A simple classification of resonant heavy-ion collisions

Volume 97B, number 1

PHYSICS LETTERS

17 November 1980

A SIMPLE CLASSIFICATION OF RESONANT HEAVY-ION COLLISIONS D. BAYE Physique ThOorique et Math~matique, CP 229, UniversitO Libre de Bruxelles, B 1050 Brussels, Belgium Received 17 August 1980

Simple diagrams based on the relative location of the effective barriers of direct channels are shown to select known resonant heavy-ion collisions. Other systems are predicted to exhibit a resonant behaviour, mainly 160 + 32S, 28Si + 32S and 28Si + 4°Ca.

An increasing number of collisions have been found to exhibit a resonant behaviour. The explanation of this behaviour requires the solution of two main problems: what is the origin of the resonances and why do only certain systems resonate? A successful answer to the latter question is expected to predict new resonant collisions. The aim of the present letter is to suggest a very simple way of selecting the resonant systems. The occurrence of resonant phenomena is associated with a weakening of absorption, or transparency, for certain partial waves. The origin of the transparency and its angular m o m e n t u m dependence are well understood if one takes into account the conservation o f good quantum numbers [ 1]. Because of the conservation of the total angular m o m e n t u m and parity, tile number of channels able to carry away the grazing wave may be relatively small for large angular momenta. Recently, models have been proposed which deduce a transparency window from low level densities in the compound nucleus [2]. However, as pointed out by Shaw et al. [3], the channels coupled directly with the entrance channel should play the major role in carrying away large angular momenta. Indeed, if the broad structures are interpreted as "molecular" states, they correspond to short interaction times. The role o f direct mechanisms should be enhanced with respect to compound-nucleus formation. The main direct channels likely to carry away large angular momenta are the few-nucleon-transfer [3] and inelastic [4,5] channels. Multi-channel microscopic calculations [5] emphasize the special importance o f inelastic

scattering in tile process of absorption. In the following. we shall simplify the discussion by considering direct channels only. The results of refs. [4,5] can be summarized in a very simple rule [5,6]. For a given partial wave, absorption is not important towards channels with an effective barrier (i.e. Coulomb plus centrifugal barrier) higtier than the elastic effective barrier. Notice that this does not mean that absorption towards channels with a lower barrier is important since we do not know the strength of the coupling between the channels. The barrier rule can be studied in a simple systematic way. Let us assume a rotational form for the effective barriers of channel c~ EJ~ = EBc~- Qc~ + lo~(lc~ + 1)kBc~ ,

(1)

where EBc~ and Q~ are respectively the Coulomb barrier and Q-value of channel a and J is the total angular momentum. In channel c~, the colliding nuclei have spins I1~ and I2c~ and several values of the relative orbital momentum l~ are possible. The most important of these values is the lowest one [4,5] l~ = J - I 1c~

I2c~ ,

(2)

which corresponds to the lowest effective barrier.(Here we assume that the value given by eq. (2) is not forbidden by parity conservation.) The rotational constant kBc~ is defined as kBc~ = h2 /21x~R2~ ,

(3)

where ~c~ is the reduced mass and RB~ the barrier 17

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PHYSICS LETTERS

radius of channel c~. Since Qa,/~c~, Ilc~ and I2a are known physical data, only EB~ a n d R B ~ need to be chosen. Let us define t#Bc~

=

C B Z l c ~Z 2c~e-/RBc ~

17 November 1980

J EB

Mg

Be+Ar --

~,Ca

(4)

and RBc~

A

=~ ~'A1/3 + A 1 / 3 ) . "B t~ 1~ 2¢~

30

These approximate forms only depend on the mass charge numbers and on two parameters C B and r B. Since the discussion is nrainly qualitative, we have chosen the rounded values 0.9 and 1.5 fm for C B and rB• The classification of the channels will be based on the location of the crossing points of the barrier curves. Let us go thoroughly into this point. The barrier curves of two channels c~ and t3 cross at Jc~ ~ [--(EB~

Qc~

EB# + Q~)/(kBc, - k B ~ ) ] 1/2 ,

(6) if we assume that all nuclei have sero spin. The crossing point does not always exist: if the absolute Coulomb barrier E B - Q is located lower for tire channel with the larger reduced mass, this channel is favoured everywhere. If the barrier curves cross, the channel with the smaller reduced mass is favoured for J < J ~ and the other one for J > Jc~#- Another instructive quantity is the crossing point between a zero-spin elastic channel c~ and one of its inelastic channels gem* ~ (ec~*/Ic~* )lac~R2Bc/h2 ,

(7)

where eu, is the excitation energy and I u , the largest channel spin. A crossing between an elastic barrier curve and one of its inelastic curves always means a stronger absorption for the grazing wave. The discussion o f (6) and (7), shows that the channels able to carry away large angular momenta are mainly more symmetrical reaction channels (i.e. with a larger reduced mass) and inelastic channels. Conditions favouring the occurrence of resonances are thus: (i) a large Q-value, (ii) a low Coulomb barrier, (iii) a large reduced mass and (iv) a large value for the lowest inelastic ratio e~,/Ic~,. Condition (ii) conflicts with condition (iii) and is often not fulfilled. Condition (i) is common to all approaches of the resonance problem [2]. On the contrary, condition (iv) is typical of "inelastic" absorption. A good candidate for the occurrence of resonances will thus involve nuclei with large 18

/

1!

(5)

o

/Be*

"~

:.A ,0

°'"'

A = Z,8

,

,0

s

Ca

_ _ 30 ~ J ~ / / / Ne+S i

1o

m

20

"=', 2s

Be F¢,

A = 60

F

....

'cY// ,

10

i,

20

2s

j

Fig. i . Barrier curves o f the main elastic (full lines) and inelastic (dashed lines) channels as a function o f J ( J + I ) for A = 40, 44, 48 and 60. The inelastic channels shown are respectively C + Si(2+), O + Si(2+), O + S(2 +) + S. The energies are de-

fined with respect to the ground state of the unified nucleus. binding energies and high excitation thresholds. In order to test the barrier rule, we have studied the crossing diagrams of 4n-nucleon systems with A = A 1~ + A 2~ ~> 36. Examples are shown in fig. 1. For the sake of clarity, a few channels only are represented: 4n-nuclei (full lines) and inelastic (dashed lines) channels. Other channels, like the nucleon-transfer channels [3], contribute to the absorption of high angular momenta but are in general less important than the inelastic channels because of a less favorable Qvalue. Before detailing the different cases, let us mention that the crossing point should be taken as indicative. Around a given crossing value, resonances should be expected in both channels. A = 36. Below J ~ 25, the best system is C + Mg; beyond this value, O + Ne is favoured. In both cases, several inelastic channels are " o p e n " i.e. are able to carry away the angular momentum. Such systems are bracketed in the summary of table 1. Resonances have been observed in both collisions [8,91. A = 40. The best channels are C + Si and O + Mg which cross near J = 27. The C + Si channel is favoured beyond J ~ 13 (see fig. 1). Absorption towards inelastic channels [mainly C + Si (2+)] is weak. The C + Si system exhibits clear resonant structures [ 1 0 - 1 2 ] . A lower 2 + excitation energy of 24Mg makes the inelastic absorption more important in the O + Mg channel. Structures have also been observed in this channel [13] A = 44. Between J ~ 14 and 18, the best channel

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Table 1 Systems likely to exhibit resonant structures according to the barrier rule. The brackets indicate less favourable cases. A

Collision

36 40 44 48 52 56 60 64 68 72 76

(C + Mg), (O + Ne) C + Si, (O + Mg) C+S,O+Si C+A~,O+S C + Ca, O + Ar, (Mg + Si) (O + Ca), Si + Si Si + S (Si+ Ar), S + S Si + Ca, (S + Ar) S+Ca (Ar + Ca)

is C + S (see fig. 1) with a very weak inelastic absorption. Back-angle structures are observed in this channel [12]. Beyond J ~ 18, the barrier rule selects the well-known O + Si collision [10] with a stronger absorption towards the O + Si* channels. A = 48. Between J ~ 15 and 18, the C + Ar channel is favoured. Beyond J ~ 18, the best candidate is O + S (see fig. 1). The first inelastic crossing is located near J = 21. This system should clearly exhibit resonant features. On the contrary, the Mg + Mg curve indicates a lack of transparency. A = 5 2 . Between J = 16 and 22, a good channel is C + Ca which exhibits back-angle anomalies [12,14]. Also good should be O + Ar especially between J ~ 22 and 25. Beyond J ~ 25, Mg + Si is a possible candidate but with a stronger inelastic absorption, the first inelastic crossing being located near J ~ 17. Recent data also show anomalies for this system [ 15]. A = 56. The best channel is Si + Si (see fig. 4 of ref. [7] ) which exhibits striking structures in its 90 ° excitation function [16]. The O + Ca collision [17] is less good but might exhibit an o d d - e v e n absorption of the grazing wave [7]. A = 60. Beyond J ~ 21, the Si + S channel is an excellent candidate (see fig. 1). The first inelastic crossing occurs near J = 28. A = 64. The best candidate is the symmetrical S + S channel which should exhibit the same striking behaviour as Si + Si. As discussed forA = 56 [7], Si + Ar might present an o d d - e v e n behaviour. A = 68. Between J ~ 31 and 47, the best channel is Si + Ca with a low inelastic absorption. Beyond J

17 November 1980

47, S + Ar is also a possible candidate but with a stronger inelastic absorption of the grazing wave. A = 72. Beyond J ~ 37, S + Ca is a good candidate. A = 76. For high J-values (>~ 45), the channel Ar + Ca might exhibit a resonant behaviour. Beyond A = 76, the barrier rule does not provide good candidates. A discussion of Ca + Ca can be found in ref. [18]. Table 1 summarizes the results of the barrier diagrams. Strikingly, table 1 contains all the systems which are known to resonate for these nucleon numbers. We thus believe that the very simple barrier diagrams contain a useful physical information and emphasize the importance of direct and inelastic channels in the resonance problem. However, the barrier rule only reflects static properties of the channels and cannot take into account the effect of dynamical couplings. Resonances might occur in less favoured channels if they are weakly coupled with the favoured ones. Table 1 predicts the existence of a number of new resonant collisions, the best candidates being O + S, Si + Ca, S + Ca and especially Si + S. The barrier rule can be applied to study other nucleon numbers and further resonant systems can be detected. As already mentioned in ref. [5], because of a large binding energy and a high inelastic threshold of the 14C nucleus, the resonant properties should not disappear if 12C is replaced by 14C in a collision. I would like to thank Drs. P.-H. Heenen and G. Reidemeister for interesting discussions about the barrier rule. References

[1 ] R.A. Chatwin, J.S. Eck, D.A. Robson and A. Richter, Phys. Rev. C1 (1970) 795. [2] N. Cindro and D. Po~anid, J. Phys. G6 (1980) 359; S.T. Thornton, L.C. Dennis and K.R. Cordell, Phys. Lett. 91B (1980) 196. [3] R.W. Shaw Jr., R. Vandenbosch and M.K. Mehta, Phys. Rev. Lett. 25 (1970)457. [4] T. Matsuse, Y. Abe and Y. Kond6, Prog. Theor. Phys. 59 (1978) 1904. [5] D. Baye, P.-tt. Heenen and M. Libert-Heinemann, Nucl. Phys. A308 (1978) 229. [6] P.-H. tleenen and D. Baye, Phys. Lett. 81B (1979) 295. [7] D. Baye and Y. Salmon, Nucl. Phys. A331 (1979) 254. [8] J.L.C. Ford Jr. et al., Phys. kett. 89B (1979) 48. [9l T.R. Renner et al., Bull. Am. Phys. Soc. 24 (1979) 843. 19

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PHYSICS LETTERS

J. Barrette et al., Phys. Rev. Lett. 40 (1978) 445. M.R. Clover et al., Phys. Rev. Lett. 40 (1978) 1008. R. Ost et al., Phys. Rev. C19 (1979) 740. S~M. Lee et al., Phys. Rev. Lett. 42 (1979) 429. T.R. Rennet, J.P. Schiffer, D. Horn, G.C. Ball and W.G. Davies, Phys. Rev. C18 (1978) 1927. [15] N. Cindro et ah, Intern. Conf. on the Resonant behaviour of heavy ion systems (Aegean Sea, Greece, 1980), contribution.

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[16] R.R. Betts, S.B. Dicenzo and J.F. Petersen, Phys. Rev. Lett. 43 (1979) 253. [17] S. Kubono, P.D. Bond and C.E. Thorn, Phys. Lett. 81B (1979) 140. [18] D. Baye and Y. Salmon, Nucl. Phys. A323 (1979) 521.