Energy dependence of the 24Mg(16O, 12C)28Si reaction

Nuclear Physics A363 (1981) 253-268 0 North-Holland Publishing Company

ENERGY DEPENDENCE J. NURZYNSKI, Department

OF THE zrMg(laO,

1*C)28Si REACTION

T. R. OPHEL, P. D. CLARK, J. S. ECK +, D. F. HEBBARD and D. C. WEISSER

of Nuclear Physics, Research School of Physical Sciences, The Australian National University, Canberra, ACT 2600, Australia

and B. A. ROBSON and R. SMITH Department

of Theoretical

Physics, Research School of Phystcal Sciences, The Australian University, Canberra, ACT 2600, Australia

National

Received 29 September 1980 (Revised 10 December 1980) Abstract:

Excitation functions for the reaction 24Mg(‘60, ‘2C)28Si(g.s., 2:) were measured at 5”(lab) in the energy range 32 < E,,, < 49 MeV. Although the resonant structure, previously observed at lower energies, becomes progressively weaker, three new correlated maxima have been observed near E, m = 37.5, 40.2 and 43.5 MeV. Angular distribution measurements at these energies yield spin assignments, from P:(cos 0) comparisons, of 27, 29 and 31, respectively. Attempts to find a consistent optical-model fit to the elastic scattering in the entrance channel and an exact finite-range DWBA tit to the four-nucleon transfer reaction in this energy range were unsuccessful. Such a failure is to be expected if strong couplings between the elastic channel and inelastic channels of either the initial or final system are important. The features of the resonance phenomena in the transfer reaction are discussed within a band crossmg model framework.

E

NUCLEAR REACTIONS 24Mg(160, i*C), E = 32.448.6 MeV; measured u(E, 0); 24Mg(160, 160) E = 37.5 40.2 MeV; measured a(0); deduced optical-model parameters. “‘&a resonances deduced J. DWBA analysis, band crossing model. 1

1. Introduction

Considerable resonance structure has now been observed in heavy-ion reactions. Typical gross structure (width N 2 MeV) is seen ‘) in the excitation functions for ’ 6O + ’ 6O elastic scattering at energies above the Coulomb barrier. Optical-model analyses “‘) of these data led to either a shallow and weakly absorbing potential (four-parameter optical model) or a surface-transparent potential (six-parameter optical model) in which the imaginary well has smaller radius and diffuseness parameters than the real well. Such potentials allow the two colliding ions to retain their individual structure during a grazing collision and give rise to the possibility + Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA. 253

254

J. Nurzynski et al. / *‘Mg( 160, 12C) 28Si

of so-called orbiting molecular states. However, because such collisions are essentially a direct process, the ions soon pass through one another so that they stay in the molecular orbit for perhaps only about f of a revolution. Quantum mechanically, such states correspond to virtual states (broad shape resonances) in the ion-ion potential. Consequently, such states give rise to broad resonance (gross) structure of the type seen in 160+ I60 elastic scattering. The surface-transparent type potentials cause the lower partial waves to be strongly absorbed while trajectories corresponding to grazing collisions are only weakly absorbed. These properties lead to two interesting phenomena: (i) diffraction effects associated with the strong absorption and (ii) the “glory” effect arising from orbiting trajectories in the weakly absorbing surface region. The model of Austern and Blair has been employed 3, to show that diffraction effects can lead to gross structure of the kind observed in the “(2 + “(2 and ’ 6O + ’ 6O inelastic scattering excitation functions. However, as pointed out by Friedman et al. 4), the presence of strong internal absorption does not exclude the possibility of resonance effects. Indeed, in a full quantum mechanical treatment, the gross structure should arise from both the diffraction and orbiting resonance effects of the optical potentials involved. The glory effect arises through interference of a normal backward scattered wave with one which has orbited through a negative deflection angle of about 180’. This leads to large back-angle elastic scattering cross sections which display large oscillatory character such as is observed ‘) in I60 + %i elastic scattering at EC,,, = 35 MeV. These large-angle oscillations often resemble the square of a Legendre polynomial, P$(cos B), suggesting a resonating partial wave of order J. The J-values extracted in this way lie close to the grazing angular momenta predicted by the appropriate surface-transparent optical potential. Since the discovery of such phenomena, several descriptions involving Regge poles 5), angular momentum dependent absorption potentials 6), resonances 7’ 8, or parity-dependent optical potentials ‘) have been proposed. All these approaches, some of which are closely related, are designed to enhance one or more partial waves close to the grazing angular momentum. In some heavy-ion reactions, the cross sections are rapidly fluctuating functions of energy and the question arises whether these are true resonance structure or simply statistical Ericson fluctuations. In some cases, e.g. for the “C+ 14N system lo, l’), statistical calculations using the Hauser-Feshbach method 12) give a good description of such fine structure provided one subtracts out the underlying gross structure. However, in several cases, e.g. the 12C + 160 system at E,.,. = 19.7 MeV, there exists a strong correlation between the excitation functions of the elastic scattering at several angles, suggesting a resonance. This resonance has a width of - 0.4 MeV and is an example of intermediate structure. Correlated structure of this kind with width - 0.1 MeV was found in the very first precision heavy-ion measurements 13) on the ’ 2C + “C reactions. In addition

J. Nurzynski

et al. 1 2’Mg( 160, 12C) 28Si

255

to the widths, the spacings of the observed resonances in this system are too small to be readily described in terms of simple shape resonances associated with quasibound states in a molecular-type potential between the two ions. Such intermediate structure has been interpreted 14*ls) in terms of “doorway” states in which the incident channel couples to another degree of freedom of the resonating system. In particular, Imanishi i4) proposed that the incident elastic channel may be strongly coupled to a channel in which one of the i*C nuclei is excited to its first 2+ state at 4.4 MeV. This concept was extended by Greiner et al. 16) in their double resonance mechanism in which a virtual state in the entrance channel is excited by a grazing partial wave and acts as a doorway state which feeds quasi-bound states in inelastic channels corresponding to excitation of collective states of the individual nuclei. In these approaches, the intermediate structure is described in terms of coupling between the elastic and inelastic channels and arises naturally in appropriate coupled-channels calculations 14*16). In order to predict the energies and spins of possible intermediate structure resonances arising from such coupling, Abe et al. “) have suggested a schematic band crossing model. In this picture, the resonance structure arises whenever two quasi-rotational bands of states in the composite system, corresponding to the elastic channel and an inelastic channel, cross one another and the molecular-type states involved are neither too narrow nor too broad. If the above intermediate structure occurs in the grazing partial waves, then similar resonance-like behaviour is to be expected, even at forward angles, in direct reactions which are strongly surface peaked. Indeed, pronounced resonance structure has been observed by Paul et al. ’ *) for the reaction 24Mg(1 60, ’ 2C)28Si leading to the ground state and 1.77 MeV 2+ state of ‘*Si at two forward angles, 0” and 11” (lab), for 23 s EC_,, s 38 MeV. The purpose of the present work is to extend these results for the 24Mg(160, 12C)28Si reaction to higher bombarding energies to determine if the strong resonance structure persists and in this way to study further the nature of the resonances in the 40Ca system. 2. Experimental method and results Thin targets (N 100 pg/cm’) were made by evaporating enriched 24Mg (99.92 % 24Mg 0.06 % 25Mg and 0.02 % *(jMg) on to a thin carbon backing (N 10 pg/cm2). These targets were bombarded with a 160 beam from the Australian National University 14 UD pelletron accelerator. The reaction products were momentum analyzed using an Enge split-pole magnetic spectrometer and were detected in a multi-electrode focal plane detector. The particles were identified using the ratio (Bp)‘/E, where Bp is the magnetic rigidity and E is the energy of the detected ions. The data were recorded event by event on to magnetic tapes using a HP-2100 data acquisition system. The incident beam intensity was recorded using a beam current integrator and two solid-state detectors at 15” and 30” were employed as

24Mg (‘“0, “C)‘*Si

(ARBITRARY

UNITS

)

Fig. 1. Excitation functions (dots) for the reaction 24Mg(160, 12C)z8Si leading to the ground state and the first excited state in **SI, t measured m the range of energies 32.44g.6 MeV (c.m.) at 5’ are compared with the excitation functions at lower energies I*) at O” (full line) and 1lo (dash-dot line). Results of ref. ‘*) are expressed here in arbitrary units. Dashed lines are used to guide the eye. Arrows indicate the energies at which angular distributions were measured.

monitors. A detailed description of the experimental equipment is given elsewhere 19). The elastically scattered 1607 ’ ions and the most intense group of “C ions ( 12C6’) from the transfer reaction have a similar magnetic rigidity. Thus the transfer reaction measurements at forward angles were carried out using a 20 cm slot in front of the focal plane detector to stop all groups of elastically scattered i60 projectiles while allowing the 12C6+ ions to enter the detector. In the energy and angular regions studied in the present experiment, the “C6+ group contains 82-98 % of the total intensity of the i2C charge state distribution. Excitation functions for the reaction 24Mg(‘60, 12C)28Si were measured at 5O (lab) in the energy range 54-81 MeV(lab) with a horizontal acceptance of 4..5”(lab). Test measurements carried out by varying the reaction angle around 5’ for various incident I60 energies indicated that the second maximum in the ground-state angular distribution was located well within the acceptance angle. Fig. 1 shows the results obtained together with earlier data 1*) taken at 0“ and 11‘(lab) for the groundstate transition at lower I40 bombarding energies. It is seen that the resonant structure becomes pro~essively weaker at the higher energies. Nevertheless, three additional correlated maxima are evident near E_,. = 37.5, 40.2 and 43.5 MeV. Angular distributions for the reactions were measured at these three energies

257

J. Nurzynski ef a/. / 2”Mg/ lbO, 12C) 2sSi

24M~

( 160, "C f*%I

O’(gs)

50

0

IO

20

40

30

60

SO

70

80

8 c m.

Fig. 2. Angular distributions (dots) for the reaction 24Mg(160, 1zC)2*Si(O*, gs.) measured at the indicated c.m. energies are compared with the DWBA calculations (full lines).

I

I

0

IO

20

30

40

so

60

70

80

ec.m.

Fig. 3. Angular distributions for the reaction 24Mg(‘6Q, 12C)28Si(2+, 1.77 MeV). See caption to fig. 2.

258

J. Nuxynski

et al. / 24Mg( 160, lzC) z8Si

24Mq C’“O,‘“O)24Mq E c,m= 37.5 MeV SETI

E, ,,,= 40.2 MeV

---

SET 6 -

I 0

I

IO

I

I

20

30

I 40

I

I

50

60

- -

SET 9 -

SET 4 -

\IlI 70

I

60

IO

I

I

I

1

1

I

I

20

30

40

50

60

70

60

8 cm

ecrn.

Fig. 4. Angular distributions for the 24Mg(‘b0, ‘60)24Mg measured at two energies (dots) are compared with the optical-model calculations. Parameter sets are listed in table 2.

and also at EC,,. = 39.0 MeV, corresponding to a trough between two maxima in the excitation functions (indicated by arrows). An acceptance angle of 1” (lab) was employed for these measurements. The results are shown in figs. 2 and 3 in which diffraction patterns are clearly evident for both transitions. However, some irregularities are present. In particular, a prominent distortion of the simple oscillatory structure occurs for the ground state distribution at 40.2 MeV in the angular range 2540’(c.m.). This irregularity is less marked for the ground-state transition at the other energies and does not occur for the transition to the 2+ state. The elastic scattering cross sections for the reaction 24Mg(160, 160)24Mg were measured at EC,,, = 37.5 and 40.2 MeV with the whole detector exposed to the reaction products. The data are shown in fig. 4. It is seen that while the 37.5 MeV measurements exhibit os’cillator structure for 8,,,. > 40”, the 40.2 MeV data are relatively structureless in this angular region. 3. Elastic scattering analysis As pointed out by Siemssen ‘O), two schools of thought exist regarding the de-

J. Nurqnskl

et al. / 24Mg( 160, l*C) 28Si

259

scription of heavy-ion scattering in terms of the nuclear optical model. The first argues that as heavy ions are complex, loosely bound particles, they must disintegrate upon impact so that only strongly absorbing potentials are acceptable. The second, however, simply attempts to determine empirically which features can, and which cannot, be consistently described by the optical model. This second approach has typically led to a range of potentials which are weakly absorbing for surface partial waves. Such “surface-transparent” potentials are often characterized by a smaller geometry for the absorption potential than the real potential i.e. r, < rR and a, < aR. Previously, elastic scattering for the “jO+ 24Mg system has been studied for various energies ranging from the Coulomb barrier up to EC,,, = 43 MeV. In general, analyses above the Coulomb barrier favour a moderately shallow real MeV), which increases linearly with energy and potential depth (V N 5 +0.5E,,,, an imaginary potential strength, which increases quadratically with energy. These and similar optical-model analyses for neighbouring mass systems have led to parameters which can be classified broadly into three groups: (i) potentials which have similar real and imaginary radii and diffusenesses (i.e. rR N r, and uR N a,) and are strongly absorbing; (ii) potentials which have similar real and imaginary potential geometries but have weak absorption strengths; (iii) potentials which have r, < rR, a, < uR and moderate absorption strength. Both potentials (ii) and (iii) give rise to surface transparency for the grazing partial waves. A selection of such potentials is given in table 1. In the present work, the potentials of refs. 21325*26) were taken as representative of the three types of interactions and were used as starting points for searches to TABLE Classification

of heavy-ion

1

optical-mode1

potentials

“)

Ref. Group (i) 100 100

1.22 1.14

0.50 0.58

30 20 Group (ii)

5+0.5E_ 7.5+0.5E, 32.8

m

1.37

0.53

1.31 1.25

0.49 0.50

O.O08E:., 0.4+0.1SE, 8.11

m

Group (iii) 35.286 17.0 ‘) Optical-model

1.307 1.26 potential

0.4926 0.42 is defined

1.242 1.20

19.67 12.48 according

to ref.

‘I).

0.204 0.25

1.307 1.26

26) 27)

J. Nurzynski

260

et al. / 24A4g( 160, ‘2C) 28S~ TABLE 2

Optical-model

potentials

for 14Mg(160,

160)24Mg

31.5 31.5 37.5 37.5 31.5

85.320 32.831 39.144 35.471 15.799

1.245 1.319 1.274 1.313 1.302

0.525 0.526 0.498 0.515 0.637

38.099 8.223 22.929 8.930 11.790

1.309 1.184 1.291 1.160 1.308

0.322 0.556 0.279 0.221 0.697

1.22 1.25 1.307 1.25 1.26

196.7 212.8 422.7 274.4 119.7

6 7 8 9 10

40.2 40.2 40.2 40.2 40.2

82.725 33.389 46.872 35.534 13.221

1.242 1.349 1.321 1.334 1.442

0.532 0.497 0.518 0.493 0.518

43.119 8.085 35.019 8.668 24.470

1.316 1.218 1.351 1.156 1.342

0.311 0.600 0.304 0.225 0.442

1.22 1.25 1.307 1.25 1.26

1063.8 1056.4 1033.4 1132.4 831.8

11

37.5-43.5

32.0

1.30

0.50

15.0

1.20

0.20

1.25

“) x2 is defined as in ref. 31).

tit the 160+24Mg elastic scattering angular distributions at EC,,, = 37.5 and 40.2 MeV. These data were analyzed using the optical-model parameter search code SOPHIE **) to obtain parameter sets for a DWBA analysis of the transfer reaction. The searches were unconstrained and each yielded a separate local x2 minimum (shown as sets l-3 and 68 in table 2 for the 37.5 and 40.2 MeV data, respectively). In attempting to fit the 40.2 MeV data with a shallow imaginary well (set 7), it was found that the imaginary diffuseness (a,) was adjusted to the unphysically large value of 1.12 fm. The set 7 shown was obtained by requiring that a, 5 0.6 fm. The 37.5 MeV data are described equally well by either a strongly absorbing potential (set 1) or a weakly absorbing surface-transparent interaction (set 2). Both sets of parameters generate the oscillatory structure of the data for 8,,,. > 40” (fig. 4). The fits obtained for the 40.2 MeV data are distinctly poorer in that all the potentials predict oscillatory structure in the differential cross section for 0,,,, > 40”, which unlike the 37.5 MeV measurements, is not observed at the higher energy. There is evidence 2g) from other data, such as few nucleon transfer reactions, for surface transparency of the optical potential with a, < aR. This was taken into account in the present analysis by using sets 2 and 7 as starting values for further searches in which the value of a, was constrained to be s 0.3 fm. The resultant potentials obtained are given in table 2 as sets 4 and 9. The predictions for these sets together with the corresponding strongly absorbing potentials (sets 1 and 6) are shown in fig. 4. It is seen that both interaction forms describe the 37.5 MeV data while neither reproduces the structureless 40.2 MeV angular distribution. In some analyses, very shallow real potentials have been found. This possibility was investigated by employing the shallow potential set of ref. *‘) (table 1) as a

J. Nurzynski

et al. / 24Mg( 160, 12C) 28Si

261

starting point for a search in which the real strength V was constrained to have a value < 20 MeV. The resultant potentials from these searches are the sets 5 and 10 of table 2. These interactions give excellent fits to the data, particularly the structureless angular distribution at 40.2 MeV. However, the shallow potential which describes the 37.5 MeV data is significantly different from that which fits the 40.2 MeV measurements. In particular, the absorption strengths are 11.79 and 24.47 MeV, respectively. Such a rapid increase in the imaginary potential is much larger than one expects even if a quadratic energy dependence 23) is assumed.

4. DWBA analysis Exact finite-range DWBA calculations were carried out for the four-nucleon transfer reaction 24Mg(160, iQz8Si leading to the ground and 1.77 MeV states of ‘*Si using the LOLA code 30). In the calculations an a-cluster transfer from 160 to 24Mg nuclei was assumed. The corresponding bound-state wave functions in the projectile and residual nuclei were calculated in a Woods-Saxon potential with radius 1.25A* fm and diffuseness 0.65 fm, the depths being adjusted to obtain the experimental u-particle separation energies in I60 and ‘%i. The number of nodes was tixed according to the rule: C2ni+li

= 2N+L,

i

where (n,, li) are the shell model quantum numbers of the individual nucleons, (N, L) describe their c.m. motion with respect to the core and OSinternal motion is assumed. In the present analysis, an attempt was made firstly to describe the reaction data for the ground-state transition at 40.2 MeV using the optical-model parameters obtained from the elastic scattering analysis. For simplicity, the same parameters were employed for both the entrance and exit channels. Such calculations gave a poor description of the measurements. However, a better fit to the angular distribution was obtained by adjusting the parameters of set 9 although similar attempts based upon parameter sets 6 and 8 were unsuccessful. Fig. 2 shows the results for these parameters (set 11 of table 2). It is seen that the calculated curve qualitatively describes the strong oscillatory character at very forward angles and the smoother angular distribution for 25’ < 8,,,, < 40”. On the other hand, parameter set 11 gives a poor description of the elastic scattering data at 40.2 MeV so that there is a consistency problem. Fig. 2 also displays results using the same parameters for several neighbouring energies. The cross sections from the LOLA calculations have been fitted to the data using the normalization factors in table 3. These numbers, which correspond to a product of spectroscopic factors, are expected to be energy independent if the DWBA theory is valid. A strong energy dependence of the spectroscopic factors,

262 TABLE3 Normalization factors “) for 24Mg( 160, ‘2C)28S~ E,(%i)

(MeV)

0.00

1.77 “) The no~alization

37.5 MeV

39.0 MeV

40.2 MeV

0.52 0.68

0.45 0.45

0.32 0.33

43.5 MeV 0.33 0.32

IS defined as in ref. 32).

as implied by these normalizations, has been previously observed for this reaction at lower energies 33). While it is possible that the observed energy dependence could be at least partially removed by allowing the optical-model parameters to be smoothly energy dependent it is unlikely that the over-all shapes of the angular distributions could be obtained at the same time. Moreover, the excitation functions calculated using such a smoothly energy-dependent potential would not exhibit rapid fluctuations of the type observed in fig. 1. Fig. 3 shows the DWBA results using parameter set 11 and the normalization factors of table 3 for the corresponding transition to the 1.77 MeV 2+ state in ‘*Si. In this case, theory and experiment are out of phase. In order to remove this gross discrepancy, large changes in the optical-model parameters, away from those that well describe the ground-state transition, would be necessary. In view of this problem and the over-all inconsistency of the optical model and DWBA analyses, it was not considered worthwhile to attach much significance to the normalization factors of table 3 or to attempt any determination of the alpha-nucleus spectroscopic factors. 5. Discussion The angular distributions for the 24Mg(160, r2C)28Si reaction to the ground state of 2sSi at E,,,. = 37.5, 40.2 and 43.5 MeV, corresponding to peaks in the excitation function for B = 5”(lab), which have been discussed above in terms of DWBA calculations, exhibit strong oscillatory structure at forward angles (Q,.,. 5 25*) which are well described by the squares of Legendre polynomials, Pj(cos 19) with J = 27, 29 and 31, respectively (see fig. 5). The more pronounced fluctuations in the excitation functions of fig. 1 at E,.,,. = 28.2, 31.2 and 34.2 MeV have been similarly described 18) with J = 21, 23 and 25, respectively. If these peaks arise from the enhancement of a single partial wave, one can use such a Legendre polynomial comparison to assign a spin J to the corresponding intermediate 40Ca system. It is interesting to note, however, that the angular distribution corresponding to the ~~~~~ in the excitation function at E,.,. = 39.0 MeV can also be well fitted by a P&&OS 8) dist~bution at forward angles. The above values of J have been deduced assuming that only a single partial

I

8

t

I

I

/

I

I

I

/

I

1

1

24Mg(‘60,‘2C)2ESi o+(g.s.)

E

= 37.5MeV

E cm = 39.OrvleV -

Eom = 40.2 MeV in

E,, =43.5MeV

P&b6~

-

-

I

0

IO

20

30

40 %n

Jo

P~Jcos0)

60

IO

L

20

P2,(COS9)

f

I

I

I

30

40

50

60

6 cm

Fig. 5. Legendre polynomial fits to the forward-angle cross sections of the experimental angular distributions measured at the indicated c.m. energies.

wave dominates the reaction. Eiowever, the deviations in fig. 5 between the PJ” fits and the data indicate that other partial waves, from either a non-resonant reaction mechanism or overlapping resonances, also contribute. Indeed, for the energy range 24 r E,.,. 6 40 MeV, a study 34) of the transfer reaction at backward angles indicates that this is the case. An angular distribution at 30.8 MeV was only satisfactorily described assuming a strong interference between an even (J = 20) and an odd (J = 23) partial wave. The corresponding angular distribution at 27.8 MeV gives, assuming a single dominant partial wave, 19 5 f 5 22, which is consistent with both the value .J = 20 deduced by Clover et al. 35) from backward angle I60 -I-24Mg elastic scattering data at 27.9 MeV and the value f = 21 obtained by

264

6. Nurqnski ef al. / 24Mg( 160, 12Cj 2sSi

Paul et al. ls) for the 28.2 MeV resonance as mentioned above. The deduction by Paul el al. 34) that a resonance with J = 20 or 22 occurs near 27.8 MeV, since the 8c.m. = 90” excitation function has a maximum at that energy, does not preclude the possibility of a strong J = 21 resonance existing at the same energy. Thus, it seems that the Legendre polynomial fitting procedure allows, at very best, a determination of the dominant J-value to f Ih, and may be less precise in those cases where there is a significant partial wave interference. For the energy region 37.5-43.5 MeV, the values of J extracted from the present experiment by optimal Pj fitting are shown in fig, 5 and are about two units greater than the grazing angular momenta (defined as the J-value of that partial wave which is 50 y0 absorbed) for both the entrance and exit channeis for potential 11 of table 2. The spin assignments for the resonant states in the intermediate 4oCa system at lower energies have shown Is) a similar correlation to the grazing angular momentum. A surface-transparent interaction of the type favoured by the DWBA analysis of sect. 4 allows the two colliding ions to retain their individual structure during a grazing collision and gives rise to the possibility of orbiting molecular states in which the two ions rotate about.the c.m. of the composite system. To examine this possibility, we show in fig. 6 a plot of the observed resonances in the 4oCa system deduced from the reactions 24Mg(160, 160)24Mg [closed circles, ref. 35)], 24Mg(160, 1*C)28Si or its inverse 28Si(‘zC, 160)24Mg [open circles, refs. 18’33-36)] as a function of J(J+ 1). The three new resonances obtained from the transfer reaction are denoted by stars. It is seen that all the resonances lie close to a straight line suggesting a rotational-like band. The gradient of the line corresponds to an effective moment of inertia of 7.08 x 10w4’ MeV * s2 and the projected J = 0 resonance lies at 29.8 MeV. These valites are in g;od agreement with the moment

0

200

400

800

IO00

J(J+r Fig. 6. Plot of observed resonance excitation energy in the 40Ca system against J(J+ 1). The assignments made in the present work are indicated by stars, those obtained from the 24Mg(‘60, ‘*Cf2*Si and 2%i(‘2C,‘60)24Mg reactions ‘*.33-36) are given by the open circles and those from the r4Mg(‘60, 160)24Mg reaction “) by the closed circles. The best fit straight line has a band-head of 29.8 MeV and an effective moment of inertia of 7.08 x 1O-42 MeV . s2.

J. ~~rz~nsk~

et al. / 2”Mg( 160, 12Cj 2sSi

265

of inertia (7.05 x 1O-42 MeV . s2) and the energy (3 I .6 MeV) for the lowest resonance (J = 0) when 24Mg and 160 nuclei are just touching one another with no relative rotational energy as predicted by Cindro and Pdcanid 37). The series of quasi-bound and virtual states (or shape resonances) of a molecularlike potential between two heavy ions are expected to form such a rotational-like band. However, DWBA calculations which employ slowly va~ing energy-dependent optical potentials to describe the initial and final elastic scattering wave functions give rise to only slowly varying excitation functions for the transfer reaction. This implies that the strong, relatively narrow fluctuations observed in the transfer reaction excitation functions do not arise from simple shape resonance effects in either the entrance or exit channels. Moreover, in the DWBA analysis, difficulties were encountered in obtaining an optical model lit to the entrance channel elastic scattering and a consistent description of the 24Mg(160, 12C)**Si (gs., 2:) reactions. A possible interpretation of these failures of the DWBA analysis will be presented here. The DWBA method of sect. 4 is not valid if strong coupling exists between the elastic channel and an inelastic channel of either the initial or final system. This is expected to occur for systems involving nuclei with low-lying collective states, which have large B(EL) matrix elements to the corresponding ground state of the nucleus (e.g. the 2+ 4.4 MeV state in 12C). It seems likely that such strong coupling effects as proposed in the double resonance m~hanism of Greiner ef al. 16), which is based upon the earlier model of Nogami and Imanishi 14), may occur in either the entrance channel (24Mg + ‘60) or exit channel ( i2C + 28Si) of a coupled-channels Born approximation (CCBA) treatment of the associated transfer reaction. These initial and final-state channel couplings may lead to strong fluctuations in the corresponding transfer reaction cross sections. To test this interpretation it will be necessary to perform coupled-channels calculations for the state vectors of the DWBA matrix elements but such a calculation is beyond the scope of this paper. However, the band crossing model of Abe er al., which has previously been used 17) successfully to predict schematically possible strong couplings of the elastic and collective inelastic channels in both the i2C + 12C and ’ 2C -I-i60 systems, allows one to predict the energy regions where such resonance structure is likely to be observed. In the band crossing picture of the coupling between an elastic channel and various inelastic channels one has several quasi-rotational bands of states in the composite system. There is the elastic band corresponding to the situation in which both ions are in their ground states with relative orbital angular momentum J. The energies of these states are given by

where f is the effective moment of inertia of the system and E, is the band head.

J. ~~r~~~sk~ et al. / 14Mg( t60, t2Cj zsSi

266

If one of the ions is excited to a state of spin Z and energy E,, there is a multiplicity of quasi-rotational bands with energies h2

KJ,,(Z)= -QL+

l)+E,+Ex,

2%’

where L = .Z--Z,.Z-Z-+-2,. . ., J+ Z and 2’ is the moment of inertia of the excited system. It is expected that f’ will be somewhat greater than $. For L < J, the excited quasi-rotational band will cross the elastic band near some angular momentum .Z = J,. If strong coupling exists between the two channels corresponding to the crossing bands, resonance effects are expected for those Jvalues close to ..Z,.Table 4 shows the band crossing predictions for various collective excitations of one of the ions for the reaction 24Mg(‘60, *2C)28Si. The results were obtained using eqs. (1) and (2) with $ = 7.08 x 1O-42 MeV . s2 and 3’ = 7.88 x 1O-42 MeV - s2 for both the 24Mg+ I60 and 12C+28Si systems. For simplicity, the value of $ is assumed to be identical in both the entrance and exit channels, which is consistent with the values (N 7 x 1O-42 MeV . s2 for both the 24Mg+ 160 and “C+ ‘*Si systems) obtained by Cindro and PoCaniC 37). Furthermore, the value of 9’ was kept constant for all the excited systems and was chosen to optimise band crossing in the region of the observed resonances. If the elastic band effective moment of inertia had been chosen for $‘, the value of J, for the aligned bands (i.e. L = J-Z) would be increased by only 2 or 3 units of angular momentum, However, the corresponding crossing angular momenta for the reTABLE4 Nucleus =Mg

State

EX

2+ 4+

1.37 4.12

6+

8.12

8+

13.21

Band (15)

JC

Possible resonances

J-2 J-4 J-2 J-6 J-4 J-8

16 23 22 27 28

J-2 J-4 J-2 J-6 J-4

12 18 25 23 28

23 + 28 23 -+ 2.5 28 -+ 30

10

7-t

15

15 -9 19 21 22 21 2s

-26 -+ 24 -v 29 --t 29

i:

1.77 4.61

6+

8.54

I60

3-

6.13

J-3

25

24 -+ 28

‘%

2+

4.44

J-2

24

23 -+ 27

28Si

10 -+ 17 II-+20

J-values for possible resonances in the reaction 24Mg(‘60, 12C)28Si are shown in last column. These correspond to the crossing, near J = J,, of the various excited quasi-rotational bands with L = J-I, J-1+2, . . . . and the elastic band. Only the ground-state rotational band (2+, 4+, .) in *“Mg and ‘%i and the lowest 3- and 2+ states in I60 and ‘Y, respectively, have been considered.

3. Nur~~nski ef al. / z4A4g( 160, 12C) lsSi

267

maining bands of table 4 would be considerably larger. The spins of possible resonances (last column) were somewhat arbitrarily determined by demanding that the separation AE of the two bands involved in the crossing should satisfy AE < 0.50 MeV for J c J, and AE < 0.75 MeV for J < J,. These values of AE were estimated from the widths of the resonance structure observed in the 24Mg+ I60 elastic channel 34). It is seen that the spins of the predicted resonances are consistent with those of the observed resonances. In order to observe a given resonance, it is necessary that it should be neither too narrow nor too broad. The shape resonances within each band are expected to increase in width with increasing excitation energy. Furthermore, microscopic calculations 38) indicate a strong parity effect may occur i.e. the effective interaction between heavy ions may depend strongly on whether the relative motion has odd or even angular momentum. This is a consequence of the Pauli principle and implies that odd and even spin resonances may have very different widths at a given excitation energy. Such a parity effect may explain the observation in the present work, based upon optimal P~(COS0) comparisons, of only odd spin resonances at the higher excitations. In summary, the resonance structure observed in the reaction 24Mg(160, 12C)28Si appears to be consistent with the predictions of a simple band crossing model, which describes schematically possible strong couplings of the elastic and collective inelastic channels in either the initial or final channels of the transfer reaction. This suggests that appropriate coupled channels calculations, which fully test this model, would be very useful. 6. Conclusion Angular distribution measurements have been made at energies corresponding to three new resonances in the 24Mg(160, ‘2C)28Si(g.s., 2:) excitation functions. Spin assignments, based upon Ps(cos 6) comparisons, yield values of 27, 29 and 31 for the observed states at EC,,. = 37.5, 40.2 and 43.5 MeV, respectively. Opticalmodel analyses of the entrance channel elastic scattering and exact finite-range DWBA calculations assuming an a-cluster transfer from 160 to 24Mg nuclei fail to provide a consistent de~ription of the measurements. This failure is interpreted as possibly arising from neglect of strong couplings between the elastic channel and inelastic channels of either the initial or final system. References 1) J. V. Maher,M. V. Sachs, R. H. Siemssen. A. Weidinger and D. A. Bromley, Phys. Rev. 188(1969) 1665

2) A. Gobbi, R. Wieland, L. Chua, D. Shapira and D. A, Bromley, Phys. Rev. C7 (1973) 30 3) R. L. Phillips, K. A. Erb, D. A. Bromley and J. Weneser, Phys. Rev. Lett. 42 (1979) 566 4) W. A. Friedman, K. W. McVoy and M. C. Nemes, Phys. Lett. 87B (1979) 179

268 5) P. Braun-Munzinger,

J. Nurzynski et a!. / 24Mg( ‘W, ’ %T) 2sSi

G. M. Berkowitz, T. M. Cormier, C. M. Jachcinski, J. W. Harris, J. Barrette and M. J. Levine, Phys. Rev. Lett. 38 (1977) 944 6) R. A. Chatwm, J. S. Eck, D. Robson and A. Richter, Phys. Rev, Cl (1970) 795 7) J. Barrette, M. J. Levine, P. Braun-Munzinger, G. M. Berkowitz, M. Gai, J. W. Harris and C. M. Jachcinski, Phys. Rev. Lett. 40 (1978) 445 8) R. E. Malmin, R. H. Siemssen. D. A. Sink and P. P. Smgh, Phys. Rev. Lett. 28 (1972) 1590 9) D. Dehnhard, V. Shkolnik and M. A. Franey, Phys. Rev. Lett. 40 (1978) 1549 10) D. L. Hanson, R. G. Stokstad, K. A. Erb, C. Olmer and D. A. Bromley, Phys. Rev. C9 (1974) 929 11) C. Glmer, R. G. Stokstad, D. L. Hanson, K. A. Erb, M. W. Sachs and D. A. Bromley, Phys. Rev. Cl0 (1974) 1722 12) W. Hauser and H. Feshbach, Phys. Rev. 87 (1952) 366 13) D. A. Bromley, J. A. Kuehner and E. Almqvtst, Phys. Rev. Lett. 4 (1960) 365 14) B. Imanishi, Phys. Lett. 27B (1968) 267; Nucl. Phys. Al25 (1968) 33 15) G. Michaud and E. W. Vogt, Phys. Lett. 3OB (1969) 85; Phys. Rev. C5 (1972) 3.50 16) W. Scheid. W. Gremer and R. Lemmer, Phys. Rev. Lett. 25 (1970) 176; H. J. Fink, W. Scheid and W. Greiner, Nucl. Phys. A188 (1972) 259 17) Y. Abe, in Nuclear molecular phenomena, ed. N. Cindro (North-Holland, Amsterdam, 1977) p. 211: Y. Kondo, T. Matsuse and Y. Abe, Prog. Theor. Phys. 59 (1978) 465: 1009; 1904 18) M. Paul, S. J. Sanders, J. Cseh, D. F. Geesaman. W. Henning, D. G. Kovar, C. Olmer and 3. P. Schiffer, Phys. Rev. Lett. 40 (1978) 1310 19) T. R. Ophel and A. Johnston, Nucl. Instr. 157 (1978) 461, ANU Report P/680 (1977) 20) R. H. Siemssen, in Nuclear molecular phenomena, ed. N. Cindro {North-Holland, Amsterdam, 1977) p. 79 21) P. Braun-Munzinger, W. Bohne, C. K. Gelbke, W. Grochulski, H. L. Harney and H. Oeschter, Phys. Rev. Lett. 31 (1973) 1423 22) I. Tserruya, W. Bohne, P. Braun-Munzinger, C. K. Gelbke, W. Grochulski, H. L. Harney and J. Kuzminski, Nucl. Phys. A242 (1975) 345 23) K. Siwek-Wilczynska, J. Wilczynski and P. R. Christensen, Nucl. Phys. A229 (1974) 461 24) R. H. Siemssen, Proc. Symp. on heavy-ion scattering, Argonne National Laboratory, March 1971, report ANL-7837, p. 145 25) J. B. Ball, 0. Hansen, J. S. Larsen, D. Sinclair and F. Videbaek, Nucl. Phys. A244 (1975) 341 26) M.-C. Lemaire, M. C. Mermaz. H. Sztark and A. Cunsolo, Proc. Conf. on reactions between complex nuclei, Nashville, 1974, vol. 1, ed. R. L. Robinson, F. K. McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam) p. 21 27) R. H. Siemssen, H. T. Fortune, A. Richter and J. L. Yntema, Proc. Conf. on nuclear reactions induced by heavy ions, Heidelberg, 1969, ed. R. Bock and W. R. Hering (North-Holland, Amsterdam) p. 65 28) G. Dehc, Phys. Rev. Lett. 34 (1975) 1468; and private communication 291 M.-C. Lemaire, M. C. Mermaz, H. S&ark and A. Cunsolo, Phys. Rev. Cl0 (1974) 1103 30) R. M. DeVries, Ph.D. Thesis UCLA (1971), unpublished 31) C. M. Perey and F. G. Perey, Atomic Data and Nucl. Data Tables 17 (1976) 1 32) R. M. DeVnes, Phys. Rev. Cs (1973) 951 33) J. C. Peng, J. V. Maher, W. Oelert, D. A. Sink, C. M. Cheng and H. S. Song, Nucl. Phys. A264 (1976) 312 34) M. Paul, S. J. Sanders, D. F. Geesaman, W. Henning, D. G. Kovar, C. Olmer, J. P. Schiffer, J. Barrette and M. J. Levine, Phys. Rev. C21 (1980) 1802 35) M. R. Clover, B. R. Fulton, R. Ost and R. M. DeVries, J. Phys. GS (1979) L63 36) S. M. Lee, J. C. Adloff, P. C. Chevallier, D Disdier, V. Rauch and F. S. Scheibling, Phys. Rev. Lett. 42 (1979) 429 37) N. Cmdro and D. PoEanic, J. Phys. G6 (1980) 359; 885 38) D. Baye, P.-H. Heenen and M. Libert-Heinemann, Nucl. Phys. A308 (1978) 229