Theoretical Study on Alpha-Decay Chains of

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1 Commun. Theor. Phys. 55 (2011) Vol. 55, No. 3, March 15, 2011 Theoretical Study on Alpha-Decay Chains of and SHENG Zong-Qiang (âñö) 1, and REN Zhong-Zhou ( ) 1,2,3 1 School of Physics, Nanjing University, Nanjing , China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy-ion Accelerator at Lanzhou, Lanzhou , China 3 Kavli Institute for Theoretical Physics China, Beijing , China (Received June 2, 2010; revised manuscript received September 1, 2010) Abstract The newly synthesized element 117 and its alpha-decay chains are systematically investigated in the framework of the relativistic mean field theory with parameter sets NL-Z2 and TMA. The ground-state properties of the superheavy nuclei on the alpha-decay chains of and are calculated. The experimental alpha-decay energies and half-lives of the two alpha-decay chains are reasonably reproduced by the model. The detailed discussions on the binding energies, alpha-decay energies, half-lives, quadrupole deformations, potential energy curves, and single particle levels of the two alpha-decay chains are made. PACS numbers: k, Dr, e, b Key words: superheavy nucleus, relativistic mean field theory, binding energy, alpha-decay energy 1 Introduction Since the theoretical prediction of the existence of superheavy island in 1960s, [1] the synthesis of new superheavy elements has been a hot topic in nuclear physics. The ultimate goal of synthesizing superheavy nuclei is to reach an expected island of stability which is located around the predicted spherical doubly magic nucleus During the period from 1995 to 1996, Hofmann et al. [2 5] successively produced the elements with Z = 110, 111, and 112 by using cold-fusion reactions at GSI in Germany. Subsequently, the elements with Z = and 118 were produced at another laboratory, Duban, in Russia by Oganessian et al. [6 13] Very recently, a new element with atomic number Z = 117 has been synthesized also at Dubna. [14] The isotopes and were produced in the 48 Ca Bk reactions. The decay mode of was detected involving six consecutive alpha decays and ending in spontaneous fission of 270 Db, and that of involving three consecutive alpha decays and ending in spontaneous fission of 281 Rg. The alpha-decay energies and half-lives of the two alpha-decay chains are measured. [14] The two decay chains involve 11 new nuclei. So far, all elements from Z = 110 to Z = 118 have been synthesized in laboratory. All these achievements increase our knowledge about the expected island of stability. The ground-state properties of all these nuclei on the two decay chains, such as binding energies, quadrupole deformations etc., are unknown. It is very significant to investigate them with theoretical models. In the past, the relativistic mean field (RMF) theory has achieved great success in describing lots of nuclear phenomena. [15 23] In Refs. [20 23], the RMF theory has been successfully used to study superheavy nuclei. In the present work, we will investigate the ground-state properties of newly discovered nuclei ( and ) and their decay chains in the deformed RMF model. 2 Theoretical Framework In the RMF approach, the nuclear interaction is usually described via the exchange of three mesons: the isoscalar meson σ, which provides the medium-range attraction between the nucleons, the isoscalar-vector meson ω µ, which offers the short-range repulsion, and the isovector-vector meson ρ µ, which gives the isospin dependence of the nuclear force. The photon A µ is included because there is the electromagnetic interaction between protons. The effective Lagrangian density is as the following: L = ψ(iγ µ µ M)ψ µ σ µ σ 1 2 m2 σ σ2 1 3 g 2σ g 3σ 4 g σ ψσψ 1 4 Ωµν Ω µν m2 ω ωµ ω µ g ω ψγµ ω µ ψ g 4(ω µ ω µ ) R µν Rµν m2 ρ ρ µ ρ µ g ρ ψγµ τ ρ µ ψ 1 4 F µν F µν e ψγ 1 τ 3 µ A µ ψ. (1) 2 Supported by National Natural Science Foundation of China under Grant Nos , , and , by 973 National Major State Basic Research and Development of China under Grant No. 2007CB and 2010CB327803, CAS Knowledge Innovation Project under Grant No. KJCX2-SW-N02, and by Research Fund of Doctoral Point under Grant No shengzongq@gmail.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 496 Communications in Theoretical Physics Vol. 55 The classical variation principle leads to the Dirac equation [ i + V (r) + β(m + S(r))]ψ i = ε i ψ i (2) for the nucleon spinors and the Klein-Gordon equations ( + m 2 σ )σ(r) = g σρ s (r) g 2 σ 2 (r) g 3 σ 3 (r), ( + m 2 ω)ω µ (r) = g ω j µ (r) g 4 (ω ν ω ν )ω µ (r), ( + m 2 ρ ) ρ µ (r) = g ρ j µ (r), A µ (r) = ej µ ρ (r) (3) for the mesons, where V (r) = g ω γ µ ω µ (r) + g ρ γ µ 1 τ 3 τ ρ µ (r) + eγ µ A µ (r), 2 S(r) = g σ σ(r) (4) are the vector and scalar potentials respectively. The equations (2) and (3) can be self-consistently solved by iteration. The details can be found in Refs. [24 25]. 3 Numerical Results and Discussions The deformed RMF codes in harmonic bases are used for calculations. The number of bases is chosen to be N f = N b = 20. This space is large enough for calculations here. The oscillator frequency is fixed as ω 0 = 41A 1/3 MeV. The pairing correlation is treated with BCS approximation, where the pairing gap is taken as n = p = 11.2/ A MeV. An axial deformation is initially assumed for the iteration in the calculations. The charge radius is taken as R c = Rp fm. In numerical calculations of the ground-state properties, the blocking effect of odd nucleon is omitted, because the quantities we concern are binding energy and alpha-decay energy. The influence of the blocking effect on the total binding energy is very small. The parameter sets NL-Z2 [26] and TMA [27] are used for calculation, which are very successful parameter sets for describing superheavy nuclei. [22 23,28] We first discuss the binding energies and alpha-decay energies of the superheavy nuclei on the two decay chains. Because there are no experimental binding energies for these nuclei, the results from the finite-range droplet model with folded-yukawa single-particle potentials (FRDM+FY) by Möller et al. are introduced for comparison. [29] The theoretical results and the experimental data are listed in Table 1. In Table 1, the calculated binding energies B, alphadecay energies Q of alpha-decay chains of and in the RMF model with parameter sets NL-Z2 and TMA, and the results from the finite-range droplet model with folded-yukawa single-particle potentials are tabulated. The last two columns show the experimental alpha-decay energies Q and half-lives T. Here, the theoretical Q is defined as the binding energy difference between the daughter nucleus and the parent nucleus via the formula: Q = B (Z 2, N 2) B (Z, N) MeV. Table 1 The binding energies B, alpha-decay energies Q and lifetimes T of alpha-decay chains and in the RMF model with parameter sets NL-Z2 and TMA. The eighth and ninth columns are the results the finite-range droplet model with folded-yukawa single-particle potentials (FRDM+FY) by Möller et al. [29] The last two columns show the experimental data from Dubna. [14] All energies are in unit of MeV and half-lives in second. Nuclei NL-Z2 TMA FRDM+FY Experiment B Q T B Q T B Q T Q T Rg Mt Bh Db Rg For binding energy, it can be seen from Table 1 that the two sets of RMF results with TMA and NL-Z2 are very close. This shows that the RMF model is very stable for calculating the nuclei in this work. The calculated binding energies with TMA are slightly larger than the ones with NL-Z2, and the results from FRDM+FY are between the ones with TMA and NL-Z2. In Ref. [22], we have calculated some superheavy nuclei whose binding

3 No. 3 Communications in Theoretical Physics 497 energies are known. We find the theoretical value with TMA sets the upper limit of the binding energy and the one with NL-Z2 sets the lower limit. It is very useful for us to predict the binding energies of the two alpha-decay chains because both calculated values with TMA and NL- Z2 are very close. The binding energies of the superheavy nuclei on the two alpha-decay chains are expected to be between the calculated values with TMA and NL-Z2. For alpha-decay energy, the RMF results with NL-Z2 and TMA are listed in the third and sixth columns in Table 1, and the results from FRDM+FY and the experimental values are shown in ninth and eleventh columns, respectively. For a clear insight into the agreement between theoretical and experimental alpha-decay energies, we plot the difference between theoretical alpha-decay energies Q theor and experimental alpha-decay energies Q expt in Figs. 1 and 2. Here the difference is defined as: Q = Q theor Q expt. Fig. 1 The difference between the Q theor for the alpha-decay chain of and the Q expt Fig. 2 The same as Fig. 1, but for the alpha-decay chain of Figure 1 shows the difference between the Q theor and the Q expt for the alpha-decay chain of It is seen from Fig. 1 that most Q are less than 1 MeV. The Q are slightly large in 282 Rg and with TMA. The largest Q is about 1.3 MeV in 282 Rg with TMA. The agreement between the theoretical results with NL- Z2 and the experimental values is similar to those of the FRDM+FY. All the three models give similar descriptions of the alpha-decay energies in and , which are slightly smaller than the experimental values. Figure 2 is the same as Fig. 1, but for the alpha-decay chain of It is seen from Fig. 2 that most of the theoretical alpha-decay energies are smaller than the experimental alpha-decay energies. Except for the result in with TMA, all Q are less than 1 MeV. We also can see that the theoretical alpha-decay energies with NL-Z2 are systematically about 0.6 MeV smaller than the experimental values. On the whole, The RMF theory with NL-Z2 and TMA and the FRDM+FY can reasonably reproduce the experimental alpha-decay energies for the two alpha-decay chains. Half-life is calculated according to the formula given by Ni et al. [30] 4(A 4) log 10 T = [2a(Z 2)Q 1/2 A + b 2(Z 2)] + c, (5) where A and Z are the mass number and proton number of the parent nucleus, and T is given in second and Q in MeV. The constants in this formula are determined as: a = , b = , and c = , and for even-n and odd-n nuclei, respectively. The calculated half-lives are listed in 4, 7, 10 columns in Table 1. It is seen from Table 1 that most theoretical halflives are bigger than experimental values. The biggest difference on the half-life is about for with TMA. It must be pointed out that it is very difficult to give an accurate description for alpha-decay half-life in theory. Sometimes, the difference between theoretical half-life and experimental one could be as large as This is because the half-life is very sensitive to the nuclear structure effect and alpha-decay energy. Next, we will discuss some ground-state properties of the nuclei on the two alpha-decay chains in RMF model. Because the two parameter sets give similar conclusions, only the results with NL-Z2 are presented here. The RMF results for the ground-state properties of the nuclei on the alpha-decay chains and are shown in Table 2. In Table 2 we listed some ground-state properties of the nuclei on the alpha-decay chains of and in the RMF model with NL-Z2. We have discussed binding energies in the previous paragraph. Now, we will focus on the question of quadrupole deformations. As the quadrupole deformations are unknown for these nuclei, we list the theoretical quadrupole deformations from Hartree-Fock-BCS (H-F+BCS) model for comparison. [31]

4 498 Communications in Theoretical Physics Vol. 55 It is seen from Table 2 that both RMF and H-F+BCS calculations show that there are prolate deformations in these nuclei. Except for the cases of and , the quadrupole deformation parameters β 2 from RMF calculations are close to those from the H-F+BCS calculations. It is well known that the deformation can increase the stability of the heavy and superheavy nuclei. As a result of the limit of the projectiles and the targets, the synthesized superheavy nuclei are usually neutron deficient and display deformation. From Table 2, it is also seen that the β 2 of all the nuclei on the alpha-decay chain of and the nuclei ( , , , 282 Rg) on the alpha-decay chain of are very large in the RMF calculations with NL-Z2. The theoretical values of these β 2 are ranging from to Therefore, these superheavy nuclei may be superdeformed nuclei. Table 2 The ground-state properties of the nuclei on the alpha-decay chains and in the RMF model with parameter set NL-Z2. The numerical results on quadrupole deformations from the Hartree-Fock Nuclear Mass Table [31] are listed in the last column for comparison. Nuclei NL-Z2 H-F+BCS B/MeV β 2n β 2p β 2 R n/fm R p/fm R c/fm β Rg Mt Bh Db Rg Fig. 3 The potential energy curves for the alpha-decay chain of as functions of quadrupole deformation in the constrained RMF model with NL-Z2. We apply the constrained RMF with NL-Z2 calculations for the nuclei on the two alpha-decay chains and the corresponding potential energy curves are plotted in Figs. 3 and 4. It is seen from Figs. 3 and 4 that the potential energy curves are more complex than those in light and medium nuclei. Because there are more neutron (proton) levels

5 No. 3 Communications in Theoretical Physics 499 in superheavy nuclei than those in light nuclei, level crossing becomes more frequent and more minimums in potential energy curves may appear. There are several minimums in the curves. The lowest one should correspond to the ground state of a superheavy nucleus, and the corresponding quadrupole deformation β 2 is the ground-state β 2. From Figs. 3 and 4, it is easy to determine the ground state for the nuclei 270 Db, 274 Bh, and in the alpha-decay chain of and in the alpha-decay chain of But for the other nuclei in the two alpha-decay chains, it is found that the energies in the minimums around β and β are very close to each other. It indicates that there may exist shape coexistence in these superheavy nuclei. Fig. 4 The same as Fig. 3, but for the alpha-decay chain of One of the advantages of microscopic nuclear models such as the RMF theory is that it can provide detailed information on single particle levels, which are very important for us to discuss nuclear structure and shell evolution. In Figs. 5, 6, 7, and 8, the single neutron and proton levels lying between 0 MeV and 10 MeV are shown for the alpha-decay chains of and The deformations of the nuclei in these figures are the ground-state deformations of them. Fig. 6 The same as Fig. 5, but for the alpha-decay chain of Fig. 5 The neutron single particle levels for the alphadecay chain of obtained by the deformed RMF model with NL-Z2. The short lines in above figures denote the energy levels. The dotted lines denote Fermi levels. Figures 5 and 6 show the neutron single particle levels for the alphadecay chains of and It is seen from Fig. 5 that the energy gap of N = 162 is very large except for Z = 117. The energy gaps of N = 162 and N = 184 are also very large in Fig. 6. It indicates that maybe N = 162 is a magic number in this region. There is no big energy gap for and , but there are several sub-shells

6 500 Communications in Theoretical Physics Vol. 55 in the neutron single particle levels of them. It is these sub-shells that enhance these superheavy nuclei stability. Figures 7 and 8 show the proton single particle levels for the alpha-decay chains of and It is clearly seen that the energy gaps for Z = 82 and Z = 90 are very large for all nuclei. 4 Summary Fig. 7 The same as Fig. 5, but for the proton single particle levels for the alpha-decay chain of Fig. 8 The same as Fig. 5, but for the proton single particle levels for the alpha-decay chain of In the present work, the newly synthesized element 117 and its alpha-decay chains are systematically studied in the framework of the relativistic mean field theory with parameter sets NL-Z2 and TMA. The ground-state properties of and and their alpha-decay chains are calculated. The calculated binding energies with TMA and NL-Z2 are very close. The experimental alpha-decay energies and half-lives of the two alpha-decay chains are reasonably reproduced by the model. Most differences between the Q theor and the Q expt are less than 1 MeV. Both RMF and H-F+BCS calculations show that all the nuclei on the two alpha-decay chains are prolate. We also apply the constrained RMF calculations for the nuclei on the two alpha-decay chains and plot the corresponding potential energy curves. It is found that potential energy curves are very complex and there are several minimums in the curves. The energies in the minimums around β and β are very close to each other for some nuclei. It indicates that there may exist shape coexistence in these superheavy nuclei. The single neutron and proton levels are also discussed. The RMF calculations for these new superheavy nuclei will be useful for future theoretical and experimental research on superheavy nuclei. References [1] S.G. Nilsson, et al., Nucl. Phys. A 131 (1969) 1. [2] S. Hofmann, et al., Z. Phys. A 350 (1995) 277. [3] S. Hofmann, et al., Z. Phys. A 350 (1995) 281. [4] S. Hofmann, et al., Z. Phys. A 354 (1996) 229. [5] S. Hofmann, et al., Rev. Mod. Phys. 72 (2000) 733. [6] Tu.Ts. Oganessian, et al., Nature (London) 400 (1999) 242. [7] Tu.Ts. Oganessian, et al., Phys. Rev. Lett. 83 (1999) [8] Tu.Ts. Oganessian, et al., Phys. Rev. C 63 (2001) (R). [9] Tu.Ts. Oganessian, et al., Phys. Rev. C 69 (2004) (R). [10] Tu.Ts. Oganessian, et al., Phys. Rev. C 69 (2004) [11] Tu.Ts. Oganessian, et al., Phys. Rev. C 70 (2004) [12] Tu.Ts. Oganessian, et al., Phys. Rev. C 72 (2005) [13] Tu.Ts. Oganessian, et al., Phys. Rev. C 64 (2006) [14] Tu.Ts. Oganessian, et al., Phys. Rev. Lett. 104 (2010) [15] S.G. Zhou, J. Meng, and P. Ring, Phys. Rev. Lett. 91 (2003) [16] S.G. Zhou, J. Meng, and P. Ring, Phys. Rev. C 68 (2003) [17] W.Z. Jiang, T.T. Wang, and Z.Y. Zhu, Phys. Rev. C 68 (2003) [18] J.Y. Guo and X.Z. Fang, Phys. Rev. C 74 (2006) [19] Z.Q. Sheng and J.Y. Guo, Commun. Theor. Phys. 49 (2008) [20] L.S. Geng, H. Toki, and E.G. Zhao, J. Phys. G 32 (2006) 573. [21] Z.Z. Ren, Phys. Rev. C 65 (2002) (R). [22] Z.Z. Ren, F. Tai, and D.H. Chen, Phys. Rev. C 66 (2002) [23] Z.Z. Ren, D.H. Chen, F. Tai, H.Y. Zhang, and W.Q. Shen, Phys. Rev. C 67 (2003) [24] P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193. [25] Y.K. Gambhir, P. Ring, and A. Thimet, Ann. Phys. (N.Y.) 198 (1990) 132. [26] M. Bender, K. Rutz, P.G. Reinhard, J.A. Maruhn, and W. Greiner, Phys. Rev. C 60 (1999) [27] Y. Sugahara and H. Toki, Nucl. Phys. A 579 (1994) 557. [28] M. Bender, Phys. Rev. C 61 (2000) [29] P. Möller, J.R. Nix, and K.L. Kratz, At. Data Nucl. Data Tables 66 (1997) 131. [30] D.D. Ni, Z.Z. Ren, T.K. Dong, and C. Xu, Phys. Rev. C 78 (2008) [31] S. Goriely, F. Tondeur, and J.M. Pearson, At. Data Nucl. Data Tables 77 (2001) 311.