-culation recently developed by Udagawa and Tamura,and is in line with the concept of a critical distance forfusion. i' The neglect of the long range part of the imagi-nary potential in the fusion calculation is more difficult tojustify, as it would seem that some depopulation of the in-cident flux due to the surface transitions caused by Wzmust occur before fusion can take place. One can arguethat if peripheral direct transitions indeed depopulate theelastic channel, they do not necessarily imply an impor-tant loss of flux for the fusion channel since they canrepresent the first steps towards a fusion event. Still it isclear that, however successful it has proved for reproduc-ing experimental fusion data, the somewhat ad hocprescription of calculating the fusion cross section as a re-action cross section using only the fusion part 8'z of theimaginary potential still awaits further justification. Wenote that for the ' C+ ' C and ' 0+ ' 0 systems, thefusion cross sections calculated by Hatogai et al. havean oscillatory structure similar to that observed experi-mentally, which is due to potential resonances in the opti-cal potential with the short range imaginary part 8'~ in

34 MOLECULAR INTERPRETATION OF THE OSCILLATIONS OF. . . 1251

contrast the cross section they calculate for the ' C + ' 0system is smooth, contrary to experiment, probably be-cause of the overlapping of the odd- and even-I resonantcontributions mentioned in the Introduction.

Proceeding along these lines, we repeated the o.+ Careaction cross section calculation after reducing the imagi-nary potential radius; good agreement with the absolutevalue of the experimental fusion cross section is obtainedfor an imaginary radius Ra ——4 fm [instead of the valueof 6 fm needed for reproducing the elastic scattering data;cf. Eq. (2.3)]. More important, oscillations with correctwidths, spacing, and peak-to-valley ratios automaticallyemerge from the calculations, as can be seen from inspec-tion of Fig. 2. The origin of these oscillations is mostclearly demonstrated by decomposing the calculatedfusion cross section O'F into its various partial cross sec-tions oF, the result of this decomposition is presented inFig. 3. It is seen that partial waves with I =2, 4, 6, 8, 10,and 12 are responsible for the six structures appearing inthe calculation. In contrast, the odd-/ partial cross sec-tions display a much smoother energy behavior. A confir-mation of the absence of the odd partial waves in thebuilding up of the oscillations is provided by the selectivesummation oP of all the odd-/ partial fusion cross sec-tions: The resulting curve, which also appears in Fig. 3, iscompletely smooth, except above about 20 MeV, whereone can discern a slight undulation due to the I =11 par-tial wave.

The sensitivity of the calculated structures on the realpotential well depth was investigated by repeating theabove calculations with values of Uo ranging from 170 to200 MeV in 5 MeV steps (the corresponding potentialswill be referred to as D170 to D200; their volume in-tegrals Jv/4A range from 331 to 389 MeVfm ); the re-sults of that numerical experiment are displayed in Fig.4(a). As expected, the calculated oscillations shift as awhole when Uo is varied. The good agreement with ex-periment obtained for Uo ——180 MeV is seen to deterioraterapidly when Uo departs from that value. On the other

2500-

1000

u ~ '0CG (o)020001950190

0185018001750170

62018065

500 MV275

018061

500

10 15 20 25E t~evI

FIG. 4. (a) Dependence of the calculated fusion cross sectionon the potential depth U0 (U0 ranges from 170 to 200 MeV in 5MeV steps). (b) Calculated fusion excitation function for vari-ous potentials belonging to the Jq/4A =350 MeU fm potentialfamily (see the text). (c) Calculated fusion excitation functionfor potentials belonging to adjacent families (see the text). Ineach part of the figure, the heavy line represents the fusion exci-tation function calculated with the D180 potential (cf. Figs. 2and 3), and the curves have been shifted vertically from thelowest one by multiples of 200 mb. The vertical dotted lines de-limit the energy range spanned by the fusion data of Eberhardet al. (Ref. 13).

1500-

e '0CO

1000-

500-

15

E~ (MeVj20

I

25

FIG. 3. Decomposition of the fusion cross section err calcu-lated with the D180 potential (heavy solid line) into partialfusion cross section contributions o~ (even I: thin solid lines;odd I: thin dashed lines); also shown is the result op of theselective summation of all odd-I partial cross sections (heavydashed line).

hand, shifting the calculated oscillatory structure by onefull "wavelength, " so that agreement with experiment isobtained with values of the angular momentum differingby two units from those given above, would require achange of the real well depth totally incompatible with asmooth extrapolation of the original potential of Delbaret al. (and it would ruin the good agreement obtained forthe elastic scattering data). It can therefore be safely con-cluded that the spins assigned above to the calculatedstructures are unambiguous.

A related point of interest is to examine to what extentthe results obtained here are peculiar to the potential in-vestigated. For that purpose we calculated the fusioncross section using two real potentials obtained indepen-dently by Gubler et al. from an extensive optical modelanalysis of their Ca(a, a) data. Both belong to the samepotential family as the potential of Delbar et al. , i.e., tothe unique family fitting the high energy data. These po-tentials have a generalized %oods-Saxon shape: one istheir potential No. 2 (which we will denote as G2) extract-ed at 31 MeV incident energy, the other is a %oods-Saxon

341252 EIDEMa»T~R* ANF. MICHEI. * +'

10W e+ Ca40

5) obtained a26.1 Me~ lail t

f 22 give essentia 4'359 Me& fm

Table 3 o Re '

l„me integrals «G2 and G5 have vo"T simphf& matters

tia sd 351 Me&fm r~p

h the same imag'na~ Parne out wit t eunctions

ca cu1 lations were caexcitationesulti. n us'on

can1 ~ above', the re S

h t calculation we ctent» . '

p 4~b). Prom tare pre presented in 'g'. 'made «a rided use '

lasticconc uinde that —Pro ' . , he low energ& ecri tion o t e

rina precise desc 'p '

artic-S P Psca e

ion excitation unetailstd dular their spins) do no i e

po ef the potential as, e.g.,'

'ff rent potential fated the usion

d"' ""dto low

h 1 1

which are equiva ena ain car-

b fo, 1 hoenergy e

gima inary part asried ou wn referable to a jus

trength in this

o h hallower potentid

hd ob lihld th of th 1 tt rnVan erp

o S Prm e value o is

ti atednorm

hwithentials have very si

onl atres ect to fusion calculations.

families also become i ecident energy.

10

Ii)

EOCD~ 10

II

Q

U

)r3

10 I

10I

15 20

E (Me V)

25

rin excitation functionan le elastic scattering exc ngcalculat wi podepths; 8'0(E ) denotes t e aelastic scattering experimenta a

III. DISCUSSION

ec' '

demonstrated that the1 edf o tto

ection it wastions arin m the calcu at

ff iM V incident energies arefunction between

to maxima12 (the last struc-Tl ——

i I S' Sture also contain s a sizea

shape resonances int eima correspond to successive

f to t8=b i: Bmding our 01SO potentia

'. 5). Introduc ion

cin their

a e eastic1'

scattering data; ther cannot1 ulation at low energyby the cacu a

xcitation function,'-t "b''E--.n--t 1 backward exci a

'

scattering is dominat y

L

—2—3—4—5—6- —7~—s—9——10—11

2m.I

I

I

l

l

Ec.m. io

'IO

20 30 ' --40

20 30 40

E IxeV)

— —17

50

shifts ansh' d bound states calculatedFIG. 6. Absolute phase sh'

with the D180 potential.

34 MOLECULAR INTERPRETATION OF THE OSCILLATIONS OF. . . 1253

pound elastic scattering at these energies.The nature of the resonances responsible for the fusion

structures can be clarified by calculating the absolutephase shifts with the real part of our potential: these arepresented in Fig. 6, together with the states bound by thepotential down to E, = —5 MeV. Inspection of thisfigure reveals the existence of three quasirotational bandswith principal quantum numbers N=2n, +I (where n,denotes the number of nodes of the radial wave function)equal to 12, 13, and 14; a fourth N =15 band of verybroad states is also apparent in Fig. 6. States of theN =14 band with l ranging from 6 to 12 are responsiblefor the structure appearing in the experimental fusion ex-citation function (the last structure is influenced by theI = 11 state of the N =15 band). Lower energy states ofthe N =15 band appear to be too broad (the correspond-ing phase shifts hardly exceed 90') to be seen in the fusionor elastic scattering excitation functions. On the otherhand, states of the N = 13 band, located at lower energies,have much too small widths to have any observable influ-ence on these cross sections (in fact these states are re-sponsible for the narrow additional oscillations appearingat the low energy end of the fusion cross section calculat-ed with the shallower D170 potential [cf. Fig. 4(a)], which

locates these states higher in energy, and thereforepredicts considerably larger widths). The decoupling ofthe %=14 positive parity band from the %=13 andN =15 negative parity bands thus appears to be an essen-tial feature of our potential for reproducing the structuresobserved in the fusion excitation function with correctspacings and widths. It is worth noting that this decou-pling emerges naturally when use is made of deep poten-tials whose form factor is not too dissimilar to that of aharmonic oscil1ator well, and does not require the intro-duction of a parity dependence in the potential As to thelower X=12 positive parity band predicted by the poten-tial, it comes in good agreement with the ground stateband of Ti, although the collapse of the higher membersof the band, which is observed experimentally, is notreproduced by the calculation. It is to be noted that thecluster band structure of Ti is a matter of controversyand that the interpretation emerging from the presentstudy disagrees with some existing attributions;" 3 fur-ther discussion of the spectroscopic aspects of our poten-tial picture of the a+ Ca system is deferred to a forth-coming publication.

To close this section we comment on the good agree-ment obtained with the experimental fusion cross section

15 1112+

10+ 10'

7~

8'

5-6+

78+ m~mm

56+14+

7:8+

56+

10+

N =114+

2+

{}+

I+ 34f

2+1

0' ¹13N=12

12+

10+

6+4+2+

4+

N= 14

12'4+

2+ 1

0+ N =175

10+ N =16

G1 D1804+

2'

0+

MV 275

FIG. '7. Comparison of the band structure predicted by potentials belonging to adjacent potential families (see the text}. The posi-tion of the quasibound states located well below their barrier was calculated using the bound-state approximation, while the positionof broader states is defined as the energy where the derivative of the phase shift with respect to energy is the largest (when the phaseshift does not pass through m/2, this position is indicated by an interrupted line). The two dashed lines delimit the energy rangespanned by the fusion data of Eberhard et al. (Ref. 13).

12S4 F. MICHEL, G. REIDEMEISTER, AND S. OHKUBO

when use is made of potentials belonging to shallower ordeeper families. To this end we have calculated the posi-tion of the bound, quasibound, and virtual states predictedby the D180 potential and by the other two potentials (Gland MV275) used at the end of the preceding section. Theresults of the calculation appear in Fig. 7; states have been

grouped into bands of constant principal quantum num-

ber N, and the states responsible for the structure seen inthe fusion excitation function are represented with thickerlines. The energy range scanned by the experimentalfusion data analyzed here appears also in Fig. 7. It can beseen that the three potentials predict essentially similar re-sults within this range. The only important difference liesin the nature of the bands supporting the various states.For example, the positive parity band responsible for thestructure in the fusion cross section, which is an N =14band when calculated with the D180 potential, becomesan N =12 or %=16 band when calculated with poten-tials of the shallower or deeper family, respectively. TheJ = 14+ state of this band (which lies outside the investi-

gated energy range) is thus missing for the shallower po-tential, while an additional J =16+ state is predicted bythe deeper one. In the same way the N =15 band, whichhas small influence on the fusion calculation (except forits J =11 member) becomes an % =13 or N =17 band,respectively, when calculated with the alternative poten-tials; in the energy region scanned by experiment, how-ever, the predicted levels have very similar energies. Incontrast, the ending of these bands at different angularmomenta has a profound influence on the elastic scatter-ing properties of the three potentials: the shallower ordeeper potential predicts the onset of the rainbow scatter-ing regime'" at too low or too high energies, respectively.Finally the two lo~er bands, which correspond to N =12and X =13 when calculated with the D180 potential, loseor gain, respectively, one state with the shallower ordeeper potential, respectively; however, since the corre-sponding states appearing above the threshold are locatedwell below their respective barriers, they have exceedinglysmall widths and their influence on fusion or elasticscattering is completely negligible.

IV. SUMMARY AND CONCLUSIONS

In this paper we have shown that a natural extrapola-tion of the potential originally used by Delbar et al. ' todescribe a-particle scattering from Ca from 24 to 166MeV provides a good description of the low energy prop-erties of the system as well: %e have succeeded in repro-ducing both the low-energy elastic scattering excitationfunctions measured by Robinson et al. ' between 12 and18 MeV at several angles (though comparison with experi-rnent is somewhat comphcated by the occurrence of Er-icson fluctuations and compound elastic scattering) andthe fusion excitation function of Eberhard et a!.' be-tween 10 and 27 MeU, including the broad oscillationsshowing up throughout that range. This was accom-plished by using a common real potential well but dif-ferent imaginary geometries for the two processes, in the

spirit of a recent work of Hatogai et al. . s The imaginaryradius we used for fusion calculations is smaller than thatneeded for describing the elastic scattering data. Thismechanism allows the calculated fusion excitation func-tion to exhibit oscillations whereas the calculated reactioncross section displays a smooth energy behavior. Theseoscillations have been shown to be due to even-I shape res-onances with spins ranging from 6 to 12 (with some ad-mixture of /=11 in the last oscillation), and they havebeen associated with molecular states belonging to an ex-cited positive parity band with N =14 in Ti. In addi-tion, the same real potential supports another band of pos-itive parity states with X =12, whose energies are in goodagreement with those of the experimental ground stateband of Ti. It was checked that the properties of thestructures observed in the calculated fusion cross section(and in particular their spins) do not depend critically inthe investigated range on the particular potential used-nor on the potential family —provided use is made of apotential giving a good description of the low energy elas-tic scattering data.

A single real potential thus proves capable of describingin a consistent fashion —and with a minimum number ofvarying parameters —a wide range of properties of thea+ Ca system, extending from negative up to high exci-tation energies. Although such unified potential descriptions have been known for a long time for nucleon-nucleus systems (see, e.g., Refs. 37 and 38), they haveremained scarce for composite particle scattering; in thiscontext, mention should be made of the systematic localpotential study recently performed for the a+'60 sys-tem."

As was discussed in the preceding section, the decou-pling between positive and negative parity bands, whichautomatically emerges from a potential description mak-ing use of deep real potentials, is a decisive feature for acorrect description of the fusion oscillations observed inthe data analyzed here. Although the common practicehas been to use real potentials of shallow type for describ-ing light heavy-ion elastic scattering, it appears that thelocal potentials which are equivalent to the resonatinggroup nonlocal potentials are necessarily very deep(the depth of the potential recently derived by Wada andHoriuchi for the ' 0+ ' 0 system is about 400 MeV).Therefore solution of the ' C+ ' 0 puzzle mentioned inthe Introduction could require explicit use of such deeppotentials, although the importance of parity dependenceeffects' should not be overlooked, since these are expect-ed to be important for nearly symmetric systems.

ACKNOWLEDGMENTS

One of the authors (S.O.) is grateful to Professor R.Ceuleneer for having made possible his stay at the Univer-sity of Mons. He is also thankful to Professor D. M.Brink for useful discussions on fusion reactions. F.M.and G.R. are very indebted to Professor R. Ceuleneer forhis constant interest and for his careful reading of ourmanuscript. S.O. is grateful to the Institut Interuniversi-taire des Sciences Nucleaires for financial support.

34 MOLECULAR INTERPRETATION OF THE OSCILLATIONS OF. . . 1255

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