(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.
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
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
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.
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].
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
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
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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