- Email: [email protected]
Influence of entrance channels on the formation of superheavy nuclei in massive fusion reactions
Nuclear Physics A 836 (2010) 82–90 www.elsevier.com/locate/nuclphysa
Influence of entrance channels on the formation of superheavy nuclei in massive fusion reactions Zhao-Qing Feng ∗ , Gen-Ming Jin, Jun-Qing Li Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China Received 16 November 2009; received in revised form 12 January 2010; accepted 12 January 2010 Available online 26 January 2010
Abstract Within the framework of the dinuclear system (DNS) model, the production cross sections of superheavy nuclei Hs (Z=108) and Z=112 combined with different reaction systems are analyzed systematically. It is found that the mass asymmetries and the reaction Q values of the projectile–target combinations play a very important role on the formation cross sections of the evaporation residues. Both methods to obtain the fusion probability by nucleon transfer by solving a set of microscopically derived master equations along the mass asymmetry degree of freedom (1D) and distinguishing protons and neutrons of fragments (2D) are compared with each other and also with the available experimental data. © 2010 Elsevier B.V. All rights reserved. Keywords: DNS model; Production cross sections; Mass asymmetries; Reaction Q values
1. Introduction The synthesis of heavy or superheavy nuclei (SHN) has obtained much progress experimentally with fusion–evaporation reactions [1,2]. Combinations with a doubly magic nucleus or nearly magic nucleus are usually chosen owing to the larger reaction Q values (absolute value). Neutron-deficient SHN with charged numbers Z=107–112 were synthesized in cold fusion reactions for the first time and investigated at GSI (Darmstadt, Germany) with the heavy-ion accelerator UNILAC and the SHIP separator [1,3]. Experiments on the synthesis of superheavy element 113 in the 70 Zn+209 Bi reaction have been performed successfully at RIKEN (Tokyo, Japan) [4]. However, it is difficult to produce heavier SHN in the cold fusion reactions because * Corresponding author. Tel.: +86 931 4969215; fax: +86 931 8272100.
E-mail address: [email protected] (Z.-Q. Feng). 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.01.244
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
83
of the smaller production cross sections at the level of 1 pb and even below 1 pb for Z>113. The superheavy elements Z=113–116, 118 were assigned at FLNR in Dubna (Russia) with the double magic nucleus 48 Ca bombarding actinide nuclei [5–7], in which more neutron-rich SHN were produced and identified by the consecutive α decay. Further verification of the element Z=114 in the 48 Ca+242 Pu reaction was reported in a recent experiment performed at Berkeley (California, USA) [8]. New heavy isotopes 259 Db and 265 Bh have also been synthesized at HIRFL in Lanzhou (China) [9]. Description of the formation of SHN in massive fusion reactions is of importance for understanding the mechanism of SHN synthesized in the cold fusion and the 48 Ca induced reactions and also for predicting optimal combinations and excitation energies to produce neutron-rich SHN close to the ‘island of stability’ in the near future. In this work, we use a dinuclear system (DNS) model [10,11], in which the barrier distribution of asymmetric Gaussian form in the capture and fusion process of two colliding nuclei is introduced in the model, and the nucleon transfer is coupled to the relative motion by solving a set of microscopically derived master equations by distinguishing protons and neutrons. The paper is organized as follows: in Section 2 we give a simple description of the further modified dinuclear system (DNS) model. Calculated results of fusion dynamics and SHN production are given in Section 3. In Section 4 conclusions are discussed. 2. Description of the DNS model The dinuclear system is a molecular configuration of two touching nuclei which keep their own individuality [10,12]. Such a system has an evolution along two main degrees of freedom: (i) the relative motion of the nuclei in the interaction potential to form the DNS and the decay of the DNS (quasi-fission process) along the R degree of freedom (internuclear motion), (ii) the transfer of nucleons in the mass asymmetry coordinate η = (A1 − A2 )/(A1 + A2 ) between two nuclei, which is a diffusion process of the excited systems leading to the compound nucleus formation. In this concept, the evaporation residue cross section is expressed as a sum over partial waves with angular momentum J at the center-of-mass energy Ec.m. , σER (Ec.m. ) =
Jmax π h¯ 2 (2J + 1)T (Ec.m. , J )PCN (Ec.m. , J )Wsur (Ec.m. , J ). 2μEc.m.
(1)
J =0
Here, T (Ec.m. , J ) is the transmission probability of the two colliding nuclei overcoming the Coulomb potential barrier in the entrance channel to form the DNS. The PCN is the probability that the system will evolve from a touching configuration to the compound nucleus in competition with quasi-fission of the DNS and fission of the heavy fragment. The last term is the survival probability of the formed compound nucleus, which can be estimated with the statistical evaporation model by considering the competition between neutron evaporation and fission of the compound nucleus [10]. We take the maximal angular momentum as Jmax = 30 since the fission barrier of the heavy nucleus disappears at high spin [13] which leads to exponentially decrease of the survival probability. In order to describe the fusion dynamics as a diffusion process along proton and neutron degrees of freedom, the fusion probability is obtained by solving a set of microscopically derived master equations numerically in the potential energy surface of the DNS. The time evolution of the distribution probability function P (Z1 , N1 , E1 , t) for fragment 1 with proton number Z1 and neutron number N1 with excitation energy E1 is described by the following master equations,
84
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
dP (Z1 , N1 , E1 , t) dt = WZ1 ,N1 ;Z1 ,N1 (t) dZ1 ,N1 P Z1 , N1 , E1 , t − dZ1 ,N1 P (Z1 , N1 , E1 , t) Z1
+
N1
WZ1 ,N1 ;Z1 ,N1 (t) dZ1 ,N1 P Z1 , N1 , E1 , t − dZ1 ,N1 P (Z1 , N1 , E1 , t)
− Λqf Θ(t) + Λfis Θ(t) P (Z1 , N1 , E1 , t).
(2)
Here the WZ1 ,N1 ;Z1 ,N1 (WZ1 ,N1 ;Z1 ,N1 ) is the mean transition probability from the channel (Z1 , N1 , E1 ) to (Z1 , N1 , E1 ) (or (Z1 , N1 , E1 ) to (Z1 , N1 , E1 )), and dZ1 ,N1 denotes the microscopic dimension corresponding to the macroscopic state (Z1 , N1 , E1 ). The sum is taken over all possible proton and neutron numbers that fragment Z1 , N1 may take, but only one nucleon transfer is considered in the model with Z1 = Z1 ± 1 and N1 = N1 ± 1 [14]. The excitation energy E1 is determined by the dissipation energy from the relative motion and the potential energy surface of the DNS. The motion of nucleons in the interacting potential is governed by the single-particle Hamiltonian [10,11]. The evolution of the DNS along the variable R leads to the quasi-fission of the DNS. The quasi-fission rate Λqf and the fission rate Λfis of heavy fragment can be estimated with the one-dimensional √ Kramers formula [11,15]. The local temperature is given by the Fermi-gas expression Θ = ε /a corresponding to the local excitation energy ε and level density parameter a = A/12 MeV−1 , which has different value for each fragment of the DNS. In the relaxation process of the relative motion, the DNS will be excited by the dissipation of the relative kinetic energy. The local excitation energy is determined by the excitation energy of the composite system and the potential energy surface of the DNS. The potential energy surface (PES) of the DNS is given by CN U {α} = B(Z1 , N1 ) + B(Z2 , N2 ) − B(Z, N) + Vrot (J ) + V {α} (3) with Z1 + Z2 = Z and N1 + N2 = N . Here the symbol {α} represents the variables Z1 , N1 , Z2 , N2 ; J , R; β1 , β2 , θ1 , θ2 . The B(Zi , Ni ) (i = 1, 2) and B(Z, N ) are the negative binding energies of the fragment (Zi , Ni ) and the compound nucleus (Z, N ) calculated by the liquid drop model, CN is respectively, in which the shell and the pairing corrections are included reasonably. The Vrot the rotation energy of the compound nucleus. The βi represent the quadrupole deformations of the two fragments. The θi denote the angles between the collision orientations and the symmetry axes of deformed nuclei. The interaction potential between fragment (Z1 , N1 ) and (Z2 , N2 ) includes the nuclear, Coulomb and centrifugal parts, the details are given in Ref. [11]. In the calculation, the distance R between the centers of the two fragments is chosen to be the value which gives the minimum of the interaction potential, in which the DNS is considered to be formed. So the PES depends on the proton and neutron numbers of the fragment. In Fig. 1 we give the potential energy surface in the reaction 30 Si+252 Cf as functions of protons and neutrons of the fragments in the left panel (2D PES). The incident point is shown by the open circle and the minimum trajectory in the PES is added by the white line. The driving potential as a function of the mass asymmetry of two fragments η = (A1 − A2 )/(A1 + A2 ) (1D PES) that was calculated in Refs. [10,11] is given in the right panel and also compared with the minimum trajectory of the 2D PES shown in the left panel. In the 1D PES, we chose the way which gives the minimum value of the PES after transferring proton or neutron from the incident point. So the 1D PES only depends on the mass asymmetry of two fragments (one degree of freedom). For the system 30 Si+252 Cf,
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
85
Fig. 1. The potential energy surface of the DNS in the reaction 30 Si+252 Cf as functions of the protons and neutrons of the fragments (left panel) and the mass asymmetry coordinate (right panel).
the driving potential at the incident point in 1D PES is located at the position of the maximum value, so there is no inner fusion barrier, which results in too large fusion probability. Therefore, we will solve the master equations within the 2D PES in order to correctly get the fusion probability for those systems with the larger entrance mass asymmetries and quadrupole deformations (2D master equations). The master equations can be also solved within the 1D PES (1D master equations). Both methods give the same values for the systems with the smaller projectile-target mass asymmetries and also smaller quadrupole deformations of the initial combinations, such as the reactions involved in magic nucleus, i.e., the cold fusion reactions. The formation probability of the compound nucleus at the Coulomb barrier B and angular momentum J is given by PCN (Ec.m. , J, B) =
Z BG N BG
P Z1 , N1 , E1 , τint (Ec.m. , J, B) .
(4)
Z1 =1 N1 =1
The interaction time τint in the dissipation process of two colliding partners is dependent on the incident energy Ec.m. and the angular momentum J , which is calculated by using the deflection function method [16] and has the value of few 10−20 s. We obtain the fusion probability as (5) PCN (Ec.m. , J ) = f (B)PCN (Ec.m. , J, B) dB, where the barrier distribution function is taken as an asymmetric Gaussian form [10]. The survival probability of the excited compound nucleus cooled by the neutron evaporation in competition with fission is expressed as follows:
x ∗ ∗ Γn (Ei∗ , J ) , (6) Wsur ECN , x, J = P ECN , x, J Γn (Ei∗ , J ) + Γf (Ei∗ , J ) i i=1
∗ , J are the excitation energy and the spin of the compound nucleus, respectively. where the ECN The Ei∗ is the excitation energy before evaporating the ith neutron, which has the relation ∗ Ei+1 = Ei∗ − Bin − 2Ti ,
with the initial condition The nuclear temperature
(7)
∗ . The energy B n is the separation energy of the ith neutron. E1∗ = ECN i Ti is given as Ei∗ = aTi2 − Ti with the level density parameter a. The
86
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
Fig. 2. Comparison of the calculated evaporation residue excitation functions using the 1D and 2D master equations with the available experimental data in the reaction 48 Ca+238 U. ∗ , x, J ) is the realization probability of emitting x neutrons. The widths of neutron evapP (ECN oration and fission are calculated using the statistical model. The detailed calculations can be found in Refs. [10,15].
3. Results and discussions With the procedure introduced above, we calculated the evaporation residue excitation functions using the 1D and 2D master equations in the reaction 48 Ca+238 U as shown in Fig. 2 represented by dashed and solid lines, respectively, and compared them with the experimental data performed in Dubna [17] and at GSI [18]. The GSI results show that the cross sections of the evaporation residues in the 3n channel are smaller that the ones of Dubna at the same excitation energy with 35 MeV, which are in good agreement with our 1D calculations. In the whole range, the 2D calculations give smaller cross sections than those by using the 1D master equations owing to the decrease of the fusion probability. For the considered system, the value of the PES at the incident point is located on the line of the minimum trajectory of the PES. So the 1D master equations can give reasonable results. Both methods have the same results for the cold fusion reactions. However, for those systems with the larger mass asymmetries and the larger quadrupole deformations, e.g. 16 O+238 U, 22 Ne+244 Pu, etc., a too low inner fusion barrier appears in the 1D PES and therefore leads to a larger fusion probability. So the 2D approach is used to treat these systems. The synthesis of heavy or superheavy nuclei through fusing two stable nuclei is confined by the so-called quasi-fission process with decreasing the mass asymmetry of the colliding partners. The entrance channel combinations of projectile and target will influence the fusion dynamics. The suppression of the evaporation residue excitation functions for less fissile compound systems such as 216 Ra and 220 Th when reactions are involved in projectiles heavier than 12 C or 16 O was observed in Ref. [19]. The wider width of the mass distributions for the fission-like fragments was also reported in Ref. [20]. In Fig. 3 we calculated the transmission and fusion probabilities using the 2D master equations for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe which lead to the formation of the same compound nucleus 272 Hs. The larger transmission probabilities were found in the reactions 64 Fe+208 Pb and 136 Xe+136 Xe owing to the larger Q values (absolute values). Smaller mass asymmetries of the two systems result in a decrease of the fusion proba-
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
87
Fig. 3. Calculated transmission and fusion probabilities as functions of the excitation energies of the compound nucleus for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe.
Fig. 4. Comparison of the calculated evaporation residue cross sections in 1n–5n channels using the 2D master equations for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe.
bilities. The evaporation residue excitation functions in 1n–5n channels are shown in Fig. 4. The competition of the capture and the fusion process of the three systems leads to different trends of the evaporation channels. The 3n and 4n channels in the reaction 34 S+238 U, 1n and 2n channels in the reaction 64 Fe+208 Pb are favorable to produce the isotopes 269,268 Hs and 271,270 Hs. The isotopes 60 Fe and 62 Fe are feasible to be accumulated experimentally, and the cross sections are 1.3 pb and 4.9 pb in the 1n channel at the excitation energy 12 MeV, and 1.8 pb and 7.2 pb in the 2n channel at the excitation energies 24 MeV and 23 MeV, respectively. Although the system 136 Xe+136 Xe consists of two magic nuclei, the higher inner fusion barrier decreases the fusion probability and enhances the quasi-fission rate of the DNS, hence leads to the smaller cross sections of the Hs isotopes. The upper limit of the cross sections for evaporation residues σ(1−3)n 4 pb was reported in a recent experiment [21], which are much lower than the ones predicted by the fusion by diffusion model [22]. In the DNS model, the larger mass asymmetry
88
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
Fig. 5. The same as in Fig. 4, but for the reactions 30 Si+252 Cf, 36 S+250 Cm, 40 Ar+244 Pu and 48 Ca+238 U leading to the formation of the element Z=112.
favors the nucleon transfer from the light projectile to the heavy target, and therefore enhances the fusion probability of two colliding partners. Combinations with the larger mass asymmetry and the larger Q values (close to shell closure) are suitable to synthesize SHN because of the larger cross sections. The superheavy element Z=112 was synthesized at GSI with the new isotope 277 112 by the cold fusion reaction 70 Zn+208 Pb [23] and also fabricated with more neutron-rich isotopes 282,283 112 in the 48 Ca induced reaction 48 Ca+238 U [17,18]. Shown in Fig. 5 is the calculated evaporation residue excitation functions in the reactions 30 Si+252 Cf, 36 S+250 Cm, 40 Ar+244 Pu and 48 Ca+238 U, which lead to the production of new isotopes of the element Z=112 between the cold fusion and the 48 Ca induced reactions. It is clear that the 2n, 3n and 4n channels in the reaction 30 Si+252 Cf at excitation energies 38 MeV, 41 MeV and 46 MeV, respectively, and 4n channel in the reaction 36 S+250 Cm at excitation energy 45 MeV have the larger cross sections to produce the isotopes 278–280,282 112. But the isotope 252 Cf is difficulty to be constructed as a target experimentally. If using the isotope 249 Cf irradiated with the 30 Si, the calculated production cross sections are 3.1 pb, 20.8 pb and 14.6 pb in the 2n, 3n and 4n channels at excitation energies 38 MeV, 41 MeV and 46 MeV, respectively. Reaction mechanism in the synthesis of heavy and superheavy nuclei should be investigated experimentally in the near future, such as the entrance mass asymmetry, reaction Q value, initial deformations of two colliding partners, etc., which are important not only for understanding the influence of the quasi-fission and the shell closure on the formation of heavy and superheavy nuclei, and also being helpful for improving theoretical models.
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
89
4. Conclusions In summary, we systematically investigated the entrance channel effects in the formation of SHN using the DNS model. The systems with the larger entrance mass asymmetry and the larger reaction Q value (absolute values) can enhance the capture and fusion probabilities of two colliding nuclei, and consequently increase the production cross sections of SHN. Calculations were carried out for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe, which lead to the same compound nucleus formation. The 2n, 3n and 4n channels in the reaction 30 Si+249,252 Cf, and the 4n channel in the reaction 36 S+250 Cm are favorable to synthesize new isotopes of the element Z=112 at the stated excitation energies. Acknowledgements We would like to thank Prof. Werner Scheid for carefully reading the manuscript. This work was supported by the National Natural Science Foundation of China under Grant Nos. 10805061 and 10975064, the special foundation of the president fund, the west doctoral project of Chinese Academy of Sciences, and the major state basic research development program under Grant No. 2007CB815000. References [1] S. Hofmann, G. Münzenberg, The discovery of the heaviest elements, Rev. Mod. Phys. 72 (2000) 733–767; S. Hofmann, New elements-approaching Z=114, Rep. Prog. Phys. 61 (1998) 639–689. [2] Yu.Ts. Oganessian, Heaviest nuclei from 48 Ca-induced reactions, J. Phys. G 34 (2007) R165–R242; Yu.Ts. Oganessian, Synthesis and decay properties of heaviest nuclei with 48 Ca-induced reactions, Nucl. Phys. A 787 (2007) 343c–352c. [3] G. Münzenberg, Discoveries of the heaviest elements, J. Phys. G 25 (1999) 717–725. [4] K. Morita, K. Morimoto, D. Kaji, et al., Experiment on the synthesis of element 113 in the reaction 209 Bi(70 Zn,n)278 113, J. Phys. Soc. Jpn. 73 (2004) 2593–2596. [5] Yu.Ts. Oganessian, A.V. Yeremin, A.G. Popeko, et al., Synthesis of nuclei of the superheavy element 114 in reactions induced by 48 Ca, Nature 400 (1999) 242–245; Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Synthesis of superheavy nuclei in the 48 Ca+244 Pu reaction, Phys. Rev. C 62 (2000) 041604(R). [6] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Experiments on the synthesis of element 115 in the reaction 243 Am(48 Ca,xn)291−x 115, Phys. Rev. C 69 (2004) 021601(R). [7] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Synthesis of the isotopes of element 118 and 116 in the 249 Cf and 245 Cm+48 Ca fusion reactions, Phys. Rev. C 74 (2006) 044602. [8] L. Stavsetra, K.E. Gregorich, J. Dvorak, et al., Independent verification of element 114 production in the 48 Ca+242 Pu reaction, Phys. Rev. Lett. 103 (2009) 132502. [9] Z.G. Gan, Z. Qin, H.M. Fan, et al., A new alpha-particle-emitting isotope 259 Db, Eur. Phys. J. A 10 (2001) 21–25; Z.G. Gan, J.S. Guo, X.L. Wu, et al., New isotope 265Bh, Eur. Phys. J. A 20 (2004) 385–387. [10] Z.Q. Feng, G.M. Jin, F. Fu, J.Q. Li, Production cross sections of superheavy nuclei based on dinuclear system model, Nucl. Phys. A 771 (2006) 50–67. [11] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Formation of superheavy nuclei in cold fusion reactions, Phys. Rev. C 76 (2007) 044606; Z.Q. Feng, G.M. Jin, M.H. Huang, et al., Possible way to synthesize superheavy element Z=117, Chin. Phys. Lett. 24 (2007) 2251–2254. [12] G.G. Adamian, N.V. Antonenko, W. Scheid, et al., Treatment of competition between complete fusion and quasifission in collisions of heavy nuclei, Nucl. Phys. A 627 (1997) 361–378; G.G. Adamian, N.V. Antonenko, W. Scheid, et al., Fusion cross sections for superheavy nuclei in the dinuclear system concept, Nucl. Phys. A 633 (1998) 409–420. [13] P. Reiter, T.L. Khoo, T. Lauritsen, et al., Entry distribution, fission barrier, and formation mechanism of 254No, Phys. Rev. Lett. 84 (2000) 3542–3545.
90
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
[14] M.H. Huang, Z.G. Gan, Z.Q. Feng, et al., Neutron and proton diffusion in fusion reactions for the synthesis of superheavy nuclei, Chin. Phys. Lett. 25 (2008) 1243–1246. [15] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Production of heavy and superheavy nuclei in massive fusion reactions, Nucl. Phys. A 816 (2009) 33–51; Z.Q. Feng, G.M. Jin, J.Q. Li, Production of new superheavy Z=108–114 nuclei with 238 U, 244 Pu and 248,250 Cm targets, Phys. Rev. C 80 (2009) 057601; Z.Q. Feng, G.M. Jin, J.Q. Li, Production of heavy isotopes in transfer reactions by collisions of 238 U+238 U, Phys. Rev. C 80 (2009) 067601. [16] J.Q. Li, G. Wolschin, Distribution of the dissipated angular momentum in heavy-ion collisions, Phys. Rev. C 27 (1983) 590. [17] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Measurements of cross sections and decay properties of the isotopes of elements 112, 114 and 116 produced in the fusion reactions 233,238 U, 242 Pu and 248 Cm+48 Ca, Phys. Rev. C 70 (2004) 064609. [18] S. Hofmann, D. Ackermann, S. Antalic, et al., The reaction 48 Ca+238 U → 286 112∗ studied at the GSI-SHIP, Eur. Phys. J. A 32 (2007) 251–260. [19] A.C. Berriman, D.J. Hinde, M. Dasgupta, et al., Unexpected inhibition of fusion in nucleus–nucleus collisions, Nature 413 (2001) 144–147; D.J. Hinde, M. Dasgupta, A. Mukherjee, Severe inhibition of fusion by quasifission in reactions forming 220 Th, Phys. Rev. Lett. 89 (2002) 282701. [20] R.G. Thomas, D.J. Hinde, D. Duniec, et al., Entrance channel dependence of quasifission in reactions forming 220 Th, Phys. Rev. C 77 (2008) 034610. [21] Yu.Ts. Oganessian, S.N. Dmitriev, A.V. Yeremin, et al., Attempt to produce the isotopes of element 108 in the fusion reaction 136 Xe+136 Xe, Phys. Rev. C 79 (2009) 024608. [22] W.J. Swiatecki, K. Siwek-Wilczynska, J. Wilczynski, Calculations of cross sections for the synthesis of superheavy nuclei in cold fusion reactions, Int. J. Mod. Phys. E 13 (2004) 261–267. [23] S. Hofmann, V. Ninov, F.P. Heßberger, et al., Production and decay of 269 110, Z. Phys. A 350 (1995) 277–280.
Influence of entrance channels on the formation of superheavy nuclei in massive fusion reactions Zhao-Qing Feng ∗ , Gen-Ming Jin, Jun-Qing Li Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China Received 16 November 2009; received in revised form 12 January 2010; accepted 12 January 2010 Available online 26 January 2010
Abstract Within the framework of the dinuclear system (DNS) model, the production cross sections of superheavy nuclei Hs (Z=108) and Z=112 combined with different reaction systems are analyzed systematically. It is found that the mass asymmetries and the reaction Q values of the projectile–target combinations play a very important role on the formation cross sections of the evaporation residues. Both methods to obtain the fusion probability by nucleon transfer by solving a set of microscopically derived master equations along the mass asymmetry degree of freedom (1D) and distinguishing protons and neutrons of fragments (2D) are compared with each other and also with the available experimental data. © 2010 Elsevier B.V. All rights reserved. Keywords: DNS model; Production cross sections; Mass asymmetries; Reaction Q values
1. Introduction The synthesis of heavy or superheavy nuclei (SHN) has obtained much progress experimentally with fusion–evaporation reactions [1,2]. Combinations with a doubly magic nucleus or nearly magic nucleus are usually chosen owing to the larger reaction Q values (absolute value). Neutron-deficient SHN with charged numbers Z=107–112 were synthesized in cold fusion reactions for the first time and investigated at GSI (Darmstadt, Germany) with the heavy-ion accelerator UNILAC and the SHIP separator [1,3]. Experiments on the synthesis of superheavy element 113 in the 70 Zn+209 Bi reaction have been performed successfully at RIKEN (Tokyo, Japan) [4]. However, it is difficult to produce heavier SHN in the cold fusion reactions because * Corresponding author. Tel.: +86 931 4969215; fax: +86 931 8272100.
E-mail address: [email protected] (Z.-Q. Feng). 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.01.244
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
83
of the smaller production cross sections at the level of 1 pb and even below 1 pb for Z>113. The superheavy elements Z=113–116, 118 were assigned at FLNR in Dubna (Russia) with the double magic nucleus 48 Ca bombarding actinide nuclei [5–7], in which more neutron-rich SHN were produced and identified by the consecutive α decay. Further verification of the element Z=114 in the 48 Ca+242 Pu reaction was reported in a recent experiment performed at Berkeley (California, USA) [8]. New heavy isotopes 259 Db and 265 Bh have also been synthesized at HIRFL in Lanzhou (China) [9]. Description of the formation of SHN in massive fusion reactions is of importance for understanding the mechanism of SHN synthesized in the cold fusion and the 48 Ca induced reactions and also for predicting optimal combinations and excitation energies to produce neutron-rich SHN close to the ‘island of stability’ in the near future. In this work, we use a dinuclear system (DNS) model [10,11], in which the barrier distribution of asymmetric Gaussian form in the capture and fusion process of two colliding nuclei is introduced in the model, and the nucleon transfer is coupled to the relative motion by solving a set of microscopically derived master equations by distinguishing protons and neutrons. The paper is organized as follows: in Section 2 we give a simple description of the further modified dinuclear system (DNS) model. Calculated results of fusion dynamics and SHN production are given in Section 3. In Section 4 conclusions are discussed. 2. Description of the DNS model The dinuclear system is a molecular configuration of two touching nuclei which keep their own individuality [10,12]. Such a system has an evolution along two main degrees of freedom: (i) the relative motion of the nuclei in the interaction potential to form the DNS and the decay of the DNS (quasi-fission process) along the R degree of freedom (internuclear motion), (ii) the transfer of nucleons in the mass asymmetry coordinate η = (A1 − A2 )/(A1 + A2 ) between two nuclei, which is a diffusion process of the excited systems leading to the compound nucleus formation. In this concept, the evaporation residue cross section is expressed as a sum over partial waves with angular momentum J at the center-of-mass energy Ec.m. , σER (Ec.m. ) =
Jmax π h¯ 2 (2J + 1)T (Ec.m. , J )PCN (Ec.m. , J )Wsur (Ec.m. , J ). 2μEc.m.
(1)
J =0
Here, T (Ec.m. , J ) is the transmission probability of the two colliding nuclei overcoming the Coulomb potential barrier in the entrance channel to form the DNS. The PCN is the probability that the system will evolve from a touching configuration to the compound nucleus in competition with quasi-fission of the DNS and fission of the heavy fragment. The last term is the survival probability of the formed compound nucleus, which can be estimated with the statistical evaporation model by considering the competition between neutron evaporation and fission of the compound nucleus [10]. We take the maximal angular momentum as Jmax = 30 since the fission barrier of the heavy nucleus disappears at high spin [13] which leads to exponentially decrease of the survival probability. In order to describe the fusion dynamics as a diffusion process along proton and neutron degrees of freedom, the fusion probability is obtained by solving a set of microscopically derived master equations numerically in the potential energy surface of the DNS. The time evolution of the distribution probability function P (Z1 , N1 , E1 , t) for fragment 1 with proton number Z1 and neutron number N1 with excitation energy E1 is described by the following master equations,
84
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
dP (Z1 , N1 , E1 , t) dt = WZ1 ,N1 ;Z1 ,N1 (t) dZ1 ,N1 P Z1 , N1 , E1 , t − dZ1 ,N1 P (Z1 , N1 , E1 , t) Z1
+
N1
WZ1 ,N1 ;Z1 ,N1 (t) dZ1 ,N1 P Z1 , N1 , E1 , t − dZ1 ,N1 P (Z1 , N1 , E1 , t)
− Λqf Θ(t) + Λfis Θ(t) P (Z1 , N1 , E1 , t).
(2)
Here the WZ1 ,N1 ;Z1 ,N1 (WZ1 ,N1 ;Z1 ,N1 ) is the mean transition probability from the channel (Z1 , N1 , E1 ) to (Z1 , N1 , E1 ) (or (Z1 , N1 , E1 ) to (Z1 , N1 , E1 )), and dZ1 ,N1 denotes the microscopic dimension corresponding to the macroscopic state (Z1 , N1 , E1 ). The sum is taken over all possible proton and neutron numbers that fragment Z1 , N1 may take, but only one nucleon transfer is considered in the model with Z1 = Z1 ± 1 and N1 = N1 ± 1 [14]. The excitation energy E1 is determined by the dissipation energy from the relative motion and the potential energy surface of the DNS. The motion of nucleons in the interacting potential is governed by the single-particle Hamiltonian [10,11]. The evolution of the DNS along the variable R leads to the quasi-fission of the DNS. The quasi-fission rate Λqf and the fission rate Λfis of heavy fragment can be estimated with the one-dimensional √ Kramers formula [11,15]. The local temperature is given by the Fermi-gas expression Θ = ε /a corresponding to the local excitation energy ε and level density parameter a = A/12 MeV−1 , which has different value for each fragment of the DNS. In the relaxation process of the relative motion, the DNS will be excited by the dissipation of the relative kinetic energy. The local excitation energy is determined by the excitation energy of the composite system and the potential energy surface of the DNS. The potential energy surface (PES) of the DNS is given by CN U {α} = B(Z1 , N1 ) + B(Z2 , N2 ) − B(Z, N) + Vrot (J ) + V {α} (3) with Z1 + Z2 = Z and N1 + N2 = N . Here the symbol {α} represents the variables Z1 , N1 , Z2 , N2 ; J , R; β1 , β2 , θ1 , θ2 . The B(Zi , Ni ) (i = 1, 2) and B(Z, N ) are the negative binding energies of the fragment (Zi , Ni ) and the compound nucleus (Z, N ) calculated by the liquid drop model, CN is respectively, in which the shell and the pairing corrections are included reasonably. The Vrot the rotation energy of the compound nucleus. The βi represent the quadrupole deformations of the two fragments. The θi denote the angles between the collision orientations and the symmetry axes of deformed nuclei. The interaction potential between fragment (Z1 , N1 ) and (Z2 , N2 ) includes the nuclear, Coulomb and centrifugal parts, the details are given in Ref. [11]. In the calculation, the distance R between the centers of the two fragments is chosen to be the value which gives the minimum of the interaction potential, in which the DNS is considered to be formed. So the PES depends on the proton and neutron numbers of the fragment. In Fig. 1 we give the potential energy surface in the reaction 30 Si+252 Cf as functions of protons and neutrons of the fragments in the left panel (2D PES). The incident point is shown by the open circle and the minimum trajectory in the PES is added by the white line. The driving potential as a function of the mass asymmetry of two fragments η = (A1 − A2 )/(A1 + A2 ) (1D PES) that was calculated in Refs. [10,11] is given in the right panel and also compared with the minimum trajectory of the 2D PES shown in the left panel. In the 1D PES, we chose the way which gives the minimum value of the PES after transferring proton or neutron from the incident point. So the 1D PES only depends on the mass asymmetry of two fragments (one degree of freedom). For the system 30 Si+252 Cf,
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
85
Fig. 1. The potential energy surface of the DNS in the reaction 30 Si+252 Cf as functions of the protons and neutrons of the fragments (left panel) and the mass asymmetry coordinate (right panel).
the driving potential at the incident point in 1D PES is located at the position of the maximum value, so there is no inner fusion barrier, which results in too large fusion probability. Therefore, we will solve the master equations within the 2D PES in order to correctly get the fusion probability for those systems with the larger entrance mass asymmetries and quadrupole deformations (2D master equations). The master equations can be also solved within the 1D PES (1D master equations). Both methods give the same values for the systems with the smaller projectile-target mass asymmetries and also smaller quadrupole deformations of the initial combinations, such as the reactions involved in magic nucleus, i.e., the cold fusion reactions. The formation probability of the compound nucleus at the Coulomb barrier B and angular momentum J is given by PCN (Ec.m. , J, B) =
Z BG N BG
P Z1 , N1 , E1 , τint (Ec.m. , J, B) .
(4)
Z1 =1 N1 =1
The interaction time τint in the dissipation process of two colliding partners is dependent on the incident energy Ec.m. and the angular momentum J , which is calculated by using the deflection function method [16] and has the value of few 10−20 s. We obtain the fusion probability as (5) PCN (Ec.m. , J ) = f (B)PCN (Ec.m. , J, B) dB, where the barrier distribution function is taken as an asymmetric Gaussian form [10]. The survival probability of the excited compound nucleus cooled by the neutron evaporation in competition with fission is expressed as follows:
x ∗ ∗ Γn (Ei∗ , J ) , (6) Wsur ECN , x, J = P ECN , x, J Γn (Ei∗ , J ) + Γf (Ei∗ , J ) i i=1
∗ , J are the excitation energy and the spin of the compound nucleus, respectively. where the ECN The Ei∗ is the excitation energy before evaporating the ith neutron, which has the relation ∗ Ei+1 = Ei∗ − Bin − 2Ti ,
with the initial condition The nuclear temperature
(7)
∗ . The energy B n is the separation energy of the ith neutron. E1∗ = ECN i Ti is given as Ei∗ = aTi2 − Ti with the level density parameter a. The
86
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
Fig. 2. Comparison of the calculated evaporation residue excitation functions using the 1D and 2D master equations with the available experimental data in the reaction 48 Ca+238 U. ∗ , x, J ) is the realization probability of emitting x neutrons. The widths of neutron evapP (ECN oration and fission are calculated using the statistical model. The detailed calculations can be found in Refs. [10,15].
3. Results and discussions With the procedure introduced above, we calculated the evaporation residue excitation functions using the 1D and 2D master equations in the reaction 48 Ca+238 U as shown in Fig. 2 represented by dashed and solid lines, respectively, and compared them with the experimental data performed in Dubna [17] and at GSI [18]. The GSI results show that the cross sections of the evaporation residues in the 3n channel are smaller that the ones of Dubna at the same excitation energy with 35 MeV, which are in good agreement with our 1D calculations. In the whole range, the 2D calculations give smaller cross sections than those by using the 1D master equations owing to the decrease of the fusion probability. For the considered system, the value of the PES at the incident point is located on the line of the minimum trajectory of the PES. So the 1D master equations can give reasonable results. Both methods have the same results for the cold fusion reactions. However, for those systems with the larger mass asymmetries and the larger quadrupole deformations, e.g. 16 O+238 U, 22 Ne+244 Pu, etc., a too low inner fusion barrier appears in the 1D PES and therefore leads to a larger fusion probability. So the 2D approach is used to treat these systems. The synthesis of heavy or superheavy nuclei through fusing two stable nuclei is confined by the so-called quasi-fission process with decreasing the mass asymmetry of the colliding partners. The entrance channel combinations of projectile and target will influence the fusion dynamics. The suppression of the evaporation residue excitation functions for less fissile compound systems such as 216 Ra and 220 Th when reactions are involved in projectiles heavier than 12 C or 16 O was observed in Ref. [19]. The wider width of the mass distributions for the fission-like fragments was also reported in Ref. [20]. In Fig. 3 we calculated the transmission and fusion probabilities using the 2D master equations for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe which lead to the formation of the same compound nucleus 272 Hs. The larger transmission probabilities were found in the reactions 64 Fe+208 Pb and 136 Xe+136 Xe owing to the larger Q values (absolute values). Smaller mass asymmetries of the two systems result in a decrease of the fusion proba-
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
87
Fig. 3. Calculated transmission and fusion probabilities as functions of the excitation energies of the compound nucleus for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe.
Fig. 4. Comparison of the calculated evaporation residue cross sections in 1n–5n channels using the 2D master equations for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe.
bilities. The evaporation residue excitation functions in 1n–5n channels are shown in Fig. 4. The competition of the capture and the fusion process of the three systems leads to different trends of the evaporation channels. The 3n and 4n channels in the reaction 34 S+238 U, 1n and 2n channels in the reaction 64 Fe+208 Pb are favorable to produce the isotopes 269,268 Hs and 271,270 Hs. The isotopes 60 Fe and 62 Fe are feasible to be accumulated experimentally, and the cross sections are 1.3 pb and 4.9 pb in the 1n channel at the excitation energy 12 MeV, and 1.8 pb and 7.2 pb in the 2n channel at the excitation energies 24 MeV and 23 MeV, respectively. Although the system 136 Xe+136 Xe consists of two magic nuclei, the higher inner fusion barrier decreases the fusion probability and enhances the quasi-fission rate of the DNS, hence leads to the smaller cross sections of the Hs isotopes. The upper limit of the cross sections for evaporation residues σ(1−3)n 4 pb was reported in a recent experiment [21], which are much lower than the ones predicted by the fusion by diffusion model [22]. In the DNS model, the larger mass asymmetry
88
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
Fig. 5. The same as in Fig. 4, but for the reactions 30 Si+252 Cf, 36 S+250 Cm, 40 Ar+244 Pu and 48 Ca+238 U leading to the formation of the element Z=112.
favors the nucleon transfer from the light projectile to the heavy target, and therefore enhances the fusion probability of two colliding partners. Combinations with the larger mass asymmetry and the larger Q values (close to shell closure) are suitable to synthesize SHN because of the larger cross sections. The superheavy element Z=112 was synthesized at GSI with the new isotope 277 112 by the cold fusion reaction 70 Zn+208 Pb [23] and also fabricated with more neutron-rich isotopes 282,283 112 in the 48 Ca induced reaction 48 Ca+238 U [17,18]. Shown in Fig. 5 is the calculated evaporation residue excitation functions in the reactions 30 Si+252 Cf, 36 S+250 Cm, 40 Ar+244 Pu and 48 Ca+238 U, which lead to the production of new isotopes of the element Z=112 between the cold fusion and the 48 Ca induced reactions. It is clear that the 2n, 3n and 4n channels in the reaction 30 Si+252 Cf at excitation energies 38 MeV, 41 MeV and 46 MeV, respectively, and 4n channel in the reaction 36 S+250 Cm at excitation energy 45 MeV have the larger cross sections to produce the isotopes 278–280,282 112. But the isotope 252 Cf is difficulty to be constructed as a target experimentally. If using the isotope 249 Cf irradiated with the 30 Si, the calculated production cross sections are 3.1 pb, 20.8 pb and 14.6 pb in the 2n, 3n and 4n channels at excitation energies 38 MeV, 41 MeV and 46 MeV, respectively. Reaction mechanism in the synthesis of heavy and superheavy nuclei should be investigated experimentally in the near future, such as the entrance mass asymmetry, reaction Q value, initial deformations of two colliding partners, etc., which are important not only for understanding the influence of the quasi-fission and the shell closure on the formation of heavy and superheavy nuclei, and also being helpful for improving theoretical models.
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
89
4. Conclusions In summary, we systematically investigated the entrance channel effects in the formation of SHN using the DNS model. The systems with the larger entrance mass asymmetry and the larger reaction Q value (absolute values) can enhance the capture and fusion probabilities of two colliding nuclei, and consequently increase the production cross sections of SHN. Calculations were carried out for the reactions 34 S+238 U, 64 Fe+208 Pb and 136 Xe+136 Xe, which lead to the same compound nucleus formation. The 2n, 3n and 4n channels in the reaction 30 Si+249,252 Cf, and the 4n channel in the reaction 36 S+250 Cm are favorable to synthesize new isotopes of the element Z=112 at the stated excitation energies. Acknowledgements We would like to thank Prof. Werner Scheid for carefully reading the manuscript. This work was supported by the National Natural Science Foundation of China under Grant Nos. 10805061 and 10975064, the special foundation of the president fund, the west doctoral project of Chinese Academy of Sciences, and the major state basic research development program under Grant No. 2007CB815000. References [1] S. Hofmann, G. Münzenberg, The discovery of the heaviest elements, Rev. Mod. Phys. 72 (2000) 733–767; S. Hofmann, New elements-approaching Z=114, Rep. Prog. Phys. 61 (1998) 639–689. [2] Yu.Ts. Oganessian, Heaviest nuclei from 48 Ca-induced reactions, J. Phys. G 34 (2007) R165–R242; Yu.Ts. Oganessian, Synthesis and decay properties of heaviest nuclei with 48 Ca-induced reactions, Nucl. Phys. A 787 (2007) 343c–352c. [3] G. Münzenberg, Discoveries of the heaviest elements, J. Phys. G 25 (1999) 717–725. [4] K. Morita, K. Morimoto, D. Kaji, et al., Experiment on the synthesis of element 113 in the reaction 209 Bi(70 Zn,n)278 113, J. Phys. Soc. Jpn. 73 (2004) 2593–2596. [5] Yu.Ts. Oganessian, A.V. Yeremin, A.G. Popeko, et al., Synthesis of nuclei of the superheavy element 114 in reactions induced by 48 Ca, Nature 400 (1999) 242–245; Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Synthesis of superheavy nuclei in the 48 Ca+244 Pu reaction, Phys. Rev. C 62 (2000) 041604(R). [6] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Experiments on the synthesis of element 115 in the reaction 243 Am(48 Ca,xn)291−x 115, Phys. Rev. C 69 (2004) 021601(R). [7] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Synthesis of the isotopes of element 118 and 116 in the 249 Cf and 245 Cm+48 Ca fusion reactions, Phys. Rev. C 74 (2006) 044602. [8] L. Stavsetra, K.E. Gregorich, J. Dvorak, et al., Independent verification of element 114 production in the 48 Ca+242 Pu reaction, Phys. Rev. Lett. 103 (2009) 132502. [9] Z.G. Gan, Z. Qin, H.M. Fan, et al., A new alpha-particle-emitting isotope 259 Db, Eur. Phys. J. A 10 (2001) 21–25; Z.G. Gan, J.S. Guo, X.L. Wu, et al., New isotope 265Bh, Eur. Phys. J. A 20 (2004) 385–387. [10] Z.Q. Feng, G.M. Jin, F. Fu, J.Q. Li, Production cross sections of superheavy nuclei based on dinuclear system model, Nucl. Phys. A 771 (2006) 50–67. [11] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Formation of superheavy nuclei in cold fusion reactions, Phys. Rev. C 76 (2007) 044606; Z.Q. Feng, G.M. Jin, M.H. Huang, et al., Possible way to synthesize superheavy element Z=117, Chin. Phys. Lett. 24 (2007) 2251–2254. [12] G.G. Adamian, N.V. Antonenko, W. Scheid, et al., Treatment of competition between complete fusion and quasifission in collisions of heavy nuclei, Nucl. Phys. A 627 (1997) 361–378; G.G. Adamian, N.V. Antonenko, W. Scheid, et al., Fusion cross sections for superheavy nuclei in the dinuclear system concept, Nucl. Phys. A 633 (1998) 409–420. [13] P. Reiter, T.L. Khoo, T. Lauritsen, et al., Entry distribution, fission barrier, and formation mechanism of 254No, Phys. Rev. Lett. 84 (2000) 3542–3545.
90
Z.-Q. Feng et al. / Nuclear Physics A 836 (2010) 82–90
[14] M.H. Huang, Z.G. Gan, Z.Q. Feng, et al., Neutron and proton diffusion in fusion reactions for the synthesis of superheavy nuclei, Chin. Phys. Lett. 25 (2008) 1243–1246. [15] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Production of heavy and superheavy nuclei in massive fusion reactions, Nucl. Phys. A 816 (2009) 33–51; Z.Q. Feng, G.M. Jin, J.Q. Li, Production of new superheavy Z=108–114 nuclei with 238 U, 244 Pu and 248,250 Cm targets, Phys. Rev. C 80 (2009) 057601; Z.Q. Feng, G.M. Jin, J.Q. Li, Production of heavy isotopes in transfer reactions by collisions of 238 U+238 U, Phys. Rev. C 80 (2009) 067601. [16] J.Q. Li, G. Wolschin, Distribution of the dissipated angular momentum in heavy-ion collisions, Phys. Rev. C 27 (1983) 590. [17] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Measurements of cross sections and decay properties of the isotopes of elements 112, 114 and 116 produced in the fusion reactions 233,238 U, 242 Pu and 248 Cm+48 Ca, Phys. Rev. C 70 (2004) 064609. [18] S. Hofmann, D. Ackermann, S. Antalic, et al., The reaction 48 Ca+238 U → 286 112∗ studied at the GSI-SHIP, Eur. Phys. J. A 32 (2007) 251–260. [19] A.C. Berriman, D.J. Hinde, M. Dasgupta, et al., Unexpected inhibition of fusion in nucleus–nucleus collisions, Nature 413 (2001) 144–147; D.J. Hinde, M. Dasgupta, A. Mukherjee, Severe inhibition of fusion by quasifission in reactions forming 220 Th, Phys. Rev. Lett. 89 (2002) 282701. [20] R.G. Thomas, D.J. Hinde, D. Duniec, et al., Entrance channel dependence of quasifission in reactions forming 220 Th, Phys. Rev. C 77 (2008) 034610. [21] Yu.Ts. Oganessian, S.N. Dmitriev, A.V. Yeremin, et al., Attempt to produce the isotopes of element 108 in the fusion reaction 136 Xe+136 Xe, Phys. Rev. C 79 (2009) 024608. [22] W.J. Swiatecki, K. Siwek-Wilczynska, J. Wilczynski, Calculations of cross sections for the synthesis of superheavy nuclei in cold fusion reactions, Int. J. Mod. Phys. E 13 (2004) 261–267. [23] S. Hofmann, V. Ninov, F.P. Heßberger, et al., Production and decay of 269 110, Z. Phys. A 350 (1995) 277–280.