Progress of Theoretical and Experimental Physics, Volume Advance Article (5) – May 8, 2018

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Prog. Theor. Exp. Phys. 2018, 053D02 (13 pages) DOI: 10.1093/ptep/pty049 The triaxiality and Coriolis effects on the fission barrier in isovolumic nuclei with mass number A = 256 based on multidimensional total Routhian surface calculations 1 1 1,∗ 2 3,4 Qing-Zhen Chai , Wei-Juan Zhao , Hua-Lei Wang , Min-Liang Liu , and Fu-Rong Xu School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China Center of Theoretical Nuclear Physics, National Laboratory for Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China E-mail: wanghualei@zzu.edu.cn Received January 8, 2018; Revised March 18, 2018; Accepted March 28, 2018; Published May 8, 2018 ................................................................................................................... The triaxiality and Coriolis effects on the ﬁrst ﬁssion barrier in even–even nuclei with A = 256 have been studied in terms of the approach of multidimensional total Routhian surface calculations. The present results are compared with available data and other theories, showing a good agreement. Based on the deformation energy or Routhian curves, the ﬁrst ﬁssion barriers are analyzed, focusing on their shapes, heights, and evolution with rotation. It is found that, relative to the effect on the ground-state minimum, the saddle point, at least the ﬁrst one, can be strongly affected by the triaxial deformation degree of freedom and Coriolis force. The evolution trends of the macroscopic and microscopic (shell and pairing) contributions as well as the triaxial ﬁssion barriers are brieﬂy discussed. ................................................................................................................... Subject Index D11, D12, D13 1. Introduction Spontaneous ﬁssion dominated by the barrier size and shape is one of the important decay modes of a heavy or superheavy nucleus. The ﬁssion barrier is a critical quantity for understanding nuclear ﬁssion properties. Moreover, the formation probability of a heavy or superheavy nucleus in the heavy-ion fusion reaction is directly related to such a barrier, which is a decisive quantity in the competition between neutron evaporation and ﬁssion of a compound nucleus during the cooling process. For example, a 1 MeV change of the ﬁssion barrier may result in several orders of magnitude difference in survival probability. Though the ﬁssion phenomenon has been interpreted by the theory of barrier penetration [1] for many years, it is still rather difﬁcult to give an accurate description. Both theoretically and experimentally, considerable effort has so far been made to investigate the ﬁssion problem. For instance, some empirical barriers, including the double-humped ones in heavy actinide nuclei (i.e., up to Cf) have been determined [2,3]. Also, the experimental evidence for the presence of triple-humped ﬁssion barriers has been pointed out by Blons et al. in Ref. [4]. On the the- oretical side, several types of models are widely adopted to study the ﬁssion barriers, including the macroscopic–microscopic (MM) models [5–16], nonrelativistic energy density functionals based on zero-range Skyrme and ﬁnite-range Gogny interactions [17–20], the extended Thomas–Fermi © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. plus Strutinsky integral (ETFSI) methods [21,22], and covariant density functional theory (CDFT) [23–26]. It is found and explained that in the available actinide region the ﬁrst and second ﬁssion barriers are strongly affected by the triaxial and octupole deformation degrees of freedom, respec- tively, whereas the third ﬁssion barrier may have an asymmetric, pear-like octupole deformation [4,27–30]. Very recently, the ﬁssion barrier of the heaviest actinide nucleus No has been measured to be 6.0 ± 0.5 MeV at spin 15 and, by extrapolation, 6.6 ± 0.9 MeV at ground state by analyzing the distribution of entry points in the excitation energy versus spin plane [31]. By employing the fusion– 208 48 254 evaporation reaction Pb( Ca,2n), the high-spin states of the nucleus No were populated up to 208 50 22 [32]. Similarly, using the Pb( Ti,2n) reaction channel and with the help of ﬁssion tagging, an extension of the recoil-decay tagging technique [33,34], the rotational band up to spin 20 of the ﬁrst superheavy nucleus Rf has, in a real sense, been built experimentally in recent years 256 254 256 [35]. Naturally, the adjacent even–even No of No and the superheavy nucleus Rf outside the actinide nuclei are expected to be good candidates for studying the static and dynamic ﬁssion barriers. Moreover, in this A = 256 isovolumic chain, the high-spin states in Fm have been observed and the nucleus Cf has already been synthesized experimentally [36–38]. With these facts and a natural extension in mind, we have systematically performed the multi- dimensional total Routhian surface (TRS) calculations for the A = 256 isovolumic chain ranging 256 256 from Cm to Sg, focusing on the triaxiality and Coriolis effects on the ﬁrst ﬁssion barrier. Note that these even–even isovolumic nuclei have already been (or will possibly be) synthesized experi- mentally [37,38], and the theoretical framework of the TRS approach is based on the MM models which usually have very high descriptive power as well as simplicity of calculation, even though they are relatively “old”. Along this special “path”, it may reveal a helpful law to some extent, which is expected to provide a valuable reference for the possibly experimental candidates. Indeed, it is found that there exist similar properties of the ﬁssion barriers, as discussed below. In our previous work, we have also investigated the effects of triaxial deformation degree of freedom on nuclear ground and/or yrast states [39–41] and the octupole effects on outer ﬁssion barriers [42] based on a similar method. In addition, by the present work we will try to ﬁnd the possible model discrepancies, especially extrapolating toward the superheavy region, and to further develop it. For instance, the different polarization effects and functional forms of the densities may appear in the superheavy region. These can be naturally incorporated within the self-consistent mean-ﬁeld calculations, whereas in the MM models preconceived knowledge about the expected densities and single-particle potentials is often required [43], indicating that the model parameters, even the Hamiltonian, may be reformulated [44–46]. The rest of this article is organized as follows. The theoretical framework and some related details of the numerical calculations in this work are described in Sect. 2. The calculated results are shown and discussed in Sect. 3. Finally, Sect. 4 gives a brief summary for the present work. 2. Theoretical descriptions The multidimensional TRS calculation is based on the theoretical framework of the MM model and cranking shell model [47,48]. Such an approach has been widely used to study nuclear structural phenomena, especially under rotation. It has several standard components, which can be easily found from the literature. We would like to brieﬂy outline the calculation procedure and give the reader some necessary references. 2/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. The total Routhian, namely the “energy” in the rotating body-ﬁxed system, is the sum of the energy of the non-rotating state and the cranking contribution. In general, the total Routhian is written as ω ω=0 ˆ ˆ E (Z, N , β) = E (Z, N , β) ω ω ω ω ω ω ω=0 ˆ ˆ ˆ ˆ +[ | H (Z, N , β) | − | H (Z, N , β) | ], (1) ω=0 where the energy of the non-rotating state E consists of a macroscopic part and a ﬂuctuating microscopic one. The last term in the square brackets indicates the contribution due to rotation. In this work, the macroscopic energy is obtained from the standard liquid-drop model (LDM) [49] with the parameters used by Myers and Swiatecki. Since our attention is on the difference between the points on the calculated potential energy surface, e.g., the energy difference (ﬁssion barrier) between the minimum and saddle point, the energy relative to that of a spherical LD is adopted, which can be calculated by [49–51] (0) ˆ ˆ ˆ E (Z, N , β) ={[B (β) − 1]+ 2χ [B (β) − 1]}E , (2) LD s c where the relative surface and Coulomb energies B and B are functions only of nuclear shape, s c (0) depending on the collective coordinates {α }. The spherical surface energy E and the ﬁssility λμ s parameter χ are dependent on Z and N . Though such a sharp-surface LD model does not consider the surface diffuseness and the ﬁnite range of the nuclear interaction, it provides a rather good description for nuclear properties. The microscopic correction part contains a shell correction δE and a pairing correction δE , shell pair namely, ˆ ˆ ˆ E (Z, N , β) = δE (Z, N , β) + δE (Z, N , β). (3) mic shell pair The shell and pairing corrections at each deformation point are calculated in terms of the Strutinsky [52] and Lipkin–Nogami (LN) methods [53]. The Strutinsky smoothing is performed with a six-order 1/3 Laguerre polynomial and a smoothing range γ = 1.20 ω , where ω = 41/A MeV. The LN 0 0 method avoids the spurious pairing phase transition encountered in the traditional Bardeen–Cooper– Schrieffer (BCS) calculation. Not only monopole but also doubly stretched quadrupole pairings are considered. The monopole pairing strength, G, is determined by the average gap method [54], which has 1 1 2 2 2 2 = ρ In( y + + y ) − In( y + + y ) , (4) 2 1 2 G 1 G G 2 − N +N −1 tot 1 where ρ is the average level density of doubly degenerate levels, y = , and − N +N tot 2 rB rB 2 s s y = . is the effective-interaction pairing gap, = , = , where 2 G G G 1/3 1/3 n p N Z B is the relative surface energy and r = 3.30 MeV. The quadrupole pairing strengths are obtained by restoring the Galilean invariance broken by the seniority pairing force [48,55]. In the pairing win- dows, dozens of single-particle levels, the respective states (e.g., half of the particle number Z and N ) just below and above the Fermi energy, are included empirically for both protons and neutrons. Finally, the pairing energy according to the LN method is given by [53], 2 4 2 2 E = 2v e − − G v + G − 4λ u v , (5) LN k 2 k k k k G 2 k k k 3/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. where v , e , and represent respectively the occupation probabilities, single-particle energies, and pairing gap. The extra Lagrange multiplier λ is the particle number ﬂuctuation constant. It should be noticed that the quantity + λ is identiﬁed with odd–even mass differences. Note that during the process of microscopic calculations, the single-particle energies and wave functions are obtained by solving the Schrödinger equation of the stationary states for an average nuclear potential of Woods–Saxon (WS) type including a central ﬁeld, a spin–orbit interaction, and the Coulomb potential for the protons [50]. The universal WS parameter set is adopted, that is: Radius parameters: r (p) = 1.275 fm, r (n) = 1.347 fm, r (p) = 1.320 fm, r (n) = 1.310 fm. 0 0 0-so 0-so Central potential depth parameters: V = 49.6 MeV, κ = 0.86. Spin–orbit potential strength constants: λ(p) = 36.0, λ(n) = 35.0. Diffuseness parameters: a (p) = a (n) = a (p) = a (n) = 0.70 fm. 0 0 0-so 0-so The deformed WS potential is generated numerically at each (β , γ , β ) deformation lattice. The 2 4 Hamiltonian matrix is built by means of the axially deformed harmonic oscillator basis with the principal quantum number N ≤ 12 and 14 for protons and neutrons, respectively. Then its eigenvalues and eigenfunctions are calculated in terms of a standard diagonalization procedure. The rotational contribution, as seen in Eq. (1), can be calculated by solving the cranked Lipkin– Nogami (CLN) equation, which will take the form of the well-known Hartree–Fock–Bogoliubov-like (HFB) equations [56], that is, (e − λ)δ − ω(j ) − Gρ + 4λ ρ U − δ V = E U , α αβ x αβ 2 αβ βk αβ ¯ k αk ¯ βk α ¯ β (6) ∗ ∗ (e − λ)δ − ω(j ) − Gρ + 4λ ρ V + δ U = E V , α αβ x αβ αβ 2 αβ βk k α ¯ k ¯ βk α ¯ β where = G κ , λ = λ + 2λ (N + 1), and E = ε − λ . Further, ε is the quasi-particle αα ¯ 1 2 k k 2 k α>0 energy and α (α ¯ ) denotes the states of signature r =−i (r =+i), respectively. The matrices ρ and κ correspond to the density matrix and pairing tensor. While solving the HFB equations, pairing is treated self-consistently at each frequency ω and each grid point in the selected deformation space. After the numerically calculated Routhians at ﬁxed ω are interpolated using a cubic spline function between the lattice points, some nuclear structure properties including the ground-state equilibrium shape, saddle points, ﬁssion paths, and so on can be analyzed based on the TRS. Finally, it should be pointed out that in the actual calculation, the Bohr shape deformation parameters [57] and the Lund convention [58] are adopted for convenience. The Cartesian quadrupole coordinates X = ◦ ◦ β cos(γ + 30 ) and Y = β sin(γ + 30 ) were used, where the parameter β speciﬁes the magnitude 2 2 2 of the quadrupole deformation, while γ speciﬁes the asymmetry of the shape. Of course, accordingly, ◦ ◦ the β value is always positive and the triaxiality parameter covers the range −120 ≤ γ ≤ 60 . The 4/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. (a)(b) Fig. 1. (a) Calculated three-dimensional potential energy surface in (β , γ , β ) deformation space for the 2 4 selected nucleus Gd, which has been minimized with respect to the deformation parameter β at each (β , γ ) grid point. The red and blue lines respectively represent the paths along the axial ﬁssion trajectory and triaxial part of the ﬁssion valley which has the lowest energy. (b) Similar to the lines in (a), calculated one-dimensional potential energy curves with (blue curve) and without (red curve) triaxial deformation as a function of quadruple deformation β for Gd. Note that at each given β point, the energy has been minimized 2 2 with respect to β (and γ for the blue curve) deformations. ◦ ◦ ◦ ◦ ◦ ◦ three sectors [−120 , −60 ], [−60 ,0 ], and [0 ,60 ] obviously represent the same triaxial shapes at the ground state but respectively represent rotation about the long, medium, and short axes at non-zero rotational frequency. 3. Calculations and discussions Our previous studies [40,41] indicate that the triaxial deformation degree of freedom can to some extent decrease the equilibrium deformation minimum at both the ground state and the high-spin state. Though the triaxiality characteristics, i.e., the quasi-γ bands [38], chiral doublets [59], wobbling motion mode [60], and signature splitting or inversion [61] have been systematically observed in experiments, the quest for ground-state γ -rigid nuclei is still a great challenge. For instance, the nuclei near the neutron number N = 76 are suggested to be more γ -rigid, with equilibrium deformation ◦ ◦ parameters γ around ∼20 –30 [41,62], whereas the triaxial energy minima are rather shallow, generally less than 0.5 MeV. To give a better display, taking the Gd nucleus as an example, Fig. 1 shows the calculated three-dimensional potential energy surface and the triaxial and axial one-dimensional potential energy curves for this nucleus, which has the biggest energy difference, ∼430 keV, between the axial and triaxial energy minima. However, comparing with the effect on ground-state minimum, the triaxiality may have a strong inﬂuence on the static ﬁssion barrier in heavy and superheavy nuclei. Almost all state-of-the-art theoretical approaches [63,64] show the inclusion of the triaxial deformation degree of freedom can improve the descriptive accuracy of the inner ﬁssion barriers in actinide nuclei. A similar model test is necessary before studying the A = 256 nuclei. The actinide nucleus Cf, which is closest to the present candidates and has the available ﬁssion barrier from experiment, is selected here to examine the validity of our calculations. As shown in Fig. 2(a), it can be clearly seen that the triaxial-deformed ◦ ◦ saddle with γ 17 has a lower energy than the axial saddle (with γ = 0 ). The optimal ﬁssion path will go along the triaxial energy valley (namely, the blue line) instead of the axially symmetric 5/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. (a)(b) Fig. 2. Similar to Fig. 1, but for the actinide nucleus Cf, paying attention to the ﬁssion barrier. For convenience of description, the energy curves are normalized with respect to the ground-state energy. The experimental and several other theoretical ﬁssion barriers are shown at the corresponding saddles for com- parison. Note that all the symbols correspond to the triaxial saddle except for the circle (HFBCS), which corresponds to the axial saddle. Further details are given in the text. trajectory as marked by the red line. In Fig. 2(b), the two paths are shown as one-dimensional potential energy curves. One can see that the inclusion of the triaxial γ deformation can lower the height of the ﬁrst ﬁssion barrier by more than 2.5 MeV, in good agreement with the experimental data. The ground-state minimum is not affected by the triaxiality. For further comparison, several available results of ﬁssion barriers, calculated by the fold-Yukawa (FY) single-particle potential and the ﬁnite- range liquid-drop model (FRLDM) [8] (here, we use the abbreviation FFL for FY+FRLDM), and the Hartree–Fock–BCS (HFBCS) [19], Skyrme–Hartree–Fock–Bogoliubov (SHFB) [65], and extended Thomas–Fermi plus Strutinsky integral (ETFSI) methods [21], are simultaneously shown. It is worth noting that the FFL calculation is based on macroscopic–microscopic models, similar to the present method, allowing fast calculations with the inclusion of most of the important shape degrees of freedom; the HFBCS and SHFB calculations are self-consistent methods in which many more shape degrees of freedom can naturally be included; whereas the ETFSI method combines the advantages of the above two method types. It is found from Fig. 2(b) that both the calculated axial and triaxial ﬁssion barriers qualitatively agree with each other. It seems that our calculation shows a high descriptive power, at least in this nucleus, which gives us some conﬁdence in the following calculations. Based on a similar method, Table 1 shows the calculated static ﬁssion barriers B for the isovolu- mic A = 256 nuclei, together with the available FFL [8], HFBCS [19], and ETFSI [21] results for comparison. One can see that the triaxial ﬁssion barriers are in agreement with each other to a large extent, lower than the axial barriers (namely, the HFBCS values) as expected. In order to further evaluate our calculations and collective properties in these nuclei, we also present several theoretical β deformation values [66–68] and the empirical R [69,70] and P factors [71]. Actually, we can 2 4/2 notice that though the experimental β values for these nuclei have been scarce to date, these theo- retical results can simultaneously denote the well-deformed properties. The slight underestimation of the WS-type calculation for β has been analyzed by Dudek et al. in Ref. [72], and a corrected formula (e.g., for protons: β 1.10β − 0.03(β ) ) is suggested by studying the relationship 2 2 between the WS potential parameters and the nucleonic density distributions. The energy ratio R 4/2 + + (≡ E(4 )/E(2 )) can usually provide a basic test of the axial assumption. It can be seen that two 1 1 256 256 available data points for Fm and Rf are almost equal to the ideal value of 3.3, corresponding 6/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. Table 1. Our calculated results (TRS) for the static ﬁssion barrier B and ground-state deformation parameter β for six selected A = 256 nuclei. The β values of the FY+FRDM (FFD) [66], HFBCS [67], and ETFSI 2 2 [68] calculations and the B inner ﬁssion barriers of the FY+FRLDM (FFL) [8], HFBCS [19], ETFSI [21] are given for comparison. Two available empirical quantities, R [69,70] and P factor [71], are also shown to 4/2 evaluate nuclear collective properties. B (MeV) β f 2 Nuclei R P 4/2 ∗ † TRS FFL HFBCS ETFSI TRS FFD HFBCS ETFSI Cm 4.7 — — 4.7 0.22 0.23 0.25 0.25 — 8.84 Cf 4.5 4.8 7.9 5.2 0.22 0.24 0.25 0.25 — 9.90 Fm 5.0 5.1 8.3 5.5 0.23 0.24 0.24 0.27 3.32 9.33 No 5.6 5.9 8.7 5.6 0.24 0.25 0.27 0.27 — 8.40 Rf 5.4 6.3 — 5.3 0.24 0.25 0.27 0.27 3.36 7.22 Sg 4.8 5.3 — — 0.24 0.25 0.25 0.27 — 6.00 The barriers correspond to the axial ﬁssion barrier, different from other theoretical values. † ◦ The calculated ground-state |γ | values of these nuclei are less than 1 , indicating axially deformed shapes. to the axially symmetric rotor [69]. As another empirical quantity, the P factor, which is deﬁned N N p n by P ≡ (where N and N are the numbers of valence protons and neutrons, respectively), p n N +N p n is sensitive to nuclear collectivity and deformation [71]. The study by Casten et al. in Ref. [73] indicates that the critical point of the P factor of the transition to deformed shape is no more than 5. From Table 1, it can be clearly seen that the P factors of these nuclei are obviously larger than the critical value (generally between 4 and 5), undoubtedly showing the appearance of large collectivity and agreement with the present calculations. Of course, such well-deformed shapes are, in principle, favorable to the present investigation of the Coriolis effect. Figure 3 shows the one-dimensional potential energy curves, which respectively go through the axial and triaxial ﬁssion paths for six even–even A = 256 nuclei. From this ﬁgure, it is easily found that the static equilibrium shapes of these nuclei are deformed, and even the superdeformed minima 256 256 are favorable, e.g., in Rf and Sg. However, the study indicates the observed rotational band in Rf is a normal deformed band [74]. Interestingly, one can see that the properties of inner ﬁssion barriers and the triaxiality effect on them are rather similar in these isovolumic nuclei, though there is a slight difference. It seems that with increasing Z number, the ﬁssion barrier becomes narrower 256 256 and narrower. There is no doubt, similar to Fm and Rf, the investigation of the high-spin bands in other A = 256 nuclei is possible and deserved due to the similar barrier structures and scarce data. In addition, one can see that other theoretical results of static ﬁssion barriers, including the axial and triaxial ones, actually support our calculations. In order to further understand the mechanism of barrier formation, as an example we display the calculated proton and neutron single-particle levels near the Fermi surface at three typical defor- mation points for No. As is well known, the minima and saddle points on the nuclear energy or Routhian surface can be attributable to shell effects, whose microscopic mechanism originates from the nonuniform distribution of the single-particle levels in the vicinity of the Fermi surface [52]. A minimum will generally correspond to a region of low level density, e.g., a region with a large energy gap, whereas a saddle point usually occurs in the vicinity of level crossings, the region of high level density. The energy difference between the ground-state minimum and the corresponding saddle point, namely the ﬁssion barrier, is certainly related to the level density near the Fermi level. From Fig. 4, one can see that at the ground-state deformation point there is the lowest single-particle level 7/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. 256 256 Fig. 3. Similar to Fig. 2(b), but for six even–even A = 256 nuclei ranging from Cm to Sg. Several available theoretical values are shown at the positions of the corresponding saddles. (a)(b) Fig. 4. Calculated proton (a) and neutron (b) single-particle levels near the Fermi surface for No at three typical deformation grid points, namely the ground state (GS) minimum, axial saddle (AS), and triaxial saddle (TS). The red lines indicate the Fermi energy levels. density near the Fermi level, indicating a large negative shell correction energy, whereas the highest neutron level density near the Fermi level appears at the axial saddle point, possibly corresponding to a positive shell correction energy. The corresponding level density at the triaxial saddle has an intermediate value that can be used to understand the decreasing ﬁssion barrier due to the triaxiality. Compared with the study of a static ﬁssion barrier, the investigation of a dynamic ﬁssion barrier is relatively scarce. As mentioned above, the origin of the barrier formation can be related to the shell effect of the single-particle levels, which may be strongly modiﬁed by the Coriolis force. Indeed, 8/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. (a)(b) Fig. 5. (a) Calculated LN pairing gaps of proton (upper) and neutron (lower) for six even–even A = 256 256 256 nuclei ranging from Cm to Sg at ground state, compared with the FY+FRDM calculations [75] and experiments. Correspondingly, the insets indicate the monopole pairing strengths for protons (G ) and neutrons (G ). (b) Similar to (a) but for the proton and neutron LN pairing gaps under rotation. See text for more details. under rotation, once the single-particle Routhians (in the intrinsic body-coordinate system, the single- particle “energy” is usually called “Routhian”) are rearranged, the ﬁssion barrier will be changed correspondingly. In addition, similar to the effect of magnetic force on the electron Cooper pair in a metal, the Coriolis force acts with a different sign on nucleons moving in time-reversed single- particle orbitals and tends to decouple the pairing correlations (certainly affecting the microscopic pairing corrections). In Fig. 5(a), we show the proton and neutron pairing gaps calculated in terms of the LN pairing model, together with the monopole pairing strengths (note that the quadrupole −4 ones are very small, 1 keV fm ), for these isovolumic nuclei. For comparison, the theoretical FY+FRDM calculations [66] and the experimental results extracted from experimental masses [76] based on fourth-order ﬁnite-difference expressions [77] are simultaneously shown, which at least show that the present calculations are acceptable. It is found that the pairing strengths approximately equal 85 (65) keV for protons (neutrons), indicating a decreasing trend in this heavy mass region. For instance, the typical values of such strength parameters are respectively about 165 (145) keV and 140 (105) keV for protons (neutrons) in the A 130 and 160 mass regions [48]. From Fig. 5(b), one can see that a decreasing evolution trend of the proton and neutron pairing gaps appears, as expected, and the unrealistic pairing collapse in the usual BCS calculation is avoided. Such a reasonable pairing model will be of importance for present work. Figure 6 shows the calculated deformation Routhian curves with and without the inclusion of the triaxial deformation degree of freedom at several selected rotational frequencies for even–even 256 256 A = 256 nuclei ranging from Cm to Sg. Since the present investigation just focuses on the ﬁssion barriers, these curves are normalized to the corresponding normal-deformed minima at both ground states and non-zero rotational frequencies. It can be seen that not only the axial ﬁssion barriers but also the triaxial ones have decreasing trends with increasing frequencies. Taking the triaxility 9/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. Fig. 6. Similar to Fig. 3, but including the cases at several selected rotational frequencies ω = 0.00, 0.10, 0.20, and 0.30 MeV/, approximately corresponding to spins I = 0, 4, 12, and 30 (with slight differences for different nuclei). effects into account, it seems that the ﬁssion barriers are too small to block the nuclear ﬁssions effectively after the rotational frequency ω = 0.3 MeV. In order to understand well the macroscopic and microscopic (shell and pairing) contributions of the calculated ﬁssion barriers and their evolutions, especially at critical deformation points, i.e., the minima and saddle points, we give the corresponding histograms of the more “real” triaxial ﬁssion barriers in Fig. 7 for these even–even A = 256 nuclei. It can be seen that both static and dynamic ﬁssion barriers have the dominated microscopic shell-correction components, except for the Cm nucleus. The ﬁssion barrier and its shell-correction contribution reduce with increasing rotational frequency. The contribution for the barrier from the microscopic pairing correction does not have the decreasing trend with rapid rotation, which is different with the energy of the pairing correction at a ﬁxed deformation point (e.g., at the ground state or saddle point). It may be positive or negative, but is generally smaller than the shell-correction contribution. In Cm, the macroscopic energy still plays the most important role in the formation of the ﬁssion barrier. The irregular properties at ω = 0.30 MeV/ may originate from the large structure change, e.g., due to the bandcrossing. Note that the triaxial deformation Routhian curves, as can be seen in Fig. 6, are rather ﬂat near the triaxial saddles in general. Of course, to an extent such histogram analysis is crude since the position of the 10/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053D02/4994006 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053D02 Q.-Z. Chai et al. Fig. 7. The calculated total energy (the height of the ﬁssion barrier) and its macroscopic and microscopic shell 256 256 and pairing contributions for the ﬁssion barrier in six even–even A = 256 nuclei ranging from Cm to Sg at the selected rotational frequencies ω = 0.00, 0.10, 0.20, and 0.30 MeV/. saddle, which has a relatively large uncertainty due to the large “softness,” can affect the present calculations, but the conclusions are valid. 4. Summary To summarize, the axial and triaxial ﬁssion barriers for six even–even nuclei with mass number 256 256 A = 256 ranging from Cm to Sg have been investigated by using the pairing self-consistent total Routhian surface approach in multidimensional (β , γ , β ) deformation space. The calculated 2 4 ground-state deformation parameter β and static ﬁssion barrier are in good agreement with available experimental data and other theoretical calculations. The present results show that by allowing for triaxial deformation, the height of the inner ﬁssion barrier may be reduced considerably in these 256 256 A = 256 nuclei (e.g., more than 3 MeV in No and Sg), indicating a possibly signiﬁcant increase of the penetration probability in the spontaneous ﬁssion process. The contribution of the microscopic correction in the ﬁssion barrier are dominant, except for the Cm nucleus. With increasing rotational frequency, both the axial and the triaxial ﬁssion barriers show a decreasing trend. The histograms of the dynamic triaxial ﬁssion barriers together with their macroscopic and microscopic components show that the microscopic shell corrections play an important role during the evolution process. 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Progress of Theoretical and Experimental Physics – Oxford University Press

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