Extension of the orbiting-cluster concept to fp shell systems

Nuclear 0

Physics

North-Holland

A433 (1985) 531-549 Pubhshing

Company

EXTENSION

OF THE ORBITING-CLUSTER TO fp SHELL SYSTEMS D. POCANIC

RudJer BoSkovtd Institute,

CONCEPT

+ and N. CINDRO 41001 Zagreb,

Received

24 April

(Revised

19 July 1984)

Croatia,

Yugoslavia

1984

Abstract: The orbiting-cluster model (OCM) of resonances in heavy-ion reactions is reviewed in the light of recent experimental results. Model predictions given earlier are compared with known data on resonances; good agreement is found. Slight modifications are introduced in the model making possible its application to a wide ensemble of heavy-ion systems in the composite mass range A = 22 to 72. Predictions of resonant behaviour are given for a number of systems. Limitations of the model are discussed.

1. Introduction

A marked shift to heavier masses has occurred in the investigation of resonances in heavy-ion reactions with the advent of new or upgraded heavy-ion experimental facilities in recent years. The most outstanding results in this field have been the discoveries of narrow, high-spin resonances in the 2*Si+ 28Si [refs. i* ‘)I and 24Mg + 24Mg [ref. “)I systems. These measurements have induced considerable interest in studying neighbouring heavy systems. On the other hand, there has been extensive experimental activity focused on promising lighter systems in which resonances were not previously found or on extracting more detailed information about known resonances. While resonances in lighter systems, particularly in 12C+ 12C, have been extensively studied theoretically, so far there have been few attempts to give a detailed theoretical interpretation of resonances observed in heavy systems (A > 40). This can hardly be surprising, as the complexity of the most adequate models [the “direct-excitation” models developed by the Frankfurt4), Kyoto ‘) and Giessen 6, groups] undoubtedly leads to lengthy calculations when applied to heavier systems. Thus, up to now the latter field has largely been left to phenomenological or schematic interpretations. These arguments have prompted us to review the concepts and predictions of the orbiting-cluster model (OCM) of resonances published earlier ‘). In fact, recent experimental results on resonances in heavy-ion reactions have lent additional ’ Presentaddress: Department

of Physics,

Stanford 531

University,

Stanford,

California

94305, USA.

532

D. PoZaniC, N. Cindro / Orbiting-clusterconcept

credibility to the orbiting-cluster quasimolecular picture. Encouraged by the favourable comparison with experiments, we have sought to test the applicability of the OCM to heavy composite systems belonging in the fp shell. Thus, in the following sections we shall briefly summarize the earlier predictions of the OCM, compare them with experiment, introduce necessary minor modifications in the model procedure and apply it to a wide ensemble of heavy-ion systems in the composite mass range A = 22-72. 2. The OCM: basic concepts, predictions and comparison with experimental results Since the concepts and the method of the orbiting-cluster model were presented in detail in ref. 7), here we give only a brief outline. The model is based on the following assumptions: (i) Resonances are due to quasimole~ular con~gurations of orbiting-cluster nature. (ii) The broad rotational states are fragmented into narrower resonances of intermediate width through coupling to intrinsic and collective degrees of freedom (the direct-excitation mechanism). (iii) Quasibound orbiting-cluster states, which manifest themselves as narrow resonances, represent doorway states to the more complex configurations of the composite system. The third assumption makes possible the use of the molecular-window concept proposed by Greiner and Scheid in 1971 [ref.*)], which links the observability of resonances to regions of phase space characterized by relatively low spreading into compound-nucleus states (fusion). This condition will be ful~lled whenever the spreading width rL = 2~~(CNlV~e1)~2p,,(E, J)

(1)

is relatively small. The whole procedure of the OCM consists in calculating r1 along predicted molecular rotational bands for various nuclei. The key problem is, of course, the choice of suitable parametrizations for rotational bands and f’. The rotational bands were parametrized according to assumption (i), so that all the energy available after overcoming the Coulomb barrier (E,) goes into rotation (orbiting) of the two barely-touching entrant-channel nuclei. Thus, E(J) = E,+E,+

Ii2

-J(J+ 2%md

l),

(2)

where E(J) denotes the excitation energy of the composite system corresponding to rotational states with angular momentum J, E, the binding energy of the projectile

533

D. PoEaniC, N. Cindro 1 Orbiting-cluster concept

and target in the compound nucleus, and #mod the model moment of inertia equal to that of two touching spheres rotating around their common centre of mass:

I mod

(A+-!-Af)“]ri x 1O-44 MeV

.s’.

(3)

(re = 1.3 fm has been used in the calculations.) On the other hand, the spreading width Tit given in eq. (I), was calculated by using the Gilbert-Cameron parametrization of the compound-nucleus levei density *), and by adopting an empirical expression for the squared matrix element. The latter quantity was assumed to decrease exponentially with compound-nuclear mass and, within the window region, to be independent of other physical quantities, such as excitation energy and spin for a given compound nucleus. This radical assumption could be accepted in a first approximation because of a previous observation that contours of constant level density p&&f) coincide approximately with resonant bands in most resonating systems 9*7). Such constant densities, labelled pwlndow,were in fact used to calculate rl. However, as absolute scaling for Ti cannot be reliably deduced, relative values r’/r& were calculated. These values were obtained by setting Ti along the i2C+“C resonant window approximately equal to unity and scaling Ti values for other systems accordingly. Although the model assumptions arc rather schematic and the procedure simple, the comparison with experiment has been altogether favourable’). Most encouraging has been the fact that the model has been able to account for both observation and absence of resonances in a number of systems, as well as to give roughly correct predictions. We shall thus briefly review the calculations of the OCM reported in ref. ‘) in view of the new experimental data obtained since. Table 1 summarizes the bandheads and moments of inertia for resonant bands predicted by the OCM and compares them with values deduced from data on eight target-projectile combinations for which such systematics of resonance data are possible. The good agreement observed over a wide mass region confirms that the gross properties of the resonant phenomena under study are adequately described by a simple rotor relation of molecular type, as in eq. (2). This statement is best illustrated in fig. 1, where the calculated bands are compared in the E* versusJ(J+ 1) plane with known resonance data for nine composite systems. It is still of more interest to survey in full the predicting power of OCM calculations over all available data. Fig. 2 summarizes the calculated values of ri/r&c for heavy-ion systems which have so far been experimentally investigated for resonance observation [only those with composite mass above A = 22 were considered since OCM calculations as formulated in ref. ‘) cannot be applied to lighter systems]. Evidently, the sustained interest over 25 years has resulted in a large number of studied systems ; 34 different projectile-target combinations are reported here. We

D. PoEaniC, N. Cindro / Orbiting-cluster concept

534

TABLE 1 Effective moments

values (deduced from of inertia for resonant

Entrance channel

Composite system

available data) and model calculations [eq. bands in heavy-ion systems with estabhshed f el‘

f mod

(2)] for bandheads and sequences of resonances

S’” 0

(1O-42 MeV. s*) ‘*c+‘%z ‘2C+ ‘60 tzc+rso t2C +24Mg ‘2C+28Si 160+14Mg r60+%i 28Si+28Si

24M g 28Si F% 36Ar ‘Wa I

44Tt “Ni

t4C+ t60, 12C+20Ne, resonances with assigned

2.6 * 0.2 3.3kO.4 - 4.4 - 5.7 I { I 8 12.7

3.1 4.0 4.4 5.6 6.5 7.1 8.1 12.8

12C+32S and 24Mg+24Mg are not included, spins in these systems to form bands reliably.

P”

= E, + E,

‘(MeV) 19 25.5 30 30 29 29 29 40 as there

20.6 25.4 32.1 28.4 27.1 31.6 28.8 39.2 are too few known

have classified them into four categories: (i) systems with unambiguously established resonant behaviour, i.e. those for which resonances with determined spins have been observed, marked by full circles ; (ii) systems displaying pronounced resonantlike behaviour, short of having firm spin assignments, marked by open circles ; (iii) systems for which the existing evidence for resonances is either weak, controversial, or incomplete, marked by open triangles; (iv) systems for which the prevailing evidence points against a resonant interpretation of the data (no resonances observed: full triangles). While the first class of systems is fairly well established, the dividing lines between the second, third and fourth groups cannot always be drawn easily ; for some systems, this classification may be subject to personal opinion. Nevertheless, we consider the classification presented in fig. 2 to be accurate to within one neighbouring category (0, 0, n and A) for each system. A striking feature of fig. 2 is the remarkable overall agreement of the OCM results with the experimentally observed behaviour of the 34 systems reported. Indeed, in spite of the rather large scatter of values of T1/T& for various systems, and notwithstanding the experimental uncertainties just mentioned, the correlation of resonance observation with relatively low values of rl is essentially upheld over a wide mass, energy and angular momentum region. In particular, the resonant systems (full circles) are fairly well described by I’i/r& 5 1. In ref 7, an empirical upper limit for resonance observability was set to z 9 (broken line in the figure), on the basis of resonances observed in rV& i2C + i*O. The resonant systems discovered recently, i4C + 160 [ref. lo)], and 24Mg + 24Mg [ref. ‘)I, 28Si + 28Si [refs. l- ‘)I “C + 2oNe [ref. “)I.

D. PoEaniC, N. Cindro / Orbiting-cluster

0

concept

535

‘“Cl 100

200

100

300

60

30 20

100

0

300

50

300

100

600

80

400

700

80

40 60 60 30

0

300

600

500

I500 J(J+I)

Fig. 1. Comparison of known data on resonances in heavy-ion reactions for nine systems with molecular bands predicted by the OCM. [References on experimental data are given in the text and ref. “‘).I Full circles and squares represent isolated resonances with established spins. Open circles and squares represent resonantlike structures or strong indications for resonances with known dominant partial wave. (Mass number not specified for nuclei with N = Z.)

approximately was predicted at the time. The second width I’l. All

follow the above description. We stress that resonance observation in ref. 7, for the first three systems ; 24Mg + 24Mg was not considered class of systems is also characterized by low values of the spreading of these systems have recently been studied experimentally with

536

D. PoEaniC, N. Cindro / Orbiting-cluster

^

O.'O O*Ne

0.

t

$e!Y elk

’ ‘k.0

concept

26..

Mg+Si

’

o*o Si+Si z o*ca

0.01c

Fig. 2. Calculated values of relative spreading widths T1/T& [ref.‘)] for 34 experimentally studied heavy-ion systems. [References on experimental data are given in the text and ref.“).] Experimental evidence on resonant behaviour is classified as follows : systems with clearly established resonances (full circles), pronounced resonantlike behaviour short of firm spin assignments (open circles), weak, controversial or incomplete evidence for resonances (open triangles) and prevailing evidence against resonant interpretation (full triangles). Nuclei with unmarked mass have N = Z.

results strongly supportive of a resonant or resonantlike interpretation of data: 9Be+13C [ref.“)], 12C+14N [ref.13)], 12C+“N [ref.14)], 12C+170 [ref.“)], 13C+ 160 [ref. “)I, 160+ 160 [refs. 16-19)], ‘60+20Ne [ref. ‘“)I, 24Mg+28Si [ref.21)] and ‘60+40Ca [ref.22)]. While 12C+14N, “C+“O and 24Mg+28Si were not considered in ref. 7), the remaining six heavy-ion systems were explicitly cited as candidates for resonance observation in the same paper. It must be said, however, that experimental indications in favour of resonances existed earlier for 13C+ 160 [ref.23)] and 160 + 160 [ref. 24)]. It is conceivable that for systems with non-zero entrance-channel spin, such as gBe + 13C, “C + 14,15N, “C + “0 and 13C + 160 definite spin assignments are not likely to be obtained. On the other hand, sysiems such as 0 +Ne, Mg+ Si and 0 + Ca have been studied rather scantily so far; nevertheless, existing data look very promising and may soon place these systems in the resonant class. The absence of a well-established resonance band in the heavily studied 160 + 160 system, however, remains a disappointment not only for the OCM, but also for all other resonance models. We shall comment on this problem in more detail in sect. 4.

D. PoEaniC, N. Cmdro / Orbiting-cluster concept

Finally, 12C+26Mg

we note systems,

that

the

for which

l”B+14N, there

13C+13C,

is strong

i4N+

experimental

531

14N,

180+180

evidence

against

and a

resonant interpretation 25), are all characterized by relatively large values of the spreading width r’. Thus, the OCM as presented in ref. ‘) appears to have successfully singled out the physical quantities most relevant to a qualitative description of the quasimolecular resonant phenomena. Furthermore, the resulting parametrization has provided a practical procedure for obtaining useful guidelines on the observability. of resonances in a number of systems. The surprisingly good of such calculations concurrence simple with experimental behaviour, demonstrated so far, has encouraged an extension of the model to heavier systems but has also raised the question of the correct interpretation and true significance of the model procedure and predictions. The latter certainly must not be overestimated.

3. Modifications

of the model procedure

When applying the orbiting-cluster calculations to heavier nuclear systems, certain questions and difficulties inevitably arise ; among them are the following: (i) the use of Gilbert and Cameron level densities 8, (including shell and pairing corrections) at nuclear excitations as high as 70 MeV ; (ii) the determination of pwindow for heavy systems ; (iii) the lack of reliable values for mass excesses in heavy compound nuclei with N =: Z far away from the stability line. In order to try to assess the extent to which shell structure is preserved at high excitations [problem (i)], let us consider two extreme cases within the present study, the light, tightly bound “C+ “C + 24Mg system (E,+E, z 20 MeV) and the heavy, loosely bound 35C1+37C1 -+ 72Se system (E,+ E, z 45 MeV). Comparably high grazing angular momenta for the two compound nuclei, i.e. those corresponding to half the limiting angular momenta J, (discussed below) are J = 1222 for C +C and J = 34h for Cl +Cl. For these grazing angular momenta, 24Mg is excited to E* x 32 MeV and “Se around E* cz 59 MeV. However, in both cases, the average energy per valence nucleon is similar, about 4 MeV. The regime of excitation and degree of preservation of nuclear structure therefore does not appear to change drastically over the mass region of interest. Thus, once accepted for lighter systems, the parametrization of Gilbert and Cameron ostensibly need not be discarded for collisions of heavier nuclei, such as Si, S and Ca. The second problem, (ii), is illustrated in fig. 3, where values of compoundnucleus level densities along molecular windows predicted by the OCM are shown for three resonant composite systems, 24Mg(‘2C + 12C), 40Ca(‘2C + 28Si,

538

D. PoTaniC, N. Cindro / Orbiting-cluster concept

CN DENSITIES (kcV-‘I ALONG MODEL BANDS 0.L t

I 0.04 I

m

1

1

I

1

1

0

I

,

,

10

l(h)

2o

0

20

I LOO02000 1000 100 200 loo-,

, 0

,

(

, 20

,

,

wmwl , / LO

,

,

,

J/h1

Fig. 3. Calculated compound-nucleus densities along bands predicted by the OCM for 24Mg(‘2C + r2C), 40Ca(‘2C + 28Si, I60 + 24Mg) and 56Ni(28Si + “Si). Vertical arrows denote the limiting angular momenta J, for the compound nuclei, calculated following Cohen, Plasil and Swiateckiz6). Shaded areas of angular momentum show domains of experimentally observed resonances.

160+24Mg) and 56Ni(28Si+ 28Si). Clearly, for heavy nuclei, where broad angular momentum regions are populated (up to- 50h), one can no more observe the near constancy of P&E, J) along the model bands. Instead, the relatively rapid change of pcN with increasing energy and spin makes it more difficult to define a single average or charactristic value like pwindow. (The change is more rapid for symmetrical than for asymmetrical systems. The smaller grazing moments of inertia associated with the latter lead to steeper molecular bands, more in line with contours of constant pcN for a given compound nucleus.)

D. PoEaniC, N. Cindro 1 Orbiting cluster concept

539

In order

to remove this difficulty, we have modified the model approach, rep1acing Pwindow by the maximum value of the compound-nucleus level density along the model band pp”. In this way, instead of calculating the expected average spreading along the predicted band for a given system, we now focus on the expected maximum value of the spreading width. These uppermost calculated values of Ti (least favourable for resonance observation) are compared for various systems. Although they exhibit a systematic behaviour similar to that of the average values, their advantage is the unique way in which they are defined. Hence, the model again relies on the relative spreading width as the main quantity governing the observability of resonances; this quantity is now defined as (4) with (rl)max = 2rr((CNJVle1)12p~. Since R,"?" takes into account only the highest calculated values of level densities along molecular bands, it is of interest to estimate the lower limit of level density and spreading expected for each heavy-ion system. In general, the minimum compound-nucleus level density along a given molecular band is expected to occur for the highest angular momentum J, which the corresponding compound nucleus can sustain before spontaneously breaking up. (Exceptions are very asymmetric heavy-ion systems, such as 12C + 28Si, which display a minimum in P,-- for angular momenta slightly lower than J,, cf. fig. 3.) In this work we have used the rotating liquid-drop model of Cohen, Plasil and Swiatecki 26) to estimate the limiting angular momentum J,. Finally, the third problem, (iii), concerns the increasing uncertainty associated with values of mass excess for heavy.N = Z composite nuclei [in this work the standard mass-excess compilation of Wapstra and Bos was used27)]. However trivial this problem may seem, for CItype nuclei (N = Z) it has limited the present study to systems with Z s 36 (Kr), since the uncertainty associated with extrapolating values of mass excess for heavier systems would render OCM calculations pointless. Although a-type systems are generally prime candidates for resonance observation owing to their simple structure and relatively low binding of the composite nucleus, the above limit does not seriously impair the model applicability as 40Ca + 4oCa is the heaviest stable combination of this type. 4. Extension of the model: results and discussion The present study encompasses 92 different projectile-target combinations. Among them we have tried to include all those which have been experimentally investigated, as well as most of the available stable beam-target combinations of interest. The results are given in table 2 and illustrated in fig. 4.

540

D. PoEaniC, N. Cindro / Orbiting-cluster

concept TABLE Summary

Composite system

=Mg =Mg ‘sMg =A1 27A1 % *%i

“Si

32S 34S

38K %a 42Ca

46Ti 4’V +Zr

Entrance channel ‘Be+13C “‘B + “C ‘lB+‘%Z l”B+14N 12C+1ZC 12C+‘3C 12C+14C 13c+13c 14C + 14C 10B+160 12C+14N “C+“N ‘%+14N ‘%I+ 160 14N+14N l*c+“o ‘%+ 160 14N+15N lzc+l*o 13c+“o Y+‘60 “N+“N 1ZC+20Ne ‘60+ 160 14C+*‘Ne 160+1*0 ‘80 + ‘so 1oB+24Mg 12C+24Mg 160+“Ne 1oB+28Si 14N+*‘Mg 12C+*8Si 160 + 24Mg 14C+28Si 160+26Mg =O + 24Mg ‘60+27AI “O+*‘Al ‘*O+*‘Al l*c+=s 160+28Si “Ne + 24Mg 23Na+*3Na 19F+ *sSi Z3Na+ 24Mg 160 + 32s 20Ne+28Si 24Mg + 24Mg

of the extended .A”) (MeV)

E,+Ec (MeV)

aFG “) (MeV-‘)

27.1 23.0 23.9 35.4 20.7 23.0 25.8 29.1 27.5 26.9 22.8 24.9 30.8 25.4 41.7 29.6 28.8 33.5 32.1 35.2 31.1 33.1 29.3 27.5 36.0 35.2 39.7 32.9 28.4 31.7 31.0 31.5 27.0 31.6 33.5 32.8 39.0 30.6 36.1 39.2 26.8 28.9 35.2 43.6 38.3 38.6 31.7 35.5 36.7

2.64 2.64 2.76

3.13 3.13 3.68

2.88

3.32

5.13

3.00

3.85

2.46

3.12

4.08

4.26

3.36

4.00

4.13

3.12

3.65

0

3.24

3.45

1.80

3.36

3.05

3.89

3.48

3.57

2.09

3.60

3.81

3.76

3.84

3.39

3.29

4.08

4.12

3.49

4.32 4.08

4.88 3.72

3.66 0

4.32

4.03

3.48

4.56

4.29

0

4.80

5.00

3.87

5.04

6.58

3.47

5.16 5.28 5.40

6.55 6.97 7.59

1.64 0 1.44

5.28

5.46

3.37

5.52

6.93

3.17

5.64

6.79

1.44

5.76

6.00

2.79

(M:;‘-

‘)

5.13 0 2.67

D. PoEaniC, N. Cindro / Orbiting-cluster concept

541

2 OCM

J,? (h) 21.6 21.4 22.6 23.5 24.7 25.8 27.9 25.6 26.8 21.6

28.8

30.0

31.6 34.1 36.4 33.6 35.6 37.6 39.4

42.0 42.6 43.9 45.1 43.0 45.7 46.2

46.6

calculations R,I

pw(J,)‘) (MeV-‘) 300 330 550 4300 39 530 1100 3100 1500 4000 720 500 2600 130 8600 2600 1700 6300 6000 1.2 x lo4 3000 5500 1300 420 3.3 x lo4 1.9 x 104 3.7 x 105 5.6 x lo4 5900 5600 2.6 x 10’ 4.7 x lo4 4.2 x lo4 4.3 x lo4 5.2 x lo6 1.8 x lo6 1.1 x 10’ 1.6 x lo6 4.2 x 10’ 2.2 x lo8 2.1 x 105 6.6 x lo4 2.7 x lo* 9.2 x 10’ 3.0 x 10’ 2.1 x 10’ 9.6 x lo6 1.2 x 106 1.3 x 106

1920 2590 5790 3.5 x lo4 312 6250 1.5 x lo4 4.7 x lo4 2.2 x lo4 3.1 x lo4 8080 5080 3.0 x lo4 1000 8.3 x lo4 2.6 x lo4 2.0 x 104 8.1 x lo4 6.3 x lo4 1.6 x lo5 4.5 x lo4 8.4 x lo4 8880 4870 4.8 x 10’ 3.7 x 105 9.2 x lo6 1.8 x lo5 3.1 x lo* 8.8 x lo4 4.9 x lo5 5.0 x 105 1.4 x 10s 7.8 x lo5 6.8 x 10’ 4.8 x 10’ 5.7 x lo8 3.8 x 10’ 1.7 x lo9 1.3 x 10’0 4.1 x 105 8.6 x lo5 9.4 x lo6 7.8 x lo9 1.4 x lo9 1.6 x lo9 1.0 x 10’ 4.3 x 10’ 6.7 x 10’

InPid 1

14 18 28 113 1.0 14 22 69 14 46 12 5.0 30 0.66 55 12 9.0 36 19 48 14 25 1.2 0.66 30 23 260 11 0.87 2.5 6.3 6.4 0.84 4.5 180 126 1500 68 2000 1.1 x lo4 0.49 1.0 11 4200 520 570 2.6 11 16

Resonance observation

possible not observed observed possible

possible possible possible observed not observed possible possible observed observed observed likely

not observed possible observed likely possible possible observed observed not observed

observed observed possible

likely possible observed

542

D. Po?aniC, N. Cindro 1 Orbital-cluster concept TABLE 2

Composite system “Cr slMn ‘*Fe s4Fe

58Ni

s9CU 60Zn

T% 12Se ‘*Kr

Entrance channel 24Mg+26Mg 23Na+28St 24Mg+27Al r*C f4’Ca 24Mg +**St ‘*C f4*Ca r4C + 40Ca ‘9F+35Cl 23Na+31P 24Mg +31P *‘Al+*sSi r4N + 4ZCa *‘Al + *?Si 180+39K 160 + 40Ca 24Mg +32S *sSi + *sSi 160+42Ca ‘so + 40Ca 26Mg f3*S 27Al+3’P **Si + 3oSi *‘Al + 32S 28Si+31P 24Mg +36Ar 28Si+32S 28Si+ 33S 2sSi+34S **Si+ 36S *rp+**s 24Mg +40Ca 28Si+ 36Ar 32S+32S **s+=s 32s + 34s 32s + 36s 3*s+*‘cI %S+*sc1 28Si+ 40Ca 32S+ 36Ar 35c1+37cl 32S+40Ar 32S+40Ca 36Ar + 36Ar

(MeV- ‘)

a’)

A”) (MeV)

6.00

6.54

2.89

6.12

6.29

1.54

6.24

6.02

3.08

6.48

6.13

2.84

6.60

5.17

1.30

6.72

5.51

0

6.84

5.95

1.27

6.72

4.67

2.50

6.96

5.44

2.47

7.08

5.58

1.27

7.20

5.81

2.33

7.32 7.44 7.68 7.56

6.39 6.94 8.03 7.35

1.06 2.35 2.56 1.29

7.68

7.81

2.65

7.80 7.92 8.16

8.34 8.94 9.94

1.36 2.77 2.86

8.28

10.37

1.50

8.16

7.99

2.84

8.64

11.33

2.93

8.64

9.43

2.67

Es+.& (MeV)

aFG“) (MeV-‘)

41.6 37.1 40.3 31.8 37.7 35.9 42.4 48.3 46.4 41.8 41.7 41.2 43.2 46.9 37.9 41.7 39.2 40.3 47.8 45.4 46.4 42.3 42.7 40.2 40.6 38.3 40.0 41.0 41.9 39.6 39.0 37.6 37.8 39.0 40.7 44.7 42.1 40.9 35.9 37.0 45.3 45.0 35.7 36.6

“) aFG = 0 124, Fermi-gas level-density parameter; a = aobcrvsd when available; “) JL = limiting angular momenta for compound nuclei, ref. 26).

otherwise

a = a,,

“) CN level densities calculated using the Gilbert-Cameron parametrization *) along OCM rotational bands of the band, i.e. J = Jr. d, RFfX = relative maximum spreading width (eq. (4)).

D. PoEaniC, N. Cindro / Orbiting-cluster

543

concept

(continued)

PJJL)‘)

J,“) (h) 49.3 49.6 49.9

52.1

52.9 54.3 55.6 53.0

55.9

56.0 56.0 51.4 58.8 61.5 58.8 58.7 58.8 61.6 64.3 64.3 61.1 66.9 63.3

[Gilbert

and Cameron

(eq. (1)): py

“)I

= maximum

Illax

max d

E

Pw ) (MeV-‘)

4-J

(MeV-‘) 2.0 X 10’ 4.3 x lo6 1.3 x 10’ 1.4 X lo6 1.7 X lo6 1.1 x lo* 2.6 x 10s 2.2 x 10s 5.2 x 10’ 2.0 x lo6 1.6 x lo6 1.1 x 10s 8.1 x lo6 2.6 x 10s 9.0 x lo5 4.2 x lo5 1.5 x lo5 1.6 x 10’ 7.2 x 10’ 6.7 x lo6 8.6 x lo6 2.2 x lo6 5.7 x lo6 2.4 x lo6 5.2 x lo6 1.5 x lo6 1.4 X 10’ 3.8 x 10’ 3.9 x lo* 6.8 x 10’ 1.9 X 108 5.4 x 10’ 4.7 X 10’ 3.6 x 10’ 9.5 x 108 2.7 x 10” 3.5 X 10’0 1.8 x 10”’ 4.1 x 10’ 4.3 x 10’ 3.3 X 10” 3.4 x 10” 3.2 x lo8 4.0 x lo8

1.4 x lo9 2.4 x lo8 7.7 x lo8 1.1 x 10’ 8.4 x 10’ 7.6 x 10’ 7.4 x lo8 5.0 x lo9 2.5 x lo9 6.3 x 10’ 6.1 x 10’ 2.4 x lo8 3.8 x lo8 3.1 x lo9 3.2 x lo6 9.6 x 10’ 4.3 x 106 6.0 x 10’ 6.4 x lo8 2.7 x lo8 3.7 x lo8 1.0 x lo8 2.4 x lo8 1.0 x lo8 1.5 x lo8 6.8 x 10’ 8.5 x lo8 2.8 x lo9 4.4 x 10’0 6.3 x lo9 8.0 x lo9 4.3 x lo9 4.7 x lo9 4.2 x 10” 1.6 x 10” 7.2 x 10” 9.7 x 1or2 5.5 x 10’2 2.5 x lo9 4.1 x 10’ 1.4 x lOI 1.2 x 1014 4.0 x 10’0 6.1 x 10”

160 18 58 0.55 4.2 1.7 17 110 59 0.98 0.94 2.5 4.0 22 0.035 0.10 0.045 0.28 3.1 1.3 1.8 0.47 0.77 0.33 0.33 0.14 1.2 2.7 20 4.1 3.5 1.9 2.1 12 33 660 600 340 0.23 0.37 2600 2300 0.75 1.2

; A = pairing energy correction, values along

rotational

bands;

)

Resonance observation

possible likely likely hkely possible

likely likely likely likely likely likely likely likely likely likely likely likely likely likely likely likely likely possible likely likely likely likely possible

likely likely

likely likely

; E,,, = E-A.

taken

from ref. ‘)

p,(J,)

= values corresponding

to the upper

limit

D. PoEaniC, N. Cindro / ‘Orbiting-cluster

544

II

1o13-

I

0

1

I’

I1

‘I’

1

concept “I

If

“IT

xx x

( MeV-’ I

$n””

w

10’2 :

:

lo” -

30

LO

50

:

:

:. x

: .**

.’

60

:

/,

:

,

.f

.-

/

,

”

m ACN

Fig. 4. Calculated maximum values of compound-nucleus level densities along bands predicted by the OCM, py, for about 90 different heavy-ion combinations. Classification of experimentally investigated systems (circles and triangles, full and open) same as in fig. 2; crosses: composite systems not yet experimentally investigated. Full line: reference trend characteristic of resonant systems, used to derive the empirical dependence of I(CNIVle1)1* on A,,; dashed and dotted lines: values of py 10 and 100 times larger, respectively, than the reference resonant trend.

Fig. 4 shows the calculated values of py” versus A,, for most of the systems treated in this study (missing are only a few heavy systems with pr” values outside the scale in the ligure). The most important feature of fig.,4 is that, if the exponential rise with A,, is removed, the behaviour of p?” is very similar to that of r1 for all experimentally studied systems shown in fig. 2. Thus the OCM approximation, which assumes an exponentially decreasing ((CNJVlel)12 with A, is essentially upheld through the fp shell. In fact, it was this result which made possible the extension of the model calculations with only slight modifications,

D. PoEaniC, N. Cindro / Orbiting-cluster

discussed pattern

in the preceding of calculated

values

section. of py”

It is also of interest for heavier

545

concept

systems,

to note

the oscillatory

emanating

from

the

combined effects of binding energies and shell structure. Table 2 summarizes the relevant information and results on all heavy-ion systems considered in this work. The most important quantities determining the observability of resonances are the binding and Coulomb energies, E,+Ec (bandhead of the quasimolecular band) and the compound-nucleus level-density parameter a. A relatively low threshold for the quasimolecular configurations (E,+E,) and a relatively low CN level density parameter a (as compared with the structureless Fermi-gas value, uro) reduce the density of the competing compoundnucleus levels and thus the spreading width rl. Of somewhat lesser consequence are the pairing correction A and the model moment of inertia for the incoming channel, which determines the slope of the calculated rotational-band. The main results of the model calculations for each heavy-ion system are the values of pr” in table 2 we have and RF?‘, discussed in the preceding section. For comparison, also included values of p&J,), compound-nucleus level densities at the upper limit of the molecular window, which usually represent the minimum values p$“. As mentioned earlier, we have calculated the limiting angular momenta J, by using the liquid-drop model of Cohen, Plasil and Swiatecki2’j). From the standpoint of resonance observation, the results presented in table 2 indicate a number of promising heavy-ion combinations in the fp shell. Among them, the best candidates are, as expected, cc-type systems such as 48Cr(O+S), 52Fe(C+Ca), 56Ni(0+Ca, Mg+S), 60Zn(Si+S, Mg+Ar), 64Ge(Si+Ar, S+S, Mg+Ca), candidates

68Se(Si + Ca, S + Ar) and 72Kr(S + Ca, Ar + Ar). However, equally likely appear to be 54Fe(‘2C+42Ca), 55Co(24Mg+ 31P, 27A1 +28Si),

56Co(14N +42Ca, 27A1 + 29Si), 58Ni(28Si + 3oS, I60 + 42Ca. 26Mg + j2S, 27A1 + 31P, 61,62Zn(28Si+33,34~) 180+40Ca), 59Cu(28Si + 31P, 27A1 + 32S), and 63Ga(31P+32S). There are also a number of systems with RF? 5 20, for which resonance observation is predicted to be possible. but less likely. At this point the results of OCM calculations need to be critically examined and interpreted. Although the overall correlation of resonance observation with relatively low spreading (RjY m the present formulation) is established beyond reasonable doubt, there remain several exceptions and important open questions. The most prominent exception is the 160 + 160 system, mentioned in sect. 2, which is characterized by low relative spreading RF?’ = 0.66 and yet fails to display a clear-cut resonant behaviour, as observed, for instance, in “C+ “C and 12C + 160. In fact, studies of the resonant behaviour of L6O+ I60 via the a exit channel i6-18) h ave shown that the resonant amplitude (which is undoubtedly present) competes strongly with the background Hauser-Feshbach amplitude within the same near-grazing angular-momentum window. This interference appears to be the main cause preventing clear observation of resonant structure. A similar conclusion concerning the interference of the resonant and direct processes

546

D. PoEaniC, N. Cindro / Orbiting-cluster concept

was reached by Singh et al. 24) in their study of the 12C exit channel. It seems that for 160 + 160, the OCM as well as other quasimolecular models underestimates the average coupling matrix element ((CNIVle1)1’, which may be increased for reasons of nuclear structure and phase-space limitations. It should be noted that the dominant spins of the resonantlike structures in “jO+ I60 follow the band predicted by the OCM fairly closely, as shown in fig. 1. On the other hand, the “C+ 20Ne system exhibits resonant scattering at large angles, with resonant angular momenta several units of h smaller than the grazing values ii). Thus the nature of these structures may be of non-molecular origin. Similarly interesting are the 3oSi(‘2C + “0, 14C + 160) and 4*Cr(24Mg + 24Mg) systems which display resonant behaviour, yet are characterized by fairly high relative maximum spreading, RF? z 15-20. Although the spreading widths [calculated following eq. (l)] are somewhat smaller for energies and spins where resonances are observed experimentally, these systems (especially 12C + i80) set an empirical maximum on the relative spreading compatible with resonance observation. The occurrence of resonances at energies lower than predicted by the OCM (particularly in 24Mg+ 24Mg, cf. fig. 1) may be related to ground-state deformations of the two nuclei in the entrance channel. The most serious shortcoming of the present model is certainly the complete neglect of the escape width into direct channels Tf, a quantity generally considered to gain importance with increasing mass and energy of interacting nuclei. In fact, schematic models of resonances have been proposed solely on the basis of rT [refs. 28*““)I. Predictions of these models, particularly those of Baye 28) for u-type systems with AcN 2 36, are in overall agreement with OCM calculations, as the same physical quantities (low E, + Ec and simple nuclear structure, e.g. of a-type) reduce both r1 and P. On the other hand, in a recent study Cseh 30) has found, by using two different types of analysis, that for J” = 2+, 4+, 6+ and 8+ resonances in 12C+ i2C, r1 is about 10 times larger than Tf. It would be very interesting to perform similar analyses for heavier systems ; however, more complete experimental resonance parameters are required for that. In the light of the present discussion, the question of the content and significance of OCM predictions arises naturally. Addressing this question, we should stress that the model represents a parametrization of rl, i.e. the imaginary part of the optical potential in the near-grazing region for various heavy-ion combinations. Imposing the condition that r1 be relatively small near grazing is thus equivalent to selecting surface-transparent heavy-ion interaction potentials. In this sense, the orbiting-cluster model should not be narrowly regarded as a model of quasimolecular resonances, but rather as one of surface transparency, which is a necessary precondition for formation of resonant or resonantlike states. Indeed, it has been observed that surface-transparent heavy-ion potentials may lead to a variety of mutually related phenomena, among them resonances. In that respect, the observation of resonant

D. PoZaniC, N. Cindro / Orbiting-cluster concept

states in heavy systems as Si + Si and Mg + Mg has also been tentatively

541

attributed

to high-spin fission isomers 2*3). The present model obviously cannot distinguish beween such similar reaction mechanisms. While this work is concerned with relatively light nuclear systems (in the sd and fp shells), the important question of the upper mass limit for observation of nuclear molecular phenomena remains open. In fact, recent theoretical interpretations of experimental studies of the electric decay of the vacuum in supercritical fields indicate that giant nuclear molecules may have been formed with Z h 180-190 in near-barrier scattering of very heavy nuclei, such as U + U and U + Cm [ref. 3 ‘)I. It is indeed interesting to apply simple phase-space arguments of the OCM to heavy nuclear systems. For example, the scattering of two double-magic nuclei, 4sCa+208Pb 9 at energies near the Coulomb barrier, would correspond to very low compound-nucleus excitation energies, close- to the ground state of 256N~. Clearly, observation of nuclear quasimolecular phenomena, if at all possible in this system, would be favoured for relatively central collisions around the Coulomb barrier. In the end, we note that the OCM is not the only schematic model linking resonance observation with low compound-nucleus spreading; the model proposed by Thornton et al. 32 re ies on a similar approach, but makes use of a different parametrization of p,‘v and ” neglects the role of the coupling matrix element [eq. (l)]. The model has been applied only to systems with mass lighter than A = 30. However, it seems improbable that the model of Thornton et al. could predict resonances resonance

in heavy systems such as Si +Si, bearing in mind the upper observation pcN 2 25 levels/MeV, given in their paper.

limit for

5. Conclusions On the basis of available experimental evidence it may be concluded that the orbiting-cluster concept, as presented and applied in the present study, provides a reasonable parametrization of the quantities most relevant to the occurrence of surface transparency in the interaction of nuclei ranging from Be to Ca. As the observation of quasimolecular resonances in heavy-ion reactions is closely related to the latter phenomenon, our calculations have so far demonstrated considerable predicting power. Nuclear-structure effects (especially on the coupling strength), deformations of nuclear shape and competition of open direct channels may prevent or partly obscure observation of isolated, well-defined resonant states in some systems favoured by general predictions and, conversely, enhance such an observation in some less favoured systems. However, the latter effects have not proved strong enough to destroy the general correlation of evidence for resonances and relatively low compound-nucleus spreading established by the orbiting-cluster calculations over the set of 34 experimentally studied heavy-ion combinations.

548

Thus, in the absence

D. PoEaniC, N. Cindro / Orbiting-cluster concept

of more refined

of a speculative character, may resonances in heavy-ion reactions

models,

the present

be regarded as a guideline in the fp shell.

calculations,

although

for observability

of

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