Binding energies, α-decay energies, and α-decay half-lives for heavy and superheavy nuclei

Available online at www.sciencedirect.com

Nuclear Physics A 893 (2012) 13–26 www.elsevier.com/locate/nuclphysa

Binding energies, α-decay energies, and α-decay half-lives for heavy and superheavy nuclei Dongdong Ni a,∗ , Zhongzhou Ren a,b,c a Department of Physics, Nanjing University, Nanjing 210093, China b Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China c Kavli Institute for Theoretical Physics China, Beijing 100190, China

Received 31 May 2012; received in revised form 20 August 2012; accepted 21 August 2012 Available online 29 August 2012

Abstract Combined with the new table of atomic mass evaluation (AME) 2011, the binding energies and α-decay energies for heavy and superheavy nuclei are investigated based on the previous studies. First, we use the new data from the AME2011 to check the reliability of the previously proposed formulas. It is found that there is good agreement between the calculated results and new experimental data, showing the good predictive power of the formulas. Then, a new formula with fewer adjustable parameters is presented for binding energies, and it shows better agreement with the available experimental data. However, it is still hard to use it to accurately calculate the α-decay energies of superheavy nuclei with N  160 and Z  106. A new formula of α-decay energies is proposed as well, where the shell corrections are better described and the Z = 108 proton shell effect is added. The experimental Qα values are reproduced with improved accuracy especially for heavier nuclei. This improvement is of great importance for precise predictions of α-decay half-lives for unknown superheavy nuclei. Moreover, the new Qα formula and the theoretical FRDM95 and KTUY05 masses are used to determine the Qα values of superheavy nuclei with Z = 108–120, and their α-decay half-lives are calculated using the generalized density-dependent cluster model (GDDCM). The agreement between theoretical results and known experimental data is discussed, together with the deformed shell effects in the superheavy mass region. © 2012 Elsevier B.V. All rights reserved. Keywords: Superheavy nuclei; Binding energy; α-Decay energy; Half-lives

* Corresponding author. Tel.: +86 025 83596499; fax: +86 025 83326028.

E-mail address: [email protected] (D. Ni). 0375-9474/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysa.2012.08.006

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1. Introduction Research on superheavy elements (SHE) concerns a series of open questions in contemporary nuclear physics, such as localization of “island of stability” and upper end of the nuclide chart. In the last decade, considerable progress has been achieved in synthesizing new superheavy nuclei in laboratories. Heavy elements up to Z = 118 have been successfully synthesized [1–6], and the attempt to hunt for new elements Z = 119, 120 has been made at the SHIP and TASCA facilities at GSI. The experimentally measured quantities for superheavy nuclei are generally nuclear binding energies (or nuclear masses), α-decay energies, and α-decay half-lives, which are basic quantities for SHE investigations. The first quantity is determined by direct mass measurements and by nuclear decay and reaction energy measurements, while the last two quantities characterizing α-decay chains are closely associated with identification and knowledge of synthesized elements and nuclides. The binding energy and α-decay energy pose a tough test for various nuclear mass models and the half-lives provide unique information on stability of nuclides. Furthermore, information on shell closures can be extracted from systematic behavior of these quantities. For example, the deformed N = 152 and N = 162 neutron shell closures and the deformed Z = 108 proton shell closure were verified by systematic analysis of the available experimental data. In a word, precise measurements of these quantities could help us understand nuclear structure properties in the superheavy mass region and guide theoretical physicists to improve their models for superheavy nuclei. From the theoretical viewpoint, various approaches have been devoted to give precise descriptions of nuclear binding energies and then α-decay energies owing to their great importance in basic nuclear physics. Some major approaches can be emphasized, such as the finite-range droplet macroscopic model combined with the folded-Yukawa single-particle microscopic model [7], the Hartree–Fock–BCS method [8], the Hartree–Fock–Bogoliubov method [9], and the Koura– Tachibana–Uno–Yamada mass formula [10]. All these calculations concentrate on a wide range of nuclei in the whole mass region and give similarly good descriptions of nuclear masses with the root-mean-square (rms) deviations of 0.37–0.74 MeV. The shell structure of superheavy nuclei was also investigated based on the self-consistent nuclear calculations of binding energies [11], where the possible closed shells for superheavy nuclei were explored within various parameterizations of relativistic and nonrelativistic mean-field models. Besides, there are some simple mass formulas with different considerations [12–14], which generally reproduce the available experimental binding energies with the rms deviations of 1.0–2.0 MeV. In some cases, in order to calculate the binding energies of nuclei in certain mass region accurately and conveniently, some local studies were performed based on mass systematics. For example, Dong and Ren [15] focused on heavy and superheavy nuclei with Z  90, N  140 and proposed a local formula of binding energies. Later an improved version was given [16], where 117 binding energies of heavy and superheavy nuclei were reproduced with the rms deviation of 0.105 MeV. Based on the formulation of binding energies, two formulas of α-decay energies were also developed for superheavy nuclei [17,18]. Such straightforward studies are useful for ongoing experiments on superheavy nuclei, because they could help experimentalists quickly know the expected properties of nuclei in question before experiments and check the measured data after experiments. Stimulated by the special interest in superheavy nuclei, α-decay has also been extensively investigated from semiclassical to microscopic calculations and from spherical to deformed systems. The available theoretical models can be divided into three groups. The first group is the phenomenological analysis [19–22], where some crucial properties are interpreted and minor influences are wrapped into adjustable parameters. The second group is based on the

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Wentzel–Kramers–Brillouin (WKB) approximation, such as the cluster model [23,24], generalized liquid drop model (GLDM) [25], density-dependent cluster model (DDCM) [26], super asymmetric fission model (SAFM) [27], unified model for α-decay and α capture (UMADAC) [28], Coulomb and proximity potential model (CPPM) [29], density-dependent M3Y (DDM3Y) effective interaction [30], mean-field potentials obtained from the Skyrme–Hartree–Fock model [31] and from the Dirac–Brueckner–Hartree–Fock model [32], and so on. The last group is the quantum mechanics treatment, where one-channel or coupled-channels Schrödinger equations are exactly solved for quasibound states [33–38]. The systematic calculations of nuclear masses mentioned above mainly used the experimental data of the Atomic Mass Evaluation (AME) in 1995 [39] and 2003 [40] to determine their adjustable parameters. Since the latest publication of AME in 2003, a large number of new experimental data have been accumulated thanks to the rapid development of experimental equipments and technologies such as Penning traps and storage rings. These new data are mainly for neutron-deficient nuclides and superheavy nuclei. Recently the new mass table called AME2011 has been made available [41], where more than 100 new masses are given for the first time and many masses are measured with improved accuracy. This provides an excellent opportunity to further investigate the binding energies and α-decay energies of heavy and superheavy nuclei. On the one hand, we can employ the new data to test the reliability of the previous formulas proposed by Dong and Ren [16,18]. Then improvements are made by analyzing the new mass table, that is, fewer adjustable parameters in the formulas and better agreement with the available experimental data. On the other hand, we can combine the reliable evaluations of α-decay energies with the well-established generalized density dependent cluster model (GDDCM) to make good predictions of α-decay half-lives for still unknown superheavy nuclei and investigate possible shell closures in the superheavy mass region, which will be useful for future experimental studies. The objective of this work is to give a straightforward and natural interpretation of the rich experimental data of heavy and superheavy nuclei, including binding energies, α-decay energies, and α-decay half-lives. 2. Binding energies of heavy and superheavy nuclei The local formula of binding energies for heavy and superheavy nuclei was proposed by Dong and Ren, which is written as [16] B(Z, A) = av A − as A2/3 − ac Z 2 /A1/3 − aa (A/2 − Z)2 /A + ap A−1/2 + a6 |A − 252|/A − a7 |N − 152|/N + a8 |N − Z − 50|/A.

(1)

This formula was derived based on the experimental data of 117 nuclei with Z  90 and N  140, which are available in the AME2003 table [40]. Since the publication of AME2003, many new data have been measured for the first time or with higher precision. In particular, superheavy elements up to Z = 118 have been successfully synthesized and new neutron-deficient nuclides have been produced. According to the recently released mass table AME2011 [41], there are 134 experimental data available for nuclei with Z  90 and N  140, where 18 new data appear for the first time and the old experimental data for 265 Sg is replaced by the estimated value. This poses a good test of the reliability of the formula (1). We use the formula (1) to calculate the binding energies of 18 new nuclei. If the calculated values agree well with the new data within allowed errors, then the formula has good reliability for theoretical predictions. Table 1 shows the results calculated from the formula, compared with the new data from the AME2011 table.

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Table 1 Reliability analysis of the binding energy formula (1) proposed in Ref. [16]. The binding energies calculated with the formula (1) are compared with 18 new data from the new mass table AME2011, showing the good reliability of the formula (1). Nucl.

Bexp (MeV)

235 Am

1777.811 1781.862 1788.551 1802.791 1802.440 1847.585 1858.002 1867.917 1877.883

236 Cm 237 Cm 239 Cm 240 Cf 247 Es 249 Fm 251 Md 253 No

Bcal (MeV)

Nucl.

Bexp (MeV)

Bcal (MeV)

1777.863 1781.965 1788.600 1802.698 1802.404 1847.678 1858.089 1867.961 1878.095

255 Lr

1887.657 1893.929 1897.096 1904.693 1906.334 1913.033 1915.678 1933.238 1950.099

1887.736 1894.126 1897.320 1904.890 1906.377 1912.928 1915.654 1933.384 1950.516

256 Lr 257 Rf 258 Rf 259 Db 260 Db 261 Sg 265 Hs 269 Ds

As one can see, there is good agreement between the theoretical values and new experimental data, showing the good reliability of the formula (1). In addition, we also use the formula (1) to evaluate the binding energies of 134 nuclei emerging in the AME2011 table. The standard deviation from the experimental binding energies is given by  134 1/2    2 i i Bexp σ2 = − Bcal /134 = 0.113 MeV, (2) i=1

and the average deviation is 134   i B σ  =

exp

 i  /134 = 0.091 MeV. − Bcal

(3)

i=1

Another tough test of the formula (1) is the evaluation of α-decay energies, Qα (Z, A) = B(Z − 2, A − 4) + B(α) − B(Z, A). This test is particularly important for superheavy nuclei, because the binding energies of superheavy nuclei are often absent in experiments but their α-decay energies are known. One can calculate the α-decay energies with the formula (1) and compare them with the experimental Qα values. Fig. 1(a) shows the comparison of the calculated Qα values with the experimental data for 196 α emitters with Z  90 and N  140. The experimental Qα data are all taken from the new mass table AME2011 [41]. As one can see, there are large deviations from the experimental Qα values in the heavier mass region of N  160 and Z  106, showing the poor extension of the formula (1). The standard deviation of the calculated Qα values from the experimental data is given by  196 1/2  

2 i i Qα (exp) − Qα (cal) /196 σ2 = = 0.288 MeV, (4) i=1

and the average deviation is σ  =

196    i Q (exp) − Qi (cal)/196 = 0.209 MeV. α

α

(5)

i=1

Since some new data of binding energies emerge in the AME2011 table, we shall perform a new analysis of the known binding energies including some new data. After analyzing the

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Fig. 1. Comparison of the Qα values calculated with the binding energy formula with 196 experimental Qα values from the new mass table AME2011. Calculations are performed (a) with the previous binding energy formula (1) and (b) with the new binding energy formula (6), respectively. In both cases, the unsatisfactory description is seen in the heavier mass region (N  160 and Z  106).

contributions of the different terms in the formula (1), we find that the term |A − 252|/A has so minor contribution to the binding energy that one can neglect it. In addition, for the better description of the additional binding energies of nuclei near the line N − Z − 50 = 0 [16], we slightly modify the last term in the formula (1) as (|N − Z − 50| + |N − Z − 52|)/(2A). With these in mind, the improved version of the binding energy formula is written as B(Z, A) = av A − as A2/3 − ac Z 2 /A1/3 − aa (A/2 − Z)2 /A + ap A−1/2   − a6 |N − 152|/N + a7 |N − Z − 50|/A + |N − Z − 52|/A /2.

(6)

This formula is essentially macroscopic, because the microscopic shell and pairing corrections are wrapped into some phenomenological terms as in the basic approach of Myers–Swiatecki. Note that the microscopic treatment of these corrections has been achieved using microscopic shell models. This is worth us further investigation. The best-fit parameters of the formula (6) to 134 binding energies are as follows: av = 15.60448, as = 16.98337, ac = 0.70614, aa = 96.45512, a6 = 17.26970, a7 = 18.32109, and ap = 12.935 for even–even nuclei, ap = 3.101 for even–odd nuclei, ap = 0.115 for odd–even nuclei, ap = −7.554 for odd–odd nuclei. The standard and average deviations of the present calculations for 134 nuclei are, respectively, 0.103 MeV and 0.081 MeV, which are separately reduced by about 9% and 10% with respect to the results reported in Eqs. (2) and (3). In order to gain clear insight into the agreement between experiment and theory, we illustrate in Fig. 2 the deviation of the calculated binding energies from the experimental data as a function of the neutron number of nuclei. As can be seen, the deviations (Bcal − Bexp ) are generally within the range of −0.2–0.2. The slightly large deviation appears at N = 156, corresponding to the nucleus 264 Hs. This can be easily understood as the existence of the Z = 108 proton shell closure which is not considered at this stage.

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Fig. 2. Deviation of the binding energies calculated with the new formula (6) from the experimental data versus the neutron number of nuclei.

Next, calculations of α-decay energies are performed using the new formula (6). The standard and average deviations of the Qα values are, respectively, 0.288 MeV and 0.209 MeV, keeping the same as the previous results Eqs. (4) and (5). The detailed results are shown in Fig. 1(b). One can see that there still exist large deviations in the mass region (N  160 and Z  106), as shown in Fig. 1(a). To overcome this, we need to improve the systematics of binding energies for superheavy nuclei. Unfortunately, there are quite few data in the superheavy mass region. According to the AME2011 table 269 Ds is the heaviest nucleus whose binding energy is experimentally known. It is therefore difficult to improve the binding energy formula such as inclusion of the N = 162 and Z = 108 shell corrections. Along with the development of experimental technologies and equipments, more and more data of nuclear binding energies will be accumulated in experiments. And it would be of significant interest to further improve the analysis of nuclear binding energies and obtain more information on nuclear shell effects in the future. 3. Alpha-decay energies of heavy and superheavy nuclei Besides the binding energy formula for heavy and superheavy nuclei, the α-decay energy formula was also presented based on the liquid-drop model by Dong and Ren [18], where the shell effects at N = 152 and N = 162 were described by the Mexican hat wavelet functions and 170 α-decay energies from the AME2003 table were analyzed. Since 2003, many new nuclides have been produced by the fusion-evaporation reactions thanks to the world-wide effort to synthesize new nuclides toward the heavier mass region and the drip lines. And these nuclides are generally identified by α-decay chains or investigated by an experimental combination of α-decay and γ emission. In terms of the new mass table AME2011, there are 196 α-decay energies for nuclei with Z = 90 − 118 and N  140, where 37 new data are added and 11 old data are deleted with respect to the AME2003 table. The modifications are mainly concentrated on the superheavy nuclei ranging from Z = 101 to Z = 118. Following the procedure presented for nuclear binding energies, we first check the reliability of the formula proposed in [18] by comparing the Qα values calculated from the formula with 37 new data. Table 2 displays the comparison of the calculated results with 37 new data which are not available in the AME2003 table. One can see that the theoretical results are in good agreement with the new data, showing the good predictive power of the formula. Then, the Qα values of 196 α emitters are calculated using the formula reported in [18], and they are compared with the experimental Qα values from the AME2011 table in Fig. 3(a). In contrast to the Qα values calculated

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Table 2 Reliability analysis of the Qα formula proposed in Ref. [18]. The Qα values calculated with the formula are compared with 37 new data from the new mass table AME2011, showing the good predictive power of the formula. exp

Nucl.

Qα

235 Th

3.606 6.576 6.256 7.074 6.770 6.540 8.160 7.461 8.764 7.708 7.963 8.752 7.572 8.816 8.556 9.336 9.193 9.501 10.400

235 Am 236 Am 236 Cm 237 Cm 239 Cm 242 Es 247 Es 247 Md 249 Fm 251 Md 251 No 253 Md 254 Lr 255 Lr 256 Db 258 Rf 258 Db 260 Bh

(MeV)

exp

Qcal α (MeV)

Nucl.

Qα

3.255 6.866 6.669 7.341 7.148 6.735 8.144 7.257 8.478 7.625 8.014 8.551 7.875 8.871 8.804 9.769 9.059 9.625 10.509

261 Sg

9.714 9.403 11.059 9.623 8.700 9.064 9.440 9.733 11.478 10.725 10.847 10.459 9.130 10.783 9.879 9.767 10.455 11.183

263 Sg 263 Hs 268 Hs 269 Sg 270 Bh 271 Hs 273 Hs 274 Rg 277 Ds 278 Rg 281 Cn 282 Rg 282 113 285 113 286 113 289 115 293 117

(MeV)

Qcal α (MeV) 9.822 9.502 10.704 10.049 8.938 9.402 9.860 9.614 11.198 10.095 10.544 10.439 9.429 10.882 10.084 9.870 10.577 11.176

with the binding energy formula which show slightly worse agreement with the experimental data for superheavy nuclei with N  160 and Z  106, the present results are in fair agreement with the experimental data. As shown in Fig. 3(a), the largest derivation occurring at N = 162 corresponds to the α emitter 270 Hs. This is attributed to the combined effects of the Z = 108 proton shell and the N = 162 neutron shell [11]. Note that the Z = 108 shell effect is not taken into account in the formula. The standard and average deviations of the Qα values for 196 α emitters are, respectively, 0.225 MeV and 0.179 MeV. In addition, we would like to point out that the shell corrections for binary processes such as nuclear fusion, fission, cluster radioactivity, and α-decay have been calculated using appropriate two-center shell models [42,43]. For example, Gherghescu et al. [42] developed the deformed two-center shell model and calculated the shell corrections with the Strutinsky method. They investigated the shell effects in both fission and fusion reactions, showing the good applicability of the deformed two-center model. As we all know, the α-decay energy is a crucial quantity, which has considerable effects on α-decay half-lives. In the following, we will focus on 196 Qα data from the AME2011 table and improve the Qα formula as much as possible, which could lay the groundwork for further researches on α-decay half-lives. Two improvements are made in the present study. One is that the shell corrections are better simulated by the function (1 − |t/σ |) exp[−(t/σ )2 /2] instead of the Mexican hat wavelet function. It is seen in Fig. 3(a) that the α-decay energies are overestimated and then underestimated away from N = 162 as one uses the Mexican waves. By contrast, when we use the new function instead, such an unwelcome behavior becomes obviously weak, as shown in Fig. 3(b). The other is that the Z = 108 shell correction is introduced into the formula in a similar manner so that the large deviation occurring

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Fig. 3. Comparison of the α-decay energies calculated with the Qα formula with 196 experimental Qα values from the new mass table AME2011. Calculations are performed (a) with the previous Qα formula presented in Ref. [18] and (b) with the new Qα formula (7), respectively. The results in case (b) show better agreement with the experimental data especially in the heavier mass region.

around Z = 108 is reduced to some extent. The improved version of the Qα formula is written as    

 N − 154  Z a2 Z N −Z 2   Qα (Z, A) = a1 + 1/3 + a3 1/3 1 − + a5 1 −  + a4 3A A a6  A A     

2 

 N − 164  1 N − 154 2  exp − 1 N − 164 + a7 1 −  × exp − 2 a6 a8  2 a8   

2 

 Z − 110   exp − 1 Z − 110 + a9 1 −  . (7) a10  2 a10 Through the least-square fit to 196 α-decay energies, we obtain a set of parameters as follows: a1 = −35.69441, a2 = 53.82437, a3 = 2.79849, a4 = −95.09205, a5 = 0.48556, a6 = 5.62902, a7 = 0.85566, a8 = 7.48830, a9 = 0.39216, and a10 = 2.28683. The standard and average deviations of the present Qα calculations for 196 α emitters are, respectively, 0.208 MeV and 0.166 MeV. Compared with the old Qα formula, the Qα calculations are improved by about 8%. To show this, the agreement between the calculated results and experimental data is illustrated in Fig. 3(b), compared with the previous results shown in Fig. 3(a). One can see that the results calculated with the new Qα formula show better agreement with the experimental data than those with the old Qα formula. This feature is particularly obvious in the heavier mass region and significant for reliable Qα predictions of still unknown superheavy nuclei. It should be noted again that the experimental Qα values are well reproduced in the mass region of N  160 and Z  106, as shown in Fig. 3. This is in great contrast to the results shown in Fig. 1, where the Qα values are calculated with the binding energies of nuclei.

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4. Predicted α-decay half-lives within the GDDCM According to the above discussions and comparison, the new Qα formula (7) not only shows the best agreement with the available experimental data but also works well for the heavier mass region. In this section, we will combine it with the generalized density-dependent cluster model (GDDCM) to evaluate the α-decay half-lives of superheavy nuclei. The validity of the GDDCM has be tested for a wide range of nuclei [34,35], including exotic α decays around the significant N = 126 shell closure, α-decaying isomers, and some newly synthesized superheavy nuclei. In all these cases, the good agreement between experiment and theory is achieved, which gives us confidence in reliable predictions of α-decay half-lives for still unknown superheavy nuclei. With the GDDCM, instead of the WKB barrier penetration probability, the α-decay width is computed by the overlap integral of the quasibound-state wave function, the scattering-state wave function, and the difference of potentials involved [34,35,44,45]: 2 ∞ 

4μ   p Γ = Pα 2  F (r) VN (r) + VC (r) − VC (r) unj (r) dr  , (8)  h¯ k  0

where Pα is the preformation factor of the α√cluster in the decaying nucleus, the wave number k is determined by the expression k = 2μQα /h¯ , the regular Coulomb wave function F  (r) is associated with the scattering-state wave function ϕ (r) by the relationship ϕ (r) =

2μ/(π h¯ 2 k)F (r)/r, VC (r) and VN (r) are respectively, the Coulomb and nuclear potentials p between the α cluster and the core nucleus during the tunneling, VC (r) is the point-like Coulomb potential corresponding to the α particle far away from the residual daughter nucleus, and unj (r) is the quasibound state wave function describing the α cluster quasibound to the core in the decaying nucleus. The microscopic potentials VN (r) and VC (r) are numerically constructed using the double-folding integral of the realistic nucleon–nucleon (N N ) interaction with the nuclear density distributions [34,35]. The quasibound state wave function unj (r) is obtained by numerically solving the Schrödinger equation,

 h¯ 2 d 2 ( + 1)h¯ 2 − + V (r) + V (r) + (9) unj (r) = Qα unj (r). N C 2μ dr 2 2μr 2 Note that the quantum numbers n of the wave function are determined by the Wildermuth condition, which accounts for the main effect of the Pauli exclusion principle. The details of the double-folding potential and quasibound state wave function can be found in Refs. [34,35]. The α-preformation factor Pα concerns the structure part of α-decay. Microscopic calculations of Pα are usually complicated and difficult due to large configuration spaces and insufficient knowledge of the clusterization process [46,47]. Fortunately, the available theoretical and experimental studies [48–50] suggest that the Pα factor does not change much in the open-shell region for heavy and superheavy nuclei. It is therefore applicable and convenient to take the constant Pα factor for one kind of α emitters, as shown in Refs. [23,26,31,34]. For the sake of consistence, the values of Pα remain the same as in the previous systematic calculations [34], which avoids introducing additional adjustable parameters into the GDDCM. Ultimately, the decay half-life is calculated by the expression T1/2 = h¯ ln 2/Pα Γ . Let us perform a systematic investigation on α-decay in the superheavy mass region within the GDDCM framework. Since the α decays of superheavy nuclei with Z = 101–109 around the N = 152 shell gap were investigated in Ref. [51], here we focus our attention mainly on the

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Fig. 4. (Color online.) Calculated α-decay half-lives as a function of the neutron number of parent nuclei for even-Z α emitters with Z = 108–118. Calculations are performed with different Qα values, which are separately obtained from the FRDM95 mass model, the KTUY05 mass model, and the new Qα formula (7). The available experimental α-decay half-lives are also shown by open stars, which are taken from Refs. [3–5,54].

superheavy nuclei with Z = 108–120 and N = 156–180. As far as we know, the decay spectroscopy of all these nuclei is not made possible using common experimental technologies and even some of them are still not produced in experiments. Therefore, the assumption of favored α transitions (i.e.,  = 0 decays) is adopted for all the superheavy nuclei under investigation. Along with the development of experimental technologies, decay schemes together with rotational bands and isomeric states will be established for superheavy nuclei in the future and then more precise α-decay studies of them will be achieved. For example, the decay schemes of 253 No and its daughter 249 Fm have recently been established thanks to the enhanced sensitivity and increased efficiency of the experimental equipments [52]. In the calculation of favored α transitions, the only input quantity is the α-decay energy Qα . Here the theoretical Qα values are used because the experimental Qα values are not available in most cases. The Qα values are calculated with the Qα formula (7), which allows us to make more reliable Qα predictions for heavier α emitters as compared with the binding energy formulas (1) and (6). In addition, two mass models are employed to calculate the Qα values as well, Qα (Z, A) = Mth (Z, A) − Mth (Z − 2, A − 4) − 2.425 MeV, where Mth are calculated atomic mass excesses in MeV. One is the finite-range droplet model (FRDM95) [7], which consists of a macroscopic droplet term and a microscopic shell-plus-pairing correction. The macroscopic term is calculated using the finite-range droplet model and the microscopic correction is calculated using the folded-Yukawa single-particle model. The other is the Koura–Tachibana–Uno–Yamada

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Fig. 5. (Color online.) Same as in Fig. 4, but for α emitters with Z = 120.

(KTUY05) formula [10], which includes a gross term, an even–odd term, and a shell term. The gross term is calculated based on the Weizsäcker–Bethe mass formula and the even–odd and shell corrections are calculated using the single-particle potential and BCS-type pairing. Very recently, these two mass models have been used to determine the released energy of possible heavy-particle radioactivity from superheavy nuclei with Z > 110 [53]. Figs. 4 and 5 show the α-decay half-lives calculated with the different Qα values as a function of the neutron number of α emitters for even-Z superheavy nuclei with Z = 108–120. Note that the y axis in the figures is related with the atomic number of α emitters. For comparison the available experimental α-decay half-lives from Refs. [3–5,54] are shown by open stars. The significant result emerging in the figures is the indication of nuclear shell closures. To our knowledge, up to now there has been no consensus among various theoretical models on the exact position of the neutron shell closure in the region of study [11]. But there has been clear evidence for the enhanced nuclear stability at N = 162 in experiments [2]. As one can see, the results of FRDM95 clearly demonstrate the existence of the N = 164 neutron shell closure especially around Z = 108–112. The results of the Qα formula suggest the existence of the N = 162 shell closure, but this indication is less obvious than that of FRDM95. For the results of KTUY05, there is no explicit information on the N = 164 or 162 neutron shell effects in the Z = 108–116 isotopic chains, while the N = 164 neutron shell effect is shown in the Z =118, 120 isotopic chains. In addition, one can also notice that for the Z = 114 isotopic chain the half-lives calculated with the FRDM95 and KTUY05 masses are apparently longer than the results calculated with the Qα formula in the neutron-rich region. This can be explained if there is additional nuclear stability around Z = 114 in the FRDM95 and KTUY05 mass models. Indeed, Z = 114 is predicted to be magic in terms of these two mass models. At present, the experimental verification of this is still not available. As compared with the known experimental α-decay half-lives, it is found that the calculated α-decay half-lives with the Qα formula show good agreement with the experimental data (shown in open stars). The largest deviation occurs at N = 161 in the Z = 108 isotopic chain. This is attributed to the slightly large deviation in the Qα value or/and the structure effect of α transitions (i.e., unfavored transitions), because the α emitter 269 Hs locates very near the predicted doubly closed shells Z = 108 and N = 162. Besides, one can see that the calculated half-lives exhibit an

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Fig. 6. (Color online.) Same as in Fig. 4, but for odd-Z α emitters with Z = 109–119.

increasing tendency with the increase of the neutron number, no matter whether the Qα values are calculated with the FRDM95 masses, or the KTUY05 masses, or the Qα formula presented in this work. This reflects that there is clear enhancement of nuclear stability against α-decay as the nuclide becomes more and more neutron-rich, which is quite consistent with the experimental analysis of Oganessian et al. [5]. In similar fashion, the calculated α-decay half-lives of oddZ superheavy nuclei with Z = 109–119 are illustrated in Fig. 6, and the similar conclusions could be deduced from systematic trends of the isotopic chains. It is expected that the present predictions of α-decay half-lives would be useful for ongoing or future experiments on synthesis of new superheavy elements and nuclides. 5. Summary In summary, based on the newly released mass table AME2011, we have presented in this paper a detailed investigation of the binding energies, α-decay energies, and α-decay half-lives for heavy and superheavy nuclei with Z  90 and N  140. As a first step, we select new data from the AME2011 table and then use them to check the reliability of the previous formulas proposed by Dong and Ren. The calculated results are in good agreement with the new data, showing the good predictive power of the formulas. Based on this, we analyze the new experimental data and make some improvements in systematics of binding energies and α-decay energies. The new formula of binding energies is proposed, which has fewer adjustable parameters and better reproduces the available experimental data. The rms deviation from the experimental binding energies for 134 nuclei is 0.103 MeV. However, as the previous formula, the new formula is still hard to

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well reproduce the α-decay energies of superheavy nuclei with N  160 and Z  106. Considering the crucial role of α-decay energies in the calculations of α-decay half-lives, the new Qα formula is presented as well, which shows better agreement with the experimental Qα values especially in the heavier mass region. The rms deviation from the experimental Qα values for 196 nuclei is 0.208 MeV, which is reduced by 8% as compared with the previous Qα calculations. Finally, we combine the new Qα formula with the generalized density-dependent cluster model to evaluate the α-decay half-lives of superheavy nuclei with Z = 108–120. For comparison other nuclear mass models such as FRDM95 and KTUY05 are employed to calculate α-decay halflives as well. By analyzing the calculated half-lives, the underlying shell effects are discussed. It would be the most welcome to compare the present results with future experimental measurements, and this would provide a reference for us to improve α-decay studies for superheavy nuclei. Acknowledgements We thank G. Audi for providing helpful discussions during his visit to Nanjing University. This work is supported by the National Natural Science Foundation of China (Grants No. 10735010, No. 10975072, No. 11035001, and No. 11120101005), by the 973 National Major State Basic Research and Development of China (Grants No. 2007CB815004 and No. 2010CB327803), by CAS Knowledge Innovation Project No. KJCX2-SW-N02, by Research Fund of Doctoral Point (RFDP), Grant No. 20100091110028, and by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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