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(ricevuto il 21 Luglio 1997; approvato il 15 Ottobre 1997)

Summary. — A new approach, the dinuclear system concept (DNS-concept), was

elab-orated for the description of the process of complete fusion of nuclei. The DNS-concept has revealed the fusion barrier of a new type and the competition between complete fusion and quasi-fission in reactions with massive nuclei. The DNS-concept has been used for the analysis of reactions of superheavy element (SHE) synthesis.

PACS 25.70.Jj – Fusion and fusion-fission reactions. PACS 01.30.Cc – Conference proceedings.

1. – Introduction

A new approach for the description of complete fusion of nuclei is suggested [1]. It is based on the information about the interaction of two nuclei in deep inelastic collisions which has been obtained in the study of deep inelastic transfer reactions. The term ”DNS-concept” emphasizes the main idea of the new approach: the content of complete fusion is the formation and evolution of the dinuclear system.

Owing to the DNS-concept, it was revealed a new type of fusion barrier, the inner fu-sion barrierB

 fus

, and an intensive competition between complete fusion and quasi-fission in reactions between massive nuclei. The models of the competition between complete fu-sion and quasi-fisfu-sion are suggested. The application of the DNS-concept to the analysis of the synthesis of heavy and superheavy elements is demonstrated.

2. – The main features of the DNS-concept

The main idea of the DNS-concept is the assumption that complete fusion and deep inelastic transfer reactions are similar nuclear processes. Both of them are realized

ac-( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

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1128 V. V. VOLKOV, G. G. ADAMIAN, N. V. ANTONENKO, E. A. CHEREPANOVandA. K. NASIROV

Fig. 1. – Illustration of the compound nucleus formation in the framework of the MDM and DNS-concept.

cording to similar scenarios. The scenario of the complete fusion process is as follows. At the capture stage, after full dissipation of the collision kinetic energy the dinuclear sys-tem (DNS) is formed. The complete fusion process is the DNS evolution that proceeds via nucleon transfer, shell by shell from one nucleus to the other one. The DNS nuclei retain their individuality throughout their way to the compound nucleus. This peculiarity of the DNS evolution is the consequence of the shell structure of the nuclei. Figure 1 illustrates the compound nucleus formation process in the framework of the macroscopic dynamical

Fig. 2. – The potential energy of the DNS formed in the110

Pd +110

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Fig. 3. – Two ways of evolution for the DNS.

model (MDM) [2] and the DNS-concept.

The DNS-concept gives the possibility to reveal two important peculiarities of com-plete fusion of massive nuclei: appearance of the specific inner fusion barrierB

 fus

and competition between the complete fusion and quasi-fission channels in the DNS which is formed in the capture stage. Figure 2 shows the potential energyV(Z ;L)of the DNS

formed in the110

Pd +110

Pd reaction [1]. The initial DNS is similar to a gigantic molecule. To form the compound nucleus, the DNS should overcome the potential barrierB

 fus

. This barrier is the result of the endothermic character of the evolution from the injection point of the reaction to the Businaro-Gallone point (the top of the barrierB

 fus

).

In reactions with massive nuclei the initial DNS has two ways of evolution (fig. 3) [3]. One of the ways brings the DNS to the compound nucleus. This way requires the overcoming of the barrierB

 fus

. The other way leads to the symmetric configuration of the DNS and to a decay of the system in two fragments, i.e. to quasi-fission. Quasi-fission requires the overcoming of the quasi-fission barrierB

qf. The statistical nature of the DNS

evolution gives rise to the competition between the complete fusion channel and the quasi-fission channel. On the basis of the DNS-concept a model of the competition between complete fusion and quasi-fission has been created for symmetrical nuclear reactions [1]. Using this model we have calculated the evaporation residue cross-section 

ER in the

reaction110

Pd +110

Pd. The calculation of

ERhas been made also by using the MDM. In

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1130 V. V. VOLKOV, G. G. ADAMIAN, N. V. ANTONENKO, E. A. CHEREPANOVandA. K. NASIROV

Fig. 4. – Calculated ER

(E)for the 110

Pd +110

Pd reaction by using the MDM and DNS-concepts. The solid squares represent the experimental data [4].

3. – Analysis of reactions of superheavy element synthesis

3.1. The minimum of the excitation energy of compound nuclei. – According to the DNS-concept the minimum excitation energy of the compound nucleus coincides with the height of the Businaro-Gallone point (fig. 5). It means that the minimum of the excitation energy of the compound nucleus is determined by the shape of the potential energy curve. The main part of the excitation energy of the compound nucleus which has to be formed is received during the DNS descent from the Businaro-Gallone point. However, the fate

Fig. 5. – Minimum of excitation energy of the DNS and the compound nucleus in the complete fusion.

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Fig. 6. – Excitation energy of the compound nuclei with Z =102-114 in cold fusion reactions. Experimental data for (HI,1n) reactions,calculated data according to the DNS-concept. a) The deformation of the heavy nucleus in the DNS and b) the deformation of the heavy and light nuclei in the DNS are taken into account.

of the DNS itself is determined at the approaching of the system to the Businaro-Gallone point. At this evolution stage the DNS excitation energy is the lowest and the DNS is cold. This peculiarity of the DNS evolution requires some modifications in calculating the potential energyV(Z ;L). Instead of the liquid-drop masses the real masses taken from

the tables were used for the DNS nuclei. The deformation of the DNS nuclei was taken into account. The deformation of the heavy nucleus was taken in the ground state, the deformation of the light nucleus in the 2+

state. Figure 6 shows the experimental data for the excitation energies of the compound nuclei of elements from 102 to 112 produced in cold fusion reactions and the minimal excitation energy calculated within the framework of the DNS-concept. The calculated data based on the surface frictional model [5] are also indicated. In MDM the excitation energy increases to 50–250 MeV as a result of a very high extra-extra push [6].

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1132 V. V. VOLKOV, G. G. ADAMIAN, N. V. ANTONENKO, E. A. CHEREPANOVandA. K. NASIROV

Fig. 7. – Potential energy of the DNS which is formed in four production reactions of the246 Fm compound nucleus.

3.2. The role of quasi-fission in the reactions of the SHE synthesis. – According to the DNS-concept the production cross-section of SHE is determined by the following expres-sion:  ER = c P cn W sur ; where

cis the capture cross-section, P

cnthe probability of the compound nucleus

forma-tion in the competiforma-tion with quasi-fission,W

surthe survival probability of the deexciting

compound nucleus. The values of

cand and W

surmay be calculated using existing

the-Fig. 8. – Probability for the formation of the246

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Fig. 9. – Production cross-section of244

Fm in the reactions with40 Ar+208

Pb and76 Ge+170

Er. The experimental data for2nare shown by solid dots and those forcnin the reaction with

40 Ar by solid squares [9]. The curves are the results of calculations. The compound nucleus formation cross-section

cnis calculated according to the DNS-concept and with the optical model.

oretical models. But there is no theoretical model for calculatingP

cn. We suggest two

models for calculatingP

cn. In the first one the Monte Carlo method is used. The second

one is based on the Kramers approach to the solution of the Fokker-Planck equation. In the first model certain simplifications of the DNS evolution process have been in-troduced [7]. From any configuration the DNS may pass to the neighbouring one inZ

andN only. This means that one proton and one or two neutrons are transferred from

one nucleus to the other. The cluster transfer is excluded. The probability of the nucleon transfer is proportional to the DNS level densities in the neighbouring configuration. The level density is determined by the DNS excitation energy [8]. A large number of the DNS trajectories in theZ andAspace reduce to one trajectory which goes along the bottom

of the potential energy valley. The calculation of the DNS evolution process is carried out using the Monte-Carlo method for different angular momentaL. It is assumed that the

DNS which crosses over the maximumV(Z ;L)goes irreversibly into the complete fusion

channel. The DNS which has reached the symmetric configuration irreversibly proceeds into the quasi-fission channel. The model was tested in the calculation of the production cross-section of244

Fm in four reactions with different charge and mass asymmetries [7]. Figure 7 shows the potential energy of the DNS which is formed in these reactions. The injection point of the reactions are indicated. The probabilities of complete fusionP

cnare

presented in fig. 8. These data demonstrate a powerful influence of quasi-fission onP cn

in symmetric nuclear reactions. Using this model it was possible to reproduce the exper-imental values of the production cross-sections for244

Fm in the reactions with40

Ar and

76

Ge ions [9] (fig. 9) and to explain the absence of the effect in the reactions with86

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1134 V. V. VOLKOV, G. G. ADAMIAN, N. V. ANTONENKO, E. A. CHEREPANOVandA. K. NASIROV

Fig. 10. – The potential energy of the DNS formed in the reaction136

Xe +136

Xe. The injection point of the reaction is indicated by the arrow.

136

Xe ions.

In the second model of competition between complete fusion and quasi-fission the DNS evolution is considered under the condition of high viscosity [10]. The evolution is de-scribed by two collective variables:andR.=(A

1 ,A 2 )=(A 1 +A 2

)describes the mass

asymmetry of the DNS,Ris the internuclear centre-to-centre distance. The stationary

solution to the Fokker-Planck equation is used in the form suggested by Kramers for the description of the DNS evolution. The stationary probability of the flow through the inner fusion barrierB

 fus

and the quasi-fission barrierB

qfdefines the value P

cn. The model was

tested for nuclear reactions in whichP

cncan be calculated using experimental data for the

evaporation residue cross-section.

This model was used for the calculation ofP

cn in the reactions of cold synthesis of

elements from 104 to 114. The results of the calculations are presented in the talk of Antonenko at this Seminar (see this issue and ref. [10]). The calculations indicate that in the cold synthesis quasi-fission is the main factor responsible for the decrease of the SHE production cross-section with increasing atomic number.

3.3. Perspectives of using symmetric nuclear reactions for the SHE synthesis. – There are large negativeQ-values in the symmetric reactions between massive nuclei. This fact

allows one to hope that weakly excited nuclei of SHE will be produced. Within the frame-work of the DNS-concept the fusion of two136

Xe nuclei into the nucleus272

108 was con-sidered. Figure 10 shows the potential energyV(Z ;L = 0) of the DNS formed in this

reaction. One can see that the inner fusion barrierB  fus

has a value of about 30 MeV. The initial DNS must have an excitation energy of not less than 30 MeV to overcome this bar-rier. The compound nucleus will have nearly the same excitation energy. With increasing

Zof the compound nucleus,B  fus

also increases. This means that it is not possible to pro-duce a cold nucleus of SHE in symmetric nuclear reactions. The symmetric massive DNS has a negligible small quasi-fission barrier, which makes the quasi-fission channel strongly dominate over the complete fusion channel. TheP

cnvalue in the reaction 136

Xe +136

Xe is equal to a few units of 10,9

[9]. Therefore, the analysis made by the DNS-concept demon-strates that symmetric or near-symmetric reactions are not feasible for the synthesis of SHE.

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CHEREPANOVE. A., NASIROVA. K., PERMJAKOVV. P. and VOLKOVV. V., Phys. Lett. B,

319 (1993) 425; Phys. Rev. C, 51 (1995) 2635.

[2] SWIATECKIW. J., Phys. Scr., 24 (1981) 113. BJORNHOLMS. and SWIATECKIW., Nucl. Phys. A, 319 (1982) 471; BLOCKIJ. P., FELDMEIERH. and SWIATECKIW. J., Nucl. Phys. A, 459 (1986) 145.

[3] VOLKOVV. V., CHEREPANOVE. A., ANTONENKON. V. and NASIROVA. K., in Low Energy Nuclear Dynamics, St.Petersburg 1995, edited by YU. OGANESSIAN, R. KALPAKCHIEVA and W.VONOERTZEN(World Scientific, Singapore) 1995, pp. 336-344.

[4] MORAWEKW., ACKERMANND., BROHNT., CLERCH.-G., GOLLERTHANU., HANELT E., HORZM., SCHWABW., VOSSB., SCHMIDTK.-H. and HESSBERGERF. P., Z. Phys. A,

341 (1991) 75.

[5] FR¨OBRICHP., Phys. Rep., 116 (1984) 337.

[6] POPEKOA. et al., Report at Deutsche Physikalische Gesellschaft, 1997.

[7] CHEREPANOVE. A., VOLKOVV. V., ANTONENKON. V. and NASIROVA. K., Heavy Ion Physics and its Application, Lanzhou, China, 1995, edited by Y. X. LUO, G. M. JINand J. Y. LIU(World Scientific, Singapore) 1996, pp. 272-282.

[8] AYIKS., SCHURMANN¨ B. and N¨ORENBERGW., Z. Phys. A, 277 (1978) 299.

[9] G¨AGGELER H., SIKKELAND T., WIRTH G., BRUCHLE¨ W., B¨OGL W., FRANZ G., HERRMANNG., KRATZJ.V., SCHADEL¨ M., S ¨UMMERERK. and WEBERW., Z. Phys. A,

316 (1984) 291.

[10] ADAMIANG. G., ANTONENKON. V., SCHEIDW. and VOLKOVV. V., to be published in Nucl. Phys..

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