Fusion Dynamics at the Extremes

45

55

Figure 4. The same as in Fig. 3 but for the ""Xe^Kr.xn) [12] and ^Ztf'^Sn.xn) [25] reactions leading to the 216Th* compound nucleus.

The indicated values of the fusion probability were deduced with the use of survivabilities (determined mainly by the kf value) obtained in our analysis of the excitation functions for ER produced in "asymmetric" reactions leading to the same CN [2,6,13]. The deduced "empirical" values of the fusion probability for considered "symmetric" systems are in quantitative agreement with the calculated ones obtained in the framework of the two-center shell model approach [27]. Thus, the highest value of P^ obtained for the 86Kr + 136Xe system seems to be reasonable from the point of view of compactness of the initial configuration, low inner barrier (connected with the transfer of mass) and the shortest way to the saddle point of the forming compound nucleus. In the two nearly symmetric 86Kr + 130Xe and 124Sn + 92 Zr systems leading to the production of the 216Th compound nucleus, the Pfus value is noticeably higher for the latter one in accordance with the calculations [27]. The less symmetric '"V + l70Er system was not considered there, but the consideration of close systems shows the existence of an additional inner barrier in touching configuration of similar nuclei [27]. It has to be overcome so that the complete fusion of nuclei could proceed. In the experiment, we observe the suppression of fusion at energies slightly above the Bass barrier [21]. Pfus grows with the energy as an integral error probability function (Fig.3), as it was observed elsewhere in ref. [25]. In contrast to nearly symmetric systems ([13,25] and Fig.4), the absolute value of Pfus reaches the unity at energies well above the nominal fusion barrier [21].

143

4.

Reactions leading to very heavy and superheavy ER

We applied the same approach to the analysis of similar data on the production of very heavy ER [12,13]. We fitted the calculated excitation functions for ER to the available experimental data for the evaporation channels of complete fusion reactions induced by heavy ions on the deformed and spherical target nuclei (see Table 2 in ref. [13]). For the reactions with the deformed nuclei we obtained a good fit to the data using the liquid-drop fission barriers of 10-30% higher than the nominal ones, with an exception of the reactions induced by 34S [17]. The strong reduction of the kf value (up to zero) for these reactions was insufficient to reproduce the obtained values of cross sections. For the reactions induced by projectiles from 16 0 to 48 Ca on spherical target nuclei we obtained a good fit to the data with the fission barrier of 10-30% lower than the nominal ones. For 5OTi+208Pb we had to use kf = 0.2 [12] to reproduce the data [14]. For the reactions leading to the heavier CN (with heavier projectile) we obtained a strong overestimation of the calculated cross sections for ER, even with kf = 0. Examples of comparing the measured excitation functions with the results of calculations are shown in Fig.5 for the 12Ca + ^ C m [28], 48Ca + 208Pb (see refs. in [13]) and 58Fe + 208Pb [14] reactions. The first two reactions lead to the same compound nucleus 256No*. If it is assumed that the survivability of nuclei produced in the 48Ca + 208Pb reaction is the same as in 12C + 244 Cm, we come to a conclusion on existing some suppression of fusion in the first more symmetric combination. This suppression increases with an increase in the energy (compare a4n < 3 nb for 48Ca + 208Pb and a4n = 150 nb for 12C + 244Cm at the same excitation energy of 36 MeV). The same is observed with an increase in the symmetry in the entrance channel of the reactions (at a transition from 48Ca to heavier projectiles). to5 lO3 T • X •

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144

We normalized the experimental cross sections to the calculated ones using the nominal values of liquid-drop fission barriers (kf =1.0) and fixed parameters of the nuclear potential. As a result, we have deduced the values of the fusion probability, Pfus, for the reactions leading to the very heavy nuclei (see Table 2 in ref. [13]). Searching for regularities in the behavior of Pjus, we have presented it as a function of the mean arithmetic fissility, xma [29], as is shown in Fig. 6 (right panel). Different atomic masses [30-33] could be used in the calculations for ZCN^108 keeping in mind the neglected values of the LD fission barriers for these nuclei and the fact that the actual fission barriers were determined by the shell corrections (the difference between empirical or calculated and LD masses of nuclei). Comparing different mass tables [31-33] with the empirical values [30] (AW'95 in figures) in the region nearest to SHE (from Db to element 110) we have found that the masses of Spanier and Johannson [31] (SJ'88 in figures) have the minimal values of the root mean square deviations (RMSD) compared with the other considered mass tables (left panel of Fig.6). Further, we considered the production of SHE using these tables, the Myers and Swiatecki masses based on the Thomas-Fermi model (TF-96 in figures) [32] and the finite-range droplet-model masses (DM-95 in figures) [33]. The TF-96 masses yield the values of RMSD comparable with the SJ'88 ones, whereas the DM-95 masses give a general underestimate of the values which leads to the corresponding overestimate of the absolute shell correction values.

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The calculations for the production of the heaviest nuclei were performed with the fixed values of the main parameters (V0= 40 MeV/fm, a(r0) = 4.0% for the 48Ca induced reactions on actinide target nuclei and V0 = 45 MeV/fm, o(r0) = 3.0% for the cold fusion reactions and kf = 1.0 for both the types). The calculation results for some reactions are shown in Fig.7 together with the estimated values of the produc-

145

tion cross sections derived from experiments [14,15,18,19]. As it was noted earlier [13], the increasing survivability (at /%, = 1.0) for the ER from Z = 110 to Z = 114 produced in the 48Ca induced reactions and for the ER from Z = 1 0 8 t o Z = 116 produced in the cold fusion reactions is the result of the growing shell corrections for the nuclei comprising the decay chains of the corresponding excited CN. Quantitatively, the growth of survivability becomes clear in the framework of the constanttemperature approximation, for which the relative probability of the neutron evaporation is defined as Tjrf ~ exp((Bf - Bn)/T), i.e., by the difference between the fission barrier {Bj) and the neutron separation energy (Bn), or about the same as the difference between the absolute shell correction value and Bn (see Fig.6 in ref.[13]).

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5.

Summary

•

The collective enhancement in the nuclear level density is clearly manifested in the complete fusion evaporation reactions leading to the production of nuclei near N = 126 at a transition from the deformed (prolate) nuclei to the spherical ones. This is not evident at the transition from the spherical nuclei to the deformed (oblate) ones. Further investigation is needed to clarify the situation in this region. Comparing the production of R n - P a ER in the asymmetric and nearly symmetric combinations leading to the same CN we come to a conclusion about a strong effect of the entrance channel on the fusion probability. It manifests itself most clearly in the reactions leading to Th CN. Thus, in the reaction 86 Kr + 136 Xe the fusion probability is about the same as in the asymmetric combination leading to the same deformed compound nucleus 222Th*, whereas for the nearly symmetric combinations leading to the spherical compound nucleus 216Th* this value is very low. Development and application of advanced models considering the fusion process dynamically is important for understanding the effect of the entrance channel on the fusion probability.

•

•

Considering the production of the heaviest ER in the asymmetric reactions, we also had to use the fusion probability to reproduce the experimental data. The scaling of this value allowed us to develop an effective semi-empirical approach to the estimates of the production cross sections for SHE in the 48 Ca induced and cold fusion reactions. We conclude that an increase in the survivability due to the growth of the shell corrections compensates a drop of the fusion probability that leads to about the same production cross sections for SHE.

This work was performed partially under the financial support of RFBR, contract No. 99-02-16447, and INTAS, contract No. 991-1344.

147

References 1. Blann M. and Komoto T. T., Phys. Rev. C 26 (1982) 472. 2. Vermeulen D., etal, Z. Phys. A 318 (1984) 157. 3. Reisdorf W., Z. Phys. A 300 (1981) 227; Reisdorf W. and Schadel M., Z. Phys. A 343 (1992) 47 and references therein. 4. Andreyev A.N., ef al, Nucl. Phys. A 620 (1997) 229. 5. Andreyev A.N., etal, Nucl. Phys. A 626 (1997) 857. 6. Bogdanov D.D., et al, Phys. of Atomic Nuclei 62 (1999) 1931. 7. Newton J.O., Particles & Nuclei 21 (1990) 821. 8. Ignatyuk A.V., Particles & Nuclei 16 (1985) 709. 9. IljinovA.S., etal, Nucl. Phys. A 543 (1992) 517. 10. Quint A.B., et al, Z. Phys. A 346 (1993) 119. 11. Schmidt K. -H. and Morawek W., Rep. Prog. Phys. 54 (1991) 949. 12. Sagaidak R.N., et al, In Heavy Ion Physics. VI Intern. School-Seminar, Dubna, 1997, ed. by Yu.Ts. Oganessian and R. Kalpakchieva (WS, 1998) 323. 13. Sagaidak R.N., et al. In Intern. Conf. on Nucl. Phys. Nuclear Shells-50 Years, Dubna, 1999, ed. by Yu.Ts. Oganessian and R. Kalpakchieva (WS, 2000) 199. 14. Hofmann S., Rep. Prog. Phys. 61 (1998) 639. 15. Oganessian Yu.Ts. et al, Eur. Phys. J. AS (1999) 63. 16. Oganessian Yu.Ts. etal, Nature 400 (1999) 242. 17. Utyonkov V.K., et al, in Heavy Ion Physics VI Intern. School-Seminar, Dubna, 1997, ed. by Yu.Ts. Oganessian and R. Kalpakchieva (WS, 1998) 400. 18. Oganessian Yu.Ts. et al, Phys. Rev. Lett. 83 (1999) 3154. 19. Oganessian Yu.Ts. et al, Phys. Rev. C, to be published (2000). 20. Popeko A.G., et al, Nucl. Instr. and Meth. In Phys. Res. B 126 (1997) 294. 21. Bass R., in Deep Inelastic and Fusion Reactions with Heavy Ions. Berlin, 1979, ed. by von W. Oertzen, Led. Notes on Phys. 117 (1980) 281 (Springer, 1980). 22. Cohen S., Plasil F. and Swiatecki, Ann. Phys. (NY) 82 (1974) 557. 23. Ignatyuk A. V., Istekov K. K., Smirenkin G. N., Yad. Fiz. 37 (1983) 831. 24. JunghansA. R., etal, Nucl. Phys. A 629 (1998) 635. 25. Sahm C. -C, et al, Nucl. Phys. A 441 (1985) 316. 26. Andreyev A. N., et al, Phys. of Atomic Nuclei, 60 (1997) 1. 27. AdamianG. G., etal, Nucl. Phys. A 646(1999) 29. 28. Sikkeland T., etal, Phys. Rev. 172 (1968) 1232. 29. Swiatecki W. J., Nucl. Phys. A 376 (1982) 275. 30. Audi G. and Wapstra A. H., Nucl. Phys. A 595 (1995) 509. 31. Spanier L. and Johansson S. A. E., At. Data Nucl. Data Tables 39 (1988) 259. 32. Myers W. D. and Swiatecki W. J., Nucl. Phys. A 601 (1996) 141. 33. Moller P., et al, At. Data Nucl. Data Tables 59 (1995) 185. 34. Ninov V., et al, Phys. Rev. Lett. 83(1999)1104. 35. Hofmann S., et al, GSI Scientific Report 1999 (GSI2000-1) 7.

148

DECAY PROPERTIES OF SUPERHEAVY ELEMENTS (THEORY AND EXPERIMENT) YU.TS. OGANESSIAN Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, 141980Dubna

Discussing today the problem of the existence of "islands of stability" of superheavy elements (SHE), which originates from the shell effects in the heavy nucleus, we assume that for superheavy nuclei the probability for a-decay is higher than that for spontaneous fission. These assumptions are not being discussed, but are simply accepted as an experimental fact, since sequential a-decay chains have been detected for the superheavy nuclei. It should be noted, however, that here the high stability relative to spontaneous fission was predicted in the macro-microscopic theory within the approach described in Refs. [1,2] for the even-even nuclei in a wide range of Z and A. No calculations are known for T s f , in particular none have been performed in the model of Hartree-Fock-Bogoliubov (HFB), nor in the relativistic mean field theory (RMF) The availability of such data would be of great interest, since in the definite region of Z and N the quantities T a and T sf become comparable. In the sequential oc-decays, observed experimentally, such a situation corresponds to the nuclei in the end of the decay chain, which is terminated by spontaneous fission. We expect from theory not only calculations of the nuclear masses in the ground states, but also a scenario of their decay determined by the competition between the a-decay and spontaneous fission. So far this has been done only by A.Sobiczewski [1], R.Smolanczuk [2] and their colleagues for even-even nuclides. Now we shall limit our considerations to the properties of superheavy nuclei in their ground states. It should be noted that in the experiment the difference in mass is measured for neighbouring nuclei, which are apart merely by AZ = 2 and AN = 2. In other words the decay energy Q a and the probability for the given decay, viz. the partial half-life T a , are measured. The values of T a and T s f obtained for all known up till now isotopes with Z > 110* are presented in Fig. 1. As it can be seen, the increase of the neutron number of the nuclei significantly enhances their stability. For the isotopes of elements 110 and 112, where the comparison can be made using the experimental * In this and in the following figures, the data on the decay of the 292116 nucleus, obtained recently in the 48Ca+248Cm-reaction [3], are included.

149

Figure 1. Calculated and experimental a-decay (dashed lines) and spontaneous fission (solid lines) halflives for even-even isotopes of elements with Z = 110-116. The calculated values are from Refs. [1,2]. The symbols with error bars denote the experimental data. The open symbols refer to even-odd nuclei, the black symbols - to even-even nuclei. The squares denote spontaneous fission, the circles - a-decay.

values, the increase of the neutron number by AN = 8 a.m.u. leads to increase in T a by 5-6 orders of magnitude. For two isotopes, viz. Z = 110, N = 170 [4] and Z = 112, N = 171 [5], also the values of T s f have been obtained. For the even-even nucleus 280110 the partial half-life relative to spontaneous fission turned out to be 3 orders of magnitude higher than the calculated one. In spite of all the difficulties and uncertainties in the calculations of T s f , connected with calculations of the probability of tunneling through the fission barrier, it should be pointed out that the stability of SHE can indeed be much higher than it follows from the macromicroscopic theory. A similar conclusion can be drawn about the a-decay of even-even nuclei with Z = 112, 114 and 116. Here the experimental values of T a by about an order of magnitude exceed the calculated values; for the nucleus 280110 the difference amounts to more than 2 orders of magnitude. Disregarding these disagreement, the experimental results well reflect the predicted by theory trend of increasing the stability of superheavy nuclei when approaching the closed neutron shell N = 184. Let us now compare the decay energies Q a , which are determined experimentally with high precision (AQa < 0.5%). Unfortunately, the majority of synthesized nuclei are even-odd isotopes, obtained in the cold fusion reactions 208 Pb(62,64Ni,n) and 208Pb(70Zn,n) [6] or in reactions with actinide targets in the 48 ( Ca,3n)-channel [5,7].

150

Exceptions are only the even-even nuclei in the decay chain 8 114- V 8 4 1 1 2 ^ -> , which, as it will be shown below, can serve 116as a good test of the theoretical predictions. So as not to complicate the picture, here only three versions of the calculations of Q a are presented for all the nuclei, which have been synthesized so far (Fig. 2). 2

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Figure 2. Partial half-lives for ct-decay of even-even isotopes with Z = 106-116. The black squares are experimental data. The solid lines and the open symbols denote the calculated values, obtained as follows: a) in the macro-microscopic model (YPE+WS) [1,2], b) in the Hartree-Fock-Bogoliubov model (HFB+SLy4) [8], c)in the relativistic mean field theory (RMF+NL-Z2) [9].

151

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Figure 3. LogT„ (s) as a function of Q„ (MeV), obtained in ref. [2] using the formula of Viola and Seaborg with constant coefficients, giving the best agreement for all known even-even nuclei heavier than 208Pb for which Q a and T„ have been measured. The black squares denote the experimental data for isotopes with Z > 100. The symbols with error bars refer to isotopes of elements 112, 114 and 116, produced in the reactions 48Ca + 244Pu [4] and 48Ca + 24SCm [3].

Let us consider, first of all, the even-even nuclei for which the a-decay is connected with the transition between the ground states of the parent and daughter nuclei: Q a = M(Z,A) - M(Z-2,A-4). For these nuclei, as it could be expected, the basic rule for a-decay (the Geiger-Nuttal relation), relating the energy and probability of a-decay for a given Z, is valid (Fig. 3). In Fig. 4, the values AQa = Qa(th) - Qa(exp) are presented for all known eveneven isotopes with Z = 106-116, covering the mass region A = 260-292. The quantity AQa, determined by comparing the experimental values with calculations, carried out in different models, amounts to about 0.8 MeV. It follows that a lot can still be done by the theoreticians so as to improve the calculations. It seems to me that the disagreement stems rather from historical than from physical reasons. The parameters in the calculations have been chosen so as to obtain the best description of the experimental data available in the region of transactinide nuclei: 256Rf (b a ~ 0.3%), 260Sg (b„ ~ 50%) and 264Hs (one event only). With the production of new, heavier nuclides the discrepancy between theory and experiment considerably increased. From this point of view, it seems that it is desirable to widen the collection of even-even nuclei. One way is to use cold fusion reactions, where 207Pb can be used as target material. Indeed, in the reactions 207Pb(62,64Ni,n) and 207Pb(70Zn,n), eight new even-even isotopes with Z = 106 -=- 112, close to the deformed shells at Z = 108

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and N = 162, can be obtained. On the other hand, in the bombardment of targets of 48 235,236JJ w ^ Ca ions it is possible, in principle, to synthesize isotopes of element 112 and their decay products - the isotopes with Z = 108-110 in the transition region of nuclei between the closed neutron shells at N = 162 and N = 184. Here, it should be kept in mind that in this region the nuclear shape is changed from deformed to spherical one. For the odd nuclei the situation is complicated by the presence of hindrance factors due to the difference in spin and parity of the ground states of the parent and daughter nuclei. It is necessary in this case to perform calculations of the states, whose quantum characteristics define the selection rules in oc-decay. As it follows from the calculations (HFB+SLy4) [8], shown in Fig. 5 for the even-odd nucleus 289114, the oc-decay of the parent and daughter nuclei originates from the excited states with strong selectivity determined by their spins. Although the calculations agree well with the experimental data, it is not difficult to imagine how sensitive the a-particle spectrum is to the parameters used in the calculation. In addition, the isomeric transitions in the case of even-odd nuclei with Z = 108, discovered in the experiment where cold fusion reactions were used, complicate the task of the theory. It seems obvious that the request of the experimentalists has its good reasons. At present, experiments on the synthesis of SHE are being carried out at GSI (Darmstadt), FLNR (Dubna), LBL (Berkeley), RIKEN (Saitama), GANIL (Caen). In these laboratories, six kinematical separators are in operation with sophisticated

153

>

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140 144 148 152 156 160 164 168 172 176 180 182 Neutron number

Figure 5. Calculated and experimental values of Q a , obtained for even-odd nuclides: a) the level scheme and decay energies of the nucleus 289114, calculated in the framework of the (HFB+SLy4) model [7], b) the values of Q a (open circles), calculated in the framework of the (RMF+NL-Z2) model [9]. The black squares denote the experimental values.

detector arrays, suitable for registration of the rare events corresponding to the synthesis and decay of superheavy atoms. Many of these setups are to be upgraded so as to increase the sensitivity of the experiments on the synthesis of SHE to production cross sections at the level of 0.2 pb. By means of effective work and good organization of the collaborations between the different groups, we can expect considerable progress in the understanding of the formation and properties of nuclei in the new region of superheavy elements. References 1. 2. 3. 4.

Z.Patyk, A.Sobiczewski. Nucl. Phys. A533, 132 (1991). R.Smolanczuk. Phys. Rev. C56, 812 (1997). Yu.Oganessian et al. Phys. Rev. C (to be published). Yu.Ts.Oganessian, V.K.Utyonkov, Yu.V.Lobanov, F.Sh.Abdullin, A.N.Polyakov, I.V.Shirokovsky, Yu.S.Tsyganov, G.G.Gulbekian, S.L.Bogomolov, B.N.Gikal, A.N.Mezentsev, S.Iliev, V.G.Subbotin,

154

5.

6. 7.

8. 9.

A.M.Sukhov, O.V.Ivanov, G.V.Buklanov, K.Subotic, M.G.Itkis, KJ.Moody, J.F.Wild, N.J.Stoyer, M.A.Stoyer, and R.W.Lougheed. Phys. Rev. C62, 041604(R)(2000). Yu.Ts.Oganessian, A.V.Yeremin, G.G.Gulbekian, S.L.Bogomolov, V.I.Chepigin, B.N.Gikal, V.A.Gorshkov, M.G.Itkis, A.P.Kabachenko, V.B.Kutner, A.Yu.Lavrentev, O.N.Malyshev, A.G.Popeko, J.Rohac, R.N.Sagaidak, S.Hofmann, G.Munzenberg, M.Veselsky, S.Saaro, N.Iwasa, K.Morita. Eur. Phys. J. A5, 63 (1999). Z.Hofmann, G.Munzenberg. Rev. Mod. Phys. 72, 733 (2000). Yu.Ts.Oganessian, V.K.Utyonkov, Yu.V.Lobanov, F.Sh.Abdullin, A.N.Polyakov, I.V.Shi-rokovsky, Yu.S.Tsyganov, G.G.Gulbekian, S.L.Bogomolov, B.N.Gikal, A.N.Mezentsev, S.Iliev, V.G.Subbotin, A.M.Sukhov, G.V.Buklanov, K.Subotic, M.G.Itkis, K.J.Moody, J.F.Wild, N.J.Stoyer, M.A.Stoyer, R.W.Lougheed. Phys. Rev. Lett. 83, 3154 (1999). S.Cwiok, W.Nazarewicz, P.H.Heenen, Phys. Rev. Lett. 83, 1108 (1999). M.Bender, Phys. Rev. C61, 031302(R) (2000).

155 S E M I - B U B B L E S A N D B U B B L E S : A N E W K I N D OF SUPERHEAVY NUCLEI K. DIETRICH Physik-Department, Technische Universitdt Munchen, D-85747 Garching, Germany

1 In this contribution, I report on work in collaboration with K. Pomorski 1 ' 2 performed on the basis of the semi-phenomenological method of Strutinsky, and on subsequent work in collaboration with J. F. Berger and J. Decharge 3 which was based on the Hartree-Fock-Bogoliubov (HFB) method with Gogny's effective nucleonnucleon interaction. To start with let us remember that bound states of baryons exist in essentially two forms with different upper mass limits and different mechanisms of stability or decay. Neutron stars, on the one hand, bound on account of gravitation, with an upper critical mass limit given by the collaps to a black hole, and atomic nuclei (and hyperfragments), on the other hand, bound by the strong nuclear interaction with upper mass and charge limits given by the different decay modes, i. e. a-decay, /3 ± -decay, and nuclear fission. Here and henceforth, we speak of a "metastable" nucleus, if it is stable with regard to the emission of a single neutron or proton, but unstable with respect to other decay channels with a measurable lifetime. The lifetime of the heavy actinides and of the superheavy (SH) nuclei is known to be limited by a-decay and by nuclear fission with life-times varying over many decades. Examples: IfUue ->• 9o4r/ii44 + a(4, 5 x 10 9 a) ^4C/i56^/l+/2(60,5d) lt2Amli7 -> /1 + / 2 (14ms) The reason for the instability versus a-decay and fission, and thus for the existence of upper bounds for the mass and charge of atomic nuclei, is of course the Coulomb repulsion between the protons and the fact that the nuclear forces are not strong enough to bind pure or almost pure neutron matter. How could a nucleus reduce its Coulomb energy? Obviously this is possible by increasing the nuclear radius. This, however, implies that either a bubble is formed which implies the formation of a second, "inner" surface, or the nuclear density drops below the saturation density in the central part of the nuclear volume which leads to a semi-bubble. The formation of a bubble instead of a compact nucleus had been suggested by several authors quite some time ago 4,0 and was investigated on the basis of

156 different theories (liquid drop (LD) model without 6,7 and with shell corrections 8,9 , Hartree-Fock (HF) method 1 0 ). For historical reasons we mention that the first investigation of a nuclear bubble by H. A. Wilson in 1946 was a purely classical discussion of the vibrations of a spherical shell of nuclear matter in an attempt of explaining the energies of certain low-lying nuclear excitations. Most of the investigations were performed for proton and neutron numbers of known nuclei with the result that bubble formation did not occur for these normal nuclei. Nuclei which are larger than the known species were investigated in the references 7,9,10 , but not in a systematic way. The first systematic analysis of nuclear bubbles for a wide range of proton (Z) and neutron (N) numbers was performed in ref. 1 and ref. 2 on the basis of the LDM with shell corrections. According to the LD theory, bubbles have a lower energy than compact normal nuclei, whenever the reduction of the (positive) Coulomb energy through bubble formation is larger than the increase of the (positive) surface energy which is due to the formation of an additional inner surface and the increase of the outer surface energy. More precisely 1 ' 2 , this is the case whenever the fissility parameter

x = Ecb{Ro) = 0

'

2Es(Ro)

z2/A

m {

(ZyA)crit

'

is larger than 2,02 Zo>2,02

(2)

In eq. (1), Ecb{Ro) and Es(Ro) are the Coulomb and surface energy for a normal compact nucleus of radius Ro, resp. At the same time, the LDM predicts all spherical bubbles to be unstable with regard to nuclear fission1'2. Metastable nuclear bubbles can only be obtained by shell effects1,2. Their lifetimes are determined by a-decay and by spontaneous fission. The fission barrier heights are roughly equal to the shell correction energy at the spherical shape 1,2 \ The shell correction energies (and thus the fission barrier heights) were somewhat overestimated in ref. 1, 2 because we used infinite square-well and oscillator potentials instead of a finite single particle potential for calculating the shell correction energy. Selfconsistent HFB calculations based on the Gogny interaction 3 lead to fission barriers for magic bubbles of about (3 - 4)MeV. The pre-dominant decay channel for the bubbles is thus nuclear fission. The calculation within the pure LDM is particularly simple: The difference F between the energy ELD{R\,RI) of the bubble and the energy ELD(RO) of the compact normal nucleus in units of the surface energy Es{Ro) depends only on one size variable and on the fissility parameter XQ. Choosing the dimensionless size variable (R2 = radius of inner surface) V2 := g

(3)

one has w(v

y)

ELD(R2,R1)-ELD(Ro)

_

AELD

157 The function F(V2,X0) is shown in fig. 1 for different values of the fissility parameter X0. For XQ > 2,02, the condition of stationarity

has two physical solutions which correspond to the minimum of F(V2,XQ) and the top of the barrier between the compact (V2 = 0) and the bubble-shaped nucleus. This is shown in fig. 2. Both figures are taken from ref. 2. The shell correction energy Esc can be written as a function of the singleparticle energies Ev and the smoothed (n„) and unsmoothed (n,,) occupation numbers in the following way ESc

= ^Ev{nv-nv)

(6)

It is this shell-correction energy which may stabilize the spherical bubble-shaped nucleus if one or both of the two types of nucleons correspond to a closed shell configuration 1 , a . Of course, the magic numbers for bubbles are different from the ones of the compact normal nucleus. Level schemes Ev (/) as a function of the ratio a r e s n o w n m re f = (Sf) f- 1>2 f° r various phenomenological potentials. We note in passing that the spin-orbit splitting is smaller in the case of bubbles as compared to normal compact nuclei because the contributions originating from the inner and outer surface are of opposite sign. This effect may even invert the order of the levels with the higher (j = l + 1/2) and lower (j = / — 1/2) total angular momentum j . Unfortunately, bubbles are obtained as ground states only for Z > 220 and N > 550 which implies that we have no chance to ever make them on our earth. Bubbles and nuclei of even more complicated topology may however exist in the crust of neutron stars. Having found the bubbles, one can surmise that there are nuclei with a reduced, but still finite nuclear density in the central region of the nucleus. We shall call them "semi-bubbles". Guided by the results of the phenomenological theory we performed extensive microscopic calculations 3 based on the HFB-theory with the effective nucleonnucleon interaction of Gogny. The search for bubble and semi-bubble solutions can be simplified by applying an external Gaussian-shaped repulsive central potential

1 _- 2 AaVG = Aa • - j - ^ e ^

(7)

which serves to push the nucleons away from the centre of the nucleus. The parameter "a" is the radius of the inner zone and the Lagrange parameter Aa may be chosen such that the average nr Na = (\VQ\) of nucleons in the central region of radius "a" has a given value. The microscopic theory can then be formulated as the variational principle 3 6($l&-iLnN-nv2-\JrGl9>=Q

(8)

where $ is the general Hartree-Bogoliubov vacuum state. By solving the variational HFB equations obtained from (8) for a network of iVa-values one can find out

158 whether bubbles or semi-bubbles exist as minima states for given neutron and proton numbers. In (8), H is the many-body Hamiltonian, N and Z are the number operators for neutrons and protons, and £Jn,/up are the corresponding chemical potentials. These stationary solutions correspond to \a = 0, i. e. the constraint does not modify the physical solution but only serves to find it. The solutions may be spherically symmetric or they may belong to a deformed intrinsic state. The stability versus fission is investigated as usually by introducing additional constraints related to the quadrupole (Q20) and octupole (Q30) operators 3 . Semi-bubbles are obtained as ground states for proton-rich superheavies with a proton nr Z > 150. Shell-effects may lead to semi-bubbles even at smaller charge numbers. The lightest double-magic semi-bubble is obtained 3 for Z = 120 and N = 172. This result has later on also been obtained for various Skyrme interactions and also for the relativistic nuclear shell theory (see ref.11 and several contributions to this meeting). We have presented more detailed results in the figures 3-5, which are taken from ref. 3, whereas the figures 1 and 2 are taken from ref. 1, 2. In fig. 3 we show the single-particle energies of protons and neutrons as a function of an effective variable / ( ( r 2 ) ) defined by f({r2))

= Ri •

V)

-Ri+b

(9)

In eq. (9), ((r 2 )) 1 / 2 is the r. m. s. radius of the nucleus subject to the constraint (Va) = Na

(10)

whereas Ri is the r. m. s. radius of the semi-bubble solution (or of the compact normal nucleus if no semi-bubble solution exists). The variable f{{r2)) is chosen in this way in order to exhibit more clearly the dependence of the s. p. energies on the r. m. s. radius (r 2 ), which was found to change very slowly in the vicinity of the solution. The parameter b in eq. (9) is chosen to be b = 0,3 fm. At the semi-bubble solution, we have (r 2 ) = .Ri i. e. f = R\. The numbers appearing in the shell gaps of fig. 3 represent the number of nucleons which occupy the s. p. energies below the gap. The larger the value of f{{r2)), the more the s. p. potential adopts the form of the "wine-bottle potential" which has played the role of a precursor of the shell-model in the thirties. The magic numbers are seen to depend on / ( ( r 2 ) ) and in particular one recognizes that Z = 120 and N = 172 form a double magic shell closure for a semi-bubble. This is the lowest semi-bubble we found which owes its existence to the two-fold shell effect. The dependence of the binding energy (H) and of the central density p(0) on the variable / ( ( r 2 ) ) is shown in fig. 4 for 3 different SH nuclei. It is of particular interest to consider the value of p(0) at the minimum of the potential energy: Thereby it is seen, that (Z = 120, N = 172) corresponds to a semi-bubble, [Z = 274, N = 626) corresponds to a true bubble, and (Z — 126, N = 204) represents a normal compact nucleus.

159

0

0.2

0.4 0.6 0.8

1

Figure 1. Difference F = Ai,£i/Es(Ro) of the LD energy of a bubble of reduced inner radius V2 and the LD energy of a compact spherical nucleus of radius RQ.

Figure 2. The function vz(Xo) defining the stationary points of the energy F(v2; Xo) as a function of the fissility parameter Xo-

160 120„.

Nrutrwu

Figure 3. Single-particle energies (eigenvalues of the submatrix h of the HFB Hamiltonian) for protons and neutrons in the semi-bubble 29212O172 as functions of f({r2)). Numbers indicate how many protons can fill the level scheme starting from the lowest (la) single-particle state. The j-value of each level is indicated as follows: no symbol = 1/2, plus = 3/2, asterisk = 5/2, circle = 7/2, x = 9/2, square = 11/2, triangle = 13/2, diamond = 15/2, open cross = 17/2. Even / positive parity (resp. odd / negative parity) levels are drawn as solid (resp. dashed) curves.

HRt energf A u<0)

.:uok "*'»». j i

'

!

;

\! !

Figure 4. Binding energy (solid curve, 1. h. ordinate) and central density p(0) (dotted curve, r. h. ordinate) for 2 9 2 1 2 0 m , 330126204 and 900274626 as functions of /((r 2 )) defined in Eq. (9). The p(0) scale indicated on the right is the same for all three plots.

v - V t f ^ / P .. .

Wx:A\ \A \ \ \

5] K\!['A' >. \

'. *, > ,(\

Figure 5. Ground state configurations of studied nuclei in the N — Z plane. True bubbles, semibubbles, and ordinary nuclei are represented as open circles, triangles, and dark circles respectively. The inset displays typical density profiles corresponding to the three configurations. Contours of equal chemical potentials /xp for protons (short-dash) and tin for neutrons (long-dash) are drawn when positive or zero. Below (resp. above) the line of/3-stability \iv (resp. /j.n) is negative.

161 Finally, in fig. 5 we show a survey of the regions in the (N, Z)-plane where compact nuclei, semi-bubbles, and bubbles exist as ground states. The straight lines in fig. 5 represent the location of isobars. As one increases the charge along an isobaric line the ground state eventually becomes a semi-bubble or even a bubble. Unfortunately, the nuclei become rapidly unstable against the emission of a proton on the proton-rich side or of a neutron on the n-rich side of the line of metastability. Thus there is no chance that true bubbles can ever be made in a laboratory. On the other hand, there is a chance that the lightest semi-bubbles may one day be found experimentally. References 1. K. Dietrich, K. Pomorski, Phys. Rev. Lett. 80, 1 (1998), 37, (received 25.04.97) 2. K. Dietrich, K. Pomorski, Nucl. Phys. A 627 (1997), 175 (received 09.06.97) 3. J. Decharge, J. F. Berger, K. Dietrich, M. S. Weiss, Phys. Lett. B 451 (1999) 275 (received 08.01.99) 4. H. A. Wilson, Phys. Rev. 69 (1946) 538 5. J. A. Wheeler, notebook (1950) 6. H. A. Bethe, P. J. Siemens, Phys. Rev. Lett. 18 (1967) 704 7. W. Swiatecki, Phys. Scr. 28 345 (1983) 8. C. Y. Wong, Aun. Phys. 77 (1973) 279 9. A. Lukasiak, A. Sobiczewski, Contribution to the Proc. of the International Symposium on Superheavy Elements, Lubbock, Texas, March 9-11, 1978, Phys. Lett. 46 B (1973) 291 10. X. Campi, D. W. Sprung, Phys. Lett. 46 B (1973) 291 11. M. Bender, K. Rutz, P.-G. Reinhard, J. A. Maruhn, W. Greiner, P. R. C 60, 034304 (1999)

162

Reaction Theory for Synthesis of the Superheavy Elements Yasuhisa Abe Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto E-mail: [email protected]. ac.jp The dynamical reaction theory for synthesis of the superheavy elements is briefly reminded and the status of its theoretical developments is presented. A few remarks are also given. 1

Introduction

The existence of the superheavy elements (SHE) is a long-standing prediction based on our common knowledge on the stability of the atomic nuclei, where microscopic effects of nucleonic motions, particularly the shell structure effects play an important role in addition to the average nuclear properties which are described by the liquid drop model.1 Although there are still certain ambiguities concerning where is the center of the stability in the superheavy region of the nuclear chart, it is sure that there exists a stable island in the very heavy unknown region of the chart. By the phenomenological single particle model, for example, it is predicted to be around Z= 114 and N=184, i.e., there the additional binding energy, so-called the shell correction energy is at maximum among the neighbouring nucleides of the region.2 It, however, is rather peculiar that there has been no reaction theory proposed for how to synthesize them. Of course, the most usual way is to use heavy-ion (HI) fusion reactions. But mechanisms of Hi-fusion reactions in massive systems are not yet known well, while in lighter systems they are well understood. Once we have a reasonable theory which enables us to predict Hi-fusion probability, i.e., formation probability of the compound nuclei, then, residues cross sections could be predicted by the use of the statistical theory of decay, based on the concept of the compound nucleus of N. Bohr.3 The formula for residue cross sections is given as <Jres{Ec.m.) = £ E ( 2 J + 1) • Pf'us(Ec.m.)

• PsJurv(Eex)

(1)

, where Pjus{Ec.m.) and P^urv(Eex) denote the fusion probability as a function of incident energy and the survival probability as a function of excitation energy of the compound nucleus. The fusion probability P / u s is considered to be expressed by the product of the contact probability Pcont and the formation probability Pform, i-e., Pjus = Pjont • Pform- That is, we have to take into

163

Reaction Processes

oo

o-Q /

Reseparation (Quasi-Fission)

Binary Processes (DIC)

C.N.

1 (SHE)

Spontaneous decays (a, fission)

Figure 1: Reaction processes for SHE synthesis is schematically displayed from an encounter of incident heavy ions to SHE residue. It should be noted that fusion is achieved by two successively processes; approaching phase and dynamical evolution of composite system.

account the whole reaction process from encounter of heavy ions to evaporation residues, as schematically shown in Fig. 1. The two factors in Eq. (1) have opposite energy-dependences, which will be discussed below more in detail. The statistical theory of decay of the compound nucleus is wellknown and well established, though there are ambiguities in parameters, for example in socalled level density parameter in practice. Since decay modes are dominantly neutron emission and fission, the survival probability is expressed as follows,

pj

/ p \_

sur,A e ;

*

rra(-feex) Ti(Eex) + rj(Eex)

(2)

, where r „ and Tf denotes probabilities of neutron emission and of fission decay, respectively. The former is given by Weisskopf formula 4 while the latter by Bohr-Wheeler one,5 or Kramers one 6 including friction effects. For cases with excitation energy higher than In, 2n, .... emission thresholds, we have to repeat the factor given in Eq. (2) with corresponding excitation energies. In realistic calculations to be compared with experimental energy spectra, cascade

164

type computer codes are used, taking into account kinetic energy distributions of neutron emitted at each step. In the superheavy mass region, the system formed with a certain excitation immediately undergoes fission, because of the extremely low or almost no barrier against fission. That is, the survival probability is a rapidly decreasing function of excitation energy. Therefore, the excitation energy of the compound nuclei initially formed should be as low as possible. This is called as the cold fusion path. For that purpose, one has to choose favourable combinations of projectile and target nuclei, taking into account reaction Q-value. Let us take an example of synthesizing the element Z=118 with neutron number N=176. In Fig. 2, excitation energies of the compound nuclei formed at the interaction barrier top 7 are shown over all the possible combinations of projectiles and targets. As is readily seen, there are two distinct minima around Kr + Pb and Xe + Gd combinations, the former of which is just the system that Berkeley group 8 made experiments with. Then, they are surely favourable in the survival probability PSUrv(Eex). But there is the other factor Pfua(Ecm) in Eq. (1) which determines residue cross sections. Naturally, it increases as incident energy increases, firstly for getting rid of Coulomb barrier and secondly for overcoming the fusion hindrance,9 but it automatically gives rise to an increase of excitation energy of the compound nuclei formed, which results in a decrease of the survival probability. Therefore, one has to compromise two conflicting factors, i.e., has to find an optimum incident energy to maximize the final residue cross sections. As mentioned above, there is so-called fusion hindrance in massive systems, which is clearly exemplified by the extra-push or the extra-extra-push energy 10 necessary for the system to fuse into compound nuclei. It is known to be more pronounced in systems with larger values of Zp*Zt, which is readily seen in the schematic Fig. 3 for the formation probability. So, the combination corresponding to the first minimum would be unfavourable and even more to the second one, though more favoured in excitation energy as stated above. On the other hand, if we employ an incident channel with a large mass asymmetry such as 4 8 Ca+ 2 4 4 Pu, we almost need not suffer from the fusion hindrance as is seen in Fig. 3, which indicates much larger formation probability Pform, thereby larger fusion probability Pfus than the cold fusion path. There, however, is a problem. The compound nucleus formed is in high excitation which is understood from Fig. 2. (this tendency is general in the liquid drop model, though there are modulations due to the shell structure effects.), which usually indicates extremely small survival probabilities. But there is a remedy. As is understood from Eq. (2), if we can make the neutron emission width to be larger, the survival probability could remain to be not extremely small. Furthermore, it has a non-linear effect, i.e., if the cooling by neutron

165 Z=118 N=176 30

25

20

E*

15

10

5

0

-5

-10 20

25

30

35

10

45

50

55

60

Zp

Figure 2: Excitation energies of compound nuclei formed at Bass barrier are given over possible projectiles. Neutron numbers of projectiles and targets are optimized, i.e., are chosen so as to give the lowest value for excitation energy. Two minima are essentially due to shell effects of the projectiles and targets. Hot fusion path corresponds to the smaller Zp side.

emissions is accelerated by the larger neutron decay width, the shell correction energy of the compound nucleus is restored rapidly, which gives rise to an additional stability and thereby to a smaller fission decay width. This makes the survival probability much larger, which would result in an appreciable residue cross section.11 For that, neutron rich targets and projectiles are necessary. Secondary neutron-rich beams could be useful. This is so-called hot fusion path. Therefore, in order to find the optimum condition, the whole reaction process is to be studied over all the possible combinations of projectiles and targets which can reach the superheavy nucleides.

166

V

LOM(R.<*) 244

\

—

••- ; •

*•; •

J

<<

\

S f>

Pu + 4f fca

r H.I

1.0 **• ^ 0

S. P. : Saddle point Re

: Contact distance

^ ~ > ^ 0

208

Pb +86Kr \

B. G. : Businaro-Callone pc a k

Figure 3: Characteristic feature of liquid drop energy surface is schematically shown. It is crucial that distance between contact point and conditional saddle varies as mass-asymmetry a changes. In larger a (like 2 4 4 Pu+ 4 8 Ca), they are very close, while in smaller a, i.e., near the mass-symmetry (like 2 0 8 Pb+ 8 6 Kr), they are far apart and in addition, height of conditional saddle relative to contact is larger. The system has to overcome it under strong energy dissipation in order to fuse into spherical compound nucleus or around.

2

D y n a m i c a l evolutions into the compound nucleus

In heavy ion collisions, it is wellknown that the dissipation of the incident energy is very strong, so the energy is expected to be almost completely damped around the contact of the two nuclear matter (an example will be discussed later). This is consistent with the fusion hindrance observed in experiments, or with extra-push or extra-extra-push energy required, because additional energies are necessary for the system to be driven toward the spherical compound nucleus configuration even after the Coulomb barrier is passed over. The amount of the energies required is very large for the cases that there is an internal barrier, so-called conditional saddle point between the pear-shape

167

configuration with the mass-asymmetry close to the incident projectile- target combination and the spherical configuration of the whole system. For, the system has to climb up the saddle under strong damping of kinetic energy into thermal energies of nucleons. Phenomenologically, it is well described by the dissipation-fluctuation dynamics, say, by Langevin equation for the relative motion or collective motions of nuclei.12

(3)

£ = - * " £ + *(*)

at oq m ,where 7 is a friction coefficient and R(t) its associate random force. R(t) is assumed to be Gaussian and Markovian, and to satisfy the dissipationfluctuation theorem, i.e., {R{t) • R{t')) = 2 • 7 • T • 6(t — t') where ( ) denotes an average over all the possible realizations and T does the temperature of the nucleons. The other quantities are defined as usual. In one-dimensional case with an inverted parabolic potential barrier, i.e., U = — | m w 2 g 2 , we can readily write down an analytic expression 13 for the probability for formation of the compound nucleus by passing over the saddle, which is given by

P

-I

f

(

P + P'

2w K + B'\ T

(4)

, where /? is the reduced friction 7 / m and fi'=y/P2 +4.w2. B denotes the saddle point height from the contact configuration, i.e., ^mw1 • q% with q0 being the initial value at the contact configuration, while K the initial kinetic energy. The critical energy Kc necessary for Pform being equal to 1/2, i.e., the extra-push energy necessary for passing over the saddle from the contact is calculated with Eq. (4),

Kc^-^ff-B

(»)

In the weak friction limit, j3 — 0, Kc ~ B which is trivial, while in the overdamped limit, /? » 2-w, Kc ~ (P/w)2-B ~ 10-B with One-body model.14 The problem remains is an extension to the multi-dimensional cases for the purpose of applications to realistic cases, but it is not easy to obtain such a concise expression as Eq. (5). Therefore, we have to solve the problems , i.e., the multi- dimensional Langevin equation numerically, where the inertia mass used is that of the irrotational flow of the nuclear matter, and the friction is that of One-body model or the two-body viscosity. From the analyses of fission time scale, we now know that the One-body friction model is more suitable than the other. 15 As stated above, realistic calculations of energy-dependence of fusion

168

probabilities are being made with Langevin, Smoluchowski Eqs. The latter Eq. is only suitable for the cases of complete dissipation of incident energy, i.e., collective momenta are completely in equilibrium and then evolutions of coordinates, i.e. diffusions are treated. 16 This is more or less consistent with the results obtained in the calculations of the approaching phase discussed below. But if the dissipation is not complete, one has to employ the former one, i.e., Langevin Eq. or equivalent Kramers Eq. The latter is practically not easy in the cases of more than two degrees of freedom which are required for fusion process. Therefore, we employ the former, i.e., the multi-dimensional version of Eq. (3). Examples of preliminary results obtained with 3-dimensional Langevin equation including the target deformation in addition to the distance between two ions and the mass asymmetry are given in Fig. 4. It would be appreciated that the optimum cross section and the optimum energy of 4 8 Ca+ 2 4 4 Pu system are surprisingly well predicted. The results were reported by T. Wada at the Conf. on Nuclear Fission, Slovakia, Oct. 1998 17 , two months before the first successful experiment was done at Dubna, Dec. 1998.18 3

Recent developments

The special aspect in the superheavy mass region is, as mentioned above, that there is almost no barrier at least at the beginning of the reaction where the shell correction energy is not yet fully restored due to excitation, and therefore Bohr-Wheeler formula may lose its validity. One can employ again Langevin or Smoluchowski Eq., but it is time-consuming in numerical calculations. Another way would be to employ the method of the Mean-First-Passage-Time 19 which gives an average decay time from a certain region to its outside and therefore gives a fission decay probability. The mean passage time over XQ is given as follows, <

t

>= ^ T

f ° dyeu^'T Jo

•f dze-u^'T y_oo

(6)

In cases with temperature T 3> U(z), the expression is approximately reduced to < t >~ ^ f ° dyeuM/T • r dye-u^'T (7) J- Jo Jo , which can be evaluated with the assumption of parabolic shapes for both around the ground configuration and around the fission barrier with the curvatures WQ and ws, respectively,

169

Evaporation Residue Cross Section o 1 258Fm+32Si •

2250Cm+44Ar

n 3244Pu+48Ca •

4 238 U+ 52 Ti

A 5234Th+56Cr

0

20 40 60 Excitation Energy E* (MeV)

Figure 4: Preliminary results of excitation functions of residue cross section of Z=114 are shown for several incident channels. The results for 2 4 4 P u + 4 8 C a (lOpb at E* ~ 35MeV) turned out to be in a surprisingly good agreement with the experiment done at Dubna later. The steep slopes in the low energy side is due to the contact probability PCOnt which is approximated here simply by parabolic barrier penetration factor for Bass potential, while those in higher energy side due to the survival probability which is approximately calculated by the use of the time-dependent average temperature. For the detail, see Refs. 11 and 17.

170

< t >^ JlL. . eBf/T

(8)

w0 -ws , where Bf denotes a fission barrier. The expression Eq. (8) just corresponds to Kramers formula for the overdamped limit. As given in Eq. (6), the method just requires numerical integrations of potentials, so it is practical. A simple schematic example for fission is studied, for cases with fission barrier to those with no barrier, generated by the change of angular momentum. 20 The former cases coincide well with the results obtained by numerical integrations of Smoluchowski Eq., while the latter cases tend to coincide with the classical traveling time of the distance, which could be conceived. One more development is about the approaching phase up to contact of incident heavy ions, i.e., contact of two nuclear surface, after which dynamical evolutions of shapes of the composite system are described by the multidimensional Langevin equation as discussed above. In other words, results of the analyses of the approaching phase provide precise initial conditions for subsequent dynamical evolutions of the whole system. It would be worth to remind once more that in our view point, fusion process is divided into two part, and the fusion probability is expressed by a product of the contact probability PCont m the approaching phase and the formation probability Pform in the passing-over of the conditional saddle, including a possible fast-fission decay process going on later. In order to facilitate the analyses, we employ the surface friction model.21 The classical equation of motion for the radial momentum is given as follows, d

-^=-f-KA

+

RT(t)

(9)

at or m , where the potential U consists of a sum of the Coulomb Uc and the nuclear UN potentials, including a centrifugal force for a finite orbital angular momentum. The radial friction is taken to be Kr = fcr° • ( ^ \

\K0r=

3.5 x 10-™s/MeV

(10)

as is usual in the surface model. The last term in the right hand side of Eq. (9) Rr(t) is the random force satisfying the dissipation-fluctuation theorem with the friction Kr. As the nuclear potential we employ that of Gross and Kalinowski 22 which is essentially a folding potential. A preliminary result is shown in Fig. 5 2 3 where the heat-up process, i.e., time-dependence of the temperature is taken

171

into account. We can see that the incident energy is fully damped at the contact. 86,,

208,-,.

Kr +

R12

Pb,

E^ = 403 M «

io 20

>. ,i

CD O

tu J£

2

«l

39

^,.4lOMaV

1M

20

<0

W

W

1<M

ail

+t

«i

*>

lot

30

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*.

«

ica

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CO **•

:

HiL~ X

•G

®

...J

50

ICO

0 L. CD

St-

«o

t_ O «

,.9< *•:

E»=4l7hteV

t\

Hife. *

it)

SO

;

|


litt

E^, = 424 MeV

sj

ao

»

en

si

ca

1

xci

CD

+J

a> m JO

|

i»j

angular momentum

angular momentum

Figure 5: Energy properties of contact configuration are given for 8 6 Kr+ 2 0 8 Pb system. The right panels show the kinetic energy distributions for several incident energies above the barrier top. It is readily seen that the average kinetic energies are zero over all the angular momenta and the incident energies. Actually, momentum distributions are Gaussian, though not shown here. Therefore, the relative motion is completely damped and is in equilibrium with the nucleonic heat bath. The left panels show the corresponding excitation energies. It should be noticed here that these features are common in the superheavy element region, but are not in lighter systems.

4

Remarks

For quantitative predictions of residue cross sections of the superheavy elements, we need to know precisely physical parameters such as inertia masses,

172

friction coefficients for collective degrees of freedom as functions of excitation or temperature. Especially it is the case for the cold fusion path. So far, we have employed One-body model for friction which is quite strong and temperature-independent, although friction should tend to zero in the low excitation limit. Microscopic theories such as Linear Response Theory 24 predict energy-, or temperature-dependences, which should be somehow incorporated into the phenomenological analyses of the reactions. The same applies to the inertia mass tensor. One more point about the limitation of the classical model. Since the probabilities obtained by the classical fluctuation-dissipation dynamics are very small in most cases, the dissipative quantum tunneling probability,25 which is expected to be small, should also be investigated seriously in both fusion and fission processes. Although there stiR exist several insufficient points in the theoretical treatments and several uncertainties in physical parameters, it can be well claimed that we are approaching to the goal of a quantitative prediction of cross sections for the synthesis of the superheavy elements by the dynamical reaction theory. The author appreciates long time collaborations with Konan group, which have given rise to many fruitful results and to new insights on the subject. He also likes to thank Drs. D. Boilley and B. Giraud concerning the collaboration on the analytic expressions for passing-over probability, and Dr. G. Kosenko concerning the analyses of the contact dynamics that are being done, supported by RIKEN. Acknowledgments should be addressed also to Drs. S. Yamaji and P. Moller for their courtesies of providing their computer programs. At last but not at least he likes to thank the experimental groups at Dubna, GSI, GANIL and RIKEN for having provided opportunities for discussions and new experimental information promptly, which have stimulated the theoretical studies enormously. References 1. see for example, a review article by S. Hofmann, Rep. Prog. Phys. 6 1 , (1998) 639. 2. P. Moller, J.R. Nix, W.D. Myers and W.J. Swiatecki, The Atom, and Nucl. Data Tables 59 (1995) 185. 3. N. Bohr, Nature 137 (1936) 344. 4. V. Weisskopf, Phys. Rev. 52 (1937) 295. 5. N. Bohr and J.A. Wheeler, Phys. Rev. 56 (1939) 426. 6. H.A. Kramers, Physica VIIA (1940) 284.

173

7. R. Bass, Nuclear Reactions with Heavy Ions (Springer, 1980) Chapt. 7. 8. V. Ninov et at., Phys. Rev. Letters 83 (1999) 1104. 9. see review article by K.-H. Schmidt and W. Morawek, Rep. Prog. Phys. 54 (1991) 949. 10. W.J. Swiatecki, Phys. Scripta 24 (1981) 113 S. Bjornholm and W.J. Swiatecki, Nucl. Phys. A391 (1982) 471. 11. Y. Abe, Y. Aritomo, T. Wada and M. Ohta, J. Phys. G23 (1997) 1275.

Y. Aritomo, T. Wada, M. Ohta and Y. Abe, Phys. Rev. C59 (1999) 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25.

796. Y. Abe, C. Gregoi're and H. Delagrange, J. Phys. 47 (1986) C4-329. P. Probrich and S.Y. Yu, Nucl. Phys. A477 (1988) 143. Y. Abe, D. Boilley, B. Giraud and T. Wada, Phys. Rev. E61 (1999) 1125. J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A. Sierk and W.J. Swiatecki, Ann. Phys. 113 (1978) 330. T. Wada, Y. Abe and N. Carjan, Phys. Rev. Letters 70 (1993) 3538. Y. Abe, S. Ayik, P.-G. Reinhard and E. Suraud, Phys. Reports C275 (1996) 49. T. Wada, T. Tokuda, K. Okazaki, M. Ohta, Y. Aritomo and Y. Abe, Proc. 4th Intern. Conf. on Dynamical Aspects of Nuclear Fission, Slovakia, Oct. 1998 (World Scientific, 2000), P. 77. Yu. Oganessian et a l , Nature 400 (1999) 242 and Phys. Rev. Letters 83 (1999) 3154. C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1990) Chapt. 5. T. Ichikawa and T. Wada, private communication. P. Probrich, B. Strack and M. Durand, Nucl. Phys. A406 (1983) 557. see also the next Ref. 22. D.H.E. Gross and H. Kalinowski, Phys. Reports 45 (1978) 175. G. Kosenko and Y. Abe, to be published. S. Yamaji, F.A. Ivanyuk and H. Hofmann, Nucl. Phys. A612 (1997) 1 H. Hofmann and F.A. Ivanyuk, Phys. Rev. Letters 82 (1999) 4603. H. Grabert, P. Schramm and G.-L. Ingold, Phys. Reports 168 (1988) 115.

174

THE DINUCLEAR SYSTEM CONCEPT FOR THE COMPLETE FUSION PROCESS V.V.VOLKOV Flerov Laboratory

of Nuclear Reactions, JINR, Dubna,

Russia

The development of different approaches to the description of the complete fusion process is discussed. A characteristic of the Dinuclear System Concept (DNSC) for the complete fusion is given. The DNSC revealed two important features of the complete fusion of massive nuclei: the existence of inner fusion barrier - B ' ^ and the competition between complete fusion and quasi-fission channels. The DNSC was applied to the analysis of reactions used for the synthesis of superheavy elements (SHE) including the shell effect in the mass distribution of the quasi-fission products.

1

Introduction

The main problem of the complete fusion process is the mechanism of the compound nucleus formation. There are two principal difficulties in solving this problem: a closed character of the complete fusion process and a very high complexity of the theoretical analysis of the reconstruction of two initial nuclei into a compound nucleus. Fusing nuclei do not send any signals which allow one to reveal the mechanism of compound nucleus formation. Experimentalists detect products of compound nucleus decay. But it is well known that a compound nucleus forgets the history of its formation. For the space of more than 40 years of the study of this fundamental nuclear process theorists have developed different approaches for its description. Their origin is connected with the development of the experimental study of the complete fusion process. 2

2.1

Three approaches to the description of the complete fusion process

The first approach

Consideration of the compound nucleus formation mechanism is omitted. In the first approach the compound nucleus formation mechanism was not considered at all. It was believed that it is Natur's own business. In the early experiments rather light heavy ions: 12C, 14N, 16 0, 20Ne were used. In the reactions with these ions the capture of a projectile by a target nucleus inevitably leads to the formation of a compound nucleus. The compound nucleus production cross section acn was equal to the capture cross section ac: CTcn = 0- c .

(1)

175

Efforts of theorists were directed to creating of models for the calculation of the capture cross section ac. The optical model [1], the model of the critical distance [2] the surface friction model [3] and the model with "pocket" in nucleus-nucleus potential V(R) [4] were formulated. In all these models the critical angular moment - £a is the most important characteristic of the complete fusion process. The compound nucleus production cross section acn is defined by ratio:

*a=x%2'tl{2e + l)Tt,

(2)

1=0

where T,, is the penetration factor of the entrance barrier. 2.2

The second approach

The complete fusion process is described on the basis of the liquid drop model. The use in experiments of more massive heavy ions such as 40Ar, 84Kr revealed the new nuclear processes: fast fission and quasi-fission. These processes were realized also in collisions with angular momentum £ j < £CI. The main postulate of the first approach: acn = a c was violated. The capture may be realized but the compound nucleus is not formed. A new approach was proposed by W.Swiatecki. It was the macroscopic dynamical model (MDM) [5]. The main peculiarity of this model was the description of the whole history of the fusion process starting from the contact between nuclear surfaces and endings with the compound nucleus formation. However, in the MDM the essential simplifications of the fusion process were made. The real nuclei which are build from protons, neutrons and have the shell structure were substituted by drops of viscose nuclear liquid. The success of the liquid drop model in the interpretation of the fission process inspired hope for success of its application to the description of the fusion process. After all, fission and fusion may be considered as direct and reverse processes. The MDM introduced new characteristics of fusion of massive nuclei: extra push - Ex and extra-extra push - Exx. The fusion of two nuclei in a mononucleus is possible if the bombarding energy E; is larger than B c + Ex, (Bc is Coulomb barrier), Ej > B c + Ex. For the production of a compound nucleus the E; must be larger that Bc + E B . In the MDM the fusion of two nuclei is not equal to the formation of a compound nucleus. The MDM was very popular among experimentalists. The terms extra push and extraextra push have firmly entered the physical language. However, serious difficulties arose in attempts to use the MDM for the description of the reactions used for the synthesis of SHE [6]. From our point of view these difficulties of the MDM are caused by the simplification of the reality used in this model. The substitution of real nuclei by drops of hypothetical nuclear liquid radically changes the picture of the complete fusion process.

176

2.3

The third approach

The complete fusion is the process of formation and evolution of the dinuclear system. The third approach was proposed at Dubna [7]. This approach has received the name of the Dinuclear System Concept (DNSC) [8]. The participants of its development are: G.G.Adamian, N.V.Antonenko, E.A.Cherepanov, A.K.Nasirov, W.Scheid and V.V.Volkov. The basic idea of the DNSC is the assumption that the complete fusion and deep inelastic transfer reactions are similar nuclear processes. What does this assumption give us? In contrast to fusion and fission, DITR are open reactions. They provide unique information on the interaction of two nuclei which appear to be in close contact after full dissipation of the collision kinetic energy. It is this unique information that has been used for revealing the mechanism of compound nucleus formation. According to the DNSC, the main features of the fusion process are the following. • At the capture stage, after full dissipation of the collision kinetic energy a dinuclear system (DNS) is formed. • Complete fusion is an evolutionary process in which the nucleons of one nucleus gradually, shell by shell, are transferred into another nucleus. • The nuclei of the DNS retain their individuality until the end of the fusion process. This important peculiarity of the DNS evolution is the consequence of the shell structure of nuclei. Fig.l illustrates the principal difference between the pictures of the compound nucleus formation process offered by the first, second and third THE FIRST APPROACH: The optica! model, the model of critical distance, the surface friction model. The "black box" or collapse.

CO...?...-0 THE SECOND APPROACH: The macroscopic dynamical model. Fusion of two nuclear liquid drops.

oooo€) THE THIRD APPROACH: The dinuclear system concept. Conservation of nuclear individualities.

<XXK)€) Figure 1. Schematic illustration of the compound nucleus formation mechanism in the frames of different approaches to the description of the complete fusion process.

177

approaches. In the first approach the restructuring stage of process is characterized as a "black box", or collaps. In the second approach fusing nuclei drops rapidly lose their individuality as a results of the neck formation. Complete fusion is a dynamical process which is developing in the deformation space. In the DNSC the fusing nuclei retain their individuality until the end of the fusion process. Complete fusion is mainly a statistical process which is developed along the mass-asymmetry coordinate of the system. 3

Peculiarities of complete fusion of massive nuclei within the framework of the DNSC

The DNSC reveals two important peculiarities of complete fusion of massive nuclei: appearance of a specific inner fusion barrier B*faS and competition between the complete fusion and quasi-fission channels in the DNS formed at the capture stage. The asymmetric initial DNS has two ways of evolution. It may increase its charge asymmetry and after overcoming the barrier - B*^ transform into a compound nucleus. Or it may evolve into a symmetric form. In the symmetric form the Coulomb repulsion between the DNS nuclei reaches its maximum value and the DNS decays into two nearly equal fragments. It means that quasi-fission occurs. In the quasi-fission the DNS has to overcome the quasi-fission barrier Bqf. The competition between complete fusion and quasi-fission in the DNSC arises naturally as a consequence of the statistical nature of the DNS evolution. As is known, the DNS evolution is determined by the potential energy of the system, which is a function of the charge (mass) asymmetry and the collision angular momentum. Fig.2 shows potential energy V (Z, L) of the DNS formed in V(Z,L) (MeV)

30

2a 20 15 10 5 0

0

20

40

60

80

Z

Figure 2. The potential energy of the dinuclear system, formed in the reaction

no

Pd + "°Pd.

178

the reaction 110Pd + ll0 Pd. The figures on the curves indicate the collision angular momentum L. The potential energy is normalized to the potential energy of the compound nucleus, which is taken as zero. The initial DNS is situated in the minimum of the potential energy. It looks like a gigantic nuclear molecule. To realize complete fusion and to form a compound nucleus, the initial DNS has to overcome the potential barrier. It was called «the inner fusion barrier - B f^- The asterisk symbolizes that the energy for overcoming the barrier is taken from the DNS excitation energy - E . 4

4.1

The analysis of nuclear reactions used for the synthesis of transfermium and superheavy elements within the framework of the DNSC

The minimum of the excitation energy of compound nuclei in the cold fusion reaction

In the cold fusion reactions Pb and Bi targets are used and the products of reaction channel (HI, In) are detected. Models of complete fusion based on the first and second approaches, namely the MDM and the SFM, predict very high excitation energy [6]. According to the DNSC the minimum of the compound nucleus excitation energy is determined by the height of the inner fusion barrier B*ftS in relation to the compound nucleus. Fig.3 demonstrates the calculated data for the minimum excitation energy of the compound nuclei with Z= 102-114 [6]. So the DNSC make it possible to estimate the minimal excitation energy of the compound nucleus and optimal value of the bombarding energy in the cold synthesis of SHE. 4.2

The role of quasi-fission in the synthesis of superheavy elements

Fig.4 shows the calculated production cross section of elements 104, 108 and 110 synthesized in cold fusion reactions. The calculations were made by B.Pustylnik [9] within the framework of the first approach. The optical model was used for the calculation of the capture cross section a c . The statistical model - for the calculation of the survival probability of the excited compound nucleus Wsur. The experimental data were obtained at the GSI [10]. One can see satisfactory agreement between the calculated and experimental data for element 104. But there is dramatic disagreement in the case of elements 108 and 110. The cause of this disagreement is quasi-fission. However, in the first and second approaches the competition with quasi-fission is not taken into account. The calculations were made according to the ratio (3). In ratio (3) there is no term which reflects the influence of quasi-fission. CTER-CTC

-Wsur,

(3)

179

106

108

110

Element Number

Figure 3. Excitation energy of the compound nuclei with Z = 102-114 in cold fusion reactions. • experimental data for (HI, In) channel, O - calculated data according to the DNS-concept. a) The deformation of the heavy nucleus in the DNS and b) The deformation of heavy and light nuclei in the DNS are taken into account.

"Pb

c

10'

6 10' 10°

•

• '

' , ,

-20 -15 -10 -5

. 1 . 1 , 1 ,

• '

0

5

-20 -15 -10 -5 E'-E*

0

5

-20 -15 -10 -5

0

5

10

/MeV

Figure 4. Experimental and calculated evaporation residue cross sections in the cold synthesis of elements 104, 108 and 110. Solid squares are experimental data [10], curves are calculated data [9].

180 According to the DNSC, the production cross section of heavy elements is defined by the ratio (4): O'ER - o-c • P c n • W SI (4) Here Pcn- is the probability of the compound nucleus formation in the competition with quasi-fission. It should be emphasized that the DNSC provides a basis for the creation of a realistic models of competition between the complete fusion and quasifission channels [8, 11, 12]. Our first model of competition between complete fusion and quasi-fission was created for the symmetric nuclear reactions between massive nuclei [8]. In these reactions the initial DNS is situated in the minimum of the potential energy V (Z, J) and in the minimum of the nucleus-nucleus potential V(R), at the bottom of the potential pocket. It means that the DNS is in a quasi-equilibrium state. One can suppose that the probability of the DNS evolution to the complete fusion channel or to the quasi-fission channel is proportional to the level density of the DNS on the top of the fusion barrier and on the top of the quasi-fission barrier. The probability of the compound nucleus formation after the capture is defined by the ratio (5): P^=-^—

(5)

Using this model, we calculated the CTER value in the reaction U0Pd+110Pd. The results are presented in Fig.5 by the solid line. The squares represent the experimental data by Moravek et al. obtained at GSI [13]. The dashed line shows the results of the O"ER calculation according to the MDM [8]. 1

~ER

(mb)

L._ MDAA

Iff1

103

! DNSC • s^~^W B

105

bass

•, i

220

/

• / I' ,

240

i

I

260

280

*r

i

300

(MeV) Figure 5. Calculated crER (E) for "°Pd + "°Pd reaction by using DNSC and MDM [8]. The solid squares represent the experimental data [13].

181 s —

•

50-n

M

Cr

• r

58Fe

•

r

64Ni 62

: r \

• •

Ni

™Zn

•

™Zn

•

•

t—l

iI

-I

I

,I

Ii

•.

I

.

iI

I

I

.I

Ii

,I

"Ge

• •*>Ge II—I—I i i

I

II

.. I

I, I

I

103 104 105 106 107 108 109 110 111 112 113 114 115

Zen Figure 6. The value of Pcn in the cold synthesis of elements from 104 to 114. The calculation were made within the framework of the model of competition between the complete fusion and quasi-fission channels [12].

We have proposed two models for the asymmetric nuclear reactions. In the first model the Monte-Carlo method was used for calculating the DNS evolution [11]. In the second model [12] the evolution of the DNS was considered as a diffusion process proceeding along two collective coordinates. Diffusion along the mass asymmetry coordinate r)=(A1-A2)/(A1+A2) leads to complete fusion. Diffusion along R-coordinate leads to quasi-fission (R is the distance between the two centers of the DNS nuclei). A quasi-stationary solution to the two-dimensional Fokker-Plank equation is used for describing the DNS evolution. The parameter of the model is the DNS viscosity. The second model was applied to the calculation of Pcn in the reactions of synthesis of transfermium and superheavy element [14]. Fig.6 presents the dependence of Pc„ on the atomic number of the compound nucleus for cold fusion reactions. The calculations of the Pcn values indicate that quasi-fission is the main factor responsible for decreasing the production cross section in the cold fusion reactions. 4.3

Production cross sections in reactions used for the synthesis of heavy and superheavy elements

Using the data for Pcn, it is possible to calculate the production cross section of heavy and SHE. Dr. Evgeny Cherepanov will present in his talk the results of these calculations. I will show two figures only. In Fig.7 the filled squares represent the experimental data for aER [10], the open circles - the calculated data [15] for the production cross section of elements 102-112 in cold fusion reactions. The DNSC makes it possible to reproduce the production cross section of transfermium and

182 103 Pb,Bi(HI,1n)- reactions experiment

104

O - theory

105 xi

10*

E

10- 9

O

_J

1

1

I

I

I

I

I

70

Ge+Pb

l_

102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 118 Zen

Figure 7. The production cross section of elements with Z from 102 to 114 synthesized in cold fusion reactions. Solid squares are experimental data [10], circles are calculated data according to the DNSC [15].

superheavy elements in warm fusion reactions, in which 48Ca ions are used. Fig. 8 shows the results of calculations of the production cross section of element 114 [16] which was synthesized at Dubna in the reaction 244Pu+48Ca -»289114+3n [17]. One can see rather good agreement between the calculations and the experimental data. It should be emphasized that the calculations [16] were made before obtaining the experimental data [17]. 4.4

The shell effect in quasi-fission

In the reactions used for the synthesis of SHE the excitation energy of an initial DNS is small and one can expect the shell effect in the quasi-fission process. This effect is easier to observe in warm fusion reactions where actinide targets and 48Ca ions are used. In quasi-fission nucleons of heavy nucleus are transferred to light nucleus and the actinide nucleus is inevitably transformed into double magic nucleus 208Pb. The potential energy of the DNS for the configuration with 208Pb has a rather deep minimum. It can be see from Fig.9a). The potential barrier arises on the DNS way to symmetric configuration. This intermediate barrier may be even higher than the quasi-fission barrier Bqf. In this situation the decay of DNS into two fragments is more favorable than evolution to the symmetric configuration. In mass distribution of quasi-fission products "the lead peak" arises. "The lead peak" is very pronounced in the warm synthesis of superheavy elements [18]. I would like to emphasize that fusion models based on the first and second approaches cannot reproduce the shell effect in quasi-fission. Only the DNSC gives an opportunity to reproduce this effect.

183 cap.

r

,'"

10' 1" =

10"'

V?

n E t>

mb

r

. . • • (us.

-'

r r

/

r r

;

10* r nb

7

w r 3n

r

10" r*

4n

r

r X 2nA

10 r

//

Art®

10

20

30 40 E*,MeV

50

60

Figure 8. Calculations of the production cross section of element 114 synthesized in the reaction Ca + Pu [16]. dcup - the capture cross section, o-fcS - the cross section of the compound nucleus formation, curves with index ln-4n reflect the competition between fission and emission of different numbers of neutrons in the excited compound nucleus. The solid circle - the experimental data [17]. 244

100 «56 20p Mass Number

50

100

150

200

250

Fragment mass number

Figure 9. a) The DNS potential energy as a function of the mass asymmetry in the reaction Ca + U for L = 0. b) The spectrum of the mass distribution of quasi-fission products in the same reaction for excitation energy equal to 26 MeV.

184

Conclusion Analyzing experimental data on the synthesis of transfermium and superheavy elements within the framework of the DNSC one may come to the conclusion that today this concept gives the most realistic description of the complete fusion process. References 1. J.A.Kuehner and E.Almgvist, in Proceedings of the Third Conference on Reaction between Complex Nuclei, Asilomar, USA, 1963 (University of California Press, Berkeley, 1963), p.l 1. 2. J.Galin, D.Guerean, M.Lefort, X.Tarrago, Phys. Rev. C9, 1081 (1974). 3. D.H.E.Gross and H.Kalinowski, Phys. Lett. B48, 302 (1974); Phys. Report 45, 175 (1978). 4. H.Ngo and C.Hgo, Nucl. Phys. A348, 140 (1980). 5. W.J.Swiatecki, Phys. Scripta 24, 113 (1981); S.Bjornholm and W.J.Swiatecki, Nucl. Phys. A391, 471 (1982). J.P.Blocki, H.Feldmeier, W.J.Swiatecki, Nucl. Phys. A459, 145 (1986). 6. V.V.Volkov, G.G.Adamian, N.V.Antonenko, E.A.Cherepanov and A.K.Nasirov, IL NUOVO CIMENTO 110A, 1127 (1997). 7. V.V.Volkov, Izv. AN SSSR, Ser. Fiz. 50, 1879 (1986). V.V.Volkov, in Proceedings of International School-Seminar on Heavy ion Physics, Dubna, Russia, 1989 (D7-90-142, Dubna, 1990), p.462. 8. N.V.Antonenko, E.A.Cherepanov, A.K.Nasirov, V.P.Permjakov and V.V.Volkov, Phys. Lett. B319, 425 (1993); Phys. Rev. C51, 2635 (1995). 9. B.I.Pustylnik, in Proceedings of an 3 International Conference on Dynamical Aspects of Nuclear Fission, Casta-Papiernicka, Slovak Republic, 1996 (JINR, Dubna, 1997), p. 121. 10. S.Hofmann, Rep. Prog. Phys. 61, 639 (1998). 11. E.A.Cherepanov, V.V.Volkov, N.V.Antonenko and A.K.Nasirov, in Proceedings of International Conference on Heavy Ion Physics and its Application, Lanzhou, China, 1995 (World Scientific, Singapore, 1996), p.272. 12. G.G.Adamian, N.V.Antonenko, W.Scheid, V.V.Volkov, Nucl. Phys. A627, 361 (1997). 13. W.Morawek, T.Ackermann, T.Brohn, H.G.Clerc, V.Gollerthan, E.Hanlet, M.Horz, W.Schwab, B.Voss, K.H.Schmidt and F.P.Hessberger, Z. Phys. A341,75(1991). 14. G.G.Adamian, N.V.Antonenko, V.V.Volkov, A.K.Nasirov, E.A.Cherepanov, W.Scheild, JINR Rapid Communications X«6 [86], 39 (1997). 15. E.A.Cherepanov, V.V.Volkov, in Extended Abstracts of 1-st International Conference on the Chemistry and Physics of Transactinide Elements, Seeheim, Germany, 1999.

185 16. E.A.Cherepanov, Pramana - Journal of Physics, Indian Academy of Science 53, 619 (1999). 17. Yu.Ts.Oganessian, V.K.Utyonkov, Yu.V.Lobanov et al., Phys. Rev. Lett. 83, 3154(1999). 18. M.G.Itkis, Yu.Ts.Oganessian, E.M.Kozulin et al. 2 Intern. Conf. on Fission, St. Andews, 28 June - 02 July, 1999 (in press).

186

CALCULATION OF FORMATION AND DECAY OF HEAVY COMPOUND NUCLEI E.A.CHEREPANOV Flerov Laboratory of Nuclear reactions, Dubna, Russia E-mail: [email protected]

JINR

The report describes a method for calculating fusion and decay probabilities in reactions leading to the production of transfermium elements. The competition between quasi-fission and fussion is described on the basis of the Dinuclear System Concept (DNSC). The both competition between fusion and quasi-fission and statistical decay of heavy highly fissionable excited compound nuclei is described in an approach based on the Monte-Carlo method.

1. Introduction In the last few years in the Laboratory of Nuclear Reactions, JINR, a lot of experimental information was obtained concerning the formation of superheavy elements (e.g. [1,2,3]) as well as the mass distribution of fragments, produced in reactions which are used for the synthesis of superheavy elements [4]. In ref. [4] for the 48Ca+233U,244Pu,248Cm reactions the fragment mass distributions were measured from which follows that in all cases there is a maximum in the distribution corresponding to the doubly magic nucleus 208Pb. This peak is observed in a rather large energy range of the bombarding ions 48Ca, which was demonstrated for the 48 Ca+ 244Pu reaction. We shall try to explain this experimental fact using our approach. In the calculations of the reactions used to synthesize superheavy elements, our approach is based on three stages: the formation of the double nuclear system, the formation of the compound nucleus (in competition to quasi-fission and complete fusion) and the statistical approach for the decay of the excited compound nucleus. 2. Calculating the stage of capturing a heavy ion by a target nucleus According to the DNS concept, complete fusion occurs as follows. At the capture stage, after the kinetic energy is completely transferred into the inner degrees of freedom, a dinuclear system is formed with a capture cross section a c . The capture cross section uc is part of the complete inelastic cross section aR:

^ = [£(2/ + i)rttiu|/£(2Z + l)7UiU

(1)

where lcr is the critical angular momentum at which the heavy ion is captured and the excited dinuclear system is formed. The values for lcr were taken from empirical systematics for theCTC/CJRratio [5]. The capture cross section is described by the formula:

187

ac(E-) = x%lY(2l

+ l)T(l,ECM),

(2)

where the penetration of the potential barrier is calculated on the basis of the standard inverted parabola method and written as T{1, ECM) = {1 + exp [{2K I hw){B - E)]}'1 , where

hw = "•/

d2V(R,l)/ /dr1

(3)

E is the energy of the bombarding ion; R

the distance between the centres of the nuclei; u the reduced mass of the system; B the Coulomb barrier. In calculating the capture cross section, the Woods-Saxon potential was used as the interaction potential: 3 \ I/JI 1 + p ~ ~ )l D _ « < A"'3 , A">

exp{[/?-,0(4' +4'' )]/rf}'

(4)

whose parameters: d, the diffuseness of the potential, V0, the depth of the potential, and r0, the radius parameter, were taken from work [5], in which empirical systematics for the parameters of the potential had been found on the basis of a great number of experimental data on the capture cross section. 3. The fundamentals of the Dinuclear System Concept The motivation for the Dinuclear System Concept (DNSC) and a comparison of DNSC with existing models for the fusion of massive nuclei have been already given in our works [6,7] (see also the V.V.Volkov talk on this Worshop). Therefore here we will only discuss its main features applied to analysis of the fusion reactions used in the production of transfermium elements. In our approach, complete fusion is the final stage of the evolution of a DNS, at which all the nucleons of one nucleus will have gradually been transferred to the other nucleus. In the best known models for the complete fusion of nuclei, the production cross section for compound nuclei, aCN, is not different from the capture cross section, ac; in other words, after the capture stage, a compound nucleus is formed with 100% probability. In our approach, the complete fusion cross section,
+l

)W,ECM)

• PCN,

(5)

1=0

where PCN is the probability that complete fusion will occur. In the evolution of the DNS, each nucleus of the DNS retains its individuality; this is a consequence of the influence of the shell structure of the partner nuclei since the kinetic energy of the bombarding ion, and, thus, the resultant excitation energy, as a rule, is low in these reactions. The Macroscopic

188

Dynamic Model (MDM) [8,9] description of the coalescence of two nuclear drops doesn't take account of the shell structure of the nuclei, and complete fusion doesn't compete with quasi-fission. Those two processes are considered to be separated in energy space. An essential characteristic of a DNS that dictates the way for it to evolve is the system's potential energy V(Z,L) V(R,L) = VrfR) + Vc(R)+Vro,(R,L), (6) In this work in contrast to [6,7], we took PROXIMITY as the nuclear potential VN(R) (for details, see [9]). VN(R) = 4xyR-b-®(t), (7) where y

2~

—

•MeVfm2,

l-1.7826f~

^ = 0.9517

C ^^ p

C. =Rt(l-tf/R* R =1.28A

1/3

+....), % = slb,

s=

C r

where

\^"p

r-(CP+CT),

1/3

- 0 . 7 6 + 0.8yT , b = 1.0 index i=P or T

-1.7817 + 0.9270 • £ + 0.14300 • £2 - 0.09000 • g, 1.9475 The Coulomb energy was calculated with the following formula:

*. '"A

10 CN_

'^Mimm "••NV£r. : ?"

4 ~ 20 ° • sym 20 o Fig. 1. PROXIMITY potential as a function of the charge number of one of the fragments and the distance between the centres of the fragments. The entrance channel and probable ways for the DNS to evolve are shown with arrows.

\

189 Mr, ' f

ZjptCjrpiC-

r>Rc

( 3 - , ^ ) , r
2V

WhCre

^='c(4"+4")

The rotation energy Vrot(R,l) was calculated with the solid moment of inertia. Our estimates showed that the rotation component has a small effect on the results of calculating quasi-fission-fusion competition at ion energy close to the Coulomb barrier. Shown in Fig.l is a qualitative picture that presents three-dimensional PROXIMITY potential as a function of the distance between the centres of the nuclei and the charge number of one of the DNS fragments (calculations of DRIVING potentials were made using the NRV program [10]). On capturing a heavy ion by a target nucleus, the DNS formed will find itself in the potential pocket. Then the double system will be moving along the valley of the potential pocket both in the direction to symmetry and in the direction to decreasing the charge number of one of the fragments - in the direction to fusion. On the way, the DNS might undergo a break-up into two fragments - quasi-fission (QF), i.e. fission without a compound nucleus being produced. In the figure, the corresponding processes are indicated by arrows. In Fig. 2, shown are two profiles of the potential energy: the profiles along the minimum and the maximum of the potential surface calculated for the reaction 48Ca+244Pu. The curve V(Z,L=0) (for the magnitude of R corresponding to the pocket) has a few local minima, which reflect the shell structure in interacting nuclei. Among them most pronounced are: the minimum corresponding to the compound nucleus (Z=0) and three others, the first one being at Z of the light fragment equal to 20 and corresponding to the entrance channel (the projectile 48Ca) and the other two respectively at Z=82 (the complementary heavy fragment - the doubly magic 208Pb) and Z=52 (the complementary fragment corresponding to the magic Sn-nucleus). In this way, there is evidence of shell structure in the driving potential, which will manifest itself (as it will be shown below) in the fragment mass distributions as well. Heavy and superheavy elements (SHE) are typically produced at the ion energies that result in the minimally possible excitation energy of the compound nucleus. This ensures higher survival probability for the compound nucleus while de-excited. In Fig. 2, the DNS energy that corresponds to the minimally possible excitation energy of the compound nucleus is shown by cross-hatching. As follows from Fig. 2, while descending from the Businaro-Gallone point (B.G.) to the point of compound nucleus formation, the system will undergo most heating. Only at this stage of the DNS evolution will most of the potential energy of the dinuclear system be transformed into the heat excitation of the compound nucleus. This peculiarity of DNS evolution, characteristic of SHE fusion reactions, required the use of experimental masses to calculate the potential energy V(Z, I). In addition, on the way to the compound nucleus, the DNS faces with overcoming the inner potential

190

290

270

>

1

160

0

10 20 30 40 50 60 70 80 90 100110 120130140

Ai Fig. 2. Driving potential (lower curve) of the dinuclear system as a function of the atomic number of one of the fragments for the reaction 48Ca+244Pu.

barrier B*fus, which is the difference between the values of the potential V(Z, I) at the B. G. and at the reaction entrance point. The inner potential barrier B"fus results from nucleon transfer in a massive DNS being of an endothermic nature, which makes the system move in the direction to the compound nucleus. The movement of the DNS in the reverse direction, to more symmetry, might result in its leaving the potential pocket (with a break-up into two fragments - the movement in the direction to increasing R) after overcoming the QF barrier, which we took to be the difference between the values of driving potential for the entrance combination and the point of the break-up into two fragments. The necessary energy to overcome those barriers is drawn from the excitation energy E* of the dinuclear system, which an essential feature of our approach. A compound nucleus is unlikely to be formed if the DNS excitation energy is smaller than the value of B*fUS. The more symmetric the entrance combination of nuclei, the higher the inner fusion barrier B*fus, which the dinuclear system has to overcome on the way to the compound nucleus, and the lower the quasi-fission barrier BQF; hence, QF offers stronger competition. 4. Competition between complete fusion and quasi-fission in SHE synthesis reactions Nucleon transfer between the nuclei in a DNS being of a statistic nature, there is a possibility that the system might reach and overcome the B. G. point, and thus a compound nucleus will be formed. The alternative to that process will be a break-up of the system into two fragments (quasi-fission). To calculate the

191

probability of proton transfer from one nucleus to the other in a dinuclear system, we applied an expression from work [11] and assumed that macroscopic nucleon transfer probability P2 can be found from the microscopic probability ^ z and level density pz as P2=Azpz. The level density can be written in terms of the DNS potential energy pz=p(E-V(Z, 1)), where E* is the excitation energy of the double system. Finally, the proton captures P+ and delivery F one probabilities can be written as follows: 290

0

270

250

230

210

190

170

150

10 20 30 40 50 60 70 80 90 100 110 120 130 140

A, Fig. 3. Driving potential of the dinuclear system as a function of the mass number of one of the reaction fragments. Presented also calculations of the total spectrum in the mass fragment distribution for the reaction 48Ca+2''4Pu.

P* = h + exp (8) 1 + exp 1/2 where T=(E /a) is the nucleus temperature; a-0.093A the level density parameter. + Knowing those relative (P +P~=l) probabilities and using a random value uniformly distributed over the interval between 0 and 1, we simulate the direction for the DNS to move in: either in the direction to the symmetric system or in the direction to the compound nucleus, repeating this procedure as many times as needed to obtain the necessary statistics. Fig.3 shows the calculated results - the mass distribution of QF products for the reaction 48Ca+244Pu for E = 35 MeV. It is seen from this figure that the spectrum for the mass distribution of reaction fragments correlates with the

192

structure of the driving potential. The maxima of the mass distribution are matched by the position of the local minima of the potential. This is a reflection of the nature of our approach to calculation in the case that there exist quasi-stationary states (local minima), which gives grounds to apply the statistical model. On Fig.4 demonstrate the comparison theoretical predictions of QF mass distribution and 1U

s

?

10

• *•• 4

••••

f*

S

o o M

-3

*

-it- 10

"51 0 ,

£

I

»j|

-1

• •

•

!

-

*

la CO

ir

• T «1

1

- ,

- t 1 * • *

a

10°^ I

'

i?

. «

'*# v '\"

10'5

1

2

?;

\

i

I -6

;

10 -7

10

i

L,-,-

50

—

•>

i

• " I * !

•• r

100

•

i-

•

-

i

.' •

i

!£0

•

»- • - • " i

2C0

"••'

,_,j

250

Ai Fig. 4. The calculation of QF fragment distributions (bar graph) and experimental data [4] points.

experimental fragment mass distribution for 48Ca+244Pu reaction. From this figure see that positions of experimental data on fragment mass distributions and calculation of QF distributions are coincident. S, Survival probability for the statistical decay of compound nucleus JlL.

(Ef)*s*xr£ivai(Er,t)-w„(Er,i),

(9)

where 0CN is the production cross section for the compound nucleus. There are a few approaches to calculate the survival factor Wsur for the compound nucleus in the competition between the processes of fission and particle evaporation within the framework of the statistical model. This depends on the particular calculating algorithm used. -For the decay of a heavy highly fissionable compound nucleus, the assumption that the main process competitive to the fission is the neutron evaporation will be a good approximation for calculation of the value Wsur. In this case, there exists a simple phenomenological method for calculating the survival factor, which can be written as the product of the ratios ( r » / r«,i) = ( r » /( r » + r / ) ~ ( r n / r / ) averaged over the evaporation cascade. The magnitude of the fissionability averaged over the evaporation cascade

193

r r

-/ /Hs(r-/r/)'

(10)

is derived from experimental cross sections, o m , with the help of equation (9). It depends weakly on the values E and x. Then the survivl of the heavy nucleus in this approach can be written in an analytical form: WAE\l)~PJE\l)-fth^\ ,

(11)

where x is the number of the evaporated neutrons; Pxn the probability of evaporating exactly of x neutrons from the excited compound nucleus of energy E and angular momentum /; k the index of the evaporation cascade stage. In work [12], a great body of experimental data on few-neutron emission reactions was analyzed, and semi-empirical systematics for the value / r „ /T \ averaged over the cascade were obtained, which permitted describing the experimental data known by that time with an accuracy of up to the factor 2-^3. The value Px„ in (11) was calculated following [13]. There exist other approaches to calculating the survival of compound nuclei in the de-excitation process within the framework of the statistical model.

w

,(£*)=n k=\ V

•PX(E ) 1

(12)

tot(Ek), ' Jk

Brief mention can be made of the latest calculations of this value for the considered compound nucleus of2 No in works [14,15]. In work [14], for calculating survival the standard expressions were substituted from the statistical model for the probability of evaporating neutrons, protons, a-particles, y-quanta and fission. A system of integrals inserted into one another (due to the number of stages in the evaporation cascade) was arrived at. Then numerical integration was carried out to calculate the magnitude of the survival. Details can be found in the cited paper. In work [15], after a number of reasonable simplifications and integration, simple expressions were derived for the probability of evaporating neutrons and fission. P„(£*,e) = Cv^exp

V

(13)

I T(E )J The factor Pxn in formula (12) was calculated on the basis of Monte-Carlo method, simulating the probability of emitting neutrons of given energy. The neutron distribution over the energy e carried out of the excited compound nucleus was taken in both cited papers to be Maxwellian: In our case, we took advantage of calculating the value Wsur within the framework of the standard statistical model that takes account of evaporation not only of neutrons but also of charged particles and y-quanta. Since we study the decay of compound nuclei of considerable angular momentum, we made use of a quasi-classical version of the statistical model [16]. In this version, the probability

194

of emitting a particle v (p, a, d, t, 3He) of energy C v and angular momentum Z in a unit time from a compound nucleus of angular momentum L and excitation energy E* in the direction n is written as follows P,(T,nft) = (2i, + l ) i % - " f i ' ^ -da^HK) (14) n h PmKEm,LJ where u is the reduced mass of the system; s the spin of the emitted particle; the symbols m and d indicate the mother and daughter nuclei, correspondingly. Integrating the latter expression over die energy of the emitted particle gives the total probability and, thus, the partial width for emitting particles from the compound nucleus T„ = Pv / h. To calculate the partial width for emitting neutrons, charge particles (v = p, a, d, t, 3He), y-quanta and fission, the following expressions were applied (see, for example, in reference [17]): U B ~" Tn(E*,L)* Y / 1 ^ / amv(en)Pd(U-Bn-en)enden,

(*hyPm(u)

0

c-a,

U

3 {xhc)

pm{U)JQ V.-B,

rf(E',L)^(2npm(U)T1

f

p,{U,-BI-e)de,

0

where U is me heat energy of the mother nucleus. The cross section of the reverse reaction
'

=

l2a^-6reMS{E')l

(16)

While calculating Ff, account was taken of the fact that the rotation of the nucleus causes the fission barrier decrease: B/L)= B/0)-(Er-Ers).To calculate the level density, the known Fermi-Gas (see, for example, [20]), was used In (16), the nucleus entropy S at the excitation energy E is taken to be described by the relationship S=2aT, using the relation between the temperature and excitation energy of the nucleus E"=af; 5 is the even-odd correction. The level density parameter a=7i?go/6 is expressed as a function of the one-particle density near

195

Fermi's energy g0 =f(Ej) = const. That the influence of shell effects in level density decreases as excitation energy increases was taken account of according to the phenomenological expressionfrom[20]: a(E*) = a[l + f(E')AW/E'l (17) Here f(E )=l-exp(-yE), AW is the shell correction to the nucleus mass; a is the asymptotical value of the level density parameter in the Fermi gas model, which can be written in the general form as a =A(a+f}A"). Empirical values for the parameters a = 0.148 MeV1; 0 = -1.39Iff4 MeV'; y = 0.06 MeV1 at n=l were found in work [21] from analysis of data on the level density with regard to the contribution of the collective states to the total level density. To calculate Wsur, the fission barrier for compound nuclei should be known. The fission barrier .fly is calculated as the sum of the liquid-drop component and shell correction AW: Bf = BfD + AW. For the compound nuclei of the heavy transfermium region, the liquid-drop component is a very small quantity. For the calculation to take account of the excitation energy dependence of the shell correction, we took advantage of the following formula: Bf(E*)=BfLD-\-AWexp(-yE*). In our calculation, we used the value B/E =0) from work [22]. 6. Scheme of calculating the de-excitation process for an excited compound nucleus on the basis of the Monte-Carlo method In this work, the de-excitation process for a nucleus was calculated by applying an approach based on gambling the random value - the Monte-Carlo method. Such an approach was successfully used to calculate the decay of heavy nuclei in [23]. In our opinion, it reflects the random, statistical nature of particle evaporation or fission in the most adequate way. The angular momenta of the compound nuclei produced in a complete fusion reaction will be distributed over the value L, the vector L lying in the plane perpendicular to the ion beam. With the help of those random numbers, the value of the angular momentum and its orientation in space were simulated. Then the maximum residual energies in the different channels of the £<->* = E ' - E ^ - B , -V„;

E ^

= E'-Erot-Bf.

(18)

statistical decay were simulated For all the Ev > 0 , the type of emitted particle was also simulated - v or y-quantum - from formulae (15). After the type of deexcitation process (the decay channel) was found and if no fission occurred the characteristics of the evaporated particles or y-quanta were simulated. So their kinetic energy, carried-away orbital momentum and angle 6 were found. For a given type of particle, the values of 6V, I and cos 6 were simultaneously sampled using three random numbers and then rejection was performed with the help of a fourth random number according to the three-dimensional probability density given by the expression [24]

196

exp \2^a(E* - Ev - {L2 + I2) / 2 J + LI cos 6 / J)

W(e„ ,l„,coaO)~l-

(19)

In a coordinate frame with a Z axis parallel to L, Z\\L, the azimuthal angle of the vector I was simulated. The azimuthal angle of the escaped particle was simulated in a coordinate frame with a Z axis parallel to I , Z || T . The fission process

was

accounted

FC/ = TTfl — r /Tm

for

with

the

help

of

the

weight

functions

, which is especially convenient for highly fissionable

nuclei and significantly saves computing time. All the values so found were brought to the centre-of-mass system of the colliding nuclei, and then the characteristics of the residual nucleus were calculated

E'd=E'm-B„-ev-(L2d-l2)/2J;

Td=i:-T\

Ad = Am - Av;

Then for that residual nucleus, the maximum residual energies were calculated for all the decay channels: particle emission, y-quantum emission and fission. Among all the processes energetically allowed, the de-excitation of the nucleus was again simulated and so on for as long as Es >0. Presented in Fig.5 are the calculated results for the probability of neutron evaporation from a nucleus of 258No, which is close to our case, as a function of -

0.1

-

Survival

10°

1.0

—^°

° ^^^^ 10 US'

0

I

I

10

20

•'

30

I

I

I

I

40

50

60

70

10"

10

E* (MeV)

Fig. 5. Probability of emitting neutrons as a function of excitation energy for the compound nucleus of25gNo.

10

20

30

40

50

E', MeV

Fig. 6. Survival of the compound nucleus of No in statistical decay calculated with three different approaches (detail see in text). 2i6

excitation energy. The solid curve corresponds to calculations taking account of the energy dependence of the shell correction; the dotted curve shows calculations with the liquid-drop barrier. Also shown in the picture are the experimental values derived from the neutron evaporation cross sections by the so-called pair reaction method (for details on the pair reaction method and references see [25], from which that figure is borrowed). It is seen from the data presented in the figure, that for the larger excitation energy of the nucleus, it is necessary to take account of the shell

197 :"ca

«Ca* ! 0 8 Pb= 2 5 8 102

io

•244Pu

: «Ca. 2 «Cm .296 116

= 292 1H

10

-^3n

9

•^4n

^ r 1n\ 1011

/ 2n^ 10

r

3nV

2n

r

An J ,

|

•13

{

1n

,

I

1

i

1

10 30 40 20 30 40 50 60 E* (MeV) Fig. 7. Excitation functions for Z=102,l 14,116 compound nuclei formation (calculations) for 48 Ca+208Pb,244Pu, 248Cm reactions. Points are experimental data from [1,28]. 10

20

correction being dependent on E . For E > 35 MeV, shell effects in such a heavy nucleus as 256No die out almost completely. Shown in Fig. 6 are the calculated results for the survival of an excited compound nucleus of 256No obtained within the framework of the statistical model with three different models. The solid curve is the calculated result from this work. 10T3 1Lib *4 08 : "\

104

r

C a + 2 K Pb

^ ^ \

4e

Pb,Bi(HI,1n)-reactions • - experiment

Ca+Bi

F«W»Pb \ o ^jfJi+Pb 10* fe:

J3

10*

O - theory

O -^TH-Bi

nb

E

54

n

Cr+Pb'

10''

54

Cr+Bi

IX^+Pb 64

* ^ \

10*

I

ipb

Ni+Pb

6e

2 \ I

Zn+Pb

_

68 Zn+Bi ?wZn+Bi a

III

™Zn+Pb * \ .

icr 10 r

iff1'

g'74Ge+Pb ^kpGe+Pb O

70

Ge+Pb

r

1 0 1 2 -fb->

L

1

,

i

I

I

I

I

I

I

l

i

i

i

102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 118 Fig.8. Experimental data (black squares) and theoretical calculations (open circles) for synthesis of elements from 102 to 114 in cold fusion reactions (HI,ln). The Figure is taken from [27]. References on experimental data one can see, for example, in [27,29]

The dash curve is the calculated result from work [14]. The dotted curve is borrowed from work [26], the model used being described in work [15]. It is seen from the figure that the different calculating methods give close magnitudes for fissionability in the region of excitation energies from 13 to 35 MeV, but as energy

198

increases the magnitudes from the dotted curve begin to exceed the magnitudes from the other curves. This might be connected with the fact that the approach elaborated in [15] only takes account of the fission channel and neutron evaporation channel whereas at the energies higher than 35 MeV, the channels with evaporation of charged particles begin to manifest themselves. As one can see from Fig. 7 and 8 our approach allow us describe the formation SHE elements both for "cold" and "hot" fusion reactions. Our estimation the formation of new 116 element in 48 Ca+248Cm reaction with evaporation 3 or 4 neutrons is about 0.5pb. Summing up, in our approach based on the concept of the DNS, it is possible to describe quite well the experimental data on the formation of superheavy elements in reactions accompanied by neutron evaporation, as well as data on the fragment mass distributions resulting from the contribution of the process of quasifission in these reactions. 7. Acknowledgements The author would like to express gratitude to Prof. V.I.Zagrebaev for NRV program of calculating driving potential and the numerous useful and fruitful discussions, to Profs. V.V. Volkov and W.Scheid, Drs. A.V.Antonenko and G.G. Adamian for the cooperation, much valuable advices given during this research, to Profs. Oganessian Yu.Ts. and Itkis M.G. for the support and attention to the work, to Prof A.Sobiczewski for numerous and useful remarks concerning properties of SHE elements, to Profs. G.Munzenberg and S.Hofmann for permanent interest to present research. Also I consider it my duty to express also gratitude to Prof. M.Ohta for numerous useful discussions of the problems of survival of the compound nucleus, Drs.Y.Aritomo and R.Kalpakchieva for cooperation and help. This work was partially supported by the Russian Foundation for Basic Research (RFBR) under grant N« 00-02-17149. References 1. Oganessian Yu.Ts., Utyonkov V.K., Lobanov Yu.V., Abdullin F.Sh,. Polyakov A.N, Shirokovsky I.V., Tsyganov Yu.S., Gulbekian G.G., Bogomolov S.L, Gikal B.N., Mezentsev A.N., Iliev S., Subbotin V.G., Sukhov A.M.,. Buklanov G.V, Subotic K., Itkis M.G., Moody K.J., Wild J.F., Stoyer NJ., Stoyer M.A., and Lougheed R.W.. Phys.Rev.Lett. 83 (1999) 3154; 2. Oganessian Yu.Ts., Utyonkov V.K., Lobanov Yu.V., Abdullin F.Sh,. Polyakov A.N, Shirokovsky I.V., Tsyganov Yu.S., Gulbekian G.G., Bogomolov S.L, Gikal B.N., Mezentsev A.N., Iliev S., Subbotin V.G., Sukhov A.M., Ivanov O.V., Buklanov G.V, Subotic K., Itkis M.G, Moody K.J, Wild J.F, Stoyer N.J, Stoyer M.A, and Lougheed R.W. Laue C.A., Karelin Ye.A, Tatarinov A.N. Phys. Rev. C (2001) (accepted for publication). 3. Oganessian Yu.Ts. Preprint JINR, P7-2000-23 (2000), submitted to Yad. Fiz. 4. Itkis M.G, Oganessian Yu.Ts, Bogatchev A.A, Itkis I.M, Jandel M , Kliman J , Kniajeva G.N, Kondratiev N.A, Korzyukov I.V, Kozulin E.M, Krupa L , Pokrovski I.V, Prokhorova E.V, Rusanov A.Ya, Voskresenski V.M,

199

5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

Hanappe F., Benoit B., Materna T., Rowley N., Stuttge L., Giardina G., Moody K.J. This proceedings (Submitted to Preprint JINR). Iljinov A.S., Oganessian Yu.Ts., Cherepanov E.A. Yad. Fys. 36 (1982) 118129. (In Russian). Antonenko A.V., Cherepanov E.A., Nasirov A.K., Permjakov V.P., Volkov V.V., Phys. Lett. B319, 425 (1993); Antonenko A.V., Cherepanov E.A., Nasirov A.K., Permjakov V.P., Volkov V.V. Phys. Rev. C51, (1995) 2635. W.J.Swiatecki, Phys. Scripta 24 (1981) 113; Bjornholm S., Swiatecki W.J. Nucl. Phys. A319 (1982) 471; Blocki J., Feldmeier P., Swiatecki W.J. Nucl. Phys. A459 (1986) 145. Blocki J., Randrup J., Swiatecki W., TsangC.F. Ann.Physics, 105 (1977) 427; Myers W. D. and. Swiatecki W. J. Phys. Rew. 62 (2000). Zagrebaev V., Kozhin A. JINR Communication, E10-99-151 (1999); Zagrebaev V., Kozhin A., Denikin A., Alekseev A. http://nrv.jinr.ru/nrv/ Moretto L.G., Sventek J.S. Phys. Lett. 58B, (1975) 26. Cherepanov E.A., Iljinov A.S., Mebel M.V. J.Phys. G: Nucl. Phys. 9 (1983) 931-938. Sickeland T. Ark. Phys. 36 (1966) 539. Zagrebaev V.I. This Proceedings Ohta M. This Proceedings Ericson T., Strutinski V.M. Nucl. Phys. 8 (1958) 284. Barashenkov V.S., Toneev V.D Interaction of High energy particle with atomic Nuclei, Atomizdat, Moscow 1972, (in Russian). Dostroovsky I., Fraenkel Z., Friendlander G. Phys. Rev. 116 (1959) p.683-702. Iljinov A.S., Oganessian Yu.Ts. Cherepanov E.A. Sov. J. Nucl. Phys. 33 (4), (1981) 520-530. Iganatuk A.V. Statistical Properties of Excited Atomic Nuclei, EnergoAtomizdat Moscow 1983, (in Russian). Cherepanov E.A., Iljinov A.S., Nukleonika 25, n5/80, 1980, 611-621. Myers W.D., Swiatecki W.J. Ark. Phys. 36(1966) 343-352. Cherepanov E.A. In book of proceedings Int. Conf. "In Beam Nucl. Spectorscopy", 14-18 May 1984, Debrecen, Hungary, Pub. House of the Hungarian Academy of Sciences, 499-506. Barashenkov V.S., Jeregi F.G., Iljinov A.S., Toneev V.D. P&N 5 (1974) 469 Cherepanov E.A., Iljinov A.S., Mebel M.V. J.Phys. G: Nucl. Phys. 9 (1983) 1397-1406. Aritomo Y., Wada T., Ohta M. This Proceedings Cherepanov E.A. Pramana Journal of Physics, 53(1999)619-630. Yeremin A.V., Chepigin V.I., Itkis M.G., Kabachenko A.P., Korotkov S.P., Malyshev O.N.,Oganessian Yu.Ts.,Popeko A.G.,Ronac J.,Sagaidak R.N., Chelnokov M.L.,Gorshkov V.A.,Lavrentev A.Y. Rapid. Comm., JINR 6 (92)-98. Hofrnann S. Rep. Prog. Phys. 61 (1998) 639-689.

200

FORMATION OF S U P E R H E A V Y ELEMENTS R. SMOLANCZUK Nuclear Theory Department, Soltan Institute for Nuclear Studies, Hoza 69, PL-00-681 Warszawa, Poland and Nuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA E-mail: [email protected] Possibilities of the synthesis of elements 119, 120 and 121 are discussed. The most promising reactions are indicated.

In this contribution, we consider the so called cold fusion reactions 1,2's based on magic 2 0 8 Pb and 209 Bi targets. We discuss the possibilities for the production of heavier elements than element 118 synthesized recently at Berkeley 4 . That experiment was proposed in Refs. 5 ' 6 as a consequence of a relatively large cross section predicted in our phenomenological model 5 for the reaction 208 Pb( 8 6 Kr,ln) 2 9 3 118. The model is based on the assumption of the quantal tunneling of the fusion barrier and the statistical description of the deexcitation process of the compound nucleus. The present discussion is performed by means of the refined model. The shape of the fusion barrier around its top is assumed to be parabolic. The barrier height, as well as its curvature, is dependent on the combination of the colliding nuclei. The transmission probability through the fusion barrier is calculated by means of the WKB approximation. The neutron-to-total-width-ratio is obtained by using the statistical model formula in which we use different temperatures for the equilibrium and the saddle-point configurations. The different temperatures simulate slower neutron emission in comparison with fission for low excitations corresponding to ln-channel. The optimal bombarding energy corresponding to the maximum of the excitation function is calculated as the sum of thresholds for neutron emission and fission following neutron emission. The details of the present calculation are given in Ref.7 The calculated optimal bombarding energy in the lab system Eiab for cold fusion reactions involving magic projectiles which are simultaneously the heaviest stable isotopes of Kr, Rb, Sr and Y are given in Table 1. The choice of magic projectiles leads to larger optimal bombarding energy and, consequently, to smaller effective fusion barrier with larger transmission probability. For the reactions listed in Table 1, we obtain the transmission probability through the fusion barrier for zero angular momentum To « 1 0 - 7 ,

201 Table 1: The calculated optimal bombarding energy in the lab system Eia\, for reactions listed in the first column. The experimental bombarding energy Ef££ and the measured cross section
Reaction

208

Pb(86Kr,ln)293118 Pb(87Rb,ln)294119 209 Bi(86Kr,ln)294119 208 Pb(88Sr,ln)295120 209 Bi(88Sr,ln)296121 208 Pb(89Y,ln)296121

208

r?exp lab

Elab

a

MeV

MeV

449.1 463.6 454.6 479.3 486.0 492.9

449

"""

™±li Pb

the neutron-to-fission-width-ratio for zero angular momentum ( r „ / r / ) o « 10- 4 and the cross sections of the order of 0.1 - 1 pb. 2 0 9 Bi( 8 6 Kr,ln) 2 9 4 119, 208 Pb( 8 8 Sr,ln) 2 9 5 120 and 209 Bi( 88 Sr,ln) 296 121 are predicted to be the best reactions for the synthesis of elements 119, 120 and 121, respectively. The reaction 2 0 9 Bi( 8 6 Kr,ln) 2 9 4 119 might give the opportunity for the production of unobserved so far elements 117, 115 and 113 as decay products of 294 119. Decay properties of the nuclei constituting the predicted decay chain of 294 119 are given in Ref. 6 . Acknowledgments This work was supported by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of Nuclear Physics under U.S. Department of Energy Contract No. DE-AC03-76SF00098 and Grant of the Polish Committee for Scientific Research (KBN) No. 2 P03B 099 15. References 1. Yu.Ts. Oganessian, in Classical and Quantum Mechanical Aspects of Heavy Ion Collisions, Vol. 33 of Lecture Notes in Physics (Springer, Heidelberg, 1975), p. 221. 2. G. Miinzenberg, Rep. Prog. Phys. 51, 57 (1988). 3. S. Hofmann, Rep. Prog. Phys. 6 1 , 639 (1998). 4. V. Ninov, K.E. Gregorich, W. Loveland, A. Ghiorso, D.C. Hoffman, D.M. Lee, H. Nitsche, W.J. Swiatecki, U.W. Kirbach, C.A. Laue,

202

J.L. Adams, J.B. Patin, D.A. Shaughnessy, D.A. Strellis, and P.A. Wilk, Phys. Rev. Lett. 83, 1104 (1999). 5. R. Smolariczuk, Phys. Rev. C 59, 2634 (1999). 6. R. Smolariczuk, Phys. Rev. C 60, 021301(R) (1999). 7. R. Smolariczuk, Preprint LBNL-45287; submitted to Phys. Rev. C.

203

P R O D U C T I O N OF S U P E R H E A V Y ELEMENTS I N HEAVY I O N REACTIONS V.YU. DENISOV GSI-Darmstadt, Planckstrasse 1, D-64291 Darmstadt, Germany Institute for Nuclear Research, Prospect Nauki 47, 252028 Kiev, Ukraine E-mail: denisovQgsi.de ; [email protected] The cold fusion reactions leading to superheavy elements (SHE) with Z=104-116 have been discussed in our model recently 3 . In this paper we shortly discuss our model and extend our consideration to fusion reactions ( 86 Kr, 8 7 Rb, 88 Sr) + 208 Pb and 86 Kr + 209 Bi leading to elements with Z=118-120 and production of SHE in reactions with similar target and projectile. The available experimental cross-sections are well described.

1

Introduction

The synthesis of superheavy elements (SHEs) was and still is an outstanding research object. The properties of SHEs are studied, both theoretically as well as experimentally [1-13]. In two series of experiments the heaviest elements from 107 to 109 and from 110 to 112 were synthesized at GSI-Darmstadt by using cold fusion [1]. In cold fusion, SHEs are produced by reactions of the type X + (Pb,Bi) —> SHE + In at subbarrier energies. The excitation energy of a compound nucleus formed by cold fusion is low, » 10-20 MeV. It was measured that the center-of-mass kinetic energy of reaction partners leading to elements with Z < 112 corresponds to the fusion barrier or is even less [1]. The cross-section for the synthesis of SHEs is very small and decreases rapidly with increasing atomic number. The experimental investigation of an excitation function for the SHE production becomes increasingly difficult due to the very small cross-sections and narrow width of the excitation function [1]. The full width at half maximum of the excitation function is about 4 MeV. An appropriate theoretical models should reproduce all observed phenomena. In the Ref. 3 we present a model for the description of measured excitation functions for the SHE production. The maximum position and the width of the excitation function for cold fusion reactions X+ 2 0 8 Pb, 2 0 9 Bi leading to elements with Z=104-112 are well described in 3 , see also Figs 1,2. Within our approach 3 the process of the SHE formation goes through three stages: (i.) The capture of two spherical nuclei and the formation of a common shape of the two touching nuclei. Low-energy surface vibrations and a transfer of few nucleons are taken into account on the first step of a reaction, (ii.) The formation of a spherical or near spherical compound nucleus. (Hi.) The surviving of an

204

136w

136w

Xe+

271. ,

Xe=

Hs+n B =6MeV

130..

Xe+

136w

Xe=

265,

Hs+n

!

10 -| 58Fe+207,208.210pb=264,265,267Hs+n

J> ^

14 B =14MeV

58,-

56,!,-

210. 210r,.

Fe+ Pb: 215

,-

-HI i 220

E

Figure 1:

20S n .

Fe+ Pb: • exp -vibr+transfer vibr -transfer WKB A—vibr+transfer - vibr+transfer 225 305

310

315

(MeV)

Calculated excitation functions for the reactions

^264,265,267Hs

+

„

a n d

130,136 X e + 1 3 6 X e ^ 2 6 6 , 2 7 1 H

s +

„_

T h e

B8

Fe +

207 208 210

.

.

c o n t i n u o u s c u r v e s h o w s

pb

t h e

results for the reaction 58Fe + 2 0 8 Pb -> 265 Hs + n taking into account both the low-energy 2+ and 3~ vibrations and the neutrons transfer channels. The dotted and the dashed curve shows the results for considering solely the 2+ and 3 _ vibrations and the neutron transfer channels, respectively. The result of the one-dimensional WKB approach is shown by the dash-dotted curve. The data obtained for the reaction 58 Fe + 207 Pb -^ 264 Hs + n are represented by (A) and those for 68Fe + 2 1 0 Pb -» 267 Hs + n by (v)- In both cases only the results including vibrations and transfer are shown. The relations taking into account the channels separately are similar as in the case of 58Fe + 2 0 8 Pb. The experimental data shown here are taken from [1,2].

excited compound nucleus during evaporation of neutrons and 7-ray emission which compete with fission. A lowering of the fission barrier was taken into account, which arises from a reduction of shell effects at increasing excitation energy of the compound nucleus. One of the heaviest systems studied experimentally over a wider range of excitation energy is 5 8 Fe+ 2 0 8 Pb -» 265 Hs + n l, the data are shown in Fig. 1. The experimental data are compared with several modifications of our model. In the simplest case, using tunneling through a one-dimensional barrier

205

70

1

Zn+OTPb: • exp vibr+transfer vibr transfer WKB '°Zn+*"Pb: —4— ™Zn+"°Pb: —<3— 2, "Zrw^Pb: —O— "Zn+ *Pb: — # -

10* \ 248

252

256

260

E c m (Mev)

Figure 2: Calculated excitation functions for the reactions 66,68,70 Zn + 207,208,2io pb _^ U 2 + n. The notation for the reactions 7 0 Zn + 207,208,2l0p b corresponds to B8 Fe + 207,208,2lo pb in Fig. 1. The insert explains the assignment of the reactions to the symbols. The experimental data shown here are taken from [1].

and the WKB method, the results strongly underestimate the experimental fusion cross-sections. Better agreement is obtained, when the neutron transfer channels from lead to iron are taken into account. Similarly, the cross-sections increase by including in the calculations the low-energy 2 + and 3~ surface vibrational excitations of both projectile and target. The best results are obtained by considering transfer and vibrations simultaneously. The value of parameters and other details are presented in Ref. 3 . In our model 3 we have adjusted and other parameters, which are taken from experimental data and other calculations. Note that we able to describe data for reactions 5 8 Fe+ 2 0 7 Pb -+ 264 Hs + n (see Fig. 1) and 5 8 Fe+ 2 0 9 Bi -» 2 6 6 Mt + n (see Fig. 11 in 3 ) by using the same fitting parameters which we fixed for reaction 5 8 Fe+ 2 0 8 Pb -» 265 Hs + n. The results of the calculations for reactions between the projectiles 66,68,702n and targets 207,208,2iop^ ^ g presented in Fig. 2. The penetration through the inner barrier (the second stage of the reaction), which taken place for these reactions during shape evolution from two touching configuration of collid-

206

ing nuclei to spherical or near spherical compound nucleus 3 , is important for these reactions, because the intrinsic excitation energies £ £ N of the compound nucleus 278 112 (E£N wlO MeV at collision energy Ecm = 253.8 MeV and EQN «12 MeV at Ecm = 255.8 MeV 1 ) are less then the value of inner barrier (jBSph « 13.5 MeV 3 ). Our calculations for reaction 70 Zn + 2 0 8 Pb -+ 277 112 + n are well agree with experimental data at collision energy 253.8 MeV and overestimated data obtained in May 2000 at energy 255.8 MeV. 2

Reactions Leading to SHE with £ = 1 1 8 - 1 2 0

Recently element 118 is formed in reaction 8 6 Kr+ 2 0 8 Pb ->• 293 118+n in Berkeley 4 . This discovery stimulates various studies of reaction mechanisms leading to SHE formation. Note that the same reaction has been studied in GSI (Darmstadt, Germany) 1 , GANIL (Caen, France) 11 and RIKEN (Wako, Japan) 1 2 , but the no events related to the formation of element 293 118 was observed. The discussion of some experimental results can be found i n x . Due to these undefinite experimental situation the theoretical studies of the reaction 86 Kr+ 2 0 8 Pb -> 293 118+n is very important. It is interesting to investigate reaction 8 6 Kr+ 2 0 8 Pb -¥ 293 118+n in the framework of our model 3 . In this part we study cold fusion reactions leading to elements 118-120. Note that the reactions leading to elements with £=118-120 could be measured on various facilities in the near future. Therefore below we discuss only new results obtained for the SHE production cross section in collisions: ( 86 Kr, 8 7 Rb, 88 Sr) + 208 Pb, and 86 Kr + 209 Bi. As pointed in Introduction, the formation of a common shape of the two touching nuclei is taken place on the first stage of reaction in our model 3 . The projectile and target penetrate through the outer barrier (capture barrier) formed by both Coulomb and nuclear ion-ion interactions. The capture stage is very important for the formation of the SHE with Z<112 in cold fusion reaction in our model. However, for heavier systems with 114
207

duction and height of the interaction potential between two separate projectile and Pb or Bi target is also found in Ref. 13 by using proximity potential and in Ref. 1 by using Bass potential. The shape of a fusing system after formation of a touching configuration is elongated, asymmetric and laced. From such a configuration the system develops further to a near spherical compound nucleus and later to the groundstate, which can be deformed or spherical. We propose that this evolution of the shape is described by smooth reduction of parameters p and q in shape parametrization 3 N

R(d)=R(p,q)[l+p

£ l=2,even

N

hYtoW+q

£

A^oW],

(1)

l=3,odd

where p and q are equal to 1 at a touching point of two spherical colliding ions and the deformation parameters Pi are fixed at a touching point. The values of the deformation parameters for SHEs in the ground-state 0. The parameters p and q (0 < p < 1, 0 < q < 1) are connected to elongated and asymmetrical degrees of freedom of a nuclear shape during the formation of the spherical compound nucleus, respectively. The radius R(p,q) depends on p and q because of the volume conservation. The potential energy surface for the nucleus 294 118 formed in reaction 86 Kr + 2 0 8 Pb is presented in Fig. 3. Although the shape parametrization (1) is rather rough at a touching point, it is possible to study the inner barrier, which has to be crossed during the process of sphere formation, as shown in Fig. 3. The potential energy surface is related to the sum of macroscopic and shell correction energies. The potential energy surface is a function of parameters p and q in the deformation space pfa,
208

Figure 3: Potential energy surface as a function of the deformation parameters /3 for cold fusion reaction 8 6 Kr+ 2 0 8 Pb—> 2 9 4 118. The touching configuration of the spherical projectile and target nucleus close to the upper right corner and that of the ground-state close to bottom left. The dash line is a tunneling trajectory which is draw by eye assuming that all deformations are monotonously changes during motion to equilibrium compound nucleus shape. The ratio between even and odd deformation parameters is fixed, see text for details. The contour lines are drawn in steps of 1 MeV, the maximum values and several others are given in MeV on graph.

correction and the centrifugal potential 3 . These two effects are very important for the correct description of SHE production cross sections. We start discussion with the excitation function calculated for reaction 86 Kr+ 2 0 8 Pb -> 293 118+n, because experimental data exist for this reaction 4 . The experimental data are compared with result of our calculations in Fig. 4. We make calculations of the SHE production cross sections for different values of the inner barrier BSph = 14, 16 and 18 MeV. The value BsPh ~ 14 MeV is close to the value obtained in other model 6 . The value -Bgph « 16 MeV is near to evaluated by using Fig. 3. The reduction of the height of the inner barrier by 2 MeV from 16 to 14 MeV or from 18 to 16 MeV increases the production cross section by the one order of magnitude. As pointed in Introduction, in our model 3 we have adjusted and other parameters, which are taken from other calculations or experimental data. Here we apply the same values of fitted parameters as for elements 112 in 3 . The other parameters are taken from 14 ' 15 ' 16 ' 17 . The values of static fission

209 1000 "W"Pb-""l18+n • exp B^.IBMeV B^-MMeV •«-••-B^ISMeV Ref. [5]

100 •:

"W-Bi^ng+n — o — B >16MeV

O. 0) C

10-:

-

o

t)

d)

co en c/) o

D -

B

=14MeV

"D"Bspt.=18 —Q—Ref. [5]

M e V

"Rb + O T Pb. i ! M 119 + n — a — B -16MeV - A - B -14MeV

1i

o

•"A"'Bsp»=18MeV — A — R e f . [5]

0.1

"Sr+MPb-2"5120+n —0— B -16MeV - O - B =14MeV '"°"Bsp»=18MeV

0.01

I • " • • I • 340 345

i

350

E (MeV)

Figure 4: Calculated excitation functions for reactions 208Bi _>294119

+

n )

87Rb

+

208p b _ ^ 2 9 4 u 9

+

„

a n ( J

86

K r + 2 0 8 P b - f 2 8 3 1 1 8 + n, 8 6 K r +

8 8 S r + 2 0 8 p b __> 1 9 5 1 2 0 + n

A

decreasing

(increasing) of the barrier Bsph by 2 MeV (see text) increases (decreases) the cross-section considerably. The experimental data obtained for the reaction 8 6 K r + 2 0 8 P b -+ 2 9 3 118 + n in [4] are represented by rilled dot. The excitation functions for reactions 8 6 K r + 2 0 8 P b -+ 2 9 3 118 + n, 8 6 K r + 2 0 8 Bi -> 2 9 4 119 + n, 8 7 R b + 2 0 8 P b - > 2 9 4 U 9 + n from the Ref. [5] are also presented.

barrier for the even-even compound nuclei are taken from 5 . We evaluate fission barriers for odd and odd-odd nuclei by using fission barriers for the nearest even-even nuclei. The parameters used in our calculation for this reaction and other reactions considered below are listed in Table 1. The good description of the experimental cross section is obtained for reaction 8 6 Kr+ 2 0 8 Pb ->• 293 118+n in calculation with B S p h = 16 MeV in Fig. 4. The result of calculation for this reaction obtained in Ref. 5 is also presented in Fig. 4. The excitation function obtained in Ref. 5 overestimates experimental data and has larger width than that obtained in our model. The cross section 0.0051 pb for reaction 8 6 Kr+ 2 0 8 Pb -> 293 118 + n obtained in Ref. 7 underestimates strongly experimental value 2.2lgg p b 4 . The production cross sections

210

0.5 pb for reaction 8 6 Kr+ 2 0 8 Pb ->• 293 118 + n evaluated in Ref. 8 is close to the experimental data. Now we consider reactions leading to elements 119-120. The same isotope of element 119 may be formed in two different reactions 8 6 K r + 2 0 9 B i _> 2 9 4

U 9 + n

a n d

8 7 R b + 2 0 8 p b _> 2 9 4

n g + n

T h e r e f o r e )

it

is

v e r y

important for future experiments to know which reaction has a larger cross section. We present results for both reactions in Fig. 4. The cross sections of both these reactions are similar, because the SHE production is determinated on the second and third stages of the reaction. We make calculation for the same values of inner barrier as for reaction 8 6 Kr+ 2 0 8 Pb —>• 293 118+n. The compound nuclei formed in collisions 8 6 Kr+ 2 0 9 Bi and 8 7 Rb+ 2 0 8 Pb are the same, therefore the fission, neutron and 7 decay properties are the same for both reactions at the equal excitation energies of compound nuclei. The

Table 1: The parameters of ground-state and saddle-point properties of the compound nucleus (CN) and of evaporation residues (SHBs, after neutron emission). E^ is the binding energy of the CN [16], SE^ ( B C N ) and SEf^f ( B S H E ) are the ground state shell corrections (fission barrier heights) of the CN and SHE respectively, /32gS and /32Sadl are the ground state and saddle point deformations of the CN and SHE, iS^jjJjp is the neutron separation energy in the CN, Tsf is the spontaneous-fission lifetime of the CN in the ground-state, 7 is the shell-correction damping parameter [3] and cv is the strength factor of the nuclear heavy ions interaction potential [3].

Colliding ions CN £n b e (MeV) CN

«•£?, (MeV) /?2gs

B C N (MeV) #2sadl

Eg% (MeV) log wTst1 SHEs 6E™f (MeV) BSHE

(MeV)

7 (MeV)" 1 cv x

The TSf is in sec.

86

Kr+ 2 0 8 Pb 294llg

2081.80 -7.67 0.070 5.45 0.375 7.72 3.13 293 118 -7.82 5.18 0.13 1.05

87

Rb+ 2 0 8 Pb 294

86

Kr+ 2 0 9 Bi

88

Sr+ 2 0 8 Pb

118 -2081.48 -7.67 0.0735 5.30 0.3675 7.67 0.985

294llg

294118

-2081.48 -7.67 0.0735 5.30 0.3675 7.67 0.985

-2081.85 -6.77 0.07 5.15 0.3675 7.73 -1.16

294ng

294llg

295120

-7.89 5.125 0.13 1.05

-7.89 5.125 0.13 1.05

-7.25 5.0 0.13 1.05

211

difference between two these reactions is related only to binding energies of colliding ions. Due to this, maxima of excitation functions for these reactions occur at different collision energies, see Fig. 4. The production cross sections for reactions 8 6 Kr+ 2 0 9 Bi - • 2 9 4 119+n and Rb+ 2 0 8 Pb - • 294 119+n obtained in Ref. 5 are again much lager than results of our model, see Fig. 4.

87

The models described in Refs. 3>5>7>8>9 assume very different mechanisms of the SHE formation. Note that all these models describe well production cross sections for the SHEs with Z< 112 in cold fusion reaction. But the models in 3>5>7'8'9 predict very different results for the SHE with Z > 114. Such a difference is not surprising. The models discussed in Refs. 7 ' 8 ' 9 use a specific trajectory of the compound nucleus formation related to the concept of a dinuclear system. Within this concept the shape evolution from two touching nuclei to a spherical compound nucleus takes into account only asymmetric shape degree of freedom near touching point, because colliding ions stop near touching point, and afterward the subbarrier multinucleon transfer from projectile (light ion) to target (heavy ion) takes place. Such a mechanism of the SHE formation does not take into account the neck formation and smooth evolution of both symmetric and asymmetric shape degree of freedom as suggested in Fig. 3. Note, that the subbarrier multinucleon transfer at touching point considered in 7 ' 8 ' 9 should takes a long time due to high value of the barrier, but the existence of a dinuclear system during such a long tunneling process is questionable. The model proposed in Ref. 5 ignores both nuclear interaction between ions during the collision and shape evolution processes. The excitation functions of cold fusion reactions leading to element 120 obtained in our model for three different values of the inner barrier are also shown in Fig. 4. The tendency observed for reactions leading to elements 118 and 119 are also kept for reaction 8 8 Sr+ 2 0 8 Pb - • 195 120+n, see Fig. 4. The production cross section for the element 120 is smaller than for elements 118-119 due to smaller values of both fission barriers and ground state shell corrections, see Table 1. The main difference between reactions considered in this section is related to the properties of colliding nuclei, binding energies and decay properties. Unfortunately, the estimation of the inner barrier penetration is not very precise in our model. Due to this we can not see any difference associated with the inner barrier penetration process in various reactions.

212

Figure 5: Potential energy surface as a function of the deformation parameters 02 and 04 for cold fusion reaction 1 3 0 X e + 1 3 6 X e - » 2 6 6 H s . The touching configuration of the spherical projectile and target nucleus is close to the bottom right corner and that of the ground-state is close to 02 « 0.2 and 04 « —0.05. The ratio between deformation parameters 02, 03, 05, 06, 07, 08 and 0g is fixed at the touching point. The contour lines are drawn in steps of 2 MeV.

3

Production of SHE in Near Symmetric Heavy Ion Collisions

It is also possible to produce SHE in collisions of similar nuclei. For example, the nuclide 265 Hs can be formed in both reactions 58 Fe + 2 0 8 Pb -> 265 Hs + n and 130 Xe + 136 Xe —^265Hs + n. The excitation functions for reactions i30,i36 Xe + i36 X e _^265,27iHs + n obtained in our model for different values of the inner barrier Bsph = 6 MeV and 14 MeV are presented in Fig. 1. We can describe the touching configuration and shape evolution of near symmetric systems by using the shape parametrization 9

R(0)=R(p,q)[l+p

£

PiYtQ(4)+qfoYm{4)],

(2)

where p = q = 1 at the touching point of two spherical colliding ions and the deformation parameters Be, are fixed at the touching point. The potential energy surface for the reaction the 130 Xe+ 136 Xe-+ 266 Hs is presented in Fig. 5.

213

B.n=6MeV ,M

Sn+

,36

Xe=259Rf+n

S '->'] = o T j + 2 0 8 p b = : 5 7 R f + n

•''

*

4

i

4-''-

;

:

l

, ' B -14MeV

I

v '"

exp vibr+transfer -vlbr - - transfer - WKB 185

190

E

285

290

(MeV)

Figure 6: Calculated excitation functions for the reactions 6 0 T i + 2 0 8 P b -> 2 5 7 Rf + n and 124 Sn + 1 3 6 Xe -> 2 5 9 Hs + n. The experimental data shown here are taken from [1]. Reaction 5 0 T i + 208p b _>.257Rf + n i s discussed in detail in Ref. [3].

As mentioned before we made calculations for the synthesis of the SHE using near symmetric reactions for two different values of the inner barrier. The value of the inner barrier J3sPh = 6 MeV is close to adiabatic fission barrier, see Fig. 5. Note that the potential energy surface for 266 Hs in Fig. 5 has a strong slope from touching point (bottom right corner) to the quasi-fission direction (upper right corner). It is possible to evaluate the exact value of the inner barrier and the branching ratio between compound nucleus formation and quasifission processes by studying the quantum dynamical shape evolution from two touching nuclei to the near spherical compound nucleus. Unfortunately such calculations are not available now. Therefore we also made calculations for a higher value of the inner barrier f?sph = 14 MeV, which may simulate a stronger competition between SHE formation and quasi-fission. Note that the SHE production cross sections for near symmetric reactions is mainly related to shape evolution stage, because the height of the outer barrier (capture barrier) is relatively small (see also 1 3 ). Experimental studies of SHE formation in near symmetric collisions may be started by using the reaction 124 Sn + 136 Xe-» 259 Rf + n, because it is easy to substitute 2 0 8 Pb by 124 Sn as target. It will interesting to check our estimates, because even for the inner barrier 2?sPh = 14 MeV the SHE formation cross section obtained in our model for near symmetric reactions is larger then for cold fusion reaction leading to similar isotopes, see Figs 1,6.

214

4

Conclusions

The theoretical description of reaction processes leading to fusion of SHEs is a difficult task. The nuclei are located at the edge of stability, and the reactions are dominated by shell structure effects in projectile, target as well as in compound nucleus. In addition, shell structure is required for the calculation of the binding energy at large deformation in order to determine the inner barrier, fission barrier and shape evolution. Acknowledgments The author would like to thank S. Hofmann, V. Ninov and W. Norenberg for useful discussions. He acknowledges gratefully support from GSI. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

S. Hofmann, G. Miinzenberg, Rev. Mod. Phys. 72 733 (2000). V. Ninov, (private communications). V.Yu. Denisov, S. Hofmann, Phys. Rev. C61, 034606 (2000). V. Ninov, et al., Phys. Rev. Lett. 83, 1104 (1999). R. Smolanczuk, Phys. Rev. C56, 812 (1997); 59, 2634 (1999); C60, 021301 (1999); C 61, 011601 (R) (2000). P. Moller, et al., Z. Phys. A359, 251 (1997), and private communications. G.G. Adamian, et al., Nucl. Phys. A633, 409 (1998); G.G. Adamian,et al., LANL-Preprint-nucl-th/9911078. E. Cherepanov, Pramana 53, 619 (1999). G.S. Giardina, et al., Eur. Phys. J. A 8 , 205 (2000). Y. Aritomo, T. Wada, M. Ohta, Y. Abe, Phys. Rev. C59, 796 (1999). J. Peter, Talk on Int. Workshop Fusion Dynamics at the Extremes, Dubna, Russia, May, 2000. K. Morita, Talk on Int. Workshop Fusion Dynamics at the Extremes, Dubna, Russia, May, 2000. W.D. Myers, W.J. Swiatecki, Phys. Rev. C62, 044610 (2000). G. Audi, A.H. Wapstra, Nucl. Phys. A595, 409 (1995). W.D. Myers, W.J. Swiatecki, Nucl. Phys. A601, 141 (1996). P. Moller, et a l , At. Data Nucl. Data Tables 59, 185 (1995). S. Raman, et al., At. Data Nucl. Data Tabl. 42, 1 (1989); R.H. Spear, At. Data Nucl. Data Tabl. 42, 55 (1989); M A . Kennedy, et al., Phys. Rev. C46, 1811 (1992); J.P.M.G. Melsen, et al. Nucl. Phys. A376, 183 (1982); M.J. Martin, Nucl. Data Sheets 63, 723 (1991).

215 FUSION-FISSION DYNAMICS OF T H E SYNTHESIS OF SUPER-HEAVY NUCLEI

V.I. ZAGREBAEV Flerov Laboratory of Nuclear Reaction, JINR, Dubna, Moscow Region, Russia Abstract Fusion-fission dynamics in the reactions leading to formation of Very heavy atomic nuclei is studied. A systematic analysis of the heavy ion fusion reactions is performed within a "standard" theoretical approach without any adjustable parameters and additional simplification. Good agreement with experimental data was obtained in all the cases up to synthesis of the 106 element. A process of the compound nucleus formation, starting from the instant when two heavy nuclei touch and proceeding in strong competition with the fission and quasi-fission processes, plays an important role in the asymmetric synthesis of super-heavy elements with ZCN > 106 as well as in the symmetric fusion at ZCN ~> 90. A new mechanism of the fusion-fission process for a heavy nuclear system is proposed, which takes place in the {A\,A2) space, where Ai and Ai are two nuclei, surrounded by a certain number of common nucleons A A These nuclei gradually lose (or acquire) their individualities with increasing (or decreasing) the number of collectivized nucleons AA. The driving potential in the (Ai,A2) space is derived, which allows the calculation of both the probability of the compound nucleus formation and the mass distribution of fission fragments in heavy ion fusion reactions.

1

Introduction

T h e interest in the problem of t h e synthesis of super-heavy atomic nuclei quickened significantly within the p a s t two years. First of all, it is connected with successful D u b n a experiments on the synthesis of the 114 element isotopes with A = 2 8 8 , 289 1 a n d A = 2 8 7 2 . T h e decay chains of these isotopes d e m o n s t r a t e t h a t we have really approached the so-called "island of stability". Shortly after these experiments, the detection of nuclei with Z=118 was announced a t Berkeley in the 8 6 K r + 2 0 8 P b fusion reaction with a n unexpectedly large cross section 3 . As a result, two other laboratories (RIKEN a n d GANIL) joined t h e well-known centers in the synthesis of super-heavy elements (Berkeley, D a r m stadt, and D u b n a ) . Detailed information on t h e synthesis of super-heavy elements (SHEs) including the latest discoveries and the current s t a t u s of t h e problem can be found in the review a r t i c l e 4 . Today at all t h e above mentioned laboratories either new experiments or intensive preparatory work are

216

in progress. Theoretical support of these very expensive experiments is vital. In this connection, one should recognize the fact that we are still far from final understanding of the heavy ion fusion process. The fusion dynamics undergoes significant changes with increasing masses of compound nuclei, and the formation cross sections decrease very fast with increasing their atomic numbers. The main reason for that is the growing role of the fission channels determining not only the survival probability of a compound nucleus in the process of its cooling (emission of nucleons and 7rays), but also the dynamics of its formation in competition with the so-called quasi-fission process. The physics nature of the whole process of interaction of two heavy nuclei leading to formation of a heavy evaporation residue or two fission fragments is very complicated even at low near-barrier energies. It has been studied rather poorly, both experimentally and theoretically. As a result, not very numerous theoretical approaches to the description of the synthesis of SHEs differ from each other not only quantitatively (several orders of magnitude in the estimation of the cross sections of the same processes) but, sometimes, qualitatively, namely when contradictory physics models are used. The formation cross section of a cold residual nucleus B, which is the product of light particle evaporation and 7 emission from an excited compound nucleus C, formed in the fusion process of two heavy nuclei Ai + A2 —> C —• B + n,p, a, 7 at center-of mass energy close to the Coulomb barrier in the entrance channel, can be decomposed over partial waves and written in the following form

A B

ap+ ^ (E)

ft2 °° « f - J2 (2/ + 1)T(25,0 • Zft£l

PCN(E,

I) • PBR(C

- • B; E\l).

(1)

1=0

Here T(E, I) is the probability for colliding nuclei to overcome the potential barrier in the entrance channel and reach the point of contact RCOnt — Ri + RiPCN is the probability that the nuclear system will evolve from a configuration of two touching nuclei into a spherical or nearly spherical form of the compound mono-nucleus. In the course of this evolution the heavy system may, in principle, fall again into two fragments without forming the compound nucleus (quasi-fission) and, thus, PCN < 1- The last term in (1), PBR(C -4 B), defines the probability of producing the cold evaporation residue B in the process of the compound nucleus C decay. It has the initial excitation energy E* = E — Qgg8, where E is the beam energy in the center-of mass system, Q/«« = M{C)c2-M(A1)c2-M(A2)c2,a,ndM(C), Af(Ai) , M(A2) are the nuclear masses. In order to avoid hereinafter a confusion in terminology, we define

217

also the "capture cross section" as follow acapt{E)

= | ^ £ (2/ +

l)T(E,l).

Approximate equality in (1) reflects the fact that the whole process of the compound nucleus formation and decay is divided here into three individual reaction stages even if connected with each other but treated and calculated separately: (1) approaching the point of contact Ri + i? 2 < r < oo, (2) formation of the compound mono-nucleus A\ + A2 ->• C, (3) decay ("cooling") of the compound nucleus C. Note that different theoretical approaches are used for analyzing all the three reaction stages. However, the dynamics of the intermediate stage of the compound nucleus formation is the most vague. It is due to the fact that in a well studied case of near-barrier fusion of light and medium nuclei, when a fissility of a compound nucleus is not so high, the fusing nuclei overcoming the potential barrier form a compound nucleus with a probability close to unit, i.e., PQN = 1, Ofus = crcapt, and, thus, this reaction stage does not influence the yield of the evaporation residues at all. But in the fusion of heavy nuclei it is the channels of fission (normal and "fast") that determine to a certain extent the dynamics of the whole process, which is rather poorly studied, and that is why very much different models, sometimes opposite in their physical meaning, are used for its description. In this connection, one may single out two mutually exclusive approaches to the description of the evolution of the nuclear system starting from the moment at which two colliding nuclei touch each other and up to the moment of formation of a spherical compound nucleus or the moment of decay into two more or less equal heavy fragments (quasi-fission process). In the first approach 5,6,7 it is assumed that two touching nuclei instantly and completely lose their individualities and can be treated as one strongly deformed mononucleus which evaluates in the multi-dimensional space of deformations into a spherical compound nucleus or goes into fission channels. A similar ideology was also used in 8 , 9 for the description of the intermediate reaction stage of the compound nucleus formation in specific calculations of the cross sections for SHE production. An opposite approach has been proposed and used in 10>n>12 Here, two nuclei having passed the Coulomb barrier reach the point of contact and after that remain in this position keeping entirely their individualities and shapes. Only nucleon transfer causes subsequent evolution of the "di-nuclear system". Compound nucleus formation means complete transfer of all the nucleons from the light nucleus to the heavier one. This process competes with the nucleon transfer from the heavy nucleus to the lighter one, resulting in a subsequent

218

separation of two nuclei (quasi-fission process). The truth seems to be somewhere in the middle. It is improbable that during the whole evolution of the system starting from the touching of two nuclei and up to the formation of the almost spherical compound nucleus, all the nucleons were strictly divided into two groups, namely, the nucleons belonging only to one nucleus and moving only in the volume of that nucleus, and those belonging to another nucleus and also remaining within its volume. The process of instantaneous nucleon collectivization and formation of one very strongly deformed mono-nucleus at the moment of contact of two colliding nuclei also looks unlikely to take place. In this paper a new mechanism of compound nucleus formation is proposed. It is assumed, that a certain number of common nucleons appear when two nuclei get in contact. These nucleons move within the whole volume occupied by the nuclear system and belong to both nuclei. Henceforth the number of such collectivized nucleons increases whereas the number of nucleons belonging to each particular nucleus decreases. The compound nucleus is formed at the instant when all the nucleons find room in the volume of that nucleus. The inverse process of nucleon de-collectivization brings the system to the fission channels. 2

The stage of approaching and the capture cross section

In fact many difficulties arise both in the calculation of PCN in (1) and in the calculation of other factors. Now it is well established that in the fusion of heavy ions the barrier penetrability T(E,l) is defined not only by the height and width of the Coulomb barrier but also by the strong channel coupling of relative motion with internal degrees of freedom, which enhances significantly (by several orders of magnitude) the fusion cross section at sub-barrier energies (see, e.g., the review article 13 ). The Bass approximation of potential energy of the interaction between two heavy spherical nuclei 14 is widely used and reproduces rather well the height of the potential barrier. Coupling with the surface vibrations and nucleon transfer channels is the second main factor which determines the capture cross section at near-barrier energies 13 . In the case of rather "soft" nuclei (low values of £ 2 + ) a realistic nucleus-nucleus interaction leads to very large deformations and, thus, to a necessity of taking into account a large number of coupled channels 15 , which significantly complicates the microscopic calculation of T(E,l) and makes it unreliable. In order to take into account explicitly the effect of a decrease in the height of the potential barrier and, therefore, an increase in the penetration probability at sub-barrier energies due to dynamic deformation of nuclear surfaces, we

219

r (fa) 18 200

C)

s

5

" ^

— • B0

// t

\ X.

S" 150

,

B 0 =179MeV

Bsi=16ZM6V

!

F
r (fin)

"

u

p

Psd

n

Figure 1: Potential energy of 4 8 C a + 2 0 8 P b . Proximity potential is used for the nuclear interaction (ro = 1.15 fm, 6 = 1.0 fm), and the standard stiffness parameter is used for the deformation energy, (a) Landscape of potential surface. The incoming flux is shown schematically by the grey-shaded arrow, (b) Interaction potential of spherical nuclei and its parabolic approximation (dashed line) in the vicinity of the barrier, (c) Potential energy at the top of the two-dimensional barrier, i.e. along the dotted line passing through the saddle point (up panel).

use here the proximity potential 16 for nuclei with quadrupole deformations in a nose-to-nose geometry. A characteristic topographical landscape of the total (Coulomb, nuclear, and deformational) potential energy of the nucleus-nucleus interaction in the (r,/?)-space is shown in Fig.la, here j3 = Pi + 02- Approximating the radial dependence of the barrier by a parabola (see Fig. lb), one can use the usual Hill-Wheeler formula 17 with the barrier height modified to include a centrifugal term for the estimation of the quantum penetration proba b i l i t y of a one-dimensional potential barrier. Taking into account now a multi-dimensional character of the realistic barrier, we may use the "barrier distribution function" 18 f(B) to determine its total penetrability

T(E,l) Here

TWJB

= ffW-+ exp( s ^ 7J [s + ^£wZ(/ + l)-JB]) dB

is defined by the width of the parabolic barrier,

RB

(2)

defines a position

220

of the barrier, and the barrier distribution function satisfies the normalization condition J f{B)dB = 1. At an accurate measurement of the capture cross section acapt(E) this function can be determined experimentally 13 . In other cases we may use only available experimental experience and theoretical analysis of model systems (see, e.g. 1 5 ) . Here the asymmetric Gaussian approximation of this function was used centered at Bm — (Bo + Bs)/2, where BQ is the height of the barrier at zero deformation and Bs is the height of the saddle point. 3

Statistical decay of low excited heavy nuclei

Survival probability of the excited compound nucleus C(E*,J) in a process of cooling by means of neutron evaporation and 7-emission in the competition with fission and emission of light charged particles - A\ + A2 —> C —> B + xn + N^f - can be calculated within a statistical model of atomic nuclei 19 . The partial decay widths of the compound nucleus for the evaporation of the light particle a ( = n,p, a,...), emission of 7-rays of multipolarity L, and fission were calculated in a usual way without any simplification and using a formula for the level density PA (E*, J) proposed in 20 along with rotational enhancement factor proposed in 21 . An estimation of the total probability for the formation of the cold residual nucleus after the emission of x neutrons - C ->• B + xn + N7 - can be performed either with a simplified expression 22 PER{C -> B + xn) = ft ( i w l R ) fc=l

V

toU

k,

• Pz{E*), where Ttot = E r a + r / 4 , + T 7 is the total decay

'k

a

width, and Px (E*) is the emission probability of exactly x neutrons by the excited compound nucleus, or within numerical calculations based on the analysis of the multi-step decay cascade 2 3 , 2 4 , 2 5 . Here an explicit analytic expression is used for such probability, which takes into account directly the MaxwellBoltzmann energy distribution of evaporated neutrons PEn(C->B J Q

+ xn) =

]K(£0*,J0)P„(^o,ei)^i-1- ZOt

/ Q

-1 tOt

^f-(E*1,J1)Pn(E*1,e2)de2

J

£»-(EZ_1,Jx-i)Pn(EZ_1,ex)'GNy{EZ,Jx->g.8.)dex

Q

*• tOt

(3)

Here E^ep(k) and e^ are the binding and kinetic energies of the kth evaporated k

neutron, El = EQ - J2 [E™p{i) + ei]

1S t n e

excitation energy of the residual

«=i

nucleus after the emission of fc neutrons, Pn(E*,e)

= Cy/eexp(—e/T(E*))

is

221

the probability for the evaporated neutron to have energy e, and the normalE*-E"p

izing coefficient C is found from the condition

J Pn(E*,e)de = 1. The o quantity GJV T defines the probability for the remaining excitation energy and angular momentum to be taken away by 7-emission. 4

"Standard approach" - the borderlines of applicability

As already mentioned above, in comparatively light systems the formation of a compound nucleus occurs with a probability close to unit straight after overcoming the Coulomb barrier. Let us call this approach "standard", when in the calculation of the cross section of the evaporation residue formation expression (1) is used with PCN = 1. In this Section the standard approach is applied to the analysis of available experimental data on the synthesis of very heavy fissile nuclei in order to find the borderlines of applicability of this approach, i.e., to find the cases in which the intermediate reaction stage, i.e., the competition between compound nucleus formation and quasi-fission after two colliding nuclei touch, plays an important role and significantly decreases the yield of the super-heavy nuclei. To avoid adjustment of the calculated and experimental data by playing with parameters, the same scheme of the calculation of T(E, I) and PER(C -> B + xn) described above was used in all the cases. Besides the neutron evaporation, 7-emission and fission, the evaporation of protons and a-particles was also taken into account in the calculation of the total decay width Ttot used in the neutron cascade. Experimental nuclear masses 26 were used to determine the separation energies of all the light particles. The fission barriers of formed nuclei Bfis(A; E*, J) are the most important and most uncertain parameters of the calculation. Theoretical estimations of the fission barriers for the region of super-heavy nuclei are not very reliable yet and significantly differ from each other. To make the systematical analysis consistent, the liquid drop fission barriers 2 7 and shell corrections 28 obtained within close approaches were used in all the cases considered here. The capture cross sections acapt and formation cross sections of the evaporation residues (TBR in the reactions 1 6 O+ 2 0 8 Pb, 1 2 C+ 2 3 8 U, 4 8 Ca+ 2 0 8 Pb, and 58 Fe+ 2 0 8 Pb are shown in Figures 2,3. One can see that the standard approach with an accurate enough calculation of all the quantities describes quite satisfactorily the experimental data obtained both in the "cold" and "hot" fusion. Consideration of the dynamical deformation of nuclear surfaces allows one to reproduce correctly the capture cross section in the sub-barrier region. Note that with an increasing mass of the projectile the dynamical deformation of

222

Figure 2: The capture cross section (o) and formation cross sections for evaporation residues in the 1 6 O + 2 0 8 P b and 1 2 C + 2 3 6 U reactions. For the first reaction experimental data are f r o m 2 9 , the lowest point at E * = 1 6 MeV is from 3 0 . For the second reaction experimental data are from 3 1 (capture cross sections) and from 3 2 (evaporation residues). The dotted curves show the capture cross section calculated without dynamical deformations of nuclei, i.e., with f(B) = S(B — Bo) in (2). By the arrows are shown the positions of the Coulomb barrier at zero deformation, the Bass barrier, and the saddle point.

nuclear surfaces acquires more and more importance, the difference Bo — B$ is only 4 MeV in the case of 1 6 O+ 2 0 8 Pb and almost 18 MeV in the case of 48 Ca+ 2 0 8 Pb. Satisfactory agreement of the standard approach with experimental data was obtained also for the 5 0 Ti+ 2 0 8 Pb-> 2 5 8 Rf and 54 Cr+ 208 Pb-s> 262 Sg reactions not shown here. However, already for the element with Z=108, synthesized in the reaction 58 Fe+ 208 Pb->- 266 Hs, the standard approach significantly overestimates the cross section for the yield of evaporation residues (Fig. 3) if one uses the fission barriers calculated on the basis of shell corrections taken from 2 8 . There are two possible reasons for such overestimation. (i) Neglecting the intermediate reaction stage of the compound nucleus formation in competition with quasi-fission, i.e., a necessity of calculating and taking into account the factor PCN < 1 in the total cross section (1). (ii) Overestimation of the fission barriers of super-heavy nuclei in the calculation of the survival probability PBR{C ->• B + xn). Starting from ZCN=W6 the shell corrections given in 2 8 begin to increase due to approaching the magic shell in the region of Z=114, 7V=184, whereas the neutron separation energies

223

E* (MeV)

E* (MeV)

Figure 3: The capture cross section (o) and formation cross sections for evaporation residues in the 4 8 Ca+ 2 0 8 Pb and 5 8 Fe+ 2 0 8 Pb reactions. For the first reaction experimental data are from 33 (capture cross sections) and from 34 (cross sections of the i n channels). For the second reaction ER cross sections are from 36 .

do not decrease at least in the fusion reactions induced by the stable projectiles and targets. If the static fission barriers are defined directly by the ground state shell correction energies, as made here and in many other papers, then the survival probabilities, as calculations show, stop decreasing with increasing ZCN at ZCN > 106, while the experiments demonstrate a systematic decrease in the yield of the super-heavy evaporation residues with increasing ZCN 36Both the high probability for the system to go into the quasi-fission channels and decreasing the height of the real fission barriers, in spite of the large values of the shell correction energies near the magic shells, could be the reasons for that. To make finally sure that the stage of the compound nucleus formation and the quasi-fission process should be considered much more carefully in the synthesis of super-heavy nuclei, the symmetric fusion reactions leading to the heavy fissile compound nuclei, fission barriers of which are known much better, were also analyzed within the standard approach. Calculated and experimental cross sections for the yield of evaporation residues in the reactions ioo M o + ioo M o _ > 20o P o ( 5 ^ = 1 2 . 2 MeV), 100 Mo+ 110 Pd-s- 210 Ra {Bfia=9.8 MeV), and 1 1 0 Pd+ 1 1 0 Pd-> 2 2 0 U (Bfi8=5.2 MeV) are shown in Fig. 4. For the second and, especially, for the third reactions the calculated cross sections noticeably

224

Tso

190

200

210

220

230

240

260

Figure 4: The capture cross sections (solid lines) and formation cross sections for evaporation residues (dashed lines) in the reactions 1 0 0 M o + 1 0 0 M o - + 2 0 0 P o , 1 0 0 M o + 1 1 0 P d - > 2 1 0 R a , and l l O p d + i i o p d ^ . 2 2 0 u Experimental data are from 3 6 .

overestimate the experimental data at low near-barrier energies and rather well agree with experiments at higher energies, in the region of the 5n channel and higher. It means, that the survival probabilities are calculated quite accurately for these cases and an additional decrease in the experimental cross sections at low incident energies is most probably due to a reluctance of the two touching heavy nuclei close in masses to form a compound nucleus. They prefer to go into the initial channel or into some fission channels close to the entrance one. Note that in the case of 1 0 0 Mo+ 1 0 0 Mo a preferable fission channel of the compound nucleus - 200Po—• 9 0 Sr+ 1 1 0 Pd - needs some rearrangement of the initial di-nuclear system, whereas in the 1 1 0 Pd+ 1 1 0 Pd reaction the compound nucleus decays exactly into the same channel in which it was formed: 220U_j,110pd+110pd.

Thus, we may conclude with much certainty that in heavy ion fusion reactions the competition between the process of the compound nucleus formation and the process of quasi-fission, starting from the instant when two nuclei touch, plays an important role at ZQN > 106 in collisions of asymmetric nuclei, and already at ZCN > 90 in extremely symmetric combinations of colliding nuclei. For symmetric combinations this competition is especially noticeable at slow collisions, i.e., at near-barrier energies. 5

Nucleon transfer in heavy ion fusion reactions

To understand clearly the mechanism of the nucleon transfer and collectivization in heavy ion collisions we analyzed the behavior of nucleons during the stage of approaching within a simplified 4-body classical model consisting of

225

r (fin)

100 MO

200 (s)

Figure 5: (a) The interaction potential and relative motion trajectory for the collision of 48 C a and 2 4 8 C m at the energy jB c m =230 MeV. (b) Probabilities of the valence neutron transfer (dashed lines) and neutron collectivization (solid lines).

two heavy nuclear cores and two valence nucleons, one inside each of the nuclei at the initial moment. Realistic Woods-Saxon potentials were used for the nucleon-nucleus interaction, and the proximity potential along with phenomenological dissipative forces were used for the nucleus-nucleus interaction. Here the time evolution of the system and the probability of nucleon collectivization at different collision stages were studied. This probability can be defined in the following way. Let Ntot is the number of all the events with randomly chosen initial configurations of colliding nuclei at fixed separation energies and angular moments of nucleons and at a given initial energy of relative motion. Let by the moment t in the case of the ANi events the nucleon has passed from the projectile into the target and is inside it, and in the case of the A./V2 events the nucleon has passed from the projectile into the target but has returned, i.e., it has crossed at least twice the surface of the projectile. Then Ptr{t) = ANi(t)/Ntot is the probability of the nucleon transfer, and Pcoii(t) = AN2(t)/Ntot is the probability of the nucleon collectivization. As an example the fusion of 4 8 Ca with 248 Cm was studied at a near-barrier energy. One of the trajectories in the (r, £ c m )-space and time evolution of the probabilities for transfer and collectivization of the projectile and target valence neutrons are shown in Fig. 5. As the calculations show the probability of nucleon collectivization begins to increase immediately after overcoming the Coulomb barrier, and after the contact between the nuclear surfaces it rapidly reaches the value close to unit in the case of the nucleons of the light nucleus and a slightly less value in the case of the heavier partner nucleons. Later all the valence nucleons are moving in the volume of both nuclei, whereas the internal nucleons with lower energies remain in the volumes of original nuclei. Thus, basing on the model calculations, we may conclude that the concept

226

of a "di-nuclear system" in which two touching nuclei keep their individualities during compound nucleus formation 10 ' 11 ' 12 seems to be too simplified. 6

Collectivization and de-collectivization of nucleons as a mechanism of fusion and fission of heavy nuclei

The following mechanism can be proposed as an alternative concept of the compound nucleus formation in competition with the quasi-fission process. (1) Down to the instant of touch the nuclei keep their individualities and the potential energy of their interaction is defined in a usual manner as described above in Sec. 2. The point of contact can be defined as the sum of nuclear radii which is smaller by 1 -J- 3 fm than the radius of the Coulomb barrier and, thus, the nuclei have to overcome this barrier to reach it. (2) In the point of contact the nuclei begin to lose their individualities due to an increasing number of common nucleons A A, here Ai + A2 + A A = ACNInteraction of two touching nuclei Ai and A2 weakens with increasing the number of common nucleons AA, and their specific binding energies approach a specific binding energy of the compound nucleus. Collectivized nucleons move in the whole volume occupied by the two nuclei and have the average over Ai and A2 specific binding energy. (3) The process of compound nucleus formation in competition with quasifission occurs in the space (A± ,A2), here the compound nucleus is finally formed when two fragments A\ and A2 go in its volume, i.e., at R{A\) + R{A2) = R(ACN) = RCN or at .A/ + A2 — AjN. Let us denote these values as AfN N and A§ , see Fig. 6. For calculating the total energy of the nuclear system consisting of two nuclei surrounded by a certain number of common nucleons, the following expression can be used based on the concept formulated above and providing a continuity of the total energy at all the reaction stages beginning from the asymptotic state of two separate nuclei and up to the moment of the compound nucleus formation Vfus-fis{r VgN{r,/?,

= R(A1)+R(A2),P;A1,A2,AA)

=

AA) + B(A1) + B{A°2) - [k • Al + b2 • A2 + lCN • AA] .

(4)

Here B{A\) and B{A2r) are the binding energies of the projectile and target; bi, b2, and bcN = (bi+b2)/2 are the specific binding energies of the nucleons in the fragments A\, A2, and that of the common nucleons AA, respectively. These quantities depend on the number of collectivized nucleons. At the border line AA = 0, i.e., at Ax + A2 = ACN (see Fig. 6) b1<2 = B(A? 2 )/A? 2 = &i,a. At

227

Figure 6: Schematic view of the process of compound nucleus formation and fission in the space of A\, A2 and AA. 1 /3

1 /3

1 /3

the moment of the compound nucleus formation, i.e., at A^ + A2' < AQN (the dark area in Fig. 6) the specific binding energy of all the nucleons is the same and equal to the specific binding energy of the compound nucleus: bx = b2 — bcN = B(ACN)/ACN

= bcN- Introducing the notations AACN

=

~ A^N - A$N (see Fig. 6) and x = (AACN ~ AA)/AACN which is the parameter characterizing the remoteness of the system from the compound nucleus state, one can use a continuous approximation of &i and 62 in the intermediate region 0 < x < 1 in the following form ACN

&1,2 = bCN + (&1,2 ~ bCN) •
(5)

where
228 V f u a ^ R c n t ) <MeV)

10

20

30

40

W

BO

60

70

BO

BO

100

110

z2 Figure 7: The driving potential Vfus_fis{Zx,Ai,Z2,A2) in the nuclear system consisting of 116 protons and 180 neutrons, (a) Potential energy of two touching nuclei at A\ +A2 = AQN , i.e., along the diagonal of the lower figure, (b) Topographical landscape of the driving potential. The grey-shaded regions correspond to the lower potential energies.

can see that the shell structure, clearly revealing itself in the contact of two nuclei, i.e. at the borderline A± + A% = AQN ( Fig. 7a), is also retained at A A ^ 0 (see, e.g., the deep minima in the regions of Z = 50 and Z = 82 in Fig. 7b). From the figure it is already clear that in the synthesis of the nucleus 296 116 in the reaction 4 8 Ca+ 2 4 8 Cm, on the way from the contact point to the compound nucleus formation (dark area) the system has to decay with a large probability into the quasi-fission channels (Se+Pb, Kr+Hg) or into the channels of normal fission (Sn+Dy, Te+Gd) - shadowed regions in Fig. 7b. Knowing the driving potential V/ t i g _/j g (^i, Ax, Z 2 ,^2) and the excitation energy of the system in every point we can now determine the probability of the compound nucleus formation PCN(A° + A% -> C; I, E), being part of expression

229 (1) for the cross section of the synthesis of cold super-heavy nuclei. It can be done, for example, by solving the corresponding Fokker-Planck equation for the distribution function F(y = {Zi,Ai,Z2,A2};t) dF

d2

d =

[ t] m

~dt ~di/ ^ '

t)] +

dip W '

t ]

'

m

t)]

'

(6)

with the drift coefficient v(y, t) — - ^ —f"'-J"^v' D and the diffusion coefficient D. The probability of the compound nucleus formation is determined as an integral of the distribution function over the region R{Ai) + R(A2) < RCNSimilarly one can define the probabilities of finding the system in different channels of quasi-fission or normal fission, i.e., the mass distribution of fission fragments measured in experiments. Such calculations are in progress. 7

Conclusion

The synthesis of super-heavy easily fissile nuclei is a difficult many-sided physics problem, in which not only some quantities crucially influencing the whole process are poorly determined but also the dynamics of the process itself. For better understanding of the role of dynamical deformations and nucleon transfer in the course of overcoming the multi-dimensional potential barrier, additional experimental and theoretical investigations are undoubtedly required. Decay properties of the super-heavy nuclei and, first of all, the heights of their fission barriers are also poorly studied and, as a matter of fact, are almost free theoretical parameters in specific calculations. However, analysis of available experimental data performed employing a rather accurate theoretical approach with the use of realistic parameters and without any additional fitting of these latter shows that the fusion process of asymmetric nuclei resulting in the formation of very heavy elements up to Z — 106 differs only insignificantly from the fusion of light and medium nuclei, when a compound nucleus is formed with a probability close to unit once the nuclei have overcome the multi-dimensional potential barrier. In the asymmetric synthesis of super-heavy elements with ZQN > 106 and also in the fusion of heavy symmetric nuclei with ZCN > 90 the process of the compound nucleus formation itself plays an important role due to a strong competition with the processes of fission and quasi-fission. A new mechanism of the fusion-fission process for heavy nuclear systems has been proposed. It takes place in the space (A 1; A 2 ), where A\ and A2 are two nuclei surrounded by a certain number of common nucleons AA. These two nuclei gradually lose (or acquire) their individualities with increasing (or decreasing) the number of collectivized nucleons AA, here "individuality" means

mainly a specific binding energy of the nucleons inside the nuclei Ai and A2, which decreases when a heavy compound nucleus is being formed and increases when it undergoes fission. The corresponding driving potential has been derived. It regulates the behavior of the system in the (Ai,A 2 )-space and allows the calculation of both the probability of the compound nucleus formation and the mass distribution of fission fragments in the heavy ion fusion reactions. Acknowledgments The work was supported by the Russian Foundation for Basic Research under grant No. 00-02-17149. The author is grateful also to A.S. Denikin for numerical calculation used in Sec.5. References 1. Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, F.Sh. Abdulin, A.N. Polyakov, I.V. Shirokovsky, Yu.S. Tsyganov, G.G. Gulbekian, S.L. Bogomolov, B.N. Gikal, A.N. Mezentsev, S. Iliev, V.G. Subbotin, A.M. Sukhov, G.V. Buklanov, K. Subotic, M.G. Itkis, K.J. Moody, J.F. Wild, N.J. Stoyer, M A . Stoyer, R.W. Lougheed, Phys.Rev.Lett. 83, 3154 (1999). 2. Yu.Ts. Oganessian, A.V. Yeremin, A.G. Popeko, S.L. Bogomolov, G.V. Buklanov, M.L. Chelnokov, V.I. Chepigin, B.N. Gikal, V A . Gorshkov, G.G. Gulbekian, M.G. Itkis, A.P. Kabachenko, A.Yu. Lavrentev, O.N. Malyshev, J. Rohac, R.N. Sagaidak, S. Hofmann, S. Saro, G. Giardina, K. Morita, Nature 400, 242 (1999). 3. V. Ninov, K.E. Gregorich, W. Loveland, A. Ghiorso, D.S. Hoffman, D.M. Lee, H. Nitsche, W.J. Swiatecki, U.W. Kirbach, C.A. Laue, J.L. Adams, J.B. Patin, D.A. Shaughnessy, D.A. Strellis, P.A. Wilk, Phys.Rev.Lett. 83, 1104 (1999). 4. S. Hofmann, G. Munzenberg, Rev.Mod.Phys. 72, 733 (2000). 5. W.J. Swiatecki, Phys.Scripta 24, 113 (1981). 6. S. Bjornholm, W.J. Swiatecki, Nucl.Phys. A391, 471 (1982). 7. J. Blocki, H. Feldmeier, W.J. Swiatecki, Nucl.Phys. A459, 145 (1986). 8. Y. Aritomo, T. Wada, M. Ohta, Y. Abe, Phys.Rev. C 59, 796 (1999). 9. V.Yu. Denisov, S. Hofmann, Phys.Rev. C 61 (2000) 034606. 10. N.V. Antonenko, E.A. Cherepanov, A.K. Nasirov, V.P.Permjakov, V.V. Volkov, Phys.Lett. B 319, 425 (1993); Phys.Rev. C 51, 2635 (1995). 11. G.G. Adamjan, N.V. Antonenko, W. Scheid, V.V. Volkov, Nucl.Phys. A627, 361 (1997). 12. E.A. Cherepanov, JINR Report No. E7-99-27, 1999, Dubna.

231

13. M. Dasgupta, D.J. Hinde, N. Rowley, A.M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48, 401 (1998). 14. R. Bass, Nuclear Reactions with Heavy Ions, Springer-Verlag, 1980. 15. V.I. Zagrebaev, N.S. Nikolaeva, V.V. Samarin, Izv.AN, 61, 2157 (1997). 16. J. Blocki, J. Randrup, W.J. Swiatecki, C.F. Tsang, Ann.Phys.(NY) 105, 427 (1977). 17. D.L. Hill, J.A. Wheeler, Phys.Rev. 89, 1102 (1953). 18. N. Rowley, G.R. Satchler, P.H. Stelson, Phys.Lett. B 254, 25 (1991). 19. A.V. Ignatyk, Statistical properties of excited atomic nuclei, Energoatomizdat, Moscow, 1983. 20. A.V. Ignatyk, K.K. Istekov, G.N. Smirenkin, Yad.Fiz., 29, 875 (1979). 21. A.R. Junghans, M. de Jong, H.-G. Clerc, A.V. Ignatyuk, G.A. Kudyaev, K.-H. Schmidt, Nucl.Phys., A629, 635 (1998). 22. T. Sikkeland, A. Ghiorso, M.J. Nurmia, Phys.Rev. 172, 1232 (1968). 23. M. Blann, Nucl.Phys. 80, 223 (1966). 24. J.R. Grover, J. Gilat, Phys.Rev. 157, 802 (1967). 25. J. Gomez del Campo, Phys.Rev.Lett. 36, 1529 (1976). 26. G. Audi, A.H. Wapstra, Nucl.Phys. A595, 409 (1995). 27. W.D. Myers, W.J. Swiatecki, Ark.Phys. 36, 343 (1967). 28. W.D. Myers, W.J. Swiatecki, Table of Nuclear Masses According to the 1994 Thomas-Fermi Model, Report No. LBL-36803, Berkeley, 1994. 29. C.R. Morton, D.J. Hinde, J.R. Leigh, J.R. Lestone, M. Dasgupta, J.C. Mein, J.O. Newton, H. Timmers, Phys.Rev. C 52, 243(1995). 30. B.I. Pustylnik, L. Calabretta, M.G. Itkis, E.M. Kozulin, Yu.Ts. Oganessian, A.G. Popeko, R.M. Sagaidak, A.V. Yeremin, S.P. Tretyakova, JINR Rapid Comm. No. 6[80]-96, Dubna, 1996. 31. T. Murakami, C.-C. Sahm, R. Vandenbosh, D.D. Leach, A. Ray, M.J. Murphy, Phys.Rev. C 34, 1353 (1986). 32. T. Sikkeland, J. Maly, D. Lebeck, Phys.Rev. 169, 1000 (1968). 33. M.G. Itkis, Yu.Ts. Oganessian, E.M. Kozulin, N.A. Kondratiev, L. Krupa, I.V. Pokrovsky, A.N. Polyakov, V.A. Ponomarenko, E.V. Prokhorova, B.I. Pustylnik, A.Ya. Rusanov, I.V. Vakatov, Nuovo Cimento A l l l , 783 (1998). 34. A.V. Yeremin, V.I. Chepigin, M.G. Itkis, A.P. Kabachenko, S.P. Korotkov, O.N. Malyshev, Yu.Ts. Oganessian, A.G. Popeko, J. Rohac, R.N. Sagaidak, M.L. Chelnokov, V.A. Gorshkov, A.Yu. Lavrentev, S. Hofmann, G. Munzenberg, M. Veselsky, S. Saro, K. Morita, N. Iwasa, S.I. Mulgin, S.V. Zhdanov, JINR Rapid Comm. No. 6[92]-98, (1998). 35. K.-H. Schmidt, W. Morawek, Rep.Prog.Phys. 54, 949 (1991). 36. S. Hofmann, Rep.Prog.Phys. 61, 639 (1998).

232

FUSION AND ALPHA EMISSION WITHIN A LIQUID DROP MODEL AND HEAVIEST ELEMENT FORMATION AND DECAY G. ROYER Laboratoire Subatech, 4 rue A. Kastler, 44072 Nantes Cedex 03, France E-mail: [email protected] R. A. GHERGHESCU National Institute for Physics and Nuclear Engineering, P. O. BoxMG-6, RO-76900 Bucharest, Romania The potential energy governing the fusion and a emission processes has been determined within a generalized liquid drop model including the proximity effects, the asymmetry, an accurate nuclear radius and an adjustment to reproduce the experimental Q value. The a decay half-lives deduced from the WKB barrier penetration probability without adjustable preformation factor are in agreement with the experimental data in the whole mass range. Accurate simple formulas are proposed for logio[Ti/2 ;a (s)]. Fusion barriers leading to the heaviest and superheavy elements as well as their a emission halflives are presented and compared with recent experimental data.

1 Introduction The synthesis of very heavy nuclides has apparendy strongly progressed recentiy 13 . The observed decay mode is die a emission. The purpose of this work is to study the fusion barriers investigated in the recent experiments and to look at die a decay half-lives of the heaviest systems. Before, it will be checked that the a decay may be viewed as a very asymmetric fission within the generalized liquid drop model (GLDM) which has allowed to reproduce most of the fusion and fission characteristics * and light nucleus emission . 2 Generalized Liquid Drop Model The GLDM 5 includes an accurate nuclear radius, die mass and charge asymmetries and a proximity energy . This last term takes into account die effects of the surface tension forces between surfaces in regard in a neck or a gap. No frozen density approximation is done and the proximity function is effectively integrated in the neck and, consequentiy, die proximity energy depends on the nuclear shape and vanishes for no-necked nuclear configurations. The selected quasi-molecular shapes describe die rapid formation of a deep neck and its filling while keeping almost spherical ends (see Fig. 1).

233

This GLDM has allowed to reproduce accurately the empirical fusion barrier positions Rb and heights Eb . A recent fitting procedure has led to the following formulas: R b = ( A | / 3 + A^ 3 )(l.908-0.0857ln(ZiZ 2 ) + 3 . 9 4 / Z ^ 2 )

2.1388Z 1 Z 2 +59.427(A 1 1 / 3 +A 2 / 3 )-27.07 In -19.38 + -

Z,Z2

A J / 3 + A ; 2/ 3

V M

J

(A | / 3 + A , / 3 )(2.97 - 0.12 ln(Z! Z 2 ) )

Fig. 1. Asymmetric shape sequence leading rapidly from a sphere to two tangent spheres assuming volume conservation. 3 Alpha decay After the contact point the energy relatively to infinity is simply the sum of the Coulomb energy and the proximity energy reproduced by (see Fig. 2) 0.172 1/3 E p r o x (MeV) 0.6584A 2 ' 3 -1.38(r1/3 + 0.4692A =e 4irV -0.02548A 1 / 3 r 2 +0.01762r 3 where A is the mass of the mother nucleus and r the distance between mass centres 9 . The difference between the experimental and meoretical Q a values has been added at the sphere energy widi a linear attenuation factor vanishing at the contact point . The inadequacy of die pure Coulomb banier is clearly displayed in Fig. 3. The proximity energy lowers the banier height by several MeV and moves the top towards two separated fragments maintained in unstable equilibrium by the balance between the repulsive Coulomb forces and the attractive nuclear proximity forces.

234

Fig. 2. Proximity energy (/47tv) between two separated a particle and daughter spherical nucleus A-4 as a function of the distance between the mass centies. >

15

10 i

i

I

i

i

i

1

1

25

20

-

- 20

20

Hs-

260

*

0

Sg + a

15

-- 15

o

264

E(MeV)

3

1 1

'

-

10

; 5 •

0

W 10

I

i

15

20

25

30

r(fm) Fig. 3. a decay bairier for ~ Hs. The solid and dashed curves conespond to the barrier with and without a nuclear proximity energy term.

235

In such an unified fission model, the decay constant of die parent nucleus is simply defined as A = v0P. There is no preformation probability factor. The assault frequency v 0 has been chosen as v 0 = 1 . 0 x l 0 2 0 s - 1 . The barrier penetrability P is calculated within the general form of the action integral P = e x p j - | { * - j 2 f i [ E ( r ) - E ( R i n t ) ] d r l with E ( R m t ) - E(R 0 U t ) = Q E x p . The ability to reproduce the experimental half-lives is shown in Fig. 4. The a decay half-lives of 373 nuclei have also been determined . The rms deviation between the theoretical and experimental values oflog 10 [T 1/ 2(s)] is only 0.63. In addition to the GLDM approach, a fitting procedure from the experimental data leads to the following formula for the 373 nuclei with a rms deviation of 0.42. logio[Ti/ 2 (s)] = -26.06 - 1.114A1/6 Vz + . 1 - 5 8 3 7 Z /Qa For the subset of the 131 even-even nuclei the rms deviation between uie theoretical and experimental values is only 0.35. Within the following formula the rms is 0.285. logio[T 1/2 (s)] = -25.31 - 1.1629A176 VZ + • 1 - 586 ^-/Qa For the subset of the 106 even-odd nuclei the rms deviation is 0.71. Within the following formula the rms is 0.39. log 10 [T 1/2 (s)] = - 2 6 . 6 5 - 1 . 0 8 5 9 A 1 / 6 ^ + L 5 8 4 8 Z /Qa For the subset of the 86 odd-even nuclei the rms deviation is 0.57. The following formula leads to a rms of 0.36. logio [T1/2 (s)] = -25.68 -1.1423 A 1 / 6 Vz + - 1 5 9 2 Z /Q« For the subset of the 50 odd-odd nuclei the rms deviation is 0.99. The following formula leads to a rms of 0.35. 1.6971Z logio [T1/2 (s)] = -29.48 -1.113 A 1 / 6 Vz + /Qa 4 On the heaviest and superheavy nuclides In me fusion reactions leading to the heaviest elements the formation of the nucleon shells plays a main role to stabilize the system. The shell effects given by the asymmetric two center shell model have been added at the macroscopic GLDM 10 as well as the corrections to obtain die experimental Q value.

236 95

100

105

110

115

120

125

130 135

95

100

105

110

115

120

125

130 135

N Fig. 4. log 1 0 [T a j/2 (sjj for Hg, Pb and Po. The solid shapes and lines correspond to experimental data and the open shapes and dashed lines to the theoretical ones.

6

7

1 ' 'i

i i i i

8 1

• i •

9 i

10

i i i i i •

1

11

12

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1 1 1 1 1 1 1 1 1

240

*•.. 236

>

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236 232

CD 232 LU 228

240

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228 64. .. 208,-.,

224

Ni+

v

•

Pb- 1 1 0 X-

220

224 220

'T

. .

9

10

11

12

13

14

r(fm) Fig. 5. Macroscopic (dashed curve) and microscopic (solid one) fusion barriers versus the mass-centre distance. The vertical line gives the contact point.

237 |

1 1 1 1 |

T n

1 |

1 t

t

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185

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170

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8

10

9

320 :

86

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165

Pu— m X

160

11

12

13

CD

320

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315

LU

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310

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238

Macroscopically, a relative minimum appears at large deformations (see Fig. 5). Microscopically, for Ni + Pb the ground state is slightly deformed while a plateau lies at intermediate deformations. For Zn + Pb double humped barriers appear separated by a deep minimum and fast fission becomes an important decay channel. The moment of the neutron emission is decisive to decide between complete fusion and fast fission. For Ca + Pu, due the system asymmetry, the barrier against reseparation is high and the system will descend till the sphere configuration. The final excitation energy depends also strongly on the moment of emission of the three neutrons. For the Kr + Pb reaction, the inner barrier is the highest but a minimum still occurs near the sphere. The stability of such formed nuclear systems is another problem. The characteristics of these fusion barriers are shown in table 1. Within the GLDM, the saddle-point corresponds always to two separated sphere configurations. The often used approximation which starts the fusion process from the contact point seems rough. At the contact point the first external peak of the fusion barrier is already passed. Macroscopically, the energy of the minimum at large deformations is lower than the sphere energy. The barrier height derived from the GLDM is systematically higher than the Bass barrier and the fusion radius is lower. These two effects lead to smaller fusion cross sections within the GLDM than within the Bass model. Indeed, for most of the incident energies presently used, the reaction is a subbarrier fusion for die GLDM while it is a fusion above the barrier for the Bass model. For the Kr + Pb, the incident energy is high (321 MeV in the mass centre) and leads to a reaction well above the barrier in the two models. The predicted a decay half-lives of the heaviest elements have been calculated (see table 2) from the theoretical Q a values given by the ThomasFermi model " since it reproduces nicely the mass decrements from Fermium to Z = 112. The values given by the GLDM should give a lower limit of the true value while the ones given by die fitted formulas should not be far from the reality. If such nuclei exist, their half-lives vary from microseconds to some days. Generally, for a given element, the half-lives increase with the neutron number. The predictions of the GLDM agree (see table 3) with the apparently observed experimental data on the heaviest and superheavy isotopes except for ^85

^89

'SI

three nuclei" 114 and " 114 and at a less degree " 112. The theoretical values are respectively of 5 to 2 orders of magnitude higher than the experimental data. One possible explanation of this discrepancy is perhaps that Z = 114 is not a good magic number as some studies using Hartree-Fock theory suggest and that the Thomas-Fermi model underestimates the Q a value and, then, overestimates the half-lives. The analysis of the experimental data is also discussed 13. With the GLDM, one obtains a kinetic energy of the fragments of 240 MeV, higher than the 190 MeV apparently measured. This is independent of die magicity of the charge number.

Reaction

Experimental result

Rext

"ext

Hfus

(fm) 12.2

(MeV) -12.2

(MeV) 322.0

SSr+'gFb SKr+igPb ^Ca+^Cm

ngSh + ln

11.3

-16.8

12.0

-14.1

307.3

n • 9 •

10.0

-3.5

12.6

11.2

203.1

11.0

-14.5

11.9

-11.1

290.8

5gSh + 3n 2 ^Sh + 3n

9.8

-2.4

12.6

12.6

199.3

9.9

-3.0

12.5

11.8

199.6

9 • 28

10.6

-11.0

11.8

-6.2

275.4

9.8

-1.9

12.5

13.3

195.8

112 Sh + In

10.3

?8Ni+ f3Bi

272

i i iSh + ln

10.0

-7.4 -5.0

12.35 12.3

-0.7 3.6

260.0 247.6

3 4 is + 2 4 4 p 16 94™ 64 M ; . 208 D .

noSh + 5n

9.9 9.6

-3.6

12.2

32.8

164.4

12.3

5.5

244.7

-0.6

12.2

10.1

245.9

9.5

0.1

12.2

11.7

241.1

9.6

-1.1

12.3

10.4

232.1

2^Ca+^Pu 4 8 2 2 o Ca+ «Pu 2

^Ge+ ^Pb 48p 2 o>-d1-

238 T , 92 U

?gZn+2gPb 2

vA *

"min

(MeV) -15.0

n

gSe+'gPb

304 120

"•miu

(fm) 11.3

296 v * 116A 290 v * 116 A

284 v A * 114

jSh + 3n

82 Pb

noSh + lii

2iN1+ 8°fPb

5 2 2?Co+ »Bi

IioSh+ln ?SSh + ln

Oft+'gBi

logMt + ln

2 8 Nl+ 6

2

(

Table 1 Characteristics of the macroscopic fusion barriers leading to the heaviest nucli respectively the positions and energies relatively to the sphere of the external minimum fusion barrier. Hft,s is the barrier height of the GLDM while Rflass and Heass are the barrie Bass model. Ecm is the centre of mass energy already used in an experiment.

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PA Tf Tf m on in O IN ^r Tt- m IN i/i CN 30 ON r- o 00 CN NO rN r00 PA Tt- Tt- PA CN in rA CN NO m 1 1 1

i/i

1 ' '1 1 ON r- in o m on m NO m

m r- r» •*

rn NO1 V) PA (N --j in

PA

r— en on

Tt NO NO ON

r-

00

u,

•

a

£<

en NO NO1 PA CN in TT PA CN m1 TC1 PA PA NO1 in1 PA CN NO1 m1 PA PA NO Tt1- PA CN m

£ <""> PA

S 2 ts <

fi

"3D

T-*

"3D

-re

"3D

^H

a2

IT)

"3D

^H

i-H

^H T-<

II S)

NO

II SI

^H

A

*-H

II SI

t>

II SJ

00

II SI

^H ^H

II SI

241

Q : 12.49 [60, 300] jus Tfl,:72M.s Tgidm: 3 (is

118

116

114

112

110

Z/N

Q : 11.5 [300, 1460] |iis Tfit: 3000 us T d d m : 110 us Q : 9.53 Q : 9.55 [290, 1450] us 5.5 s T fit : 130 s Tfit: 117 s Tgidm: 5.5 s 1 eldm • ^ S Q : 10.35 Q:8.8 [0.4, 2.2] ms [20, 200] m Tfit: 130 ms Tat: 84 m Teidm: 6.3 ms Teidm: 4.5 m Q : 10.89 Q: [1.5,7.7]ms 8.75(MeV) Tfit :1.3 ms [1, 12]m Tgunj: 0.07 ms Tfit: 23 m Tgidm '• 1.5 m 167 173 171 169

Q : 9.08 [2, 23] s Tat :3000 s Tgidm: 145 s

175

Table 3: Comparison between the experimental a decay half-lives (between square brackets) of the heaviest and superheavy elements 2'3 and the predicted ones by the fitted formulas and the GLDM. 5 Conclusion The a decay process and particularly the a decay half-lives may be described within a tunneling process below a deformation barrier calculated from a generalized liquid drop model adjusted to reproduce the experimental Q a value and without introducing a preformation probability factor as for the spontaneous fission. In addition to the GLDM approach, a fitting procedure has also led to accurate formulas for the a decay half-lives. Extrapolations to very heavy and superheavy nuclides have been investigated. The theoretical half-lives are respectively of 5 to 2 orders of magnitude higher than the experimental data for the 28 114 and the 289114 nuclei apparently observed recently. This perhaps indicate that 114 is not a good magic proton number. The calculated kinetic energy of the fragments is also higher than the apparently measured one.

242

In the entrance channel, the combination of the generalized liquid drop model and the asymmetric two-center shell model shows that the formation of the nucleon shells before reaching the compound nucleus shape is essential to stabilize the formed nuclear system. The moment of emission of the excess neutrons which evacuate a part of the excitation energy is crucial to decide between complete fusion and fast fission processes. References 1. S. Hofmann and G. Mtinzenberg, GSI report 2000-02. 2. V. Ninov et al, Phys. Rev. Lett. 83, 1104 (1999). 3. Y. T. Oganessian et al, Phys. Rev. Lett. 83, 3154 (1999). 4. G. Royer, J. Phys. G, in press. 5. G. Royer and B. Remaud, Nucl. Phys. A 444, 477 (1985). 6. G. Royer, Heavy Elements and Related New Phenomena, edited by R. K. Gupta and W. Greiner (World Scientific, 1999) p 591. 7. G. Royer, C. Normand and E. Druet, Nucl. Phys. A 634, 267 (1998). 8. G. Royer and R. Moustabchir, Nucl. Phys. A, in press. 9. R. Moustabchir and G. Royer, submitted to Nucl. Phys. A. 10. R. A. Gherghescu and G. Royer, Int. J. Mod. Phys. E 9, 51 (2000). 11. R. Bass, Proc. Conf. on deep inelastic and fusion reactions with heavy ions, Berlin (Springer, 1979) p 281. 12. W. D. Myers and W. J. Swiatecki, Nucl. Phys. A 601, 141 (1996). 13. P. Armbruster, E. Phys. J. A 7, 23 (2000).

243

ON SCISSION CONFIGURATION IN T E R N A R Y FISSION

1

V. G. KARTAVENKO x'2, A. SANDULESCU2'3 AND W. GREINER 2 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow District, 141980, Russia 2 Institut fur Theoretische Physik der J. W. Goethe Universitat D-60054 Frankfurt am Main, Germany Romanian Academy, Calea Victoriei 125, Bucharest, 71102, Romania

A static scission configuration in cold ternary fission has been considered in the framework of two mean field approaches. The virial theorems has been suggested to investigate correlations in the phase space, starting from a kinetic equation. The inverse mean field method is applied to solve single-particle Schrodinger equation, instead of constrained selfconsistent Hartree-Fock equations. It is shown, that it is possible to simulate one-dimensional three-center system via inverse scattering method in the approximation of reflectless single-particle potentials.

1

Motivation

Ternary fission involving the emission of a-particle was first observed more than fifty years ago x . Emission of a-particles in the spontaneous fission of 252 Cf has also a long history of investigation experimentally 2 and theoretically 3,4 . A renewed interest in these processes arose in connection with • modern experimental techniques (7—7—7 and x—7—7 triple coincidence, (Gammasphere with 110 Compton suppressed Ge detectors), which allow the fine resolution of the mass, charge and angular momentum content of the fragments.) 5 • Recently direct experimental evidence was presented for the cold (neutronless) ternary spontaneous fission of 252 Cf in which the third particle is an a-particle 6 , or 10 Be 7 . This confirms that a large variety of nuclear large-amplitude collective motions such as bimodal fission 8 , cold binary fission s.9.10.11!12^ heavy cluster radioactivity 13>14, and inverse processes, such as subbarier fusion 15, could belong to the general phenomenon of cold nuclear fragmentation. • Cluster like models 1 6 , 1 7 were used successfully to reproduce general features of the cold ternary fragmentation. However the scission configuration has been built in fact by hands. Therefore it is actual to develop microscopical or semi-microscopical approach to this scission-point concept of nuclear fragmentation. There are

244

well developed methods to calculate, in the framework of many-body selfconsistent approach, static properties of a well isolated nucleus in its ground state. There also exists a well developed two-center shell model 18 . However, a three-center shell model has not been developed yet, except for very early steps 19 . Three-center shapes are practically not investigated, in comparison with the two-center ones. There exists the generalizations of mean-field models to the case of two-centers 20 , but a ternary configuration is out of consideration, because of uncertainties to select a peculiar set of constraints. There exist a number of calculations for nucleus-nucleus collisions in the frame of time-dependent mean-field methods, but an evolution of the cold fragmentation has not been investigated yet. Therefore, although the principal way to describe nuclear fragmentation in the framework of many-body self-consistent approach exists, it is interesting to develop other mean-field approaches to analyse these phenomena from different points of view. In this Contribution we suggest two methods to analyse a static scission configuration in cold ternary fission in the framework of mean field approach. In Section 2, starting from kinetic equation the virial theorems has been suggested to investigate correlations in the phase space. It gives possibility to formulate in future generalised set of constraints in momentum and cartesian spaces. In Section 3 the inverse mean field method is applied to solve singleparticle Schrodinger equation, instead of constrained self consistent HartreeFock equations. It makes irt possible to simulate one-dimensional three-center system via inverse scattering method. 2

Virial theorems

Within the mean-field approximation, we analyze the evolution of one-body Wigner phase-space distribution function of the full many-body wave function, following the well developed scheme using as the starting point the Vlasov equation for the Wigner phase-space distribution function 21

0/

p df

dV df _

at

m

or

or

op

with a "relaxation term" lTt\ is added to the kinetic equation to describe dissipation effects. The quantity V(r,t) is the self-consistent single-particle potential which is assumed here to be local, m is the mass of nucleon. Out of the Wigner distribution function, virials at different orders can help to extract useful physical information from the total phase space dynamics. Integrating the initial kinetic equation (1) over the momentum space with different polynomial weighting functions of the p-variable one comes, as well

245 know 2 2 ; 2 3 2 4 ) to an infinite chain of equations for local collective observables including the density, collective velocity, pressure and an infinite set of tensorial functions of the time and space coordinates, which are defined as m o m e n t s of the distribution function in the m o m e n t u m space: • the particle n(f,t) sities,

= Jdpf(r,p,t),

and the mass p(r,t)

= mn(r,t)

den-

• the collective current and velocity of nuclear m a t t e r p(r,t)u(f,t)

= /

dppf(f,p,t),

• the pressure tensor and the energy and m o m e n t u m transfer tensors of different orders ¥ij(f,t)

= —

dpqiqjf(f,p,t),

qt = Pi - mtij,

Til J

^ii..k(r,t) = ——[J

dpqiqj..qkf{r,p,t), n

n

• and the integrals related to relaxation terms / dplrei

=0,

/ dpplrei

ffi.j = ~~ /

mJ

= 0,

dpqiqjlrei,

Truncating this chain at order two in q one arrives at the "fluid dynamical" level of description of nuclear processes.

k

k

k

+ p^2eisj{2QsUj

Dt

^

\ k

+—pXj)-0,

lK

dxk

JK

dxk

(3)

'3dxk

246 i -^ / j ^ m ^ j m a M ' s ~r £im$>±js)

1 . . , , = (&* E^=^L' +

i^r)rer™Idp~qiqjIre" where the usual notation jy^ = §i + Ylukg§~ k

k

' s introduced for the operator

giving the material derivative, or the rate of change at a point moving locally with the fluid. The hydrodynamical set of Eqs. (2-5) describes an evolution of a rotating nuclear system. We consider two frames of reference with a common origin: an inertial frame, {Xi,X2,Xz), and a moving frame, (xi,X2,x3). Let 3

Xi = ]T] TijXj

i =1 x, of

be the linear transformation that relates the coordinates, X

and a point in two frames. The orientation of the moving frame, with respect to the inertial frame, will be assumed to be time dependent. Since Tij(t) must represent an orthogonal transformation, the vector

it.™

^

'

?m

represents a general time-dependent rotation of the z-frame with respect to the inertial frame. Let us define integral collective "observables" (the integrals over the whole phase space of one nucleon containing the distribution function appropriatly weighted), namely an inertia tensor Iij(t), the dynamical part of the angular momentum Li(t), the integral pressure tensor Hij(t) defined as = I dx XiXjp, Lk = ^Skij

Hij — I

(JL JJ

i± j A •

/ dx pxtUj, J

i,j

The dynamics in terms of the latter "observables" is expressed by a set of virial equations in the rotating frame 25 d2 k

+ 2^fi9 s,k

/ df puk(eiskxj •*

+SjskXi)

247

+ 2W,j - 2K,j - 2Utj —jriSisklkj

+ SjskSki) — 0,

s.k

dt

+ z2

e

*itfitfimJjm - 2 ^ ft, /

—II,j + F,j + 2^2ns(eisknkj

dfpukxs

+ e^n*,-) = l&ij,

«. = !.!«{'•>&+'»£). where the tensors of collective kinetic and potential energies, and the relaxation tensor are Kij = I dxuiUjp, M.{j = / dx I

W,j = / dx XJ——n, 1

XjXj.

The above equations constitute a formal framework within which the coupling of the deformations in the r*-space and in the p-space can be explicitly worked out. It may help to fofmulate in a future a generalised set of constrains in the phase space. 3 3.1

Inverse mean field m e t h o d The framework

Methods of nonlinear dynamics, gave yet the possibility to derive for nuclear physics unexpected collective modes, which can not be obtained by traditional methods of perturbation theory near some equilibrium state (see e.g. review 2 7 and 28 for the recent refs.). The most important reason is that the fragmentation and clusterization is a very general phenomenon. There are cluster objects in subnuclear and macro physics. Very different theoretical methods were developed in these fields. However, there are only few basic physical ideas, and most of the methods deal with nonlinear partial differential equations. One of the most important part of soliton theory is the inverse scattering method 29>30>31 and its applications to the integration of

248

nonlinear partial differential equations 3 2 . The inverse methods to integrate nonlinear evolution equations are often more effective than a direct numerical integration. Let us demonstrate this statement for the following simple case. The type of systems under consideration are slabs of nuclear matter 33 , which are finite in the z coordinate and infinite and homogeneous in two transverse directions. The wave function for the slab geomethry is 1 ^kxn(x) = -7=ipn{z)exp(ik±r), where r = (x, 2/),kj_ = (kx,ky),

h2k2 ekj_„ = —— + en,

(5)

and Q is the transverse normalization area.

H2 d2

A direct method to solve the single-particle problem (6) is to assign a functional of interaction S (usually an effective density dependent Skyrme force), to derive the ansatz for the one-body potential, as the first variation of a functional of interaction in density U{z) = U[p(z)] — SS/Sp. Then to solve the Hartree-Fock problem under the set of constraints, which define the specifics of the nuclear system. In the simplest case of a ground state, one should conserve the total particle number of nucleons (A), which is related to the "thickness" of a slab, via A = » A = {6Ap2N/7r)^3, which gives the same radius for a three-dimensional system and its one-dimensional analogue. As a result, one obtains the energies of the single particle states e n , their wave functions i>n{z), the density profile p(x) =>• p(z) No

No

A = n= l

y^gn

an = —Z^\ZF - e„),

(7)

n= l

and the corresponding single-particle potential. an are the occupation numbers, No is the number of occupied bound orbitals. The Fermy-energy ep controls the conservation of the total number of nucleons. The energy (per nucleon) of a system is given by

Finally, the set of formulas (5-8) completely defines the direct self-consistent problem. Following the inverse scattering method, one reduces the main differential Schrodinger equation (6) to the integral Gel'fand-Levitan-Marchenko

249 equation

29 30

'

roo

K(x,y) + B(x + y)+

B{y + z)K(x,z)dz

= 0.

(9)

Jx

for a function K(x,y). The kernel B is determined by the reflection coefficients R(k)(ek = h2k2/2m), and by the N bound state eigenvalues

i r°°

A B z

()

= Z2 CniKn) + ~ n=l

R{k) exp(ikz)dk,

e„ =

-h2K2n/2m.

"" ^ - ~

N is the total number of the bound orbitals. The coefficients C„ are uniquely specified by the boundary conditions and the symmetry of the problem under consideration. The general solution, U(z) = —(h2/m)(dK(z,z)/dz), should naturally contain both, contributions due to the continuum of the spectrum and to its discrete part. There seems to be no way to obtain the general solution U(z) in a closed form. Eqs. (6),(9) have to be solved only numerically. In Ref. 3 4 we used the approximation of reflectless (R(k) = 0), symmetrical (U(—z) — U(z)) potentials. This gave the possibility to derive the following set of relations

N V'n(z) = X ) ( n= l

h2 B2

Oh2

m dz2

m

N

n=\ M

l

)nl^l{z),

An(z) = C „ ( K „ ) e x p ( - K „ z ) , JV

K{z) l{z

Mnl{z)=Snl+

^ \

Cn(«„)=(2Kn|n^L±^l)1/2.

(10)

Consequently, in the approximation of reflectless potentials (R(k) = 0), the wave functions, the single-particle potential and the density profiles are completely defined by the bound state eigenvalues via formulas (5),(10). 3.2

Results and Discussion

In Ref. 3 4 a series of calculations for the different layers, imitating nuclear systems in their ground state was provided. For a direct part of the calculations by the Hartree-Fock method, the interaction functional was chosen in the form of effective Skyrme forces. The calculated spectrum of bound states was fed into the scheme of the inverse scattering method, and the relations were used to recover the wave functions of the states, the single-particle potentials, and the corresponding densities. In this note, we generalize this

250 0.25 0.2 1—1

'S ° 1 5 1J

O.I

^_^ 0.05 0.0

0.25

0.2 en

' £ 0.15

S 0.05 0.0

Figure 1. The density profiles of the light (A « 20, A W 1.0) three-levels (TV = 3, N0 = 3) model system calculated in the frame of inverse mean field method, a) the ground state (solid line); a fragmented two-center configuration (dotted and dashed lines); b) a ternary fragmentation of the system into three fragments.

method to the case of fragmented configuration, trying to imitate two- and three-center nuclear systems. We use here only the inverse mean-field scheme (10). The details of the approach and systematic calculations of fragmented nuclear systems will be provided in a forthcoming publication. In Fig.l we present the results of the calculations of three-level (iV = 3, No = 3 ) light (.A PS 20, A s=s 1.0) model system simulating the ground state (Fig. 1(a) solid line), and fragmentation of the system into two fragments (Fig. 1(a), the dotted and dashed lines). In the same figure (Fig. 1(b)) we present the fragmentation of the system into three fragments (solid and dotted lines). One

251

can see that it is possible to simulate a one-dimensional three-center system via inverse scattering method. The following conclusions can be drawn. • The density profiles, calculated in the framework of inverse method, are practically identical to the results of calculation by SHF method. These results are valid for the ground state and for the system in the external potential field. • The global properties of single-particle potentials (the depth and an effective radius) have been reproduced quite well, but the inverse method yields the quite strongly pronounced oscillations of the potential distributions within the inner region, and slightly different asymptotic tails of potential. In the framework of inverse scattering method, all bound states are taken into account in the calculation of the potential (10), but for the density distribution only the occupied states are taken into account (see Eqs. (7)). Therefore, the slope of the tails of the potential and of the density distributions will we different. • We used, the approximation of refiectionless potentials, which gave us the possibility to obtain a simple closed set of relations (10), to calculate wave functions, density distributions and single particle potentials. The omitted reflection terms {R{k) = 0)are not important for the evaluation of the density distributions, due to the fact that only the deepest occupied states are used to evaluate density distribution (see Eq. (7)). The introduction of these reflection terms will lead to a smoothing of the oscillations in the inner part of the potential and to a correction of its asymptotic behaviour. • The presented method gives a tool to simulate the various sets of the static excited states of the system. This method could be useful to prepare in a simple way an initial state for the dynamical calculations in the frame of mean-field methods. 4

Conclusions

Recent experimental progress in the investigation of cold nuclear fragmentation has made the development of theoretical many-body methods highly desirable. Modern variants of self-consistent Hartree-Fock and relativistic mean-field models give the principal way to describe nuclear fragmentation in the framework of many-body self-consistent approach. However, the generalization of these approaches to three-center case is not provided yet because of existing difficulties to select a suitable set of constraints.

252

We suggest two methods to analyse a static scission configuration in cold ternary fission in the framework of mean field approach. The virial theorems has been suggested to investigate correlations in the phase space, starting from a kinetic equation. The inverse mean field method is applied to solve singleparticle Schrodinger equation, instead of constrained selfconsistent HartreeFock equations. It is shown, that it is possible to simulate one-dimensional three-center system via inverse scattering method in the approximation of reflectless single-particle potentials. These models may be useful as a guide to understand the general properties of fragmented systems and to formulate the suitable set of constraints for the realistic three-dimensional mean field calculations of the three-center nuclear system. Acknowledgments The partial financial support by Russian Foundation for Basic Research and Deutsche Forschungsgemeinschaft is gratefully acknowledged. References 1. By I.W.Alwarez, as reported by G.Farwell, E. Segre and C.Wiegand, Phys. Rev. 71, 327 (1947) 2. Z. Fraenkel, Phys. Rev. 156, 1283 (1967) 3. Y. Boneh, Z. Fraenkel and I. Nebenzahl, Phys. Rev. 156, 1305 (1967) 4. A. Sandulescu et al, Int. J. Mod. Phys. E 7, 625 (1998) 5. G.M. Ter-Akopian et al, Phys. Rev. Lett. 73, 1477 (1994) 6. A.V. Ramayya et al, Phys. Rev. C 57, 2370 (1998) 7. A.V. Ramayya et al, Phys. Rev. Lett. 8 1 , 947 (1998) 8. E.R. Hulet et al, Phys. Rev. Lett. 56, 313 (1986) 9. F.-J. Hambsch, H.-H. Knitter and C. Budtz-Jorgensen, Nucl. Phys. A 554, 209 (1993) 10. W. Schwab et al, Nucl. Phys. A 577, 764 (1994) 11. Y.X. Dardenne et al, Phys. Rev. C 54, 258 (1996) 12. A. Sandulescu et al, Phys. Rev. C 54, 258 (1996) 13. A. Sandulescu, D.N. Poenaru, and W. Greiner, Phys. Part. Nucl. 11, 528 (1980) 14. P.B. Price, Nucl. Phys. A 502, 41c (1989) 15. P. Armbruster, Rep. Prog. Phys. 62, 465 (1999) 16. A. Sandulescu, A. Florescu and W. Greiner, J. Phys. G: Nucl. Part. Phys. 22, 1815 (1989)

253

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

F. Gonnenwein and B. Borsig, Nucl. Phys. A 530, 27 (1991) J. Maruhn and W. Greiner, Z. Physik 251, 431 (1972) J. Hahn, H.-J. Lustig and W. Greiner, Z. Naturf. 32a, 215 (1977) J.F. Berger.M. Girod and D. Gogny, Nucl. Phys. A 502, 85c (1989) E.P. Wigner, Phys. Rev. 40, 749 (1930) S. Chandrasekhar, Elliptical figures of equilibrium (Dover, NY, 1987) G. Rosensteel, Ann. Phys. NY 186, 230 (1988) E.B. Balbutsev and I.N. Mikhailov, In Collective nuclear dynamics, ed. R.V. Jolos (Nauka, Leningrad, 1990) p.3 V.G. Kartavenko et al, Phys. Part. Nucl. Lett. 1(98), 39 (2000) M.Bender et al, Phys. Rev. C 58, 2126 (1998) V.G. Kartavenko, Phys. Part. Nucl. 24, 619 (1993) V.G. Kartavenko, A. Sandulescu A. and W. Greiner, Int. J. Mod. Phys. E 8, 381 (1999) I.M. Gel'fand and B.M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat. 15, 309 (1951) V.A. Marchenko, Dokl. Akad. Nauk SSSR 104, 695 (1955) L.D. Faddev, Sov. Phys. Dokl. 3, 747 (1959) The Theory of Solitons: The Inverse Scattering Method (ed. by S.P. Novikov, in Russian) (Nauka, Moscow) 1970. P. Bonche, S. Koonin and J.W. Negele, Phys. Rev. C 13, 1226 (1976) V.G. Kartavenko and P.Madler, Izv. Akad. Nauk SSSR, Ser. Fiz. 51, 1973 (1987)

254

F U S I O N OF WEAKLY B O U N D STABLE NUCLEI - W H A T C A N W E LEARN ? M. D A S G U P T A , A . C . B E R R I M A N , R. D . B U T T , D . J . H I N D E , C.R. M O R T O N AND J.O. N E W T O N Department

of Nuclear

Research School of Physical Engineering, National University, Canberra, ACT 0800, E-mail: [email protected]

Australian

Physics,

Sciences

and

Australia

R. M. A N J O S , P . R . S . G O M E S A N D S.B. M O R A E S Instituto

de Fisica,

Universidade

Federal Flurninense, 24210-340, Brazil

Av. Litoranea,

Niteroi,

RJ,

N . C A R L I N A N D A. S Z A N T O D E T O L E D O Instituto

de Fisica,

Universidade de Sao Paulo, Caixa Postal Sao Paulo, S.P., Brazil

66318,

05315-970

Fusion excitation functions for the 7 Li + 2 0 9 B i and 9 B e + 2 0 8 P b reactions have been measured at energies around the Coulomb barrier. The fusion barrier distributions extracted from these d a t a allow reliable predictions of the expected complete fusion cross-sections. The measured cross-sections are however only ~ 70% of those expected, thus showing conclusively that complete fusion cross-sections at above-barrier energies are suppressed compared with the fusion of more tightly bound nuclei.

1

Introduction

Reactions with radioactive beams are being vigorously studied at many laboratories around the world. The interest is focussed on understanding the structure of such nuclei, and investigating the effect of their unusual properties 1 on reaction mechanisms. An understanding of these aspects is intimately related to the optimisation of the production of new nuclei far-off the line of stability. Fusion reactions involving neutron-rich 2 ' 3 ' 4 and proton-rich 5 nuclei has generated a lot of controversy and interest 6 ' 7 ' 8 ' 9 ' 10 . Some unstable nuclei have a very low energy threshold against breakup and the controversy centres around the influence of breakup on fusion. The different theoretical approaches are in disagreement regarding the relative magnitudes of enhancement, due to coupling to breakup channels, and/or suppression due to loss of incident flux. Due to the experimental situation of low beam intensities and poor beam energy resolution, fusion cross-section measurements involving halo nuclei are

255 difficult, though some results have emerged recently. On the other hand, precise measurements of fusion excitation functions are currently possible for stable nuclei, and have an i m p o r t a n t role to play in understanding the reaction mechanism of their unstable counterparts. For example, deductions about the reaction mechanism for 1 0 , 1 1 Be + 2 0 9 B i were found to be difficult 2 ' 9 until the reaction mechanism for 9 B e + 2 0 9 B i , used for comparison, was made clear 1 1 . Further, any theoretical model t h a t predicts the effect of breakup on fusion of unstable nuclei should be able to explain the measured fusion cross-sections of weakly bound stable nuclei, which also have a low energy breakup threshold, such as 6 , 7 Li and 9 B e . In this contribution we present experimental work, performed at the Australian National University, aimed at understanding the effect of breakup on the fusion of the 7 Li + 2 0 9 Bi and 9 B e + 2 0 8 P b systems. T h e breakup of 7 Li is a t w o - b o d y process, whilst t h a t of 9 B e is of three-body nature. T h e latter defies complete theoretical description at present, and was studied primarily to u n d e r s t a n d 1 1 the results 2 ' 9 of the 1 0 , 1 1 Be + 2 0 9 Bi fusion reactions. A complete theoretical description of fusion and breakup of 7 Li should, however, be achievable, and it is hoped t h a t the present d a t a will encourage such calculations. The precisely measured fusion excitation functions, presented here, allow determination of the experimental fusion barrier distribution (discussed in the next section), which dramatically reduces the uncertainty in the choice of potential and coupling parameters in the theoretical models. T h e experimental procedure used to measure the complete and incomplete fusion excitation functions is described in Sec. 3. T h e identification of complete and incomplete fusion products is described in Sec. 4. The experimental excitation functions are presented in Sec. 5 followed by a comparison with the theoretical calculations in Sec. 6. The s u m m a r y is given in Sec. 7. 2

Fusion Barrier Distributions

Experiments with stable beams have shown t h a t fusion near the barrier is dramatically affected 1 2 ' 1 3 by coupling of the relative motion to the intrinsic degrees of freedom of the interacting nuclei. In an eigenchannel approach, this coupling effectively causes a splitting in energy of the single, uncoupled fusion barrier, resulting in a distribution of barrier heights 1 4 , around the uncoupled (average) barrier. This is manifested most obviously as an enhancement of the fusion cross-sections at energies near and below the average barrier. Experimentally, a represention of the distribution of barrier heights can be obtained 1 5 from precisely measured fusion cross-sections crfU5, by taking the second derivative of the quantity E<7fus with respect to energy E. Extrac-

256 tion of meaningful values of the quantity d2(E<jfas)/dE2 requires the fusion excitation function to be measured to high precision 1 6 with typical uncertainties of 1%. T h e d a t a presented in this contribution permit the extraction of the experimental fusion barrier distribution, and we utilise this information to place constraints on the parameters of the fusion model, thus allowing a quantitative determination of the suppression of fusion at above barrier energies.

3

Experimental Procedure

T h e experiments were performed with pulsed 7 Li and 9 Be beams from the H U D t a n d e m accelerator at the Australian National University. Targets of na ' B i and enriched 2 0 8 P b S of thicknesses 1 m g / c m 2 and 340 - 400 fJ-g/cm2 respectively, were used. Two monitor detectors, placed above and below the b e a m axis, measured the elastically scattered beam particles for normalisation purposes. Aluminium catcher foils placed immediately behind the targets were used to stop the recoiling heavy reaction products. The cross-sections for the evaporation residues were determined by measuring the a-particles emitted during their subsequent decay, as detailed in Ref. [11]. The products were identified by their characteristic a-energies and half-lives, with the latter ranging from 270 nsec to 138 days. Alpha particles from short-lived activity (half-life 7\/2 < 24 min) were measured in-situ between the b e a m bursts, using an annular silicon surface barrier detector placed at a mean angle of 174° to the beam direction. Alpha particles from long-lived ( T i / 2 > 24 min) activity were measured using a silicon surface barrier detector situated below the annular counter, such t h a t the target and catcher could be placed at a close geometry to the detector after the irradiation. T h e 24 min activity ( 2 1 2 R n ) was used to normalise the solid angles of the two ct-detectors. Fission following fusion was measured during the irradiations using two large area position sensitive multi-wire proportional counters 1 7 , centred at 45° and —135° to the beam direction. Absolute cross-sections for evaporation residues and fission were determined by performing calibrations at sub-barrier energies in which elastically-scattered projectiles were detected in the two monitor detectors, the annular detector and the backward-angle multi-wire proportional counter 1 8 .

257 4 4-1

Complete and Incomplete Fusion Identification

of Complete

Fusion

Products

216

T h e compound nuclei R n and Rn, formed following the fusion of 7 Li + 2 0 9 Bi and 9 Be + 2 0 8 P b respectively, cool mainly by neutron evaporation. The cross-sections for the Rn isotopes are shown in Fig. 1(a) and (b). In addition to the a-particles from the decay of Rn nuclei, a - p a r t i c l e s from the decay of 2 1 ° - 2 1 2 p 0 and 2 1 1 ~ 2 1 3 A t nuclei were also observed. T h e Po nuclei were populated in b o t h the reactions, whilst the At nuclei were only produced in the 7 Li-induced reaction. T h e direct production cross-section for these nuclei are shown in Fig. 1(c) and (d). In principle, complete fusion followed by axn and xnyp evaporation could lead to Po and At nuclei. However the shapes of the excitation functions for these nuclei are distinctly different from those in Fig. l ( a , b ) , and are not typical of fusion-evaporation. T h e origin of Po and At yields were investigated by forming the same compound nuclei 2 1 6 R n and 2 1 7 R n at an excitation energy similar to the ones formed by the 7 Li and 9 Be induced reactions, but by using the fusion reactions 1 8 Q _J_ i98p t a n d I 3 Q _J_ 204 H g respectively. T h e production cross-sections of Po and At nuclei were found to be insignificant and the fusion cross-sections determined from the sum of the xn evaporation and fission cross-sections agreed with the predictions of a coupled-channels calculation and the Bass model 1 9 , indicating t h a t the xn evaporation yield essentially exhausts the total evaporation residue cross-section. Thus, the direct Po and At production observed in the reactions induced by the 7 Li and 9 Be beams cannot be due to complete fusion, and are attributed to incomplete fusion, as discussed in the next section. T h e observed fission cross-sections were a t t r i b u t e d to complete fusion, since fission following incomplete fusion should be negligible due to the lower angular m o m e n t u m and excitation energy brought in, and the higher fission barriers of the resulting compound nuclei. Thus, the cross-section for complete fusion, defined as the capture of all the charge of the 9 Be projectile, was obtained by summing the Rn xn evaporation residue cross-sections and the fission cross-section at each energy. 4-2

Products

of Incomplete

Fusion

T h e nucleus 7 Li has a very low threshold against breakup into lighter fragments, with its breakup into 4 He and 3 H having the lowest threshold (2.47 MeV). Therefore in its approach towards the target, the 7 Li nucleus m a y breakup into 4 He and 3 H nuclei, with one of the fragments subsequently fusing with 2 0 9 Bi to form the combined systems of 2 1 3 At or 2 1 2 P o . Similarly, for

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9

Be, breakup into 4,6 He or two a particles (or 8 Be) and a neutron is favourable compared to other light fragment partitions. Capture of one of the He nuclei by 2 0 8 Pb would lead to Po. The combined system formed though incomplete fusion will subsequently cool down by evaporation of neutrons leading to the observed Po and At nuclei. Capture of 8 Be by 2 0 8 Pb cannot be distinguished

259 from complete fusion and hence would be included in the complete fusion yield. 5 5.1

Results Complete. Fusion

Cross-sections

T h e complete fusion cross-section for 7 Li + 2 0 9 Bi and 9 B e -1- 2 0 8 P b , which, as explained in Sec. 4.1, is the s u m of the R n xn evaporation residue crosssections and the fission cross-section at each energy is shown in the upper panels of Fig. 2. 5.2

Determination

of the Average

Barrier

T h e experimental barrier distributions, d2(Ea{us)/dE2, extracted from the measured fusion excitation functions using a point difference formula, are shown in Fig. 2(c) and (d) for the reactions indicated. A c m . energy step of ~ 2 MeV was used for b o t h the reactions; for 7 Li -f 2 0 9 Bi, a step length of 3.5 MeV was used for E c . m . > 32 MeV. T h e average barrier positions obtained from the experimental barrier distributions for 7 Li + 2 0 9 Bi a n d 9 B e + 2 0 8 P b are 29.6 ± 0 . 4 MeV and 38.3±0.6 MeV respectively. T h e uncertainties were determined by randomly scattering the measured cross-sections, with Gaussian distributions of standard deviation equal to those of the experimental uncertainties, and re-determining the centroid. By repeating this process many times, a frequency distribution for the centroid position was obtained, allowing determination of the variance, and thus the uncertainty. 6

Comparison with Calculations

To predict the fusion cross-sections expected from the measured barrier distribution, realistic coupled-channels calculations 2 0 ' 2 1 were performed using a Woods-Saxon form for the nuclear potential chosen such t h a t the average barrier energy of these calculations matched t h a t measured. In the case of 7 Li + 2 0 9 Bi, coupling to the 1 / 2 - state in 7 Li at 0.48 MeV was included by approximating the coupling between the 3 / 2 _ ground-state and the l / 2 ~ state by 0 —* 2 rotational coupling 2 2 . T h e coupling strength was obtained from the experimental B(E2)1 value 2 3 . For 2 0 9 Bi, the collective 3 ~ , 5~ and the double-octupole phonon states were included in the calculation. T h e septuplet and decuplet of identified states 2 4 formed by the 3~ and 5~ collective excitations respectively, were each approximated 2 4 by a single level with a n

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energy equal to that of the centroid of each multiplet and a deformation length 24 corresponding to that of the combined states. For the 9 Be + 2 0 8 Pb reaction, couplings to the 5/2~ and 7/2~ states of the KK = 3/2~ groundstate rotational band 25 in 9 Be, and to the 3~, 5~ and the double-octupole

261 phonon 2 6 states in 2 0 8 P b , were included. Coupling strengths were obtained from the experimental ground-state quadrupole m o m e n t 2 5 of 9 Be, and experimental deformation lengths 2 7 for one-phonon states in 2 0 8 P b . Couplings to the double-octupole phonon states in 2 0 9 Bi and 2 0 8 P b were calculated in the harmonic limit. T h e results of these calculations are shown in Fig. 2 by the dashed lines. They reproduce satisfactorily the shape of the measured barrier distribution, which is nearly symmetric for 7 Li + 2 0 9 Bi and asymmetric for the 9 B e + 2 0 8 P b system. T h e area under the calculated distribution is a measure of the geometrical cross-section 7ri22, where R is the fusion barrier radius. The measured distributions show a much smaller area t h a n the calculations, despite the average barrier energies being in agreement. This disagreement is necessarily reflected in the cross-sections as well, where the calculated values are considerably larger t h a n those measured. This is in contrast to fusion with tightly b o u n d projectiles 1 6 ' 2 8 ' 2 9 ' 3 0 , 3 1 , such as 1 6 0 and 4 0 C a , where calculations which correctly reproduce the average barrier position and the shape of the barrier distribution, and hence has the right potential parameters and couplings, give an extremely good fit to the cross-sections. Agreement between the measured and calculated quantities can be achieved if the calculated fusion cross-sections are scaled by 0.73 and 0.68 for the 7 Li and 9 Be induced reactions respectively. The result of such a scaling is shown by the full lines in Fig. 2. This scaling factor will be model dependent at the lowest energies, as the calculations are sensitive to the types of coupling and their strength. However, at energies around and above the average barrier, the scaling factor is more robust to changes in the couplings or potential shape within the constraints of the measured barrier distribution. T h e suppression of complete fusion at energies above the barrier, observed in both the reactions, is attributed to the reduction in the incident flux due to the large breakup probability of 7 Li and 9 Be before they reach the fusion barrier. T h e high breakup probability is evidenced by the large incomplete fusion cross-sections which were observed in this experiment. It should be noted t h a t in this work, complete fusion is identified as the capture of all the charge of the projectile, and it includes those breakup events where all the breakup fragments are captured by the target. In addition, in the case of the 9 Be induced reaction, the complete fusion cross-sections include events where 9 Be breaks up into two alpha particles (or 8 Be) and a neutron, with the neutron escaping capture. Thus the experimental complete fusion yields m a y underestimate the breakup probability. As discussed above, complete fusion can be defined conceptually more clearly t h a n can be achieved experimentally. Thus, when comparisons between calculated and measured fusion cross-sections are made to determine

262 suppression or enhancement, care should be exercised to take into account any incomplete fusion channels which may have been included in the complete fusion cross-sections. Demanding a consistent picture of complete fusion crosssections and the experimental barrier distribution, as done in this work, is likely to give a more quantitative measure t h a n comparisions of cross-sections alone.

7

Summary

In summary, the precisely measured fusion excitation functions for 7 Li + 2 0 9 Bi and 9 B e + 2 0 8 P b , allowing determination of the fusion barrier distributions, show conclusively t h a t complete fusion at above-barrier energies is suppressed compared with the fusion of more tightly bound nuclei. T h e calculated fusion cross-sections need to be scaled by a factor 0.73 and 0.68 for the 7 Li + 2 0 9 Bi and 9 B e + 2 0 8 P b reactions respectively, to obtain consistency between the measured fusion excitation function and barrier distribution. T h e loss of flux at the fusion barrier implied by this result can be related to the observed large cross-sections for At and Po nuclei, which result from the capture by the target of the breakup fragments of 7 Li and 9 B e . T h e determination of the average barrier position has been crucial in obtaining a quantitative measure of the fusion suppression at above-barrier energies. This is possible due to the high precision measurements t h a t are currently achievable only for stable beams. Theoretical calculations, which aim to describe the effect of breakup on fusion should be able to explain the fusion d a t a obtained for these loosely bound stable nuclei. T h e additional advantage is the absence of a skin or halo in stable nuclei. These features, which can be present in unstable nuclei, are expected to result in cross-section enhancements. Their absence in stable nuclei makes it easier b o t h to isolate the effects of breakup, and to obtain a theoretical description of fusion and breakup. Thus, fusion with loosely bound stable nuclei should be excellent candidates to understand the effects of breakup on the fusion process and we hope t h a t these d a t a will stimulate quantitative theoretical calculations.

Acknowledgment s One of the authors (M.D.) acknowledges the support of a Queen Elizabeth II Fellowship of the Australian Research Council.

263 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

P.G. Hansen et al, Annu. Rev. Nucl. Part. Set. 4 5 , 591 (1995). A. Yoshida et al., Phys. Lett. B 3 8 9 , 457 (1996). J.J. Kolata et al., Phys. Rev. Lett. 8 1 , 4580 (1998). M. T r o t t a et al., Phys. Rev. Lett. 84, 2342 (2000). K.E. Rehm et al, Phys. Rev. Lett. 8 1 , 3341 (1998). N. Takigawa et al, Phys. Rev. C, 4 7 , R2470 (1993). M.S. Hussein et al, Phys. Rev. C 4 6 , 377 (1992); Phys. Rev. Lett. 72, 2693 (1994); Nucl. Phys. A 5 8 8 , 85c (1995). C.H. Dasso et al, Phys. Rev. C 5 0 , R12 (1994); Nucl. Phys. A 5 9 7 , 473 (1996). C. Signorini et al, Eur. Phys. J. A 2, 227 (1998). K. Hagino et ai., Phys. Rev. C61, 037602 (2000). M. Dasgupta et al, Phys. Rev. Lett. 82, 1395 (1999). M. Beckerman, Rep. Prog. Phys. 5 1 , 1047 (1988). M. Dasgupta et aZ., Annu. Rev. Nucl. Part. Sci., 4 8 , 401 (1998). C.H. Dasso et al, Nucl. Phys. A 4 0 5 , 381 (1982); A 4 0 7 , 221 (1983). N. Rowley et al, Phys. Lett. B 2 5 4 , 25 (1991). J.R. Leigh et al, Phys. Rev. C 52, 3151 (1995). D.J. Hinde et aZ., Nucl. Phys. A 5 9 2 , 271 (1995). C.R. Morton et aZ., Phys. Rev. C 52, 243 (1995). R. Bass, Phys. Rev. Lett. 39, 265 (1977); R. Bass, Lecture Notes in Physics 117 Springer, Berlin (1980). K. Hagino et a/., Phys. Rev. Lett. 7 9 , 2014 (1997). K. Hagino et al, Comp. Phys. Commun. 1 2 3 , 143 (1999). P. Schumacher et al, Nucl. Phys. A 2 1 2 , 573 (1973). A. Weller et a/., Phys. Rev. Lett. 55, 480 (1985). M.J. Martin, Nucl. Data Sheets, 6 3 , 723 (1991). J.P. Glickman et aZ., Phys. Rev. C43, 1740 (1991); S. Dixit et al, Phys. Rev. C43, 1758 (1991). Minfang Yeh et al, Phys. Rev. Lett. 76, 1208 (1996). M.J. Martin, Nucl. Data Sheets 4 7 , 797 (1986). J.X. Wei et al, Phys. Rev. Lett., 6 7 , 3368 (1991). C.R. Morton et al., Phys. Rev. Lett., 72, 4074 (1994). A.M. Stefanini et aZ., Phys. Rev. Lett., 74, 864 (1995). J . D . Bierman et aZ., Phys. Rev. Lett. 76, 1587 (1996); Phys. Rev. C 54, 3068 (1996).

264 SPIN AND EXCITATION ENERGY DEPENDENCE OF FISSION SURVIVAL: A NEW PROBE FOR THE FUSION FISSION DYNAMICS

S. K. HUI, A. K. GANGULY AND C. R. BHUINYA Dept. ofPhysics, Calcutta University, Calcutta N. MADHAVAN, J. J. DAS, P. SUGATHAN, S. MURALITHAR, L. T. BABY, V. TRIPATHI AND A. K. SINHA Nuclear Science Centre, Aruna AsafAli Marg, Post Box 10502, New Delhi 110067 E-mail: [email protected] A. M. VINODKUMAR L. N. L., Padova, Italy D. O. KATARIA Milliard Space Science Lab., HolmburySt. Mary, Dorking, Surrey Rh5 6nt, UK N. V. S. V. PRASAD I. K.S., K.U. Leuven, Belgium P. V. MADHUSUDHANA RAO Department of Nuclear Physics, Andhra University, Visakhapatnam RAGHUVIR SINGH Department ofPhysics and Astrophysics, Delhi University A study of formation of cold residues in fusion reactions involving heavy nuclei is a topic of great interest from a variety of exciting reasons as it offers possible clues to the mechanism for forming nuclei towards limit of stability against fission. New Aspects of competition between fission and evaporation have been explored in terms of spin population leading to the formation of evaporation residues and fission for the l9F + l75Lu system. Spin distribution is seen to be critical for understanding the role of fission dynamics in survival against fission as it reduces the dependence on the model parameters. The data seem to rule out onset of dissipation at the pre-saddle regime of the fission process and suggest that the post saddle motion may be contributing to the excess of pre-fission emissions.

1

Introduction

New features indicating deviations from the statistical model have emerged over the decade of experimentation in the decay of the heavy fused compound systems formed in heavy ion collisions. These features concern with studies with light heavy ion beams having velocities quite small compared to the Fermi velocity

265

where fusion leading to complete equilibration is expected. It is generally being understood that the observed behaviour relates to the effect of dynamical delays due to the large scale re-arrangement of nucleons that accompany the heavy ion fusion and fission processes. One normally identifies certain milestones in the evolution dynamics of such heavy nuclear systems, viz., complete equilibration stage or the Compound Nucleus formation, saddle state and nuclear scission. Emission of photons and light particles (n, p, alpha-particles) could occur throughout the entire evolution process of the heavy nuclear system with probability controlled by the statistical parameters and potential. Experimental energy-angle correlations of the emitted particles and photons are fitted through simulations of the emissions from various stages of the evolution. These simulations are helpful in de-composing the total yields into contributions from different stages of the nuclear evolution. Sensitivity of the emission probability of charged particle and energy of GDR-photons to the deformation of the system is sought in some work [1,2] so as to identify the contributions from the pre-saddle and post-saddle stages. Observed yields of neutrons for fusion-fission reaction, on the other hand, could be broken up in to pre and post scission parts although subsequent breakup of the observed pre-scission yield into a pre-saddle and a post-saddle part is difficult due to relative insensitivity of neutron emission to the shape/deformation of the evolving nuclear system. All these emissions, however indicate enhanced yields during the pre-scission stage. The extraction of the pre-saddle part of the observed excess pre-neutron multiplicity has another important implication on the saddle point temperature which sensitively controls the fission fragment anisotropy within the standard saddle point model of fission. This pre-saddle evaporation of excess number of neutrons is, for example, able to account for a puzzling problem of larger angular anisotropy for the fission fragments [3]. However, the decomposition of the pre-scission emissions has its own uncertainties as use is made of model calculations of the evolution of the hot nuclear system employing a combination of fluid dynamics and statistical techniques [4]. There are also attempts made so as to provide an alternate explanation of the observed larger fission fragment anisotropy where non-equilibration of the spin projection K along symmetry axis is considered to be responsible for enhanced fission fragment anisotropy in the heavy ion fusion fission [5,6]. The excess pre-saddle emissions of neutrons and other light particles have significant implication on the probability of fission. As the systems in consideration have large fissility and are populated at states with initial excitation energy large compared to the fission barriers, the over all fission probability is quite high. The survival against fission is less probable but a measurement of this survival probability is very sensitive to small changes in the fission probability. Such a measurement requires a determination of the fusion-evaporation residue formation cross section alongwith that of the fusion-fission cross section.

266

A measurement of residue formation cross section for fusion-evaporation reaction gets progressively more difficult as for heavy and highly excited nuclei, fission becomes dominant. Special experimental techniques are used and measurements performed so far, have concentrated on the excitation functions. The available fission evaporation competition data, so far, are therefore integrated over the entire spin range. This results in an unhappy situation where a whole range of parameter space of the statistical model calculations could be consistent with the data. Thus due to the lack of exclusive and comprehensive data and also, for want of desirable precision in nuclear parameters, no definitive conclusions have been so far been possible about the effect of dynamical delays in the fusion-fission and fusion-evaporation competition Relevant parameters regulating the evolution of the compound nucleus are spin dependent and therefore, spin gated data on compound nucleus decay should provide for a comprehensive evaluation of models and concepts involved in understanding the fusion—fission dynamics. Here we report on our measurements of spin distributions associated with evaporation residue formation for a heavy compound nucleus 194Hg* over a range of excitation energies and spins. The measurements have been performed over a range of energies around the Coulomb barrier using the 19F beam where the quasi-fission channel is expected to be absent. This permits extraction of the fission survival probability of a hot rotating heavy nucleus as a function of spin and excitation energy. The dissipation effects are expected to be already set in for the fissility and excitation energy range chosen and therefore the measurements provide comprehensive tests of the ideas underlying the phenomenon of the fission hindrance [7]. 2

Experiments and Discussion

Heavy Ion Reaction Analyzer

(HIRA)

48 mm x 48 mm 2-D Pos. Sensitive Double Sided Silicon Strip Detector Fig. 1 Schematic layout of the setup used for spin distribution measurements

267

The 19F beam was delivered by the 15UD Pelletron at Nuclear Science Centre (NSC) at New Delhi. The recoil spectrometer HIRA [8] at NSC has been used to carry out measurements of the ERs for fusion reactions leading to heavy compound systems. A 14 element BGO array mounted close to the target is used for the y-multiplicity measurement in coincidence with the ERs detected at the focal jlane of HIRA. A schematic layout of the setup is shown in Fig. 1. 1000

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l9

F+,75Lu system. Cross section

268

One of the problem in such studies with recoil separator is the clean identification of these low energy heavy residues reaching the focal plane from the low energy tail of the scattered beam background. For example, the recoil Separator FMA at ANL, USA has been used for ER-formation cross section measurements for highly fissile compound system for the reaction with 32S on 184W [9]. A time of flight measurement over a flight path of few tens of cms. was carried out using a pair of transmission and stopping detectors at focal plane. With 32S beam the quasifission component is substantial, which implies additional complexity in the analysis of the data as pointed out earlier. However, if one goes for lighter beam, the recoil energies get very low and it may be difficult to do the TOF and total energy measurement using the method adopted for FMA at ANL. At NSC, measurements are carried out for lighter beams. Measurements of the ER-formation cross sections have been made for the l9F+175Lu and !9F+'8lTa systems while the spin distributions have been studied for the former system. A measurement of the total flight time from target to the focal plane of the spectrometer HIRA was used to obtain a clean identification of the ERs. The start trigger was either obtained from the 14 BGO detector array at the target site or from the RF when pulsed beam with large pulse separation of 4 us is used. High singles count rate seen by the BGO array could be handled as detector signals were processed individually and time recorded using a multi-channel TDC. The stop signal and total energy were obtained by using a large 48 mm X 48 mm double sided silicon strip detector with 3 mm wide strips. An excellent beam background rejection was obtained in a TOF vs. total Energy plot. The charge state distribution and the energy distribution of the ERs were measured by setting the electromagnetic fields appropriately. Fig. 2 shows the experimental angular distribution of the ERs. These distributions are used to extract the efficiency of HIRA for the detection of the ERs. A more direct and accurate method involves a measurement of the singles and coincidence y-rays emitted promptly from the ERs using a high resolution HPGe detector. Such a measurement has been done in the present case. The efficiency is obtained from the ratio of the coincidence data to singles data for the identified gamma-lines and is found to be ~ 1.5 % for the 19F+175Lu system at 110 MeV beam energy. Fig. 3 shows the measured ER-formation cross section. Also shown is data from an earlier work [10]. The fission cross section data [11] exist for the '9F+175Lu and have been used to obtain the fusion excitation function in the barrier region from the ER formation cross section data from the present measurement. The fusion excitation function was fitted using the coupled channel code CCDEF by incorporating the deformation of the target. The compound nucleus spin distribution was then taken from this calculation. Standard PACE calculations were done using Sierk fission barriers with only the barrier height scaling factor adjusted so as to fit the ER-cross section data at low energies. This factor was taken as 0.87 and was not varied over the beam energy range studied. Fig. 4 shows the data for the 19F+' 'Ta

269

system where our measurements have been compared with the data from Hinde et al. [12].

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270

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271

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272

Fig. 5 shows the mean angular momentum and the mean square angular momentum obtained from the measured spin distribution and the result of the PACE calculation. The overall quality of fit is found to be good indicating that angular momentum dependence of the Sierk barrier is quite reasonable. One remarkable point about the excitation functions is that we do not see the rise in ER formation probability as expected from the fission hindrance effects as seen from the partcile emission data. The moments of the spin distribution data are also well described by using standard statistical model calculations. In order to understand the role of nuclear friction, detailed calculations have been done for the beam energy at 125 MeV [13]. The calculations were done with and without nuclear dissipation. It has been found that calculations performed with a viscosity of y=10 worsen the fit significantly while the standard Cascade calculations reproduce the observed spin dependent ER-formation cross section data extremely well. The energy regime considered here shows the additional preponderance of pre-fission neutron emission indicating the presence of dissipation effects. Thus a simultaneous fit to data on the probes for fusion fission dynamics , viz., pre-fission neutrons, ER-formation cross-section will run into conflict. It may be quite possible to consider the post saddle emission as the more likely source to the excess prefission neutrons here as that will provide a consistent explanation. At the same time, fission fragment anisotropy data will have to look for an alternate explanation different from that associated with the cool down at the saddle by pre-saddle neutron emissions and new insight may be needed , as for example, considered in the recent work [5,6]. 3.

Conclusion

A study of formation of cold residues in fusion reactions involving heavy nuclei is a topic of great interest from a variety of exciting reasons as it offers possible clues to the mechanism for forming nuclei towards the limit of stability against fission. These investigations on fission survival of hot rotating nuclei probe sensitively certain unique aspects of fusion fission dynamics and provide information which is complementary to that extracted from other probes, viz., emission of photons from GDR decay and particle evaporation. New aspects of the dynamical competition between fission and evaporation have been explored in terms of spin population leading to evaporation residues and fission. The ER spin distribution is an important additional physical parameter for a critical evaluation of the concepts underlying the description of the fusion fission and evaporation process. The data seem to rule out onset of dissipation at the pre-saddle regime of the fission process. The post saddle motion may be contributing to the excess prefission neutron emissions.

273

4.

Acknowledgements

We thank the accelerator crew of the NSC, New Delhi for efficient operation during the experiment. We are grateful to Prof. G. K. Mehta, Director NSC for his constant encouragement during the entire duration of the project. We thank Dr. Ambar Chatterjee and Dr. S. Kailas for permitting us to use their Lu target. References [I] J. P. Lestone, Phys. Rev. Lett. 70, 15(1993)2245 [2] R. Butsch, D. J. Hofrnan, C. P. Montoya, P. Paul, and M. Thoennessen, Phys. Rev. C44 4(1991) 1515 [3] C. R. Morton, D. J. Hinde, J. R. Leigh, J. P. Lestone, M. Dasgupta, J. C. Mein, J. O. Newton, and H. Timmers, Phys. Rev. C52, 1(1995)243 [4] P. Froebrich and H. Rossner, Z. Phys. A 349, (1994)99 [5] J. P. Lestone Phys. C59, 3(1999)1540 [6] D. V. Shetty et al. Phys. Rev. C58 (1998)R616 [7] M. Thoennessen and G. F. Bertsch, Phys. Rev. Lett. 71, 26(1993)4303 [8] A. K. Sinha, N. Madhavan et al.Nucl. Instr. Meth. A 339, 543 (1994). [9] B. B. Back et al., Proc. of the International Workshop on Physics with Recoil Separators and Detector Arrays, Nuclear Science Centre, New Delhi-67, 1995, page 22 [10] M. Rajagopalan, D. Logan, Jane W. Ball, Morton Kaplan, Hugues Delagrange, M. F. Rivet, John M. Alexander, Louis C. Vaz, M. S. Zisman, Phys. Rev. C 25, 5(1982)2417 [II] A. Chatterjee et al., Proc. of Dept. of Atomic Eenergy Symp. On Nucl. Phys., Vol. 39B, 170 (1996) [12] D. J. Hinde et al., Nuclear Physics A 385 (1982) 109 ). [13] M. Thoennessen, G. Gervais, National Superconducting Cyclotron Laboratory, and Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824-1321, USA : Private Communication

274 SUB-BARRIER FUSION AND MULTI NUCLEON TRANSFER IN MEDIUM-HEAVY NUCLEI F.Scarlassara, S.Beghini, G.Montagnoli, G.F.Segato Universita di Padova and INFN sez. Padova L.Corradi, A.M.Stefanini, A.M.Vinodkumar INFN, Laboratori Nazionali di Legnaro

The present contribution reports on a selection of recent measurements in the field of sub-barrier fusion with the aim to pinpoint the effect of transfer in the enhancement of fusion below the Coulomb barrier. However, the result is that probably we cannot say thefinalword.

1 Introduction It is widely accepted that nuclear fusion around the Coulomb barrier is strongly affected by coupling to inelastic channels of the projectile and target nuclei like inelastic excitations and nucleon transfer. A large body of experimental and theoretical studies leave little doubt on that, but there are open problems when one tries to reproduce precise experimental data in detail. For instance, the role of transfer is not yet clear. Evidence for its role may be inferred from correlations between the sub-barrier enhancement and transfer cross sections and Q-values, but such comparisons are not conclusive either because the data are not precise enough or because of computational difficulties, so that exact CC calculations are limited in practice to inelastic excitations. The problem is complicated by the complex nature of the transfer process as can be appreciated from recently measured multinucleon transfer reactions in 40Ca+124Sn [1] showing a rich variety of exit channels often with broad Q-value distributions. The so called "barrier distribution method" pioneered by N.Rowley [2] is based on the realization that accurately measured fusion excitation functions may display details linked to the specific coupling mechanism: they should be reproduced by a model that pretends to understand the reaction process. The experimental research performed in recent years at LNL on nuclear sub-barrier fusion is mostly concerned with two main topics: the effect of vibrational, multi-phonon couplings and the effect of transfer coupling in mediumheavy systems. The recently measured fusion reactions 40Ca + 90'96Zr [3] (see fig.l) are significant in both respects: 40Ca + 90Zr represents a typical case where fairly good agreement is found with exact CC calculations including only

275 1

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Figure 1. Excitation functions (top) and barrier distributions (bottom) of 40 Ca + 90Zr (open circles) and 40 Ca + 96Zr (fall circles). The lines show CC calculations including only inelastic excitations. In the top panel, the dashed line refers to 40Ca + ^Zr and the fall line to 40Ca + %Zr.

the inelastic excitations of 90Zr (to a large extent, 40Ca behaves here like an "inert" nucleus, which means the effect of its inelastic excitations reduces to a renormalization of the nuclear potential), however the same approach failed with 40 96 40 90 40 Ca + 96Zr The difference between Ca + Zr and Ca + Zr is striking, both in the low energy enhancement and in the barrier distribution, the latter system displaying a much larger enhancement at low energy with a broad, structureless barrier distribution. Such a failure led the authors to conclude that transfer coupling

276 was presumably responsible for the "missing cross section" at low energy as well as the barrier distribution. The claim was backed by the large difference in the transfer Q-values which clearly favours nucleon transfer with 96 Zr compared to 90 Zr (table 1). At that time, this was considered in fact the most clear cut evidence of such an influence. That was the starting point for the investigations reported here. On the one hand, there was a strong suggestion of transfer influence in 40 Ca + 96 Zr where no transfer cross sections were known, whereas large transfer cross sections, populating a large variety of exit channels had been measured for 40 Ca + 124Sn which, incidentally, is characterized by similar transfer Q-values. A two-pronged strategy involved measuring transfer cross sections for 40 Ca + 90 ' 96 Zr and fusion for 40 Ca + 124Sn in order to compare the results. Table 1. Transfer Q-values (ground state to ground state) of a few of the most significant channels of the systems considered here. The listed values are for neutron pickup for all systems except 36 S+ 90 Zr where they represent neutron stripping (stripping and pickup from the point of view of the lighter projectile). 4U

Ca+9UZr

40

Ca+ 96 Zr

4U

Ca+ 1M Sn

36

S+ 90 Zr

36

S+ 96 Zr

In 2n 3n

-3.6 -1.4 -5.9

0.5 5.5 5.2

-0.1

5.4 4.5

-2.7 -1.0 -5.7

-3.5 -2.0 -5.8

4n

-4.3

9.6

9.5

-6.2

-4.8

Table 2. Deformation lengths and energies (in MeV) of the 2 + and 3" levels of the nuclei considered in this work. The B(E3) of the 3" • 0 + of 36S is not known. Nuclide 36S 4()

Ca

yo

Zr

96

Zr

124

Sn

E

(MeV) 3.291 4.192 3.904 3.737 2.186 2.748 1.751 1.897 1.132 2.602

D 2+

D 0.16

3"

2+ 3" 2+ 3" 2+ 3" 2+ 3"

0.12 0.40 0.09 0.22 0.08 0.27 0.10 0.11

277

Another important check would be measuring fusion in systems involving the same 90 Zr and 96Zr isotopes in combination with a projectile as "inert" as 40Ca (i.e. whose excitations do not modify the barrier distribution substantially) but without the positive transfer Q-values of 40Ca + 96Zr. 36 S + 90,96Zr fitted the requirements nicely: in this case both combinations have Qvalues very similar to those of 40Ca + 90Zr. 2 Fusion of 36S+90'96Zr The idea behind this study is the following: if transfer is responsible of the effects observed in 40Ca + 96Zr we expect that 96 40 90 the barrier distribution of S + Zr should resemble that of Ca + Zr 96^ w TO 4
Monitors

beam MCP

IC

Figure 2. Schematic representation of the experimental setup used for the fusion measurements (electrostatic deflector [4]) and the transfer measurements (ToF telescope "Pisolo" with two quadrupole doublets [5]) described in this work. They are connected to the same sliding seal scattering chamber. Micro-Channel Plate detectors are used for time pick up in both arms.

278

The device, operated at 0°, separates the evaporation residues from the beam-like ions; a further separation is accomplished by means of the E-ToF method. Absolute normalization is obtained thanks to four monitor detectors placed at a small angle, which also allow to correct for small deviations of the beam. For more information on the experimental setup refer to any of our experimental works [1,3,4,6,7].

75

80 E e m (MeV)

85

70

75

80

E e m (MeV)

Figure 3. Same as in fig. 1 for 36S + *°%Zr. The inset compares 36S + w^Zx (open and full circles respectively) with 40Ca + 1,0'%Zr (open and full squares respectively in a reduced scale. Whereas all systems agree at high energy, differences emerge at low energy: 40Ca + 96Zr displays the largest enhancement, followed by 36S + %Zr while 36S + ""Zr and 40Ca + "'Zr overlap each other.

279 The result can be seen in figure 3 and seems to favor the transfer effect hypothesis: indeed the excitation function of 36S + 96Zr can be reproduced by an "exact" CC code including only inelastic excitations (CCFULL code [8]), the reduced scale comparison reproduced in the inset shows that such a system indeed falls in between 40Ca + 90Zr and 40Ca + 96Zr, while the data of 36S + 90Zr behave as 40Ca + 90 Zr both in the reduced scale comparison and in the shape of the barrier distribution. Unfortunately, the barrier distribution of 36S + 96Zr is affected by rather large error bars and it is difficult to draw a conclusion: nevertheless, it is more compatible with a two-peak structure than a broad bump. These data provide nice evidence of the importance of the octupole vibration coupling and can only be reproduced if two-phonon excitations of 96Zr are included in the calculations. Higher phonon states have only a minor effect. 3 Fusion of 40Ca+124Sn Prior to the fusion measurement, detailed information was available on the transfer cross sections of this system showing a complex pattern of exit channels with large cross sections. The transfer Q-values are also very similar to those available in 40Ca + 96Zr as can be seen in table 1. The transfer data had been reproduced in [1] within an independent particle model, sequential nucleon transfer to a good overall agreement so that reliable and consistent form factors were available for the fusion calculations. On the other hand, inelastic excitations in 124Sn are rather weak compared to 96Zr (see table 2), so one may expect the effect of transfer to show up clearly in the fusion data. The excitation function and barrier distribution [7], measured as described in sec. 2, are reported in figure 4. The main features to be observed are: • The strong enhancement seen at low energy compared not only to the nocoupling limit, but also to the CC result including the 2 + and 3" excitations of target and projectile (dashed line), including multiple phonons in 124Sn which have little effect, however. Q The rather structureless barrier distribution, quite similar to the one found in 40 Ca + 96Zr. Unfortunately, coupling of the transfer modes is not possible within the CC code used for the inelastic coupling [8], so the calculation was performed with an approximate CC program [9], after adjusting the optical potential parameters in such a way as to reproduce the exact calculation when only inelastic excitations are considered. Because of the complexity of the transfer channels and in view of the approximate nature of the code, only "effective" neutron transfer channels were included: namely, In pickup is given an effective form factor which is a weighted sum of the single particle form factors of the transfer analysis, and an effective null Q-value (the optimum value for neutron transfer). For subsequent neutron transfer such as In • 2n etc. the effective parameters were obtained according to a recipe originally developed for multiple inelastic excitations [10]. The result of coupling

280

up to 4n transfer is displayed in fig. 4 as a full line; inclusion of further neutron transfer channels like 4n • 5n does not improve the result, whereas the coupling

UO

120

130

E cm ( M e V ) Figure 4. Excitation function and barrier distribution (insert) of Ca+ Sn. The dashed lines represent CC calculations limited to inelastic excitations, whereas the full lines show calculations including up to 4 neutron transfer.

to proton transfer was not attempted due to the approximate nature of the code and the "effective" way in which the transfer channels were included. The indication, though not conclusive, seems to be again that transfer and indeed multinucleon transfer is playing an important role in the penetration of the fusion barrier. 4 Nucleon transfer in 40Ca + 90'96Zr The nucleon transfer was also measured in 40Ca + 90'96Zr [11,12]. The experiment was performed with the ToF spectrometer shown in fig. 2, with a 40Ca beam at the following energies: 152 MeV (90Zr and 96Zr), 135.5 MeV (only 96Zr) and 139.8 MeV (only 90Zr), i.e. above the barrier and 4 MeV below the nominal barrier. Figure 5a shows an example of the complete mass and Z identification which is possible with the ToF spectrometer, while in figure 5b the angle and Q-value integrated cross sections are compared for 40Ca + 96Zr and 40Ca + 124Sn. The similarity is obvious and adds to the evidence presented so far, suggesting a link between the "anomalous" fusion enhancement / broad barrier distribution and transfer coupling.

281

However, one must be very careful in drawing such conclusions for a number of reasons. In fact, none of the above arguments is conclusive, they are only clues, not proofs of the effect we are investigating. For instance: • the calculations presented so far either cannot handle the couplings of multi nucleon transfer, or are very approximate. Q The correlations with transfer cross sections and/or ground state Q-values are qualitative at best: the transfer cross section depends on the optimum Q-value as well, while the virtual channels may be as important to fusion as the

outgoing ones. •

Inelastic excitations and transfer are not independent processes and separation is therefore impossible, neither theoretically nor experimentally. One cannot disentangle the effect of transfer and excitations since they affect each other: for instance, the deformation of the nuclear surface due to vibrations modifies transfer. It is essential that all processes be treated consistently in a realistic model, as was made possible by the recent upgrading of the GRAZING code [13], the same used to analyze transfer data [1,12], that can now calculate fusion as well. Calculations performed for the systems described in this work demonstrate a good agreement both in fusion cross sections and barrier distributions (besides transfer cross sections). Surprisingly, the conclusion seems to be that the difference between 40Ca + 90Zr and 40Ca + 96Zr is to be found in the different strengths of the octupole vibrations rather than transfer: exchanging the strengths of 90Zr and 96Zr actually reverses the result: the larger enhancement and broad barrier distribution are found in the system with the larger B(E3). The program does not allow to "switch off the transfer", that is calculate the effect of "pure" inelastic excitations. The reverse, i.e. switching off the excitations is possible, but the results are meaningless since the transfer (and its effect on fusion) are not the same one would have in the real world. So maybe some ambiguity persists as to the effect of transfer, but the question takes now a different meaning. Aknowledgments We would like to thank drs D.Ackermann, M.Bisogno, J.H.He, C.J.Lin, H.Timmers and L.Zheng who collaborated on some of the research reported here. We are also grateful to prof. G.Pollarolo in particular for the fruitful discussions and use of the GRAZING code.

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References 1. L.Corradi, J.H.He, D.Ackermann, A.M.Stefanini, A.Pisent, S.Beghini, G.Montagnoli, F.Scarlassara, G.F.Segato, G.Pollarolo, C.H.Dasso, A.Winther, Physical Review C54 (1996) 201-205 2. N.Rowley, G.R.Satchler, P.H.Stelson, Physics Letters B 254 (1991) 25

283

3. H.Timmers , D.Ackermann, S.Beghini, L.Corradi, J.H. He, G.Montagnoli, F.Scarlassara, A.M.Stefanini, N.Rowley, Nuclear Physics A 633 (1998) 421 4. S.Beghini, C.Signorini, S.Lunardi, M.Morando, G.Fortuna, A.M.Stefanini, W.Meczynski, R.Pengo, Nucl. Instr. Methods A 239 (1995) R1727 5. G.Montagnoli, F.Scarlassara, S.Beghini, G.F.Segato, A.M.Stefanini, D.Ackermann, L.Corradi, J.H.He, C.J.Lin, Nuclear Instruments and Methods A, in press. 6. A.M.Stefanini, L.Corradi, A.M.Vinodkumar, Y.Feng, F.Scarlassara, G.Montagnoli, S.Beghini, M.Bisogno, Physical Review C62 (2000) 014601 7. F.Scarlassara, S.Beghini, G.Montagnoli, G.F.Segato, D.Ackermann, L.Corradi, C.J.Lin, A.M.Stefanini, L.F.Zheng, Nuclear Physics A 672 (2000) 99 8. K.Hagino, N.Rowley and T.A.Kruppa, Computer Physics Communications 123 (1999) 143 9. M.Dasgupta, A.Navin, Y.K.Agarval, C.V.K.Baba, H.C.Jain, M.L.Jhingan, A.Roy, Nuclear Physics A 539 (1992) 351 10. R.A.Broglia, C.H.Dasso, G.Pollarolo, A.Winther, Physics Reports 48 (1978) 378 11. G.Montagnoli, S.Beghini, F.Scarlassara, G.F.Segato, L.Corradi, C.J.Lin, A.M.Stefanini, Proc. Fusion 97 workshop on heavy-ion collisions at near barrier energies, South Durras, NSW, Australia, March 17-21 1997, published in Journal of Physics G 23 (1997) 1431 12. G.Montagnoli, S.Beghini, F.Scarlassara, G.F.Segato, A.M.Stefanini, L.Corradi, C.J.Lin, G.Pollarolo, A.Winther, in preparation. 13. G.Pollarolo and A.Winther, in preparation.

284

INTERPLAY BETWEEN FUSION, TRANSFER AND BREAK-UP REACTIONS AT NEAR - BARRIER ENERGIES. P. R. S. GOMES1'2, J. LUBIAN1-2, S. B. MORAES , J. J. S. SANTOS1, A.M.M. MACIEL1, R. M. ANJOS 1 , I. PADRON 12 , C. MURI1, R.LIGUORI NETO3, N.ADDED3 1. Instituto de Fisica, Universidade Federal Fluminense, Av. Litordnea s/n, Gragoata, Niteroi, R.J., 24210-340, Brazil 2.

Permatent Address CEADEN, Havana, Cuba, P.O. Box 6122.

3.

Departamento de Fisica Nuclear, Universidade de Sao Paulo, Caixa Postal 66318, Sao Paulo, S.P., 05315-970, Brazil

The role of the breakup process of stable weakly bound light projectiles and the transfer channels on the near barrier fusion reaction and elastic scattering are investigated by different approaches. From a semi-classical formalism it is discussed the conditions that should be fulfilled by a transfer channel, in order to be important as a doorway to fusion. The measurement of the elastic scattering for the 6,7Li + 138Ba and 'Be + 64Zn systems were used to study the behavior of the threshold anomaly and its consequences for the fusion process. The fusion cross sections for the 9Be + 64Zn system were compared with the ones from other similar systems where no break-up is expected. There are signatures of fusion suppression only for the 6Li + l38Ba system.

1

Introduction

It is quite alive the interest in the investigation of reaction mechanisms induced by heavy ions at near and sub-barrier energies. At low energies, the different reaction mechanisms have strong couplings that are very closely related with nuclear structure properties. Due to the strong interplay between the reaction mechanisms, when one wants to understand one of them, one has to study some others. A simultaneous analysis of different reactions and scattering processes, by using a unique potential, is one of the main goals to be achieved [1]. It is well understood that the fusion cross section enhancement at subbarrier energies, relative to the unidimensional barrier penetration model predictions, is due to the splitting and lowering of the Coulomb barrier, when collective degrees of freedom such as the permanent deformation of the nuclei and/or their surface vibrations are taken into account. The fusion enhancement may also arise by the additional attraction in the incident channel. Transfer channels may act as doorway to fusion in a complex multi-step process, where the point of noreturn from fusion may be situated at distances larger than the position of the barrier. On the other hand, transfer channels with large cross sections and taking

285

place at large distances may suppress the flux going to fusion. In this paper it is discussed the role played on the fusion, by the distance where the transfer mechanisms take place. Another related subject of recent interest is the investigation of the role of the break-up process of weakly bound nuclei on the fusion mechanism, at near and sub-barrier energies. The suitable stable projectiles for these studies are 6,7Li and 9 Be. The small separation energies of 9Be ( 9 Be -> 8Be + n - Sn = 1.67 MeV), 6Li ( 6 Li-> 4He + 2H - S a = 1.48 MeV) and 7Li (7Li -> 4He + 3H -S„ = 2.45 MeV) should favor the break-up process, but the consequence of that on other reaction mechanisms is not yet clear. The understanding of the reaction mechanisms induced by those projectiles should be important for the understanding of reactions induced by the radioactive n Li and u Be beams. Information on the role of nuclear and Coulomb break-ups or of their competition may be obtained by the study of systems from light to heavy mass targets. At the present, there are very few experimental data on this subject. The theoretical predictions are also preliminary and controversial [2-6]. They may predict either the enhancement of the fusion due to the coupling of this additional channel [4,5] or the suppression of the fusion due to the break-up [6]. The role of the breakup process on the fusion may also be different at sub-barrier and above barrier energy regimes. Fusion data for light systems (At<25) [6] show a strong suppression of the fusion for 6'7Li, 9Be induced reactions, where the ratio fusion/reaction cross sections decreases with the increase of the break-up probability. For the heavy 9Be + 208Pb and 7Li + 209Bi systems, important fusion suppression has also been observed above the Coulomb barrier [7,8]. Systems using radioactive beams have been recently studied, 911Be + 238U [9] and 91011 Be + 209Bi [10], where fusion suppression were also observed at energies above the Coulomb barrier, whereas large fusion enhancement was observed for the 6He + 209Bi system at sub-barrier energies [11]. In this paper we present and discuss results on the fusion of 9Be + 64Zn and on the elastic scattering of 6,7Li + 138Ba and 9Be + 64Zn systems, at energies close to the Coulomb barrier.

2

The coupling between fusion and transfer reactions

A difficult task in the investigation of the role of transfer channels on the fusion cross section is the derivation of the corresponding transfer form factors. However, at low energies, and for heavy systems, semiclassical approximations are suitable to be used in the description of the transfer process [12,13]. Although there are some signatures that transfer channels with large Q-values or large cross sections may couple with the fusion and contribute to its enhancement at sub-barrier

286

energies, the relation between fusion and transfer reactions is not so clear, because they take place at different distances. At energies near the Coulomb barrier, the following ansatz is made[12,13]:

F(r) = F (
(i)

where Dc is the core distance, deduced from the elastic scattering data, a is the slope factor, F0 is a normalization factor and cp(r) is a smooth function that limits F(r) towards a maximum close to the Coulomb barrier position. The slope parameters a of the form factors can be extracted experimentally from the plot of the logarithm of the transfer probability versus the distance of closest approach. The normalization factors FQ are derived from the fits of the Qintegrated experimental angular distributions at small angles (or large distances of closest approach), when these reaction mechanisms are supposed to be simple onestep processes. Then, one can derive the transfer cross sections for very backward angles. The coupled channel calculations of transfer channels with fusion can, then, be performed. Liang et al[14] have measured differential cross sections for the most important transfer channels in the 32S + 92,98,100Mo systems, at near barrier energies. Large transfer cross sections were observed for the 98,100Mo, while small values were observed for the 92Mo. The large fusion cross section enhancement for the 32S + 98,100Mo [15] could not be explained by coupling just the inelastic channels. This was possible only for the small enhancement of the 32S + 92Mo fusion excitation function. We have applied this formalism for these systems [16]. The main transfer channels for the 32S + l00Mo are the stripping of one and two protons (-lp and -2p) and the pick-up of one and two neutrons ( + In and + 2n). For the 32S + 92 Mo system the same stripping channels are important, but there are no measurements of the neutron transfer channels. For 32S + 100Mo, the +2n channel has the largest Q value, the largest form factor at the barrier and the smallest average transfer distance, < dTrans > = 1.44frn , not far from the position of the Coulomb barrier ( rB = 1.38 fin). For all the channels, there are "missing cross sections" at backward angles, although this effect is most impressive for the + 2n channel. For the 32S + 92 Mo system, this effect was found to be much smaller. Then, simplified coupled channel calculations were performed by the CCFUB code [13]. The results for the 32S + 100Mo system show some small contributions to the fusion cross section enhancement due to the coupling of the -lp, -2p and +ln channels, and a large contribution from the +2n channel, at the low energy limit. For the 32S + 92Mo system, the additional contribution to the fusion from the -lp and -2p transfer channels can hardly be distinguished from the inelastic couplings. In both systems, the experimental fusion excitation functions could be fitted, with the additional coupling of transfer channels.

287

We also studied the transfer couplings for the 160 + i44,i48,i5o,i52,i54Sm systems [17], for which the fusion excitation functions have already been explained by the coupling of just inelastic channels. Transfer angular distributions for the main channels were analyzed by the semiclassical method. Ten transfer channels were studied: stripping of one alpha and stripping of two protons for each system. For only one of them, the stripping of two protons for A = 144, it was observed a very small "missing" cross section at backward angles. For the other nine channels, good fits of the whole angular range were obtained. The reduced average transfer distances for all the channels were calculated to be at least of the order of 1.57 fm, i.e. much larger than the position of the Coulomb barrier ( rB = 1.38 fm). Simplified coupled channel calculations, including the transfer channels, were performed by the CCFUSB code. The results showed no contribution of the transfer channels to the fusion cross section enhancement. The conclusion is that the channels for which there are "missing" transfer cross sections contribute to the fusion enhancement, whereas the others give no-contribution to the fusion cross section. Figures la and lb are two examples of transfer angular distributions for which: (a) there is the "missing transfer cross section" at backward angles and (b) the whole angular distribution can be predicted by the simple semiclassical description. :

1

1

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Figure 1. Transfer differential cross sections for the (a) + 2n channel for the "S +'"" Mo system, (b) +a channel for the l 6 0 + 150Sm system. The symbols represent experimental data and the curves are obtained by the fitting of the data at the smallest angles, within the semiclassical approach described in the text. In (a) there is a large "missing transfer cross section" at backward angles. In (b) the whole angular distribution is well described by the simple semiclassical model.

Therefore, transfer reactions that occur at distances not so far from the position of the Coulomb barrier, corresponding to very steep form factors, are the natural candidates to behave as doorway to fusion. This means that the absorption of

288

flux from the elastic channel, leading to fusion, may start at distances larger than the position of the Coulomb barrier. Transfer reactions which take place at large distances are in competition with fusion, and may inhibit the fusion cross section. 3

Fusion Cross Section for 9Be + 64Zn

101-

6

* 10 2

u Fusion cross section Reaction c. s. (volume part only) Reaction c.s. (volume + surface parts) m 20

30 E

[MeV]

cm. Figure 2. Comparison of the experimental fusion cross sections (full circles) with the reaction cross sections derived from optical model analysis, for the 'Be + 64Zn system. The full stars correspond to the calculations with volume and surface parts of the imaginary optical potential, and the full triangles, with volume part only (see text in section 4, for details).

We have measured, at the 8 UD accelerator of the Universidade de S3o Paulo, the fusion excitation function for the 9Be + 64Zn system, at energies from the Coulomb barrier to 50% above this value [18]. Our group had measured before the fusion cross section for the 16 0 + 64Zn [19,20] and 14N + 59Co systems [21,22]. The latter leads to the same compound nucleus as 9Be + 64Zn. This would allow a comparison between these middle size systems and the investigation of the role of the 9Be breakup on the elastic and fusion processes. The experimental method used was the gamma ray spectroscopy method [23]. Two HPGe detectors with Compton suppressors were used, and placed at + 55° with the beam direction. The energy resolution of the detectors was 2 keV for the 1332 keV line of 60Co. Single and coincidence spectra were measured, and the cross sections were determined by the addition of the two single spectra, for each energy.

289 For each bombarding energy, in-beam and off-line decay spectra were accumulated. The number of incident particles was determined by Coulomb excitation of the thick l81 Ta backing. All the gamma lines of the spectra were identified and, in order to avoid misinterpretation of their origin, the identifications were done not just by their energies, but also by their relative intensities and shape of the excitation functions. A source of uncertainty in the cross section determination is the direct feeding of the ground state of the residual nuclei that, however, should not be important at this energy and mass range.

*•

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10 20 "c.m.-red

40 [MeV]

Figure 3. Comparison of reduced fusion excitation functions for three different systems. Be + Zn and 14 N + 59Co lead to the same compound nucleus. 'Be + 64Zn and 16 0 + 64Zn differ by the projectiles: weakly bound and strongly bound.

The total fusion cross sections were obtained by adding the cross sections of eleven evaporation channels. Figure 2 shows the fusion cross sections and the total reaction cross section, derived from fit of the elastic scattering data as described in the next section. One can see that they are quite similar in the whole energy range, leaving no room for a significant cross section for any other reaction mechanism, including the break-up. Figure 3 shows the reduced fusion excitation functions for three systems: 9Be + 64Zn, lfiO + 64Zn and 14N + 59Co. One can also notice that there

290 is no fusion suppression for the 9Be induced reaction, when compared with the other systems. However, one should have in mind that there might be the contribution of incomplete fusion, following the break-up, on the derived fusion cross section. The 9 Be break-up process is 9Be -> 8Be + n - » a + oc + n. The reason is that the gamma rays emitted by the desexcitation of the incomplete fusion residual nucleus, formed through the channel x would be the same as the ones emitted by the desexcitation of the complete fusion residual nuclei formed through the axn channel. Therefore, what was measured might be the addition of complete and incomplete fusion. However, the complete fusion evaporation channels, which could possibly be contaminated by the break-up, have small cross sections and are in good agreement with the predictions of the evaporation code PACE [24]. Therefore, the present results are a strong signature that the 9Be breakup does not suppress the fusion cross section for this system, at energies above the Coulomb barrier. 4

The elastic scattering of the 6'7Li + 138Ba and 9Be + 64Zn systems

Another approach to study the influence of the break-up on other reaction mechanisms is through the detailed analysis of elastic and inelastic scattering, at near barrier energies. The role of the break-up process on the elastic scattering process is investigated by the analysis of the behavior of the energy dependence of the real and imaginary parts of the optical potentials. A dispersion relation [25,26] associates a peak in the strength of the real part of the optical potential, V, in the vicinity of the Coulomb barrier with the decrease of the imaginary part of the potential, W, as the bombarding energy decreases towards the barrier energy. This behavior is called threshold anomaly. It has been shown [25,26] that when the coupling of the inelastic and eventual strong direct reaction channels are taken into account, the anomaly is destroyed. The presence of the anomaly can, therefore, be interpreted as the effect of the strong coupling of the elastic channel with the inelastic and/or other direct reaction channels at near barrier energies. Consequently, it may also be interpreted as a signature of the fusion cross section enhancement. The coupling to the break-up channel may contribute as a repulsive effective potential, which may exceed the attractive term arising from the inelastic coupling to bound states or be of the same order of magnitude. Therefore, for very weakly-bound nuclei, this effect might be strong enough to affect the real part of the optical potential near the Coulomb barrier in such way that it leads to the absence of the threshold anomaly in the scattering of these projectiles [27]. For the 6'7Li + 138Ba, the experiments were performed at the 8UD Pelletron accelerator of the University of Sao Paulo. The beam energies ranged from energies just below the nominal Coulomb barrier to 50% above this value. The detection system was an array containing 9 silicon surface barrier detectors. The angular

291

separation between two adjacent detectors was 5°. In front of each detector there was a set of collimators and circular slits for the definition of solid angles and to avoid slit-scattered particles. The angle determination was made by reading on a goniometer with a precision of ± 0.5°. A monitor was used for normalization purposes. The energy resolutions of the detectors were of the order of 300 - 500 keV(FWHM), good enough to resolve the elastic and inelastic peaks. The angular distribution data were taken in the range 20° < 0iab < 165°, for the lower energies and up to 90° at the higher ones. The analysis of the elastic scattering data was performed by the ECIS code [28]. The real and volume imaginary potentials were of the Woods-Saxon form. The introduction of a surface imaginary potential of derivative form was required in order to fit the elastic scattering angular distributions and shows the importance of direct reaction channels. The ambiguities of the optical potentials were minimized by taking into account the values of the potentials at the sensitivity radius, found as the positions were the different families of optical potential parameters have the same value. i

*

*

1.0

" ±-

VJ * -

7

Li 6

•

:

-it - *

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Li * - - •

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25

30 ELab[MeV]

Figure 4. Dispersion relation calculations for the 6,7Li + 138Ba systems, at the radius of sensitivity. The curves represent the dispersion relation calculations: the dashed-line for 7Li + 138Ba and the full-line, for the 6Li + 138Ba system.

Figure 4 shows the energy dependence of the potentials, at the sensitivity radii, for both systems [29]. The error bars correspond to the range of deviation of the

292 potential, corresponding to distinct set of parameters with different values of diffusenesses and roughly the same %2- The curves are the results of the dispersion relation calculation [25]. The results of the analysis show important differences in the elastic scattering of the two Li isotopes. For the 7Li scattering, the usual threshold anomaly is present, whereas there are no evidences of it for the 6Li scattering. For the deformed nucleus 7Li, the presence of the anomaly can be interpreted as the effect of the strong coupling with its first excited state, the one neutron channel and other reaction channels, leading to an attractive potential. It may also be interpreted as a signature of the fusion cross section enhancement. The dissociation energy of 7Li into 4He + 3H is 2.45 MeV, much higher than the energy of its first excited state 1/2" (0.478 MeV). On the other hand, the spherical 6Li has a dissociation energy into 4He + 2H (1.5 MeV) much smaller than the energy of its first excited state and, consequently, the probability of exciting this state is small. Therefore, for this projectile, the break-up channel is the dominant direct channel, leading to weak couplings between the elastic and inelastic or transfer channels. The role of break-up channel in the total polarization potential has to be interpreted as being repulsive, and it may exceed, or be of the same order of, the weak attractive term arising from the inelastic coupling to bound states, resulting in the vanishing of the threshold anomaly of the optical potential. Similar results were obtained for the 67 ' Li + 208Pb systems [27]. Then, it should be expected that the fusion at low energies should be enhanced for 7Li relative to 6Li, whereas quasi-elastic reactions should be much stronger for the 6Li and dominated by the break-up. Indeed, the total reaction cross sections derived from the fit of the elastic angular distribution, at low energies , are larger for the 6Li + 138Ba system. For the 9Be + 64Zn system, the same experimental set up and similar data analysis were used. Figure 5 shows the values of the real and imaginary part of the optical potential at the radius of sensitivity for the 9Be + 64Zn. In this figure, two sets of optical potentials are shown, corresponding to the following assumptions. The first one, full stars, corresponds to the calculations with imaginary volume and surface optical potentials. The second one, full circles, corresponds to the calculation with only a volume shape for the imaginary part of the optical potential. One can see from this figure that the two sets have different behaviors for low energies. The calculations with only the volume part of the optical potential shows a drop of the imaginary part for lowest energies and the increment of the strength of the real part. This fact, following the conclusion for the elastic scattering of 7Li on 138 Ba, would allow us to say that the break-up channel has no effect on the elastic scattering for this system. On the other hand, the results with both parts of the imaginary part of the optical potential show almost no anomalous energy dependence of the real and imaginary parts of the optical potential, and could allow us to say exactly the opposite, i.e. that the break-up channel is the responsible for the absence of the threshold anomaly. Both assumptions for the imaginary part of the optical potential give very good fits of elastic angular distributions for this

293

system with the same %2 - value. The only difference between them is the value of the reaction cross section for 17 MeV that is of 27 mb in the case of only volume part and 66 mb when both volume and surface parts are included. i

-r

-

i It > 4i

i

-

•

Only volume part

•

Surface and volume parts

?

i

i

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-

i

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—

i

•

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i

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Lab [MeV]

J

Figure 5- Energy9dependence of the real and imaginary parts of the optical potential, at the radius of sensitivity, for the Be + MZn system. Two different variants were used (see text for details). So, in order to draw any conclusion on the influence of the break-up process on the elastic scattering, for this system, one has to measure the fusion cross section (the most relevant channel in the reaction cross section at energies bellow and near the barrier), for this lowest energy. The simultaneous analysis of both data would enable us to conclude on the presence or not of the threshold anomaly of the optical potential for this system. 5

Conclusions

In this paper, the role of the breakup of weakly bound projectiles and the transfer channels on the fusion and elastic scattering, at energies close to the Coulomb barrier, was discussed.

294 The transfer channels that are good candidates to behave as doorway to fusion, and consequently give rise to fusion enhancement at sub-barrier energies, are the ones with steep form factors. If the transfer occurs at distances close to the Coulomb barrier position, this may be a first - step of the reaction, leading to fusion and corresponding to an additional attraction at the entrance channel. On the other hand, if the transfer occurs at large distances, the coupling to fusion is not important and the transfer mechanism might induce a fusion cross section suppression. New very precise transfer data have become available, and at the present it is required more reliable theoretical treatment of transfer form factor derivation and coupled channels calculations including several transfer channels, in order to obtain a better quantitative understanding of the role of transfer coupling on the fusion process. The optical model analysis of the elastic scattering of the Li isotopes has shown an important difference: For the 7Li scattering, the well known threshold anomaly was observed, while it was vanished out for the 6Li scattering. The interpretation is that for the 7Li there are strong couplings between the elastic channel and the first excited state of 7Li, the one-neutron transfer and other direct reaction channels, leading to an attractive polarization potential. For the 6Li scattering, the breakup is the dominant direct channel, leading to weak couplings between elastic and inelastic or transfer channels, and producing a repulsive polarization potential that compensates the weak attractive one produced by the other direct channels. For the 9Be + 64Zn system, the analysis of the elastic scattering data was not conclusive about the presence of the threshold anomaly. One needs to measure the fusion cross section at the barrier energy in order to assess the influence of the break-up channel on the elastic scattering. On the other hand, the fusion cross section results for the 9Be + 64Zn system show a strong signature that the fusion cross section is not suppressed by the 9Be break-up, at higher energies. Comparing with the results obtained for 9Be on heavy nuclei, one can conclude that the Coulomb break-up plays a major role in the reactions with heavy targets, by suppressing the fusion at energies above the Coulomb barrier, but for lighter systems, when the nuclear break-up is predominant (or at least is competing with the Coulomb breakup) and occurs at short distances, the fusion suppression effect is not important. We believe that one needs more experimental data, both for fusion and elastic scattering cross sections, with different systems, before one can have a better understanding of this fascinating subject.

6

Ackowledgements

The authors would like to thank the Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico (CNPq), the CAPES and FAPERJ for their partial financial support.

295 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

R. M. Anjos et al; Journal of Physics G 23 (1997), 1423 N. Takigawa et al; Phys. Rev. C 47 (1993), R2470. M.Hussein et al; Phys. Rev. C 46 (1992) 377. C. H. Dasso, A. Vitturi; Phys. Rev. C 50 (1994), R12. A. Vitturi et al; Phys. Rev. C 61 (2000), 037602 J. Takahashi et al; Physical Review Letters 78(1997), 30 M. Dasgupta et al; Physycal Review Letters 82 (1999), 1395 M. Dasgupta et al; to be published V. Fekou-Youimbi et al; Nucl. Phyys. A 583 (1995), 811c A. Yoshida et al; Phys. Lett B 389 (1996), 457; C. Signorini et al; European Phys. Journal A 2(1998), 227 J. Kolata et al; Physical Review Letters 81 (1998), 4580 K.E. Rehm; Annu. Rev. Nucl. Part. Sci. 41 (1991) 429. L. Corradi et al; Zeitchrift fur Physik A335 (1990) 55. J. F. Liang et al ,Phys. Rev.C 50 (1994) 1550 R. Pengo et al; Nucl. Phys. A 411 (1983) 25 P.R.S. Gomes et al; Journal Physics G23 (1997), 1315 A.M.M. Maciel and P.R.S. Gomes; Phys. Rev. C 53 (1996) 1981 S.B. Moraes et al; Physical Review C61 (2000), 64608 C.Tenreiro et al; Physical Review C 53 , (1996), 2870 C. Tenreiro et al; Proc. Workshop on Heavy Ion Fusion: Exploring the Variety of Nuclear Properties; Padova, Italia (1994), 98. P.R.S.Gomes et al; Nuclear Physics A 534 (1991),429. C. Muri et al; European Physical Journal A 1 (1998), 143. P.R.S. Gomes et al; Nuclear Instrum. and Methods in Phys. Res. A 280 (1989), 395 A. Gavron; Phys. Rev. C 21(1980), 230 (1994), 326 G. R. Satchler; Phys. Rep. 199 (1991), 147 M. A. Nagarajan et al; Phys. Rev. Lett. 54 (1985) 1136 N. Keeley et al; Nuclear Physics A 571 J. Raynal; Proc. Workshop on Applied Nuclear Theory and Model Calculations for Nuclear Technology Applications, Trieste -1988, World Scientific (1988), 506 A. M. M. Maciel et al; Phys. Review C59 (1999), 2103

296

M E C H A N I S M S OF S U B - B A R R I E R FUSION E N H A N C E M E N T

Institut

N. R O W L E Y de Recherches Subatomiques (UMR7500), 23 rue du Loess F- 67037 Strasbourg Cedex 2, France E-mail: [email protected]

It is well known that couplings to highly collective excited states, possessing either a rotational or vibrational character, can give rise to enormous enhancements of the sub-barrier fusion cross section. This phenomenon has been elucidated by the use of the technique of experimental barrier distributions in cases where sufficiently detailed d a t a have been available. It is also well understood in terms of theoretical calculations involving a limited number of coupled-channels. Experimentally, very large effects have also been unambiguously attributed to strong transfer processes. These are, however, much more difficult to handle theoretically due to the large number of final states involved. This is particularly true for positive Q-values where the most important effects are to be found, and where the transfer may resemble a collective flow of nucleonic matter. Guided by the combined experimental d a t a on fusion and transfer reactions for the same systems, a simple qualitative approach will be presented which gives rise to a rather flat barrier distribution of the type proposed by Stelson. The problem is to use the approach predictively for cases where enhancements might, for example, be exploited in the creation of superheavy elements.

1 1.1

Introduction Coupling to Rotational States

The suggestion of extracting fusion barrier distributions from high accuracy experimental cross sections 1 was soon followed by the taking of such data at the ANU in Canberra. The system first chosen was l e O + 154 Sm, discussed in Ref. 1 as a case which should be relatively easy to understand, being an inert double-closed-shell projectile on a well deformed target. The results are shown in Fig. 1. The upper panel shows the data 2 along with a fit which simply integrates the fusion cross section over all orientations of the target but includes quantal penetration effects correctly. The lower panel shows the fusion barrier distribution:

taken with a point-difference formula employing an energy interval of around 2 MeV. The solid line shows the fit of the upper panel using the same energy interval and the dashed curve shows an adiabatic coupled-channels calculation including up to the 10 + member of the ground-state rotational band of 154 Sm.

297 Clearly the macroscopic treatment (orientation as a classically collective variable) and the microscopic quantal treatment of excited states are in excellent agreement 3 if enough excited states intervene. Since the width of the distribution (difference between the highest and lowest barriers) varies as Z1Z2P2, large effects are expected for strongly deformed heavy systems. T h a t is heavy systems displaying large collectivity.

50

55

60 65 Energy (MeV)

70

F i g u r e 1: T h e cross s e c t i o n a n d b a r r i e r d i s t r i b u t i o n for l e O + 1 5 4 S m . T h e solid c u r v e t r e a t s t h e t a r g e t o r i e n t a t i o n classically a n d t h e d a s h e d c u r v e is a n a d i a b a t i c c o u p l e d - c h a n n e l s c a l c u l a t i o n i n c l u d i n g u p t o t h e 10"*" s t a t e of t h e t a r g e t .

1.2

Coupling to Vibrational

States

Similar effects are expected for collectivity arising from vibrational modes and an excellent example can be found in the 5 8 Ni + 6 0 N i system studied at the INFN Laboratory in Legnaro 4 . The results are shown by solid circles in Fig. 2.

298 Here, the theoretical interpretation was rather more taxing and involved the coupling to the single- and double-quadrupole-phonon states in both the target and projectile, including mutual excitations. In addition it is necessary in this case to take account of the large excitation energies of the phonon states. In particular these allow large Coulomb distortions before the nuclei come into contact. 1000.0 100.0 2-

.*»•»•

10.0

•«

o Ni+ Ni data • "Ni+^Ni (+0.6 MeV)

E

(MeV)

*

o

I.O 0.1

100 Figure 2: A comparison of the two systems 5 s N i + rather than transfer channels dominate here.

no 58

> 6 0 Ni shows that phonon couplings

Of course the question arises in such systems as to whether nucleon transfers will also affect the fusion enhancement. In order to address this question, the open circles in the figure compare results of 5 8 Ni + 6 0 Ni with those 5 for 58 Ni + 5 8 Ni. While the phonon excitation energies and < / 3 | y1'2 are similar for both these nickel isotopes, the transfer Q-values are very different. T h e close similarity of the cross sections and barrier distributions for the two sys-

299 terns suggests, therefore, t h a t effects are dominated by the phonon couplings and t h a t transfers are relatively u n i m p o r t a n t . 1.3

Coupling to Transfer

Channels

In order to pursue the above question of transfer effects further, the Legnaro group 6 studied the reactions 4 0 C a + 9 0 ' 9 6 Z r . This is an excellent choice of system since it is really only from studying isotopic variations t h a t such effects can be unravelled. T h e first target has a closed JV=50 neutron shell and negative transfer Q-values, whereas the second target has 6 neutrons outside the N—hQ closed shell and positive Q-values for 1- and multi-neutron transfers.

10 J

1

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C a + 90Zr

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10',2 .

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E/B Figure 3: Of the two systems 4 0 C a + 9 0 ' 9 6 Z r , the lighter-target results are well fitted by phonon couplings (solid line; the dashed line shows the corresponding no-coupling result). For the heavier target, which has many positive-Q transfer channels, such a calculation fails completely.

The cross sections are shown in Fig. 3. The solid line shows the results of a coupled-channels calculation for the 9 0 Zr target again taking into account phonon excitations, whereas the dashed line is the corresponding no-coupling result (single barrier). The full calculation again gives an excellent fit to the experimental d a t a . However, a similar calculation for the 9 6 Zr target fails completely to produce a good fit. This has been a t t r i b u t e d 6 to the large effects

300 of transfers in this system where the TV > 50 neutrons can all be transferred with positive Q. For the closed-shell 9 0 Zr target, all Q-values are negative. 2 2.1

How to Treat Stelson

Transfers?

Model

Many years ago, Stelson 7 suggested t h a t transfer was a major factor in fusion enhancement. We now know t h a t many cases can be well explained 8 without transfer effects but it is clear t h a t for the above system and many others, the basic hypothesis of Stelson is correct. Furthermore, he suggested t h a t transfers would lead to a rectangular barrier distribution with the lower-energy edge being the point at which the nuclei come sufficiently close for a collective flow of neutrons to take place, t h a t is, at a distance of closest approach where the sum of the two neutron-nucleus potentials possesses a barrier whose height is equal to the neutron Fermi energy Ep. T h e corresponding incident energy was one of the parameters of his model, the second being the width of the rectangle, whose area, being unity, defines the upper edge.

85

95 Energy (MeV)

105

Figure 4: Stelson plot and barrier distribution for

Ca +

Zr.

The phenomenological basis for this model arose from the observation t h a t for many systems, the fusion cross section has the property t h a t [Ea)1'2 is linear around the region of the unperturbed barrier. (Note t h a t (Ecr)1/2 oc

301 (E — B) over a given region leads to a constant d2(E a)/dE2.) T h e Stelson model has no real explanation for the range of energies over which the barrier distribution is constant, nor any physical mechanism for the high-energy barriers. The linearity is shown in Fig. 4 for the 4 0 C a + 9 6 Zr system. Note t h a t the linearity does indeed persist over the region where D(E) is relatively constant. 2.2

Qualitative

Quantal

Model

In the case of rotational and vibrational modes, the barriers which possess an increased height have a n a t u r a l explanation: a deformed target nucleus possesses orientations for which its surface is further from the projectile (higher barrier), as well as orientations for which it is closer (lower barrier). Obviously a vibration can also lead to the surfaces moving further apart for the same centre-to-centre separation. In a similar vein, one can envisage the transfer of a neutron from a state 4>I{T\) in nucleus 1 to a state <^>2(r2) in nucleus 2. A coherent linear superposition of these two states exists where the the neutron-matter density between the partners is increased. Conversely, for the orthogonal linear superposition, the neutron-matter density will be decreased in this region. T h e former will lead to a lowered barrier a n d the latter to a raised barrier. Transfer effects will become important for systems where the overlap of the two wavefunctions is large at the unperturbed barrier. In practise, this may not be too different from the Stelson criterion of the total neutron-nucleus barrier being of the order of E-pIn principle, we should be able to calculate transfer effects approximately in a fashion similar to inelastic excitations, introducing some coupling strengths and treating the Q-value as a, possibly negative, excitation energy. A clue on how to perform such calculations would of course be given by the actual transfer d a t a in these systems, and for t h a t reason, Montagnoli et al. 9 have performed such measurements. Fig. 5 shows the one- and two-neutron transfer cross sections for the 96 Zr target taken at 6>iab = 65° and Ecm - 107 MeV. We see t h a t in both cases, despite the positive Q-values, the d a t a are peaked at slightly negative Q and t h a t the spectrum extends over a wide range of Q-values. A possible explanation for this is the following: transfer dynamics favour a Q-value of around zero in order to give good matching in the entrance and exit channels. However, the density p of final states in the system increases with increasing excitation energy. Therefore the transfer probability should vary as J

(2)

trans ( Q ) OC p(Qsg

- Q) Gdynamic(Q),

302

800 l - n ( Ca):Qgg=+0.51MeV 2-n ("Ca): Q^=+5.53 MeV

600

§ 400 u

200

0 -10

0 Q-value

10

Figure 5: Q-value spectrum for 1- and 2-neutron transfer at #jab = 65° and Ecm — 107 MeV for the 4 0 C a + 9 6 Z r system.

where G will be symmetric and strongly peaked at Q — 0 and where p will roughly increase exponentially. This explains (a) the basic shape of the Qvalue spectrum (b) why it is peaked just below Q — 0 and (c) why positive-Q systems display much stronger effects t h a n ones with negative Q-values 1 0 . Fig. 6 shows the angle-integrated cross sections of various transfer channels for Ecm = 94.5 MeV, close to the unperturbed barrier. T h e most striking feature of these d a t a is t h a t the transfer of many neutrons is possible (thick line) and t h a t the d a t a fall off roughly exponentially with the number of neutrons transferred. It is also clear t h a t for a large number of transferred particles (AiVtrans = |AiV| + | A Z | is the minimum number of transferred particles which take one from the initial to the final mass partition), proton channels become important and can give rise to larger cross sections t h a n pure neutron channels. The above d a t a suggest the following simple phenomenological model for a coupled-channels calculation: • Approximate the Q-value spectrum by putting all the strength into a single fictitious state where the d a t a are peaked. In the current talk we

303 100

o

Figure 6: Total transfer cross sections for various channels in the Ca -fZr reaction at ECm ~ 94.5 MeV, very close to the unperturbed barrier. In all cases the proton is transferred from the 4 0 C a . Solid and dashed lines represent neutron transfer to and from the 4 0 C a respectively and AiV t r a n s = \AN\ + \AZ\.

shall take this state to have Q = 0. • Take sequential coupling to these states with the same coupling constant at each step. (Since <JAN/&AN-I & constant.) • Consider only the neutron transfer chain since it is the only one coupled strongly in the first steps. T h e simplifying features of these choices will be discussed more extensively elsewhere. This leads to the following coupling matrix:

M,sequential

=v

/ 0 1 0 0 0...\ 1 0 1 0 0.. 0 10 10.. 0 0 1 0 1.. \ 0 0 0 1 0.../

(3)

304 and a model which contains relatively few parameters. They are • the height B of the unperturbed Coulomb barrier • the surface diffuseness a of the nuclear potential • the coupling strength V and of course the number of channels N one wishes to include. Fig. 7 shows the spectrum u of eigenvalues A for different values of iV in the transfer matrix M (the heights of the bars show the weights of the corresponding barriers). It is immediately apparent t h a t these results give rise to a rather flat symmetric spectrum (as demanded by Stelson's observations) with - 2 < A < +2. N=10

N=6

I

0.5 |

0.0 '

L

N=4

1

1 N=3

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

X Figure 7: Eigenvalue spectrum of the sequential-transfer matrix for different dimensions.

Fig. 8 shows a comparison with experimental d a t a for different AT, taking B - 95.7 MeV, a = 1.45 fm and V = 3.7 MeV. It can be seen t h a t a large enhancement of a can be obtained and t h a t the results converge rapidly with increasing N. For N = 7 (i.e. including transfers of all six N > 50 neutrons from the 9 6 Zr target) a good fit to the experimental d a t a is obtained despite the simplicity of the model. The fit to the d a t a can be further improved by allowing a slight increase in the coupling strength (12%) at each stage of

305 the transfer process. The resulting cross section and barrier distribution are shown in Figs. 9 and 10 respectively, again with the above parameters. Note in particular t h a t the structures apparent in D(E) are rather well fitted for this choice of channel number.

y ^ ^

10' --

All couplings equal

1 •

10"

10"

• Data

N=l N=3 N=7

80

90

100

-

110

Energy (MeV) Figure 8: Constant transfer coupling can give large enhancements and reproduce rather well the overall shape of the experimental cross section. Note the rapid convergence with increasing N.

3

Conclusions

T h e quality of the fit to the d a t a obtained with this calculation (Figs. 9 and 10) lends credence to the underlying validity of the kind of coupling scheme t h a t we have outlined in this talk. Many questions, however, remain unanswered and many of the drastic simplifications made in the present theoretical model require detailed justification. In particular, while we have shown why postive Q-value channels are important, and how they can lead to flat barrier distributions, only finding a microscopic estimate of the effective coupling strength V will allow us to use such an approach to make predictions. This is essential for some of the heavier systems which may lead to superheavy elements, and

306

10

1 10" N=7 (12% increase) Single barrier

10"'

80

90

100 Energy (MeV)

110

120

Figure 9: Allowing a small increase of the coupling strength per step gives an excellent fit to the experimental data.

where fission processes make a detailed measurement of the experimental barrier distribution very difficult. In this domain, we are not simply interested in understanding the basic underlying reaction mechanism but are also interested in exploiting the enhancement of the fusion cross section in order to reduce the incident energy and form compound systems more likely to survive fission, thus producing more efficiently long-lived isotopes of the superheavy elements. References 1. N. Rowley, G.R. Satchler and P.H. Stelson, Phys. Lett. B 2 5 4 (1991) 25 2. J.X. Wei, J.R. Leigh, D.J. Hinde, J . O . Newton, R.C. Lemmon, S. Elfstr6m, J.X. Chen and N. Rowley, Phys. Rev. Lett. 6 7 (1991) 3368; J.R. Leigh, N. Rowley, R.C. Lemmon, D.J. Hinde, J . O . Newton, J.X. Wei, J. Mein, C.R. Morton, S. Kuyucak and A.T. Kruppa, Phys. Rev. 4 7 (1993) R437 3. M.A. Nagarajan, A.B. Balantekin and N. Takigawa, Phys. Rev. C 3 4 (1988) 894 4. A.M. Stefanini, D. Ackermann, L. Corradi, D.R. Napoli, C. Petrache,

307

400 -

> 200

a,

80

90

100

110

Energy (MeV) Figure 10: The barrier distributions corresponding to the cross sections of Fig. 9.

5. 6.

7. 8. 9. 10. 11.

P. Spolaore, P. Bednarczyk, H.Q. Zhang, S. Beghini, G. Montagnoli, L. Mueller, F. Scarlassara, G.F. Segato, F. Sorame and N. Rowley, Phys. Rev. Lett. 74 (1995) 864 M. Beckerman, Phys. Rep. 1 2 9 (1985) 145 H. Timmers, L. Corradi, A.M. Stefanini, D. Ackermann, J.H. He, S. Beghini, G. Montagnoli, F . Scarlassara, G . F . Segato and N. Rowley, Phys. Lett. B 3 9 9 (1997) 35-39; H. Timmers, D. Ackermann, S. Beghini, L. Corradi, J.H. He, G. Montagnoli, F. Scarlassara, A.M. Stefanini and N. Rowley, Nucl. Phys. A 6 3 3 (1998) 421-445 P.H. Stelson et al., Phys. Rev. C41 (1990) 1584 M. Dasgupta, D.J. Hinde, N. Rowley and A.M. Stefanini, Annu. Rev. Nucl. Part. Set. 4 8 (1998) 401 G. Montagnoli et al., J. Phys. G: Nucl. Part Phys. 2 3 (1997) 1439 E.F. Aguilera, et al., Phys. Rev. Lett. 84, 5058 (2000) T h e author is indebted to D. Hodgkinson and M.W. Kermode, University of Liverpool, for the analytic result Am:N = 2cos(rmr/(N + 1)).

308 FUSION ENHANCEMENT ABOVE THE BARRIER FOR THE 6 He + 209 BiREACTION YU.E. PENIONZHKEVICH, YU.A. MUZYCHKA, S.M. LUKYANOV, R. KALPAKCHIEVA, N.K. SKOBELEV, V.P. PERELYGIN, L.V. MIKHAILOV, YU.G. SOBOLEV Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia E-mail: [email protected] Z. DLOUHY, YA. MRAZEK, J. VINCOUR Institute of Nuclear Physics, Rez near Prague, Czech Republic N.O. POROSHIN Moscow Engineering and Physical Institute, Moscow, Russia F. OLIVEIRA DE SANTOS GANIL, Caen, France L. ROSTOV Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria N A . DEMEKHINA Yerevan Physical Institute, Yerevan, Armenia F. NEGOITA Institute of Nuclear Technology, Bucharest, Romania The experimentally measured excitation functions for the fission and 4n-evaporation channels are presented for the 6He + 20,Bi-reaction. The secondary 6He beam was produced using the standard beam-transport line of the U400M accelerator at FLNR, JINR. The comparison of the obtained experimental data with similar results for the 4He + 209Bireaction shows that in the case of the 6He + 209Bi-reaction a significant enhancement of the cross section is observed for energies above the barrier. In order to get an agreement between the experimental data and the theoretical calculations it is necessary to reduce the Coulomb barrier by 15-20 %, which corresponds to an increase of the parameter r0 of the nuclear potential up to 1.5-1.6 Fm.

309 1

Introduction

In the past few years, in different scientific centers, intensive experimental studies are performed using secondary beams formed from the radioactive products of nuclear reactions. Lately there has been a growing trend to use secondary beams as a means of investigating the interaction cross sections of these exotic nuclei with the target nuclei. These data help getting information on the structure of nuclei far from the line of stability, on the distribution of nuclear matter and on charge radii. In particular, experimental evidence was found for the existence of the neutron halo ("Li) and skin (6He) in the neutron-rich nuclei [1,2]. The mechanism of reactions induced by such nuclei has specific features due to the weakly bound valence neutrons. These features manifest themselves in the necessity to go, for the description of the experimental results, beyond the simple one-dimensional models, where the interaction between the colliding nuclei depends only on the distance between their centers. This served as a pushing factor for many studies, dedicated to the sub-barrier enhancement of the complete fusion cross section in reactions induced by neutron-rich nuclei such as 6He and 8He. In such studies the influence of low-energy oscillations of the colliding nuclei on sub-barrier fusion and on nucleon transfer has been investigated. The choice of sub-barrier fusion is based on the fact that in this case the strongest channel connected with the penetration through the one-dimensional barrier is inhibited and it becomes possible to identify it in spite of the competition from other reaction channels. However, it is evident that measurements of the fusion cross section are of interest in the region of the Coulomb barrier as well, because in this case the weakly bound neutrons in 6He, 8 He and others of this type should influence the process. This interest follows also from that the mechanism of such an influence in the region above the barrier may be different from those, which play an important role at energies lower than the Coulomb barrier. The situation above the barrier is much more complex. This is due to the exponential rise of the fusion cross section, connected with the necessity to overcome the one-dimensional barrier. It hampers the observation of the contribution from other channels to the fusion process. In this connection it is important to correctly choose the decay channel of the compound nucleus, which indicates that the complete fusion of the interacting nuclei has taken place. The choice of the target-nucleus is also very significant. One may hope that such a channel when using heavy enough, but not strongly fissile, nuclei can be the fission of the excited compound nucleus. This is determined by the noticeable contribution to fission from nuclei with large enough orbital angular momentum t. This increase in the fusion cross section, connected with the extension of the ^-feeding of the produced compound nuclei, will bring forth a considerable increase in the fission cross section. It is noteworthy that the contribution of the partial cross sections to the cross section for the formation of the compound nucleus is proportional to (2£ +1).

310 Calculations have been carried out for the reactions "Li + 208Pb and "Li + 238U [3-5]. In these papers special attention was paid to the influence on the fusion cross section of the breakup of "Li into 9Li and two neutrons in the field of the target nucleus. It was shown that breakup strongly influences the fusion cross section at energies close to the Coulomb barrier and actually decreases the fusion cross section. The fusion-fission reaction induced by secondary beams was first performed using 6He-beams [6,7]. Later measurements were continued with 6He[8-10], as well as "Be- [11] and 38S-beams [12]. The majority of experimental data indicate a rise in the fusion and fission cross sections for the reactions 6He + 209Bi and 6He + 238U. The data of Kolata et al. [8,9] on the fusion cross sections at energies below the barrier are contradictory. On one side, in the experiments, where the 6He-induced fission and fusion of Bi were studied, no enhancement in the cross section was observed compared to the predictions of the statistical model [8]. On the other hand, a strong enhancement was observed in the cross section for the 209 Bi(6He,3n)-reaction in the sub-barrier region of energies [9], on the basis of which the authors made the conclusion of the lowering of the 6He-fusion barrier by 25 %. The data obtained using "Be-beams are also ambiguous. In the reaction 38 S+181Ta enhancement in the fission cross section was observed compared to the one in 32S-induced reactions - this was interpreted as due to the reduction of the fusion barrier [12]. For these experiments, the choice of the target-nucleus is of great importance. It should not be too light, as then the fission cross section would be small and the low intensity of the 6He-beam would make the performance of such experiments too difficult. It should not be very heavy either (Uranium or heavier). The reason is that for heavy nuclei fission is the main decay mode for all values of I and it is difficult to distinguish the influence of other reaction channels on the probability of compound-nucleus formation. Most suitable targets are nuclei close to lead, and in particular 209Bi for which there also exists a detailed measurement of the excitation function of the (a,f)-reaction [13]. This is of importance in the analysis of the experimental data. An open question still remains - what is the influence of the structure of the colliding nuclei on the fusion cross section at energies above the barrier. We address this issue in the present paper. 2

Experiment

The secondary 6He-beam was produced in the reaction 7Li(35MeV/A) + Be. The primary 7Li-beam from the U400M accelerator was focused on a cooled Be-target 3 mm thick. The separation of the products produced in the target from the projectiles and the formation of the secondary beam was achieved with the help of the ion-optical system of U400M (Q4DQ-spectrometer).

311

The use of four dipole and quadrupole magnets made it possible to obtain 5-104 pps of the 6He beam. Better purification of the beam was obtained with the help of a degrader (2 mm of polypropylene) and slits located between the dipoles. The magnetic rigidity was chosen so as to achieve the best possible purification and the necessary beam energy. The spot size on the secondary target and the quality of the beam were controlled by position-sensitive parallel-plate avalanche counters and silicon detectors. The 6He-beam did not change its characteristics during a long period of measurement. It was possible to reach up to 98 % purity of the secondary beam at the used energy. The energy dispersion of the secondary beam amounted to ±0.6 MeV. The 6He-beam fell on secondary Bi-targets (about 700 ug/cm2 thick). The targets were prepared by evaporation onto polymer layers 2.5 u thick. The fission fragments were registered on-line with the help of semiconductor surface-barrier silicon detectors around the targets. The overall geometrical efficiency was 30 % of 47i. At the lower energies of the beam (< 60 MeV) the excitation function was measured using plastic track detectors similarly to the earlier experiments [6,7]. 3

Experimental Results and Analysis

Figure 1 presents as a function of energy the experimentally measured fission cross section for the reaction 6He + 209Bi (the data from [7] are also included together with the excitation function of the 209Bi(6He,4n)-reaction). For comparison the data on the 209Bi a-particle-induced fission [13] are shown as well. A difference is observed in the fission cross sections for the two reactions 4He + 209Bi and 6He + 209 Bi. In order to obtain qualitative and, particularly, quantitative information it is necessary to perform comparative analysis of the results on the two reactions. But before we do this, we have to answer the question whether all fission events in the 6 He+209Bi-reaction are the result of complete fusion of these two nuclei. Fission in this reaction, besides the complete fusion, may, in principle, arise due to the breakup of 6He into 2 neutrons and an a-particle with the consequent capture either of the neutrons or the a-particle. In the first case, the nucleus 211Bi is formed. At the maximum energy used in our experiment (about 70 MeV), the excitation energy of 2 "Bi is 33 MeV (23.4 MeV is the kinetic energy of the neutrons and 9.7 MeV is the reaction Q-value). Then only 0.0001 out of all formed compound nuclei can undergo fission, and this is a negligible amount. When the aparticle is captured (also at the maximum energy of 6He) the compound nucleus formed is 213At with an excitation energy »37 MeV (24 MeV are taken away by the neutrons and the binding energy of the a-particle in 213At is -9.3 MeV). In this case 2 % of the compound nuclei 213At may undergo fission and the cross section for such a process is expected to be less than 5 % of the measured fission cross section, which is equal to 0.8 barn. At lower energies of the 6He-beam this value is expected

312

to be even less. Therefore, it can be asserted that all registered fission events in the bombardment of 209Bi with 6He are due to the complete fusion of the interacting nuclei. 10000-

i

•

r

1000-1

100XI

E b

io=

^ * •

0,1

-i

20

1

30

1

1

40

1

1

50

°He+Bi He+Bi

4

1

1

1

60

1—

70

E*, MeV Figure 1. Fission excitation function for the reaction 6He + 209Bi - black triangles; the solid circles denote the excitation function of the 209Bi(6He,4n)-reaction, the open squares - the data of the 209Bi ainduced fission (present work and ref. [7]). The latter are in agreement with [13].

In the analysis of the experimental data we used the well-approved programme ALICE-MP [14], which is a modified version of the well-known programme ALICE. The calculation of the fission widths in this programme is based on the classical formula of Bohr-Wheeler, the calculation of the evaporation widths - on the formalism of Weisscopf-Ewing. In the calculation of the level densities the relations of the Fermi-gas model are used phenomenologically taking into account the effect of nuclear shells on the level density parameter [15]

av{E*) = 5„ {l + [l - exp(-0.054£ *)]AWy

(A,Z)/E*},

where E* is the excitation energy of the compound nucleus, AWV - the shell correction to the mass of the nucleus, formed after the emission of the particle v (a neutron, a proton or an ot-particle). The level density parameter in the fission channel af is regarded as not depending on the excitation energy and proportional to the asymptotic value of the level density parameter in the particle evaporation channel (this is the assumption

313

of the very small value of the shell correction in the saddle point). The fission barriers are calculated by the relation Bf(£) = cBLfD(£) + MVf , where c is a free parameter, defining the contribution from the liquid-drop component in the fission barrier, Bf (I) - the fission barrier in the model of the rotating liquid drop of Cohen-Plasil-Swiatecki [16], AWf - the shell component of the fission barrier of the compound nucleus, which is equal to the module of the shell correction to the mass of the ground state of the nucleus. In the analysis of the experimental data and especially in drawing the conclusions of great importance are the values of the parameters used in the calculations. In calculating the cross sections for reactions with particle evaporation and of fission cross sections for excited nuclei we use two different sets of parameters. One set defines the formation of the compound nucleus and is connected with the geometrical size of the nuclear part of the interaction potential (radius parameter r 0 ) and its shape (the diffuseness of the potential d and its depth V). Numerous calculations of compound-nucleus formation cross sections carried by us for different projectile-target combinations (the projectiles ranging from 7Li to 48 Ca) have shown that in all considered cases one can use one and the same set of parameters, viz. r0 = 1.29 Fm, V = 67 MeV, d = 0.4 Fm. The second parameter set defines the competition between the fission and evaporation channels of the formed compound nucleus. These parameters determine the level density in the fission and evaporation channels. In our calculations such parameters were the ratio of the asymptotic values of the level density in the fission and evaporation channels af I av and the free parameter c in the formula for the fission barrier. Comparing the calculated and the experimental excitation functions of the fission and evaporation reactions for a wide range of nuclei a conclusion was drawn that in all cases one can use the fixed value of afldv= 1. Moreover, the dependence of the ratio TJTf on excitation energy, obtained in 22Ne-induced reactions on 194-196'198pt [17], has shown that this very value of af lav leads to the consistency between experiment and calculations. Therefore, this parameter was also fixed. The parameter c does not have a fixed value. It changes with Z and A of the compound nucleus. The point is that it changes slowly and, more importantly, smoothly. This allows reliable determination for a definite compound nucleus by

314

extrapolating from the close-by nuclei. For the heavy Astatine nuclei the parameter c = 0.8. Thus, in the analysis of the experimental data for the 6He + 209Bi-reaction there are no free parameters. A special attention should be also paid to the quantity £a - another parameter of importance in the calculations of the fission cross section. This is the critical angular momentum that, at high enough energies (when it becomes less than £mm), determines the number of partial waves leading to the formation of the compound nucleus. As the parameters (r0,d , V ,aJay) stay constant in a wide range of nuclei and c varies smoothly with Z and A, one can use for the calculation of the cross sections of a definite reaction the values from any other reaction leading to the formation of compound nuclei lying close by. At the same time the parameter £CT takes different values for different reactions. It is noteworthy that so far there exists no reliable way of calculating £„. One may only assume, on the basis of very general conceptions about the formation of the compound nucleus, that £„ has a square-root dependence on energy as well as on the mass of the particle inducing the reaction. However, this is true only if there is no abrupt change of properties (e.g., changes in the radius of the fusion channel, not proportional to (A x going from one particle to another.

+ A^ )) when

Before analyzing the fission data, obtained in the reaction 6He + 209Bi, it was necessary to perform calculations for the reaction 4He + 209Bi for which a detailed measurement of the fission excitation function [13], as well as of the 3n-, 4n-,and 5n-evaporation channels [18] exists. The reason for this was that, first, it was necessary to check the validity of the used set of parameters and, second, to carry out a comparative analysis for the two reactions. Such an analysis would help revealing the peculiarities of the 6He + 209Bi-reaction, if they exist. The experimentally measured excitation functions of the reaction 4He + 209Bi for the (a,f)-, (a,3n)-, (a,4n)- and (a,5n)-channels, together with the results of calculations are shown in Figure 2. In the calculations, the only free parameter was £a. Agreement between the experimental and calculated fission cross section in the last point of the excitation function at E* = 69.2 MeV (Elab = 80 MeV) was obtained for the value of £cr = 35. It should be noted that £max, determined by the height of the Coulomb barrier, becomes equal to this value at Ehb = 57 MeV (E* = 46.7 MeV), while it starts to play a more or less significant role at Ejab = 68 MeV (E* = 57.5 MeV). It may be assumed that in the larger part of the considered energy range the number of partial waves leading to the formation of the compound nucleus is defined by the nuclear interaction potential. The good agreement between

315

calculations and experiment showed the possibility to use our approach to calculate reactions induced by such light particles as helium and confirmed the validity of the used set of parameters. •3n"

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He+Bi t

30

40

i 1—.

50

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60

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70

E*, MeV Figure 2. Excitation functions for the reaction 4He + 209Bi: the symbols denote the experimental data from the present work and from ref. [13,18]; the solid, dotted, dash-dotted and dashed lines are the result of calculations.

This set of parameters was then used to calculate the fission and 4n-evaporation excitation functions for the reaction 6He + 209Bi, which are presented in Figure 3, together with the experimental data. The solid line denotes the results, obtained with the standard set of parameters. It can be seen that for 6He (in contrast to 4He, for which good agreement was reached when using these parameters) the calculated excitation function is significantly lower than the experimental one, while for the 4n-evaporation channel the agreement is satisfactory. Also, the calculated and the experimental fission excitation functions are parallel in the whole energy range and do not converge when increasing the energy. It is noteworthy that in the calculations the value £a = 40 was used. For the highest excitation energy, E* = 70 MeV, the quantity £max practically coincides with this value. For this reason, it is not possible to improve the agreement by a simple increase of £CI (the increase of £0T to 50 causes a f to change in the last point by only

316

15%). Therefore the only way to increase the compound-nucleus formation cross section, and consequently the fission cross section, is to decrease the height of the Coulomb barrier. Remaining in the one-dimensional model, this can be achieved by increasing the interaction radius. Indeed, the increase of the value of rQ to 1.5 Fm or 1.6 Fm (and £CT to 50) will bring forth complete agreement between the experimental and calculated fission excitation functions. Such an increase in the interaction radius corresponds to decreasing the Coulomb barrier by 15 % (r0 = 1.5 Fm) or by 20 % (rQ= 1.6 Fm). In the meantime, the 4n-evaporation cross section changes insignificantly, as it could be expected, since the main contribution to the cross section of the evaporation reactions for energies not far from the maximum of the excitation function comes from the partial waves with relatively small values of £(£ = 30-35). 10000-g •

zc.

4n exp 4ncalc. r=1.29 4n calc r=1.5

1000-

100-

E 6

10

fission exp fiscal r=1.29 fiscalr=1.5 fiscalr=1.5 fiscalr=1.5

i

l=40 l=40. l=50 l=60

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i

30

40

50

—r~ 60

i

70

E, MeV Figure 3. Excitation functions for the reaction 4He + 209Bi: the black symbols denote the experimental data, the lines - the calculations with different values of r0 and < . For details - see the text.

Indeed, it is unlikely that the increase in the interaction radius by 15 % (or even 20 %) when going from 4He to 6He could have something to do with the

317

geometrical size of the nucleus 6He. More probably, it is connected with the enhancement of fusion above the barrier, due to the influence of other channels on the fusion process. What are these channels and what is their contribution is a matter of further investigations. However, it seems reasonable to suppose that the pair of weakly bound neutrons in 6He plays here a decisive role. 4

Acknowledgements

We thank Profs. Yu.Ts.Oganessian and M.G.Itkis for fruitful discussions in the process of performing the present work, and the U400M crew for providing the intense 7Li-primary beam. The work was carried out with partial financial support from Bulgaria and the Czech Republic in the frame of the collaboration with JINR. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Tanihata I. et al., Phys. Rev. Lett. 55 (1985) pp. 2676-2679. Tanihata I. et al, Phys. Lett. B289 (1992) p. 261. Takigawa M. et al, Phys. Rev. C47 (1993) pp. R2470-R2473. Dasso C. et al., Nucl. Phys. A597 (1996) p. 473. Hussein M et al., Phys. Rev. C46 (1992) pp. 377-379; Nucl. Phys. A588 (1995) p. 85c. Skobelev N.K. et al., JINR. Rapid Comm. 4(61) pp. 36-48 (1993). Penionzhkevich Yu.E. et al., Nucl. Phys A588 (1995) p.258; Fomichev A.S. et al., Z. Phys A351 (1995) pp.129-130. Kolata J.J. et al., Phys. Rev. C57 (1998) pp. R6-R9; P.A. De Young et al., Phys. Rev. C58 (1998) pp.3442-3444. Kolata J.J. et al., Phys. Rev. Lett. 81 (1998) pp.4580-4583; Phys. Rev. Lett. 84 (2000) pp. 5058-5061. Trotta M. et al, Phys. Rev. Lett. 84 (2000) pp. 2342-2345. Yoshida A. et al, Phys. Lett. B389 (1996) p.457. Zyromski K.E. et al, Phys.Rev. C55 (1997) pp. R562-R565. Ignatyuk A.V., Itkis M.G. et al., Yad. Fiz. 40 (1984) p. 625 (in Russian). Muzychka Yu.A., Pustylnik B.I., in Proc. Inter. School-Seminar on Heavy-ion Physics, Alushta, 1983, JINR Publ. Dept. D7-83-644, Dubna, 1983, p. 420. Ignatyuk A.V., Smirenkin G.N., Tishin A.S., Yad. Fiz. 21 (1975) p. 485 (in Russian). Cohen S., Plasil F., Swiatecki W.J., Ann. Phys. 82 (1974) p.557. Andreev A.N., Bogdanov D.D., Chepigin V.I. et al., Nucl. Phys. A620 (1997) p. 229. Rattan S.S., Chakravarty N., Ramaswami A., Singh R.J., Radiochim. Acta 55 (1991) p7.

318 T R A N S F E R , B R E A K U P , A N D F U S I O N R E A C T I O N S OF W I T H 2 0 9 BI N E A R T H E C O U L O M B B A R R I E R

fl

HE

J.J. KOLATA Physics Dept., University of Notre Dame, Notre Dame, IN 46530-5670, USA E-mail: Kolata.Wnd.edu The fusion of 6He with a 209Bi target displays a large enhancement at energies near to and below the Coulomb barrier. Recently, a 4He group of remarkable intensity, which dominates the total reaction cross section, has also been observed in the near-barrier interaction of the same system. It is argued that this transfer/breakup channel acts as a doorway state to fusion. 1

Introduction

Theoretical studies of near barrier and subbarrier fusion of the exotic "neutron halo" nucleus n L i 1^2>3>4'5 have generated a considerable amount of interest and controversy. This system contains two valence neutrons that are only very weakly coupled to a relatively tightly bound 9 Li core, and neither the n- 9 Li nor the n-n subsystems are bound. As a result, the particle stability of n L i is achieved only via three-body interactions. Systems of this kind, referred to as "Borromean" nuclei 6 , provide an unusual opportunity to study three-body forces in the nucleus. It has been known for some time that subbarrier fusion of stable nuclei can be enhanced by several orders of magnitude beyond expectations from simple onedimensional barrier penetration calculations due to couplings to internal degrees of freedom of the target and projectile 7 . This dynamical effect is a very sensitive probe of the nuclear structure of the colliding partners, which was the rationale for the studies mentioned above. Strong sub-barrier fusion enhancement, resulting in a lowering of the effective barrier by 20% or more, is a general feature of all these calculations. The role played by projectile breakup channels, which are important due to the weak binding of the valence neutrons, has generated the controversy. Several groups 2 ' 3 , 4 have reported that coupling to breakup channels reduces the fusion cross section near the barrier, leading to intriguing structure in the excitation function in this region. These calculations have been criticized by Dasso and Vitturi 5 , who report only enhancement of the fusion yield, even in the presence of strong breakup channels. Unfortunately, experimental studies of n L i fusion near the Coulomb barrier are not possible at present, due to the low flux and poor energy resolution of the available beams at these low energies. However, the 6 He nucleus, with two weaklybound neutrons around a 4 He core, is the simplest of the Borromean nuclei and its fusion with 209 Bi near the barrier has been studied 8 . Despite the weak binding of the valence neutrons, little evidence was found for suppression of fusion due to projectile breakup. However, a large enhancement of subbarrier fusion, implying a striking 25% reduction in the nominal fusion barrier, was observed. This result has recently been confirmed in a study of 6 He + 238 U fusion9. The fusion data 8 are shown in Fig. 1 in comparison with a number of different models that are appropriate for the fusion of normal nuclear systems. First (dot-dash curve), the highest-energy points

319

He+

Bii Total Fusion

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Em(MeV) Figure 1. Total cross section for 6 He+ 2 0 9 Bi fusion. The dashed curve is a PACE prediction for the total fusion cross section, the dot-dashed curve is the result of a least-squares fit to the highenergy data on a \/Ecm plot, and the dotted curve is a CCFUS calculation. The solid curve was generated in the Stelson model (see text).

were fit to a straight line on a 1/E cm plot to determine the fusion barrier height and radius (see e.g. Gupta and Kailas 10 ). A PACE prediction 11 for the total fusion cross section is the dashed curve in Fig. 1, and the dotted curve is a calculation using the code CCFUS 1 2 . None of these models provides an adequate representation of the behavior of the experimental data below the barrier. However, the solid curve in Fig. 1, which fits the data very well, was generated in the Stelson model 13 . In this model, a distribution of barriers with uniform weight extending from some threshold energy T to 2B-T (where B is the nominal barrier) is introduced. Stelson, et al. have shown that the threshold barrier T correlates with neutron binding energies (not collective properties of the participating nuclei), as would be expected if the fusion cross section in the near subbarrier region reflects neck formation promoted by "neutron flow" 13 . We therefore speculated that the explanation for the dramatic enhancement in the sub-barrier fusion cross section visible in Fig. 1 might lie in the neutron transfer channels of the 6 He-t- 209 Bi system 8 . In this talk, I will discuss the results of an experiment that was undertaken to test this speculation.

2

Experimental setup

The 6 He beam used in the experiment was produced by the TwinSol radioactive nuclear beam facility at the University of Notre Dame 14 , developed in collaboration with the University of Michigan. Two large superconducting solenoids (Fig. 2) act as thick lenses to collect and focus the secondary beam of interest onto a spot that

320

Figure 2. The TwinSot radioactive nuclear beam facility.

is typically 5 mm M l width at half maximum (FWHM). In this experiment, a maximum 6 He rate of 105 s" 1 was produced. The secondary beam was contaminated by ions having the same magnetic rigidity as the desired 6 He beam. This contamination was reduced by placing an absorber foil at the crossover point between the two solenoids, to eliminate unwanted ions from the beam via differential energy loss. The remaining contaminant ions were identiled by time-of-flight (TOF) techniques, using the time difference between the occurrence of the secondary reaction and the E F timing pulse from a beam buncher. The secondary target was a 3.2 mg/cm 2 Bi layer evaporated onto a 100 fig/cm2 polyethylene backing. The reaction events were detected with five Si AE-E telescopes placed at various angles on either side of the beam. Each telescope had an effective angular resolution of between 9°-ll° (FWHM), computed by folding in the acceptance of the collimator with the spot, size and angular divergence of the beam. A typical spectrum, taken at 22.5 MeV and an angle of 135°, is shown in Fig. 3. The elastic 6 He group is visible, along with 4 He and H. isotopes. A strong, isolated group of 4 He ions having a mean energy about 2.5 MeV less than that of the 6 He elastic group is clearly visible. This spectrum is gated by TOF so scattered 4 He ions in the secondary beam (which have an energy 1.5 times that of 6 He) have been identified and removed. The 4 He ions at lower energy, below the isolated peak, come from reactions in the backing of the target, as determined from a separate spectrum taken with a backing foil without Bi. Also visible in Fig. 3 is a 3 H group, which cannot be identified on the basis of TOF. The 209 Bi( 3 H, 4 He) reaction has a large positive Q-value and the 4 He ions in the isolated group might be coming from this reaction, but this possibility was eliminated in a separate experiment with a 3 H beam of the appropriate energy, which showed no events in this region.

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3

Results

Angular distributions obtained for the isolated 4 He group (Fig. 4) are broad and approximately Gaussian in form, with a centroid that moves backward at lower energy (Table I) as expected for a predominantly nuclear process. A striking feature of these data is the very large magnitude of the total cross section, equal to 773 mb at 22.5 MeV and 643 mb at 19 MeV. For comparison purposes, the fusion cross sections 8 measured at these energies are 310(45) and 75(17) mb, respectively. This very surprising result was confirmed by the elastic-scattering angular distributions (Fig. 5) which imply total reaction cross sections of about 1170 mb and 670 mb at the two energies, consistent with the sum of the fusion and 4 He yields within experimental error. The curves shown in Fig. 5 were obtained from optical-model fits to the data,

322 Table 1. Parameters of the Gaussian fits to the data shown in Fig. 4. Blab

(MeV) 22.5 19.0

Centroid

(deg)

FWHM

(deg)

"total (mb) 773 (31) 643 (42)

119.6 ( 5.6) 131.8 (19.7)

86.2 (2.5) 116.6 (5.3)

Table 2. Optical-model parameters used in the calculations shown in Fig. 5. The third row gives a potential determined for 4He + 209 Bi at an incident energy of 22.0 MeV 15 . In each case, the Coulomb radius was taken to be 7.12 fm. Elab

(MeV)

V

22.5 19.0 22.5

(MeV) 150.0 150.0 100.4

R(fm) 7.36 7.36 8.57

o

(fm) 0.68 0.68 0.54

W

(MeV)

27.8 "> 47.8 " ' 44.3 6>

Ri

(fm) 9.38 9.38 7.12

aj

(fm) 0.99 0.99 0.40

Vreac (mb) 1167 668 238

"' Volume imaginary potential. 6 ' Surface imaginary potential.

resulting in the parameters given in Table II. The imaginary well depth was found to be about 75% greater at the lower energy. This rapid energy dependence implies that the effective imaginary potential is not well described by a Woods-Saxon form. Also shown in Fig. 5 are optical-model predictions using parameters obtained from 4 He scattering, but with radius parameters increased to correspond to the larger size of 6 He. This illustrates the expectations for elastic scattering of a "normal" nuclear system near to and below the barrier. The predicted total reaction cross sections are 238 mb and 5.2 mb, respectively. 4

Discussion

The 4 He group seen in this experiment dominates the total reaction cross section near the barrier, so it is important to determine the reaction mechanism that accounts for its very large yield. Unfortunately, neutron transfer cannot be separated from breakup modes using only the present data. Based on the absence of events in the appropriate energy region, two-nucleon transfer to the ground state of 209 Bi (Q = +8.8 MeV) is unimportant. This agrees with a finite-range DWBA calculation for dineutron transfer; the predicted maximum yield was less than 0.1 mb/sr. Single neutron transfer followed by breakup of the remaining unstable 5 He has a very different Q-value, as does direct breakup into 4 He plus two neutrons. However, outgoing 4 He ions resulting from either of these mechanisms could be accelerated by the Coulomb field of the target to approximately the energy of the observed group, so neither process can be eliminated based solely on energy considerations. The experimental angular distributions do reveal some information about the reaction mechanism. The sideward peaking, and the fact that the maximum of the distribution shifts to a larger angle at lower energy, argue for a nuclear process. The result of a coupled-channels calculation of direct nuclear breakup at 22.5 MeV is illustrated by the thin solid line in Fig. 4. Another possibility is transfer to excited states in 211 Bi. We first assumed £ — 0 transfer of a dineutron to a state

323 100

20 100

40 •

1

60 '

1

80 100 120 © C M (de9-) •

140

160 180

i

22.5 MeV 80

-1

to !o 60 %

40

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:

s

> /

-

/ i

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K^v

i

i

i

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20 i

20

40

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60

•

i

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80 100 120 0 C M (deg.)

i

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140

i

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160 180

Figure 4. Experimental angular distributions for the 4 He group measured in this work. The solid curves are Gaussian fits to the data, with the parameters given in Table I. The thin solid curve is the result of a direct nuclear breakup calculation. The dashed curve is a calculation of transfer to a barely bound state; the magnitude of the predicted yield has been multiplied by a factor of 10 in this case.

having a binding energy of 0.1 MeV. The calculated angular distribution is shown as the dashed line in Fig. 4. The absolute yield is much too small: the theoretical prediction has been multiplied by a factor of ten. The result of a preliminary

324

10 F

T'"" " ' *""

T

• ~

I

' *

I

•

1

•

1-

19.0 MeV

0.1

0.01

_J

20

40

60

.

I

I

80 100 ©CM (deg.)

0.01

_L_

120

140

160

160

©CM (deg.)

Figure 5. The experimental elastic-scattering angular distributions. The ratio to the Rutherford cross section is compared with optical-model fits (solid curves), which yield the parameters given in Table II. The dashed curves are calculations with potentials 16 appropriate for 4 He+ 2 0 9 Bi but with a radius appropriate for 6 He. The total reaction cross section computed with this potential at 19.0 MeV is 5.2 mb. Reaction cross sections corresponding to the other curves are given in Table II.

nucleon-transfer calculation including continuum states is more encouraging. In this calculation, the valence neutron pair in 8He was transferred into a range of

325 unbound states in 211 Bi, up to 8 MeV above threshold. Under these conditions, the wave function of the valence dineutron is very extended, as there are no Coulomb or angular momentum barriers to overcome. Since the favored "Q-window" for neutron transfer is at Q ~ 0, the reaction also gains a kinematic enhancement. As a result, the predicted cross section is very large, comparable to the experimental yield, and the angular distribution is characteristic of a nuclear process and appears very similar to the dashed curve in Fig. 4. In addition, coupling to the fusion channel was included consistently, and the calculation predicts an enhancement in sub-barrier fusion comparable to that shown in Fig. 1. As to the speculation regarding "neutron flow" mentined above, the observed Q-value spectrum conclusively shows that ground-state transfer, with its high positive Q value, is unimportant. However, the positive Q value does play a role in making the continuum states in 211 Bi accessible within the preferred Q window. The transfer to these unbound states could be described as neutron flow, though transfer/breakup seems more appropriate under the circumstances. In conclusion, we have for the first time measured near-barrier and sub-barrier transfer/breakup yields for an exotic "Borromean" nucleus, e He, on a 209 Bi target. An isolated 4 He group was observed at an effective Q-value of approximately -2.5 MeV. The integrated cross section for this group is exceptionally large, greatly exceeding the fusion yield both above and below the barrier. Simultaneouslymeasured elastic scattering angular distributions require total reaction cross sections that confirm this large yield. Preliminary coupled-channels calculations suggest that the reaction mechanisms can best be described by direct breakup and neutron transfer to unbound states in 211 Bi. The latter process is enhanced by the large radial extent of the wave function of the unbound states, leading to excellent overlap with the weakly-bound valence neutron orbitals of 6 He. It also experiences a kinematic enhancement due to the fact that the large positive ground-state Q value for transfer makes the neutron unbound states accessible within the optimum "Q-window". The resulting mechanism bears some resemblance to "neutron flow" as discussed by Stelson, et a/.13. Finally, the calculations also predict an enhancement in the sub-barrier fusion yield due to coupling to the transfer/breakup channel, which strongly suggests that this is the "doorway state" that accounts for the remarkable reduction in the fusion barrier observed 8 in this system.

Acknowledgments This work was supported in part by the U.S. National Science Foundation under Grant No. PHY99-01133, and has recently been published 16 . The experiments were carried out in collaboration with groups from the University of Michigan (led by F.D. Becchetti), Hope College in Holland, MI (led by P.A. DeYoung), and ININMexico (led by E.F. Aguilera). The coupled-channels calculations were carried out by F. Nunes of the Universidade Fernando Pessoa in Porto, Portugal.

326 References 1. C. Dasso, J.L. Guisardo, S.M. Lenzi, and A.Vitturi, Nucl. Phys. A597, 473 (1996). 2. N. Takigawa, M. Kuratani, and H. Sagawa, Phys. Rev. C47, R2470 (1993). 3. M.S. Hussein, M.P. Pato, L.F. Canto, and R. Donangelo, Phys. Rev. C46, 377 (1992); ibid., Phys. Rev. C47, 2398 (1993). 4. M.S. Hussein, Nucl. Phys. A588, 85c (1995). 5. C. Dasso and A. Vitturi, Phys. Rev. C50, R12 (1994). 6. M.V. Zhukov, B.V. Danilin, D.V. Federov, J.M. Bang, I.J. Thompson, and J.S. Vaagen, Phys. Rep. 231, 151 (1993). 7. M. Beckerman, Rep. Prog. Phys. 51, 1047 (1988). 8. J.J. Kolata, et al, Phys. Rev. Lett. 81, 4580 (1998). 9. M. Trotta, et al, Phys. Rev. Lett. 84, 2342 (2000). 10. S.K. Gupta and S. Kailas, Phys. Rev. C26, 747 (1982). 11. A. Gavron, Phys. Rev. C21, 230 (1980). 12. J. Ferhandez-Niello, C.H. Dasso, and S. Landowne, Comput. Phys. Commun. 54, 409 (1989). 13. P.H. Stelson, H. Kim, M. Beckerman, D. Shapira, and R.L. Robinson, Phys. Rev. C41, 1584 (1990). 14. M. Y. Lee, et al, Nucl. Instrum. Methods in Phys. Research A422, 536 (1999). 15. A.R. Barnett and J.S. Lilley, Phys. Rev. C 9, 2010 (1974). 16. E.F. Aguilera, et al, Phys. Rev. Lett. 84, 5058 (2000).

327 S T U D Y OF S U B - B A R R I E R A N D N E A R - B A R R I E R OF HALO NUCLEI

FUSION

N . A L A M A N O S , J.L. SIDA, V. L A P O U X CEA/DSM/DAPNIA/SPhN, Saclay, 91191 Gif-sur-Yvette Cedex,

Department

INFN,

of Physics,

Laboratori

Nationali

A. PAKOU The University of Ioannina,

45110 Ioannina,

M. TROTTA di Legnaro, via Romea 4, 1-35020 Legnaro, Italy

France

Greece

Padova,

Coupled channel calculations were performed to investigate the near-barrier and sub-barrier fusion cross section of light unstable nuclei and their associate stable isotopes. A microscopic optical potential was used to generate the entrance channel potential. A rather satisfactory description of the experimental d a t a was obtained under the condition that the optical potential is reduced for the weakly bound systems. The analysis points out some complementary measurements which are necessary to obtain a better understanding of the sub-barrier fusion process involving light weakly bound systems.

1

Introduction

Several experimental and theoretical studies concerning the fusion of two asymmetric nuclei under and near the coulomb barrier were performed in the past 1 . Most of the times the results were interpreted adequately well under the context of coupled channel calculations 2 . W i t h the advent of radioactive beam facilities, the interest on such studies was renewed, aiming to reveal the structure and behaviour of halo nuclei 3 . Such nuclei present specific features like an extended neutron tail, low-lying dipole modes and very low energy thresholds for breakup. Fusion, as other reaction processes should be appreciably affected by such features. In this letter we a t t e m p t to describe into the same framework, near-barrier and sub-barrier fusion for both stable and halo nuclei. T h e first measurements with halo nuclei were visualized through the systems n B e + 2 0 9 B i 4 , 6 H e + 2 0 9 B i 5 and 6 H e + 2 3 8 U 6 . T h e d a t a are presented in Fig.l together with the d a t a of the associated stable isotopes 9 B e + 2 0 9 B i 4 , 4 H e + 2 0 9 B i 7 and 4 H e + 2 3 8 U 6 ' 8 . Cross sections are presented as a function of the energy divided by the coulomb barrier, Vt. In the present case Vb's were extracted via the relations of Christensen and Winther and are shown in Table 1. T h e presentation of all the d a t a in Fig.l facilitates the extraction of the

328 I

'

'

'

I

'

'

'

I

'

'

'

I

I

'

I

I

I

10-

46

..• •

238 He+

U

10'

-Q

E C

g

•

10

P o

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46

' He+209Bi

°2** •

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•

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c

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•

9

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,1 10

/ J

0.6

0.8

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1

l_l

L

1.2

1.4

1.6

1.8

E A/h(MeV) Figure 1: Fusion measurements for the halo systems and their associated stable isotopes

following conclusions. For energies higher than the coulomb barrier the cross sections for the fusion of 9 ' n B e and 4 ' 6 He on 2 0 9 Bi and 238TJ targets present the same behaviour. T h a t is the cross sections with halo projectiles are enhanced over the cross sections with the stable ones. On the other hand, no apparent enhancement is seen for the fusion of the 6 H e over t h a t of 4 H e on 2 0 9 Bi targets. For lower energies t h a n the coulomb barrier, the fusion cross section for the halo nucleus 6 He on 2 3 8 U and 2 0 9 Bi targets is enhanced over t h a t of 4 He, no such enhancement is observed for the fusion of the n B e on 2 0 9 Bi over t h a t of 9 Be. Into this paper we will perform a consistent analysis of all the above sys-

329 Table 1: Coulomb heights according to A. Christensen and Winther B. The BDM3Y1 potential and C. The BDM3Y1 potential reduced by 40%.

system 4 He+ 2 b 8 U 4 He+ 2 0 9 Bi 9 Be+ 2 0 9 Bi 6 He+ 2 3 8 U 6 He+ 2 0 9 Bi u Be+209Bi

Vfe(MeV) A 22.61 20.90 39.95 22.14 20.47 39.46

B 22.48±0.2 21.34±0.2 38.44±0.2 19.51±0.2 18.18±0.2 35.68±0.2

C

39.92±0.2 20.37±0.2 19.10 ±0.2 37.40±0.2

tems and we will try to unreveal new aspects in physics t h a t may emerge from these measurements. 2

T h e analysis

It is known t h a t , in general, coupled channel calculations can reproduce qualitatively and several times quantitatively the fusion results. For the stable nuclei the main ingredients of the calculations, performed with the code ECISP are the entrance channel potential and the structure of the colliding nuclei. T h e real potential is calculated within the double folding model n by using the BDM3Y1 interaction 2 2 . This interaction was found to describe rather well elastic scattering for b o t h stable and unstable n u c l e i 1 0 , 2 3 . T h e imaginary potential simulated the incoming wave boundary condition. T h e densities involved in the real double folded potential for the stable isotopes were obtained from electron scattering d a t a by adopting standard procedures n . For the radioactive nuclei shell model densities 1 2 , and H F densities 1 3 were used for 6 He and u B e correspondingly. T h e calculation for the system 4 H e + 2 3 8 U has been performed within the rotational model. Couplings to the first excited states of 2 3 8 U were considered with deformations extracted from B(E2)'s reported previously 1 4 . In addition to our previous calculation 6 we have used now not only multipolarities with A=2 but also with A=4. T h e calculation for the system 4 H e + 2 0 9 B i includes coupling to the two excited levels of 2 0 9 B i , E=0.896 MeV(A=2) and E = 1 . 6 0 8 MeV (A=3) with deformations reported in the compilation 1 6 . Finally for the system 9 B e + 2 0 9 B i , we have taken into account the excited state of 9 B e , E=2.430 MeV (A=2). T h e B(E2) for the transition to this state, was obtained

330

10-

-Q >

E

10

—•*

c o

(/) D

1 -1

10

10

10

10

0.6

0.8

1.2

1.4

1.6

1,

Ecm/Vb(MeV) Figure 2: Fusion cross section for stable nuclei.

recently by Rudchik et al. 17 . The coulomb barriers and radii of the potentials are shown in Table 1. It is obvious that potential heights of systems with radioactive nuclei present a reduction of ~ 3 MeV relatively to the heights of systems with their associated stable isotopes. This is a well known effect, and a quantitative understanding has been achieved in terms of the halo structure 18 . The consequence of a reduced height is the enhancement of the sub-barrier fusion cross sections, sometimes by several orders of magnitude. The results of the calculation for the stable projectiles are shown in Fig.2 together with the same data which are presented in Fig.l. The fits are adequately good, for both the 4 He+ 2 3 8 U and

331 4

H e + 2 0 9 B i systems. We point out the additional d a t a indicated in Fig.2 with squares, obtained previously 1 9 . These d a t a concern the I n evaporation channel of the reaction, 2 0 9 B i ( a , n ) 2 1 2 A t . T h e addition of these points make the fit excellent. On the other hand, the calculations for the system 9 B e + 2 0 9 B i , overestimate highly the results. T h e same effect was reported before for the system 9 B i + 2 0 9 P b 2 0 a n d it can be well assigned to the coupling to the continu u m . T h e nucleus 9 Be presents a very low threshold to one neutron emission ( S n = 1.67 MeV). Elastic scattering of such nuclei with a weak binding energy, has been described by Satchler and Love 1 1 , into a microscopic description with a reduced potential t h a n the one describing elastic scattering of stable nuclei. T h e effect was studied also by Sakuragi et al. 2 1 into a context of discretized coupled channel calculations and was attributed to the coupling to the continu u m due to their weak binding energy. Moreover into a recent study by Trache et al. 2 3 for elastic scattering of light elements including 9 B e , it was found t h a t the description of 9 Be can be successfully done via a potential reduced by a factor of ~ 4 0 % . It has to be pointed out here t h a t Trache et al. 2 3 have been using the microscopic potential B D M 3 Y 1 , which is- used in the present work. Into this context, it is clear to us t h a t two types of calculations can anticipate the reduction of the potential height and describe sub-barrier and near-barrier fusion. T h e first is the method of discretized coupled channel c a l c u l a t i o n s 2 1 , 24 which is probably the most accurate m e t h o d but which depends on several parameters not known for each system. T h e second one involves a reduced real potential, which is a local representation (not exact) of the discretized coupled channel calculation. It is obvious t h a t the effect of the coupling to the continuum m a y not be represented by a simple reduction of the entrance channel potential but may also affect its shape 2 7 . Adopting the over simplified point of view t h a t couplings to the continuum affect only the height of the potential and with the experience of the calculation on the case of 9 Be we have proceeded with the analysis of the unstable systems. T h e performed calculations with a standard potential (solid line) and a reduced one by 40% (dashed line) are presented in Fig.3. T h e calculations involve coupling to the excited states of the targets as before, and the following couplings to excited states of the projectiles. For 6 He we considered coupling with the first excited state at 1.87 MeV (A=2) with deformation extracted from our recent inelastic scattering results 6 H e ( p , p ' ) 6 H e 1 2 , 2 5 . For n B e the excited state E=0.320 MeV ( A = l ) with a B(E2)=0.116 e 2 fm 4 was taken into account. We have to keep in mind, t h a t the aim of the present calculations was the achievement of an unified description, in a qualitative basis, for the first subbarrier and near-barrier fusion d a t a involving halo nuclei. As it is seen from Fig.2 the systems 4 H e + 2 0 9 B i and 4 H e + 2 3 8 U need no reduction of the poten-

332

10-

10'

_Q

E

10

* • — "

c o

CO 3

1

10~

e>

11

Be + 2 0 9 Bi X10000

10

0.6

0.8

1.2

1.6

E cm /V b (MeV) Figure 3: Fusion cross section for unstable nuclei.

tial as it is expected for well bound nuclei. On the contrary, a 40% reduction is necessary to describe the systems 9 B e + 2 0 9 B i , 6 H e + 2 0 9 B i and u B e + 2 0 9 B i , although for energies well above the coulomb barrier the later system is better described with non reduced potential calculations. T h e situation is more complex for 6 H e + 2 3 8 U . Above the coulomb barrier this system is probably better described with calculations with non reduction of the potential, whereas well below the coulomb barrier the calculations fail to reproduce the d a t a . It has to be noticed however t h a t this is the first system for which sub-barrier fusion, for energies well below the coulomb barrier, has been measured. A new experiment is planned to investigate in more details sub-barrier fusion of 6 H e + 2 3 8 U .

333 From this discussion we can draw the conclusion t h a t coupled channel calculations reproduce the gross properties of near-barrier fusion involving halo nuclei. T h e agreement of the calculations with the d a t a is particularly spectacular in the case of 9 B e + 2 0 9 B i and u B e + 2 0 9 B i . 3

Conclusions

We have performed coupled channel fusion calculations for several systems with halo and their associated non-halo projectiles. A description of the weakly bound stable system ( 9 B e + 2 0 9 B i ) and the halo systems was qualitatively obtained, by making use of a reduced potential. T h e reduction of the potential can be understood in t e r m s of breakup processes due t o the weak binding energy of the stable nucleus 9 Be and the halo nuclei. T h e reduction of potential was justified before via elastic scattering of weakly bound nuclei and in particular of 9 Be on different targets. In this context elastic scattering measurements for halo nuclei are highly requested. These measurements would help to pin down a possible variation of the strength and eventually of the shape of the entrance channel potential. In general it has to be stressed out t h a t additional measurements including elastic scattering, complete fusion (without contributions due to incomplete fusion) and break-up are necessary to enlight the subject of near-barrier and sub-barrier fusion of halo nuclei. Acknowledgments One of the authors (N.A) acknowledges Dr R. Wolski for the discussions he had with him and for pointing out the reference with the 2 0 9 B i ( a , n ) 2 1 2 A t d a t a . References 1. M. Beckerman Rep. Progr. Phys. 5 1 , 1047 (1988). 2. S.G. Steadman and M. J. Rhoades-Brown, Ann. Rev. Part. Sc. 3 6 , 649 (1986). 3. P.G. Hansen et al, Ann. Rev. Part. Sc. 4 5 , 591 (1995). 4. C. Signorini et al, Eur. Phys. J. A2 2 2 7 , 157 (1998). 5. J . J . Kolata et al, Phys. Rev. Lett. 8 1 , 4580 (1998). 6. M. T r o t t a et al, Phys. Rev. Lett. 8 4 , 2342 (2000 ). 7. W . J . Ramler et al, Phys. Rev. 114, 154 (1959). 8. V.E. Viola and T. Sikkeland, Phys. Rev. 128, 767 (1962). 9. J. Raynal, Phys. Rev. C 2 3 , 2571 (1981). 10. V. Lapoux, to be published and P h D thesis, University of Orsay 1998.

334

11. G. R. Satchler and W.G Love, Phys. Reports 55, 183 (1979). 12. S. Karataglidis et al, Phys. Rev. C 6 1 , 024319 (2000); private communication. 13. H. Sagawa, Phys. Lett. B 286, 7 (1992). 14. Mc Gowan et al, Phys. Rev. Lett. 28, 1741 (1971) 15. C.E. Bemis et al,Phys. Rev. C 8, 1466 (2000). 16. Nuclear Data Sheets 63, 723 (1991). 17. A. T. Rudchik et al, Nucl. Phys. A 662, 44 (2000). 18. N. Takigawa and H. Sagawa, Phys. Lett. B 265, 23 (1991). 19. A. R. Barnett and J.L. Lilley, Phys. Rev. C 9, 2010 (1974). 20. M. Dasgupta et al, Phys. Lett. B 82, 1395 (1999). 21. Y. Sakuragi et al, Phys. Rev. C 35, 2161 (1987). 22. D.T. Khoa, G.R. Satchler and von Oertzen Phys. Rev. C 56, 954 (1997). 23. L. Trache et al, Phys. Rev. C 6 1 , 02461 (2000). 24. M. S. Hussein et al, Phys. Rev. C 29, 2383 (1984). 25. A. Lagoyiannis et al, to be published. 26. K. Hagino et al, Phys. Rev. C 6 1 , 037602 (2000). 27. V. Lapoux et al, to be published.

335

S U B - B A R R I E R F U S I O N OF DRIP-LINE NUCLEI K. HAGINO Institute for Nuclear Theory, Department of Physics, University of Washington, Box 351550, Seattle, WA 98195, USA E-mail: [email protected] A. VITTURI Dipartimento di Fisica, Universita di Padova and INFN, Padova, Italy E-mail: [email protected] We discuss the role of break-up process of a loosely-bound projectile in subbarrier fusion reactions. Coupled-channels calculations are carried out for 11 Be + 2 0 8 Pb and 4,6 He + 238TJ reactions by discretizing in energy the particle continuum states. Our calculations show that the coupling to the break-up channel has two effects, namely the loss of flux and the dynamical modulation of fusion potential. Their net effects differ depending on the energy region. At energies above the Coulomb barrier, the former effect dominates over the latter and cross sections for complete fusion are hindered compared with the no coupling case. On the other hand, at below the barrier, the latter effect is much larger than the former and complete fusion cross sections are enhanced consequently.

1

Introduction

Many questions concerning the effects of break-up process on subbarrier fusion have been raised during the last few years both from the experimental 1,2 ' 3 ' 4 ' 5 ' 6 and theoretical 7 ' 8,9 ' 10 points of view. The issue has become especially relevant in recent years due to the increasing availability of radioactive beams. These often involve weakly-bound systems close to the drip lines for which the possibility of projectile dissociation prior to or at the point of contact cannot be ignored. One interesting question is whether the break-up process hinders or enhances fusion cross sections. In addressing this, it is vital to specify which quantity is being compared with which quantity; otherwise a comparison is meaningless. The latter quantity is rather obvious in theoretical calculations, because one can artificially switch on and off couplings to the break-up channel. When we refer in this paper to that break-up enhances/hinders fusion cross sections, it will be in this sense unless we mention explicitly. As for the former quantity, there are two possibilities, namely complete fusion cross sections or the sum of complete and incomplete fusion cross sections. From studies of fusion of stable nuclei where break-up process is not important, we have learned that any coupling of the relative motion of the colliding nuclei

336

to nuclear intrinsic excitations causes large enhancements of the fusion cross section at subbarrier energies over the predictions of a simple barrier penetration m o d e l . u It may not be difficult to imagine that the same thing happens to the break-up channel as well; cross sections for inclusive processes, i.e., the sum of complete and incomplete fusion cross sections would be enhanced by couplings to the break-up channel. On the other hand, one could also argue intuitively that break-up process removes a part of flux and thus cross sections for complete fusion would be hindered. As will be shown below, fusion cross sections are determined by the competition of these two mechanisms. In passing, this sort of consideration is relevant also when one discusses experimental data and compares them with theoretical calculations. For instance, it is important to bear in mind that the recent Padova/RIKEN d a t a 3 as well as the Canberra data, 4 which used 9 Be beams, contain both the complete and incomplete fusion cross sections for the neutron break-up channel (9Be —> n + 8 Be), while they are only the complete fusion cross sections with respect to the a-break-up channel (9Be -> a + 5 He or 2a + n). In the past, different theoretical approaches to the problem have led to controversial results, not only quantitatively but also qualitatively. Hussein et al. derived a local dynamical polarization potential VDPP for break-up process and computed complete fusion cross sections as 7

*<*• = pE(2'+1)pi°)exp (Ifl^wn)'

(1)

where PF' is the fusion probability in the absence of break-up, k is the wave number in the entrance channel, and W(r) is the imaginary part of the dynamical poralization potential VDPP- The integral in the exponent is evaluated along the classical trajectory r(t), where r(t — 0) corresponds to the distance of closest approach. The idea of this formalism is that the effects of break-up can be well described in terms of the survival probability PSUrv = 1 ~ -Pbu = exp(2 J W(r) dt/h). Since W(r) is negative, the break-up process always hinders complete fusion cross sections in this formalism. Subsequently, Takigawa et al. pointed out that the classical trajectory used by Hussein et al. corresponded to the one for scatterings, and for fusion reactions the time integral should have been from - c o to 0. 8 Consequently the effects of break-up were moderated over the estimate made by Hussein et al, but the qualitative conclusion remained the same. These conclusions were later criticized by Dasso and Vitturi. They performed coupled-channels calculations by treating the break-up channels as a single state and obtained totally different results, i.e., enhancement of complete fusion cross sections due to the break-up. 9

337

What would be the origin of these apparently controversial results? We argue that the coupling to the break-up channel leads to the dynamical modification of fusion potential as in fusion of stable nuclei. At subbarrier energies, this effect is most relevant, leading to the enhancement of complete fusion cross section. Such effects are automatically included in the coupled-channels formalism. The dynamical modulation of fusion potential is related to the real part of a dynamical polarization potential, which both Hussein et al. and Takigawa et al. completely threw away. Therefore, if the real part of the polarization potential is properly taken into account, their formalism would provide an enhancement of complete fusion cross sections at energies below the Coulomb barrier. Actually, Ref. 12 shows that it is indeed the case, although their calculations are not satisfactory in a sense of Takigawa et al. 8 and a full calculation within their formalism has still been awaited. Another origin of the controversy might be the simplified assumption used by Dasso and Vitturi. As mentioned above, the entire continuum space was mocked up by a single discrete configuration, 9 and therefore the main feature of continuum couplings were not entirely included. This would underestimate the break-up effects, especially those effects, if any, which remove the incident flux. From this point of view, in this paper, we repeat similar calculations as of Dasso and Vitturi, but by considerably increasing the number of continuum states. 1 3 We also use microscopic form factors in contrast to the previous studies, where form factors were assumed to have an exponential form. These two aspects enable us to describe simultaneously the dynamics of continuum couplings inside and outside the Coulomb barrier, leading to a more conclusive result on the effects of break-up process on subbarrier fusion. In the next section, we review briefly the coupled-channels formalism and define the complete and incomplete fusion cross sections. The specific applications of the formalism to u B e + 2 0 8 Pb and 4,6 He + 238 U are given in the following section 3. 2

Coupled-Channels Formalism for Subbarrier Fusion of Drip-Line Nuclei

Our aim is to discuss effects of the break-up of the projectile nucleus on subbarrier fusion reactions by solving coupled-channels equations. The coupledchannels equations in the isocentrifugal approximation are given by 1 4 '

h2 (P

l{l + l)h2

= -Y, 771

F

nm(r)i>m(r),

T _(o),

,

ZPZTe2

_1 , . ,

(2)

338

where the angular momentum of the relative motion in each channel has been replaced by the total angular momentum 1.15 In eq. (2), V ^ ' is the nuclear potential in the entrance channel, and e„ is the excitation energy of the n-th channel. Here, we assume that n = 0 labels the ground state of the projectile nucleus and the all other n refer to particle continuum states, which are associated with the break-up channels. It is straightforward to include bound excited states of the projectile and/or the excitations in the target nucleus. Fnm are the coupling form factors, which we compute on the microscopic basis. It is to fold the external nuclear and Coulomb fields with the proper single-particle transition densities, obtained by promoting the last weakly-bound nucleon to the continuum states. More explicitely, the single-particle form factor for the promotion from the bound state (ni^ijimi) to the continuum state (i2j2m2) with continuum energy E assumes the form *lmtijimi^yEi2J2m2\.r)

=

= v^F £ ( - ! ) » * + * S(£1+£2

+ A,even) (jx±j2

V2J! + W2j2 + 1 . ^|Y+T Oi -ml32m2\L(m2 \jC°radrf

f

duR*Et2h(r')Rmeijl

- ||A0^

- r m ) ) ]j (r')VT(Vr*

/2A+1 ~^~

+ r" -

2rr'u)Px{u) (3)

The functions REt2h(r) a n d -ftm^iji ( r ) a r e the single particle wave functions for the continuum and bound states, respectively. The potential Vr is the mean field felt by the single particle due to the presence of the target, responsible of the transition, not to be confused therefore with the projectile mean field generating the single particle wave functions. This potential obviuosly involves both a nuclear and a Coulomb component. Note that for long ranged transition densities, as in the case of a weakly bound system, the resulting Coulomb formfactor will differ from the pure r _ A _ 1 form much outwards than in the traditional case of stable systems. See Ref. [16] for more details. The coupled-channels equations (2) are solved by imposing the incoming wave boundary condition (IWBC), where there are only incoming waves at fmin which is taken somewhere inside the Coulomb barrier. 14 ' 17 The boundary conditions are thus expressed as ipn{r) ->• Tn exp f -i -> HI~] (knr)6nfi

/

kn(r')dr'

1

+ RnH\+) (knr)

r < rmin,

(4)

r > rmax,

(5)

339 where /2M (v

i. < \

l{l + l)h2

(0)

is the local wave number for the n-th channel and kn = J2^{E

ZPZTe>\

- en)/h2.

H^

(+)

and # , are the incoming and the outgoing Coulomb functions, respectively. The fusion probability is defined as the ratio between the flux inside the Coulomb barrier and the incident flux. For our boundary conditions given by eqs. (4) and (5), it reads P„ =

kniT min) k

\Tn\2

(7)

for the n-th channel. Complete fusion is a process where all the nucleons of the projectile are captured by the target nucleus. We thus define cross sections of complete fusion using the flux for the non-continuum channel (i.e., n=0) a s 1 8

*CF = £ J > + DPo = £ B 2 / + 1) ^ T ^ l^ol2 •

(8)

The flux for the particle continuum channel (n ^ 0) are associated with incomplete fusion, whose cross sections we define as

«*CF 3 3.1

= £ £(2< +1) E Pn = J £(2< +1) £ ^ f 1 \Tn\2. 0) v

0

I

n#0

u

I

n^O

Results The nBe

+ 208Pb reaction

We now apply the coupled-channels formalism presented in the previous section to fusion of drip-line nuclei. We first consider the fusion reaction 11 Be + 208 Pb, where the projectile is generally regarded as a good example of a single neutron "halo". In a pure single-particle picture, the last neutron in n B e occupies the 2si/2 state, bound by about 500 keV. The strong concentration of strength at the continuum threshold evidenced in break-up reactions 19 has been mainly ascribed to the promotion of this last bound neutron to the continuum of p3/2 and P1/2 states at energy Ec via the dipole field.19,20 Since the presence of the first excited lpi/2 state (still bound by about 180 keV) may perturb the transition to the corresponding pi/2 states in the continuum, 21

340

we prefer here to consider only the contribution to the p3/2 states. The couplings to the P1/2 states are certainly not negligible, but we expect that they will simply further enhance the effects on the fusion cross section and therefore will not alter our qualitative conclusions. The initial 2si/ 2 bound state and the continuum p3/2 states are generated by Woods-Saxon single-particle potentials whose depths have been adjusted to reproduce the correct binding energies for the IP3/2 and 2sx /2 bound states. In particular, one needs for the latter case a potential which is much deeper than the "standard" one. As we mentioned in the previous section, the form factor F(r; Ec) as a function of the internuclear separation r and of the energy Ec in the continuum is then obtained by folding the corresponding transition density with the external field generated by the target. In addition to the Coulomb field, a Woods-Saxon nucleon-nucleus potential is used, with parameters of R = roA}/3, r0 — 1.27 fm, o = 0.67 fm, V = (-51 + 33 [N - Z)/A) MeV, and Vu = -0.44V. Selected cuts of the form factor F(r;Ec) are shown in Figure 1. The individual contribution arising from the Coulomb and nuclear interactions are shown separately. In Fig. 1 (a), we display the form factor as a function of r for a fixed value of the energy in the continuum (Ec being 0.9 MeV). Note the long tail of the nuclear contribution as a consequence of the large radial extension of the weakly-bound wave function, resulting in the predominance of the nuclear form factor up to the unusual distance of about 25 fm. The same reason gives rise to a deviation of the Coulomb part from the asymptotic behaviour 1/r2. Note also, due to the negative El effective charge of the neutron excitation, the constructive interference of the nuclear and Coulomb parts. In figs. 1 (b) and 1 (c), we show, instead, the energy dependence of the form factor at fixed value of r. While at large r the curves are peaked at very low energies, reflecting the corresponding B(E1), at smaller distances around the barrier, for values which are more relevant to the fusion process, the peaks of the distributions move to higher energies, in particular for the nuclear part. Hence, the sudden approximation employed in Refs. 7 and 8, where the energy of continuum states Ec was all set to be zero, is not justified. In order to perform the coupled-channel calculation, the distribution of continuum states is discretized in bins of energy, associating to each bin the form factor corresponding to its central energy. We have considered the continuum distribution up to 2 MeV, with a step of 200 keV. In this way, the calculations are performed with 10 effective excited channels. We have checked the convergence concerning the maximum energy of the continuum states included in the calculation, and found that it is rather slow. We have, however, found that our main conclusions again do not qualitatively change provided that the maximum excitation energy of the continuum states is at least 2 MeV, as con-

341

1.0 0.8

-

1 ' I ' M . — Coulomb ~ \ Nuclear _ \\\\ " Total V \ \\ \ -

// / /

0.6

•t

1

I •

-

0.41-1

•

0 . 2— - /*

£

0.0 -0.2

1 10

1

4

1

1

1

1

1

1 , 1 , "

I

20

30

40

r (fm)

1

1

i

50

I ' M r =

r

3 -''

2

b "

"""-

/ y

- 1* 11

It

—

1 II

S

1 tin

0 0.25 0.20 0.15 0.10 0.05 0.00 -0.05

|

i

1

i

1

1 ii

1r =

i

l

= 30 fm~

Y\

-

L ~'\ \ \ " i 1

0

0.5

1

-r'

i

1

i

1 i "

1.5 2 2.5 (MeV)

Figure 1: Coupling form factor F(r;Ec) associated with the dipole transition in n B e from the neutron bound state 2 s ! / 2 {Eb = - 5 0 0 keV) to the continuum state p 3 / 2 with energy E c - In (a) the Coulomb (the dash-dotted line), nuclear (the dashed) and total (the solid) form factors are shown as a function of r at the continuum energy Ec = 0.9 MeV. In (b) and (c) the form factors are shown as a function of the energy Ec in the continuum for : r =11.6 fm and r = 30 fm, respectively. ba

342

' ^ _ _ _ J ^ - J ^"-^i

1

^

^

^

i

.

'in

10" = ' 1 ' n 2 8 : Be + ° Pb 10

4^ / /

:

y /

10 Ir

/ /

:

10 30

/

/ /

Bare

- - CF + ICF

i

io rr -2

/

//

i

: «•

• i

1 IO 1

CF 1

if / //,

/

1

\

,

35 cm.

i

,

40 (MeV)

i

45

50

Figure 2: Fusion cross section for the reaction u B e + 2 0 8 P b as a function of the bombarding energy in the center of mass frame. The thin solid curve shows the results of the onedimensional barrier penetration as a reference. The solid anifl the dashed lines are solutions of the coupled-channels equations for the complete fusion ana the complete plus incomplete fusion, respectively.

sidered in this work. The ion-ion potential is assumed to have a Woods-Saxon form with parameters V0 = —152.5 MeV, r o = l . l fm and a =0.63 fm, a set that leads to the same barrier height as the Akyiiz-Winther potential. In the previous theoretical studies of fusion of drip-line nuclei, 7 ' 8 ' 9 the excitations to the "soft-dipole mode" of the projectile were also included in the calculations and the effects of the break-up were thus somewhat perturbed. In the present calculations, in order to isolate the genuine effect of the breakup process, we include only the continuum states in the coupling scheme, neglecting continuum-continuum coupling as well as other inelastic channels such as bound excited states in either reaction partner. For the same reason, we do not take into account static modifications on thenon-ion potential which may arise from the halo properties of the projectile. 22 Our calculations, therefore, only attempt to give qualitative indications. They will, however, still reveal interesting aspects of effects of couplings of the ground to continuum states on subbarrier fusion. Figure 2 shows the results of our calculations. The solid line represents

343

the cross section of complete fusion, leading to 219 Rn, while the dashed line denotes the sum of the complete and incomplete fusion cross sections. Also shown for comparison, by the thin solid line, is the cross section in the absence of the couplings to the continuum states. One can see that they enhance the fusion cross sections at energies below the barrier over the predictions of a one-dimensional barrier penetration model. Note that this is the case not only for the total (complete plus incomplete) fusion probability, but also for the complete fusion in the entrance channel. This finding supports the results of the original calculation performed in Ref. 9. As it has been emphasized there, accounting properly for the dynamical effects of the coupling in the classically forbidden region is essential to arrive at this conclusion. The situation is completely reversed at energies above the barrier. Fusion in the break-up channel becomes more important and dominates at the expense of the complete fusion. As a consequence, the cross sections for complete fusion are hindered when compared with the no-coupling case. 3.2

The ifiHe

+ 238 U reactions

We next consider the fusion reactions 4 ' 6 He + 238 U. This is partly motivated by the recent measurement by the SACLAY group. 6 They compared the fusion cross sections of these two systems and concluded that the cross sections for the 6 He projectile were systematically larger than those for the 4 He projectile, both below and above the Coulomb barrier. One might think that this conclusion is inconsistent with ours discussed in the previous subsection as well as with the experimental observation of the Canberra group 4 that the break-up hinders complete fusion cross sections at energies above the Coulomb barrier. However, the situation is somewhat more complicated since the experimental data of the SACLAY group 6 likely include both the complete and incomplete fusion cross sections, and also since the halo structure of 6 He may alter significantly the static fusion potential. We therefore decided to carry out coupled-channels calculations for these systems in order to clarify the role played by the breakup of the 6 He projectile. To this end, we describe the structure of 6 He using the di-neutron cluster model as in Ref. 23. The potential between the di-neutron and a particle is determined so as to reproduce the correct binding energy of 6 He. The ground state is assumed to be the 2s state in this potential. In the coupled-channels calculations, we include continuum states with L = 1 and L = 2 spins. To construct these wave functions, we slightly adjust the depth of the potential for L = 2 so as to reproduce the correct resonance state at 1 MeV. We include the continuum states up to Ec = 5 MeV with energy step of 0.25 MeV. As for the

344

6

238

TJ

He+

TT

U

- - - CF + ICF CF Bare 4TT

He + 25 (MeV)

238TT

U 30

35

Figure 3: Same as fig.2, but for the 6 H e + 2 3 8 U reaction. The meaning of each line is the same as in fig.2 except for the dot-dashed line, which denotes fusion cross sections for the 4 He + 2 3 8 U system.

bare potential, we use the Woods-Saxon potential with Vo = —174.05 MeV, ro=1.01 fm, and a=0.67 fm for the 4 He + 238 U system. The potential for the 6 He + 238 U system is constructed with the folding procedure using the same di-neutron cluster model. For simplicity, we neglect the deformation effects of the 238 U target which would influence the both reactions in a similar way, although one definitely has to include them when one compares theoretical fusion cross sections with the experimental data. Figure 3 shows the theoretical fusion cross sections for the 6 He + 238 U reaction and their comparison with the 4 He + 238 U system. The relation between the complete fusion cross sections (the solid line) and the bare cross sections (the thin solid line) within the 6 He 4- 238 U system is qualitatively the same as that in the 9 Be + 2 0 8 Pb system discussed in the previous subsection; the coupling to the break-up channels enhances the complete fusion cross sections at energies below the barrier and hinders them at above the barrier energies. The cross sections for the 4 He projectile are denoted by the dot-dashed line. The figure shows that the complete fusion cross sections for the 6 He + 238 U reaction are even smaller than the cross sections for the 4 He projectile at energies above the Coulomb barrier. Notice that the sum of the complete and incomplete fusion cross sections for the 6 He projectile is always

345

larger than the cross sections for the 4 He projectile, that reminisces the experimental observation by the SACLAY group. 6 It would thus be interesting, but perhaps experimentally challenging, to isolate the contribution from the complete fusion in the 6 He + 238 U system. 4

Summary

We have performed exact coupled-channels calculations for weakly-bound systems using realistic coupling form factors to discuss effects of break-up on subbarrier fusion reactions. As examples, we have considered the fusion of n B e with a 2 0 8 Pb target as well as the 4,6 He + 238 U fusion reactions. We found that the coupling to break-up channels enhances cross sections for the complete fusion at energies below the Coulomb barrier, while it reduces them at energies above. Very recently, a complete fusion excitation function was measured for the 9 Be + 2 0 8 Pb reaction at near-barrier energies by Dasgupta et al.4 They showed that cross sections for complete fusion are considerably smaller at above-barrier energies compared with a theoretical calculation that reproduces the total fusion cross section. Also, the fusion cross sections for the 4 ' 6 He + 238 U systems measured by the SACLAY group 6 seem to indicate that the break-up effects enhance fusion cross sections at energies below the barrier. These two recent experiments are in general agreement with our results. Acknowledgments We thank C.H. Dasso and S.M. Lenzi for discussions. The work of K.H. was supported by the U.S. Dept. of Energy under Grant No. DE-FG03-97ER4014. References 1. 2. 3. 4. 5.

J. Takahashi et al, Phys. Rev. Lett. 78, 30 (1997). K.E. Rehm et al, Phys. Rev. Lett. 8 1 , 3341 (1998). C. Signorini et al, Eur. Phys. J. A 2, 227 (1998). M. Dasgupta et al, Phys. Rev. Lett. 82, 1395 (1999). J.J. Kolata et al, Phys. Rev. Lett. 8 1 , 4580 (1998); P.A. De Young et al, Phys. Rev. C 58, 3442 (1998); E.F. Aguilea et al, Phys. Rev. Lett. 84, 5058 (2000). 6. M. Trotta et al, Phys. Rev. Lett. 84, 2342 (2000). 7. M.S. Hussein, M.P. Pato, L.F. Canto and R. Donangelo, Phys. Rev. C 46, 377 (1992).

346

8. N. Takigawa, M. Kuratani and H. Sagawa, Phys. Rev. C 47, R2470 (1993). 9. C.H. Dasso and A. Vitturi, Phys. Rev. C 50, R12 (1994). 10. K. Yabana, Prog. Theo. Phys. 97, 437 (1997). 11. M. Dasgupta, D.J. Hinde, N. Rowley, and A.M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48, 401 (1998); A.B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70, 77 (1998). 12. M.S. Hussein, M.P. Pato, L.F. Canto and R. Donangelo, Phys. Rev. C 47, 2398 (1993). 13. K. Hagino, A. Vitturi, C.H. Dasso, and S.M. Lenzi, Phys. Rev. C 61, 037602 (2000). 14. K. Hagino, N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123, 143 (1999). 15. K. Hagino, N. Takigawa, A.B. Balantekin, and J.R. Bennett, Phys. Rev. C 52, 286 (1995). 16. C.H. Dasso, S.M. Lenzi and A. Vitturi, Nucl. Phys. A 639, 635 (1998). 17. S. Landowne and S.C. Pierper, Phys. Rev. C 29, 1352 (1984). 18. K. Hagino and N. Takigawa, Phys. Rev. C 58, 2872 (1998). 19. T. Nakamura et ai, Phys. Lett. B 331, 296 (1994); T. Nakamura et al, Phys. Lett. B 394, 11 (1997). 20. H. Esbensen, G.F. Bertsch and C.A. Bertulani, Nucl. Phys. A 581, 107 (1995). 21. C.H. Dasso, S.M. Lenzi and A. Vitturi, Phys. Rev. C 59, 539 (1999). 22. N. Takigawa and H. Sagawa, Phys. Lett. B 265, 23 (1991). 23. K. Rusek and K.W. Kemper, Phys. Rev. C 61, 034608 (2000).

347 FUSION OF LIGHT WEAKLY BOUND NUCLEI A. Szanto de Toledo. E. Alonso, N. Carlin, R.J. Fujii, M.M. de Moura, M.G. Munhoz, F.A. Souza, A.A.P Suaide, E.M. Szanto and J. Takahashi. Instituto de Fisica da Universidade de Sao Paulo, Departamento de Fisica Nuclear Sao Paulo, S.P., Brasil

Systematic study of fusion barriers for p and light s-d shell nuclei show an increase of the value of their height as the atomic mass number decreases. Two features are observed to be associated to this effect: one, related to the binding energy of nucleon or cluster, which turns the breakup processes relevant and thus inhibiting the fusion probability; a second, associated to the decrease of the number of nucleons located in the nucleus "core" compared to the "surface". The relative competition of these two features is discussed in the present. A systematic analysis of 14 reactions and up to 50 angular distributions allowed also the extraction of the energy thresholds (Eef) for the fission channel forming the final products at the ground state ("elastic fission"). A significant suppression of the contribution of statistical processes is observed when the size of the system is reduced. This effect may be related to a relative increase of the fast and peripheral yields.

Considerable interest is being devoted to the study of fusion reactions involving light and weakly heavy ion reactions [1-5] . An anomalous decrease of the fusion cross section, with respect to the total reaction cross section, is observed at barrier energies. This feature has important consequences in reactions involving radioactive beams [2,6-8] as well as for astrophysics and superheavy nuclei studies. Recent theoretical and experimental studies have investigated the role of nuclear skin, halos and very low binding energies in the fusion and reaction cross sections. In particular a controversy on theoretical expectations about the effect of the breakup channel is open in the literature [9-13]: inhibition and/or enhancement of the fusion cross section. It is known that the coupling of collective degrees of freedom to the fusion channel usually tends to lower the effective potential barrier and consequently enhances its cross section at near and sub-barrier energies [14]. On the other hand, recent findings [2,8] in light heavy ion fusion reactions, surprisingly showed that the breakup channel is able to inhibit the fusion process, when weakly bound nuclei are involved, by increasing the probability of the projectile dissociation along the collision. A clear correlation has been observed [2] between the degree of inhibition of the fusion cross section and the binding energy of the ions involved in the collision. However, although the binding energy is an important factor, it does not appear to be the only relevant parameter. Experiments involving weakly bound nuclei [2,6,7,8] reacting with targets spanning a wide variety of sizes (atomic number) revealed that the size of the system also plays an important role. While the exact nature of the processes responsible for the inhibition of fusion cross-section, at these energies, is still unclear, recent experiments have been able to determine the

348

onset of the limitation process. It has been shown that systems with effective binding energy e < 2MeV and size Ai" 3 +A21/3 < 4.7 already display a fusion hindrance which is larger the farther the system is from these limits [2,3,8]. The open question is the relative importance of these two effects and how correlated they are. Recently fission yields emerging from light heavy-ion reactions have been identified [1,15] with intensities consistent with the Transition State Model (TSM) when new and realistic fission barriers are used [16]. In order to examine the competition between all the mechanisms through which light-ions reactions may occur, spanning from fast direct to slow compound nucleus processes, we performed a systematic data analysis of elastic fission yields observed in light heavy-ion reactions accounting simultaneously and in a consistent way for the elastic, inelastic, transfer, fusion and fission processes. •

1

'

1

i

1.2

1

1.0

H

0.8 0.6

0.8

0.4 0.2

0.6

I

•i l l

•U + »Bef0.4 0.2

H- 1, F + 1!C . 7 B- U + 'Be • -"N + ^AI C - \ ] + ,,!C J ^ C - t ^ C D-"Be + a Si K - " B + " C E-7U + UC L-^C + ^MgF- 1 'F + "F M-nB + aAl

• 1 A - ' U + 'Be

0.4

4 .

A

i 0.2 "

f

G-'WC nn

17F +

0.80 S M p b

0.78 0.76

I.78 Be + ^Bi^o. 0.76

"He + ^Pb

0.74 0.72 0.70 0.68

•

i

I

t

0.68

e (MeV) Figure 1 a): Dependence of the fusion probability Pf=oYo-r on the effective separation (Ee) for a set of reactions [2-4 and references therein], b): Dependence of the fusion probability Pf on the separation energy, used as a parameter, for certain systems found in the literature: : A = (6Li+9Be), E = ( 7 li+ 12 C) and M = (nB+27Al). H = 6He+208Pb; B = Be +208Pb [6,16]; F = 17F +208Pb [8].

The experiments were performed at the University of Sao Paulo Pelletron Laboratory. Complete angular distributions were measured for several systems at various energies using E-AE particle identification. Reverse kinematics has been used for the most mass asymmetric reactions and particle-particle coincidence has

349 been used in many cases to reduce the effect of the background in the cross section determination. The energy resolution was sufficient to separate the elastic and inelastic peaks in all the cases. For the completeness of the analysis, data for some reactions were taken from the literature when available. The data shown in figure la, taken from reference 2, present the dependence of the fusion probability P f = o f /o R (where o f and OR represent the fusion and reaction cross section at energies equivalent to twice the barrier) on the effective separation energy ee. The correlation between the fusion probability Pf and ee indicates that weakly bound nuclei present a high fusion hindrance. It is important to note that the set of data considered in figure la accounts for systems as light as ^ i + ' B e up to the medium weight ones as U B+27A1. Recent experimental data involving rare isotopes and heavy targets became available indicating different rates of inhibition factors [4,6,7,8], '

14

I

•

1 /

>

/ i

/

O i

/

/

i

L/

12

|

10 i |

J 8

i

•

•

-|

6

1

i

1

i

L

i_

3

8

( V + A," ) Figure 2: Experimental values for the reduced radius of the fusion barrier rnu = ( Z ^ e /VB) where VB represents the experimental fusion barrier extracted from the literature [3,4,5,14]. The lines represent linear fits to the low (L) and high (H) mass regions leading to the relations: in* = 1.04 (A/^+Az"3) + 3.8 fm for region H and rua = 3.72 (Ai'^W 3 ) - 9.0 fm for region L Reduced radii extracted from experimental fission barriers and compound elastic cross sections [14] are indicated by open circles.

The systematic for the reduced radius of the fusion barrier for the case of bound systems ee > 2 MeV, shown in figure 2, presents two distinct behaviors for A!1/3+A21/32:4.7 (Region H) and A! 1/3 +A 2 1/3 <4.7 (Region L)

350

The reduced radius of the fusion barrier (rfuS) is determined as the inverse of the reduced fusion barrier height (v^): rfi^Evft.r'^VB/Z.Zze 2 ]- 1 The sudden decrease of r ^ at the region L, for the lightest systems, can be understood as the onset and manifestation of an inhibition of the fusion process. Linear fits can be performed to these data and the relations r&s = 1.04 (A^+Az" 3 ) + 3.8 fm for region H (1) and rft,s = 3.72 (A! 1/3+A21/3) - 9.0 fm for region L (2) can be extracted. It can be also understood that this inhibition is caused essentially by the geometry and spatial density distribution of the nuclei. Therefore, we expect that nuclei located near the drip lines should display a "thicker skin" and therefore give rise to new phenomena. It is important to note that the same behavior is observed for the fission barriers for the elastic compound process, associated to the formation of the same compound nucleus [5]. The analysis of the shape elastic scattering has been performed in the framework of the Optical Model (OM). The real potential is given by the double-folding Model with a realistic density dependent effective interaction based on the M3Y interaction [19]. The imaginary potential used to account for the loss of incident flux is given by a Woods-Saxon [WS] volume potential. The renormalization factor (N d ) of the folding potential was kept close to unity within 15% in most of the cases. The imaginary potential had its geometry kept fixed and taken from references 2 and 5. It is important to mention that the main purpose of this work is not the extraction of unambiguous sets of OM parameters but to obtain a consistent and systematic evaluation of the yields originated from statistical processes. It is known that in the case of light heavy-ion reactions, at energies up to 2-3 times the Coulomb barrier, the fusion process carries most of the reaction flux, but at near barrier energies, direct and inelastic processes become the most important processes. Therefore it is important, in the analysis of the elastic cross section, to consider the influence of the coupling to the inelastic transitions on the elastic channel. In the present work this has been taken care of by using the Coupled Channels approach. The transition potential was given by the first derivative of the volume optical potential. It is known that some light nuclei configurations may favor the elastic transfer process which is responsible, in some cases, for the rise of the elastic scattering cross section at backangles. Such reactions present a perfect kinematics matching and therefore are expected to occur with a significant probability when the nucleon (cluster) spectroscopic factor allows. Such contributions were evaluated within the Exact Finite-Range Distorted Wave Born Approximation [(EFR)DWBA] by means of the code FRESCO [17]. After the subtraction of all these direct components from the data, an angular distribution aeiastic/0Rutherford rising at backward angles remains, normally with a smooth non oscillatory behavior following a aeiastic a l/sin6 behavior and

351 isotropic in the reaction plane. This remaining component has been analyzed within the Statistical Model framework. The Hauser-Feschbach (HF) theory has been used in this case [18] .The continuum region of the spectrum has been described by means of the Fermi-gas level density expression [18]. Transmission coefficients for the HF calculations were consistently determined, for the elastic channel, from the Coupled Channels calculations of the direct components. Furthermore, the angular momentum distribution of the compound nucleus, important in the emission of heavy clusters, has been determined by the experimental cross section for the complete fusion process [5,15]. The fitting procedure of the data has been done in three steps, using the code EOS [20]. In the first step, an Optical Model fit, considering only the forward angle data (up to two times the grazing angle ( 9 ^ ^ ) ) was performed. In a second step, the parameters obtained in the preceding procedure are used, as starting parameters, to generate within the Coupled Channels calculations, the transition potential, having three free parameters, namely Nei, W0, and the normalization factor N^i, corresponding to the transitional potential. In this step, only the first excited state of the target has been coupled to the elastic channel, y} -fits of the angular distributions were performed considering the forward angles up to 3 times the grazing angle (G^ing). In a third step, a simultaneous analysis is performed, using the Coupled Channels and Hauser-Feschbach calculations, for all the data covering the whole angular region measured. A y?-¥& is performed adjusting four parameters: Ne), W0, Njnei and the normalization factor N H F of the statistical model calculations . The angle integrated value of this statistical component is taken as the "elastic fission" cross section. Typical fits to the angular distributions are presented in figure 3. The excitation function of the "elastic fission" cross section obtained using the procedure described above, rise rapidly with energy for all the systems investigated and display a quasi linear dependence on 1/Ecm similar to the complete fusion cross section. This behavior is consistent with the interpretation of a statistical component describing the back angle elastic yields. Fits of the excitation functions to the Glas and Mosel model [5,15] allow to extract values for the " elastic fission" threshold ( E e f) which correspond to the intersection of the curve with the 1/Ecm axis (see figure 4). The "elastic fission" threshold is related to the fission barrier of the system, the Compound Nucleus Coulomb parameter %cn = Z2/A1/3 , the Coulomb barrier Vcb= Z ^ e ^ A / ' 3 +A21/3) and its Rotational energy. Due to the fact that at the threshold energy the angular momentum is relatively low, we expect that the value of F^f should be closely related to the elastic fission barrier (Vef).

352

Figure 3: Angular distributions for the 10 B+ n B, nB+12Ca, loB+160 and H B+ 17 0 elastic channels. The curves indicate the partial contributions predicted by the Coupled Channels calculations (CC) (dotted), elastic transfer (ET) (dashed), Hauser-Feschbach (HF) (dash-dot). The total elastic cross section (CC+ET+HF) is described by the solid line.

Reduced values for the energy thresholds for the "elastic fission" channels \ef=Ee/ZiZ2e2 are presented in figure 5. It is interesting to note that the value for the reduced "elastic fission" energy threshold vcf shows a similar behavior and an equivalent reduction for systems of the same size indicating that both processes are associated to statistical processes and both are suppressed for very light systems.

353 — '

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\ \

\ \ \

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• fusion • fission (x5) •ef(x1000)

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\ \

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i \

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I

\ \ \\ \ \ \ \ \

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\

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\ \ \ \ \

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• 9 B e + 10B \ m 9 Be+11 12B , •9Be+ C I A 11B + 12 C L \ D 1 8 O+ 1 0 B(*5) \ O 180+11B(*5)

\ * \ -\

'

< — 1

i i

10.5

\

•

\i

* * V V\ . Ki ,:

0.04 0.06 0.08

0.1

0.25

0.12

1/ECM(Me\T1) Figure 4: Left: -Excitation function of the "elastic fission" yields for some systems. The dotted lines represent linear fits to the data points. Right: -Excitation function for the fusion [5], fission [5] and "elastic fission" yields for the 18 0+"B reaction.

The hindrance of the statistical processes (fusion, fission and 'elastic fission' ) is observed to occur at the same mass region (A = A / ^ H ^ 1 <4.7 ) and the enhancement of the barrier is of equivalent intensity. This fact can be clearly seen when we calculate the ratio R= Ve/v^ as a function of the size of the system, as shown in reference 5, which presents a constant behavior over all the mass region investigated (even for at A< 4.7 ). This onset mass A<~ 4.7 can be related to the atomic number of nuclei for which the nuclear density distributions no longer present the volume saturation (core or Fermi type density distribution) and are dominated by the nuclear skin (surface). As a consequence, the binding energy per nucleon is below the 8 MeV/nucleon average saturation value, and the direct

354

reactions associated to peripheral collisions, for which the extension of the surface is a relevant question.

0.3

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4.5

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25 26 a

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Figure 5: Reduced values for: a) the "elastic fission" threshold vef and b) the fusion barrier vim for light systems [2,15] as a function of the size of the system (Ai1/3 + A2"3). The dots represent the results from the present work. Squares represent reduced fission thresholds values VRS- The number labels represent the following systems: 10-10B+13C 19-"B+ 1 7 0 28-12C+24Mg 1-1J+"C 2-6Li+'3C 11-"B+ 12 C 20- 12C+160 29- 12C+28Si 3-7Li+12C 12- 12C+12C 21-"B+ 18 ° 30- 14N+28Si 9 10 12 ,9 12 13 4- Be+ B 13- C+ C 22- C+ F 31 12C+35C1 5-7U+l3C 14- I0B+16O 23- 1 6 0+"F 32- "O+^Ca 35 62 6-9Be+"B 15-12C+"N 24-"F+"F 33- Cl+ Ni 7-10B+10B 16-"B+ 1 6 0 25-19F+27Al 8-10B+"B 17-10B+18O 26-19F+40Ca 9-"B+"B 18-18O+10B 27-9Be+12C

355

A first conclusion we can reach is that the fusion of very light systems is already hindered independently of their binding energies. The inhibition of the fusion process manifests itself through the increase of the effective fusion barrier with respect to a value observed for the heavier systems (region H). This increase is correlated to an increase of the surface diffuseness observed in double folding potentials for this mass region. However this extended spatial density distribution is not uncorrelated to a significant decrease of nucleon/cluster binding energy. To account for the effect of the separation energy e of the colliding nuclei on the fusion process, the breakup channel can be taken into account through the addition of the appropriate polarization potential of the optical potential in the entrance channel, in a single-channel calculation. Assuming that this polarization potential is dominated by its imaginary part, it will lead to a reduction of <JF and this reduction can be expressed as a survival probability in each partial wave, expressing the fact that only the fraction of the incident flux that remains in the elastic channel contributes to fusion. The resulting inhibited fusion cross section can then be written as

° > = - p - E ( 2 / + l ) 7 T P*, k

(3)

1=0

where k is the wave number, T°p

is the transmission coefficient for the optical

potential and Pt is the survival probability calculated according to reference 9. No free parameters were used in these calculations. The coupling coefficient (F02) can be scaled from references 9 and 2 to all the systems. In the present calculations an average value of F02 =5 has been used. The curves in figure lb represent the predicted values of the fusion probability, based on the formalism described above, as a function of the effective separation energy of the system. Three systems are taken for illustration: A = (sLi+9Be), E s (7Li+12C) and M = (UB+27A1). We already observe that for these different systems behave differently as the hindrance effect is concerned. The fact that the effect of the geometry is complementary to the one of the binding energy becomes clearer if we extend the prediction to very heavy systems. For that purpose we selected some systems like He, 9 Be, u Be, l7F +208Pb which had their fusion cross sections recently measured [4,6,7,8] (see figure lb). A systematic study of the effect of the binding energy is presented in figure 6. Due to the fact that reaction cross sections are not available for such a wide variety of systems, we calculate the Fusion Inhibition Factor (FIF) FIF=CTfe(e , F02=5 ) / o^ ( s , F 0 2 = 0 )

(4)

356

which is related to the fusion probability Pfl,Si where the reaction cross section area in the denominator is substituted by the expected fusion cross section <jfuS( 8—»°o, F02 = 0 ) when the breakup effect is not considered. Values for the fusion barriers are extracted from the linear fits presented in figure 2, which account for possible geometrical effects. Figure 6a selects the effect of the breakup process on FIF by using the fitted linear relations to the experimental fusion barriers resulting in the relations 1 and 2. Figure 6b describes the overall effect of the breakup and geometry, considering only the relation 1 in the FIF denominator for the determination of the fusion cross section (CT^S ( e , Fo2 = 0 )) extrapolating the general behavior of the heavy systems. From figure 6 we can conclude that the Fusion Probability presents a significant reduction only for light systems and that the breakup process contributes significantly essentially for those light systems with very low binding energies e < 2 MeV. It is now clear why the correlation observed in figure la cannot be extrapolated for very heavy systems, and why the inhibition in the fusion probability observed in reactions involving heavy targets, even with weakly bound rare isotopes is expected to be very moderate. This fact has to be taken into account when studies of light and rare isotopes on heavy targets are undertaken [4,6,7,8] These results have important consequences within the scope of nuclear reactions of astrophysical interest, as well as in the newly growing field of interactions with light radioactive beams. Open questions still remain to be investigated as the extent to which, incomplete fusion contributes, as well as the identification of other peripheral competing processes from the experimental point of view. More detailed comparison between the conflicting theoretical models with exclusive data are also necessary in order to identify the importance of the real polarization potential and its importance in the introduction of a sub-barrier enhancement. We also expect that the Coulomb breakup should also play an important role when heavy targets are considered. Exclusive data on breakup and incomplete fusion cross sections are also necessary. This work was supported in part by the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) and Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP), Brasil.

357

Figure 6 a): Predicted values for the Fusion Inhibition Factor (FIF) as a function of the size of the system (Ai^-fAi1*3) and the binding energy (e) based on the expressions (3) and (4) selecting me effect of the breakup channel In this case, the numerator of the FDR expression FIF= Gfa (8, F 0 2 =5) / Ota C e , F02 = 0 ) (5) is calculated using the fusion cross sections obtained with the parametrization from relations (3). The denominator is calculated using a breakup coupling strength, FQ 2 =0. The thick curves represent the intersection of the'FIF surface with the planes e= 2 MeV and (Ai1/3+A21/3) = 4.7, b): Predicted values for FIF when the breakup and geometrical effects are accounted for. In mis case, the FflP numerator is the same as in figure 6a» and the denominator is calculated using only the relation (1) , extrapolating the linear behavior observed for the heavier elements (region H). The thick curves represent the same limits as in figure 6a.

1. 2. 3. 4. 5. 6. 7. 8.

9.

10. 11. 12. 13. 14.

15.

16. 17. 18. 19. 20.

S.J. Sanders, A. Szanto de Toledo and C. Beck, Phys. Rep.311,V6(1999) J. Takahashi, M. Munhoz, E.M. Szanto, N. Carlin, N. Added, A.A.P. Suaide, M.M. Moura, R. Liguori Neto, A. Szanto de Toledo and L. F. Canto, Phys.Rev.Lett.78, 30 (1997) L. Fante Jr., N. Added, R.M. Anjos, N. Carlin, M.M. Coimbra, M.C.S. Figueira, R. Matheus, E.M. Szanto and A. Szanto de Toledo, Nucl. Phys. A552, 82 (1993). N.Dasgupta, D.J. Hinde, R.D. Butt, R.M. Anjos, A.C.Berriman, N. Carlin, P.R.S. Gomes, C.R. Morton, J.O. Newton, A. Szanto de Toledo and K. Hagino, Phys. Rev. Lett 82,1395 (!999) R.Cabezas, E.M.Szanto, N. Carlin, N. Added, A.A.P Suaide, M.M. de Moura, M.Munhoz, J. Takahashi, R. Liguori Neto, R.M.dos Anjos2, W.H.Z. Cardenas and A. Szanto de Toledo, Phys. Rev. C60, 067602 (1999) C. Signorini, J. Phys. G Nucl. Part. Phys. 234,1235 (1997) and Phys. Lett. B389, 457 (1996). J.J.Kolata et al Phys. Rev. C57, R6 (1998) and Phys. Rev. Lett. 81, 4580 (1998). K.E. Rerun, H. Ersbensen, C.L. Jiang, B.B. Back, F. Borasi, B. Harss, R.V.F. Janssens, V. Namal, J. Nolen, R.C. Pardo, M. Paul, P. Reiter, R.E. Segel, A. Sonzogni, J. Uusitalo and A.H. Wuosmaa, Phys. Rev. Lett. 81,3341(1998). L.F.Canto,R. Donangelo, P. Lotti and M.S.Hussein, Phys. Rev. C52, R2848 (1995) and M.S. Hussein, M.P. Pato, L.F. Canto and R. Donangelo, Phys. Rev. C46, 377 (1992). N. Takigawa, M. Kuratani, H. Sagawa. Phys. Rev. C47, R2470 (1993) K.Hagino, A. Vitturi, C.H. Dasso.and S.M. Lenzi, Phys. Ver. C61, 037602 (2000). C.H. Dasso,and A. Vitturi Phys. Rev. C50, R12 (1994) and C.H. Dasso, J.L.Guisado, S.M. Lenziand ,A Vitturi, Nucl. Phys. A597,473(1996). B. Sahu and C.S. Shastry Z. Phys. A359, 403 (1997). Proceedings of Fusion 97: International Workshop on Heavy-Ion Collisions at Near-Barrier Energies, Australia, March 1997. J. Phys.G 23 (1997) and references therein. R. M. Anjos,N. Added, N. Carlin, L. Fante Jr., M.C.S. Figueira, R. Matheus, H. Schelin, E.M. Szanto, C. Tenreiro and A. Szanto de Toledo, Phys. Rev. C48, R2154(1993) K.A. Farrar et al Phys.Rev. C54, 1249(1996) and references therein. J.J. Thompson, Com. Phys. Rep. 7, 167 (1988). R.G. Stokstad, Treatise on Heavy-Ion Science, Vol. 3, ed. by D.A. Bromley Plenum Press-1985. G.R. Satchler, Nucl. Phys. A279,493 (1977) J. Reynal, ECIS88, unpublished

359 LIST

SPEAKERS

ABE, Yasuhisa Yukawa Institute, Kyoto University, Japan, E-mail: [email protected]

DASGUPTA, Mahananda Australian National University, Canberra, Australia E-mail: [email protected]

ALAMANOS, Nicolas CEA, Saclay, France E-mail: [email protected]

DENISOV, Vitali INR, Kiev, Ukraine E-mail: [email protected]

ARITOMO, Yoshihiro FLNR, JINR, Dubna, Russia E-mail: [email protected]

DIETRICH, Klaus Technischen Universitat, Garching, Germany [email protected]

BENDER, Michael J.W. Goethe-Universitat, Germany E-mail: [email protected]

GOMES, Paulo R. S. Univ. Federal Fluminense, Niteroi, Brazil E-mail: [email protected]

BLOCKI, Jan Institute for Nuclear Studies, Swierk, Poland E-mail: [email protected]

GREINER, Walter Universitat Frankfurt, Germany [email protected]

BURVENICH, Thomas Universitat Frankfurt, Germany [email protected]

HANAPPE, Francis Universite Libre de Bruxelles, Belgique E-mail: [email protected]

CHEREPANOV, Evgeni FLNR, JINR, Dubna, Russia E-mail: [email protected]

HAGINO, Kouichi University.of Washington, USA [email protected]

360

HOFMANN, Sigurd GSI, Germany E-mail: [email protected]

OGANESSIAN, Yuri FLNR, JINR, Dubna, Russia E-mail: [email protected]

ITKIS, Michail FLNR, JINR, Dubna, Russia E-mail: [email protected]

OHTA, Masahisa Konan University, Kobe, Japan [email protected]

KARTAVENKO, Vladimir BLTP, JINR, Dubna, Russia E-mail: [email protected]

PENIONZHKEVICH, Yuri FLNR, JINR, Dubna, Russia E-mail: [email protected]

KOLATA, James J. University of Notre Dame, USA E-mail: [email protected]

PETER, Jean LPC Caen, France E-mail: [email protected]

MARUHN, Joachim A. J.W. Goethe-Universitat, Germany [email protected]

ROYER, Guy Laboratoire Subatech, Nantes, France [email protected] .fr

MOLLER, Peter P. Moller Scientific Computing and Graphics, Inc., USA E-mail: [email protected]

ROWLEY, Neil IReS, Strasbourg, France E-mail: [email protected]

MORITA, Kosuke RIKEN, Japan E-mail: [email protected]

SAGAIDAK, Roman FLNR, JINR, Dubna, Russia E-mail: [email protected]

361

SCARLASSARA, Fernando Dip. Fisica "G.Galilei", Padova, Italy fernando. scarlassara@pd. infh. it

VOLKOV, Vadim FLNR, JINR, Dubna, Russia E-mail: [email protected]

SINHA, Ajit Kumar NSC, New Dehli, India E-mail: [email protected]

YEREMIN, Alexandr FLNR, JINR, Dubna, Russia E-mail: [email protected]

SMOLANCZUK, Robert Soltan Inst, for Nucl. Studies, Warsaw, Poland E-mail: [email protected]

ZAGREBAEV, Valeri FLNR, JINR, Dubna, Russia E-mail: [email protected]

SOBICZEWSKI, Adam Soltan Inst, for Nucl. Studies, Warsaw, Poland [email protected]

SZANTO de TOLEDO, Alejandro Universidade de Sao Paulo, Brazil E-mail: [email protected] UTYONKOV, Vladimir FLNR, JINR, Dubna, Russia E-mail: [email protected]

VITTURI, Andrea Dip. Fisica "G.Galilei", Padova, Italy E-mail: [email protected]

363

AUTHOR INDEX

Abdullin F.Sh. AbeY. AddedN. Alamanos N. Alonso E. Anjos R.M. Aritomo Y. Baby L.T. Beghini S. Belozerov A.V. Bender M. BenoitB. Berriman A.C. Bhuinya C.R. Bogatchev A.A. Bogomolov S.L. Buklanov G.V. Burvenich T. Carlin N. Chelnokov M.I. Chepigin V.I. Cherepanov E.A. Corradi L. Das J.J. Dasgupta M. Demekhina N.A. Denisov V.Yu. Dietrich K. Dlouhy Z. Fujii R.J. Ganguly A.K. Gherghescu R.A. Giardina G. Gikal B.N. Gomes P.R.S. Gorshkov V.A. Greiner W. Gulbekian G.G.

65 123,162 284 327 347 254, 284 123 264 274 81 31,51 93 254 264 93 65 65 31,39 254, 347 81 81, 135 186 274 264 254 308 203 155 308 347 264 232 81,93 65 254, 284 81 1,31,39,243 65

Hagino K. Hanappe F. Hinde D.J. Hinde R.D. Hofmann S. Hui S.K. Iliev S. Itkis I.M. Itkis M.G. Ivanov O.V. Iwasa N. Jandel M. Kabachenko A.P. Kalpakchieva R. Kartavenko V. Kataria D.O. Kliman J. Knjajeva G.N. Kolata J.J. Kondratiev N.A. Korotkov S.P. Korzyukov I.V. Rostov L. Kozulin E.M. KrupaL. Lapoux V. Liguori Neto R. Lobanov Yu.V. Lougheed R.W. Lubian J. Lukyanov S.M. Maciel A.M.M. Madhavan N. Madhusudhana Rao P.V. Malyshev O.N. Maruhn J.A. Materna T. de Moura M.M.

335 93 254 254 81 264 65 93 65,93 65 81 93 81, 135 308 243 264 93 93 318 93 81 93 308 93 93 327 284 65 65 284 308 284 264 264 81, 135 31,39 93 347

364

Mezentsev A.N. Mikhailov L.V. Montagnoli G. Moody K.J. Moraes S.B. Morita K. Morton C.R. Mrazek Ya. Munhoz M.G. Muntian I. Munzenberg G. Muralithar S. Muri C. Muzychka Yu.A. Negoita F. Newton J.O. Oganessian Yu.Ts. Ohta M. Oliveira de Santos F. Padron I. Pakou A. Patyk Z. Penionzhkevich Yu. Perelygin V.P. Pokrovski I.V. Polyakov A.N. Ponomarenko V.A. Popeko A.G. Poroshin N.O. Prasad N.V.S.V. Prokhorova E.V. Raghuvir Singh Reinhard P.-G. Rohac J. Rowley N. Royer G. Rusanov A.Ya. Rutz K. Sagaidak R.N.

65 308 274 65,93 254, 284 81 254 308 345 21 81 264 284 308 308 254 65,81,93, 135, 148 110,123 308 284 327 21 308 308 93 65 93 81, 135 308 264 93 264 31,39 81, 135 93, 296 232 93 31 81, 135

Sandulescu A. Santos U.S. Saro S. Scarlassara F. Segato G.F. Shirokovsky I.V. Sida J.L. Sinha A.K. Skobelev N.K. Smolanczuk R. Sobiczewski A. Sobolev Yu.G. Stefanini A.M. Stoyer M.A. Stoyer N.J. Stuttge L. Suaide A.A.P. Suaide F.A. Subbotin V.G. Subotic K. Sugathan P. Sukhov A.M. Szanto de Toledo A. Szanto E.M. Takahashi J. Tripathi V. Trotta M. Tsyganov Yu.S. Utyonkov V.K. Veselsky M. Vincour J. Vinodkumar A.M. Vitturi A. Volkov V.V. Voskresenski V.M. Wada T. Wild J.F. Yeremin A.V. Zagrebaev V.I.

243 284 81 274 274 65 327 264 308 200 21 308 274 65 65 93 347 347 65 65 264 65 254, 347 347 347 264 327 65 65 81 308 264, 274 335 174 93 123 65 81, 135 215

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