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Phys. Rev. C 91 (2015) 024310 LA-UR-14-24999 Fission Barriers at the End of the Chart of the Nuclides

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Phys. Rev. C 91 (2015) 024310 LA-UR-14-24999 Fission Barriers at the End of the Chart of the Nuclides Phys. Rev. C 91 (2015) 024310 LA-UR-14-24999 Fission barriers at the end of the chart of the nuclides Peter M¨oller,1, ∗ Arnold J. Sierk,1 Takatoshi Ichikawa,2 Akira Iwamoto,3 and Matthew Mumpower4 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 3Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai-mura, Naka-gun, Ibaraki, 319-1195, Japan 4Joint Institute for Nuclear Astrophysics, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA (Dated: August 25, 2015) We present calculated fission-barrier heights for 5239 nuclides, for all nuclei between the proton and neutron drip lines with 171 ≤ A ≤ 330. The barriers are calculated in the macroscopic- microscopic finite-range liquid-drop model (FRLDM) with a 2002 set of macroscopic-model parameters. The saddle-point energies are determined from potential-energy surfaces based on more than five million different shapes, defined by five deformation parameters in the three- quadratic-surface shape parameterization: elongation, neck diameter, left-fragment spheroidal deformation, right-fragment spheroidal deformation, and nascent-fragment mass asymmetry. The energy of the ground state is determined by calculating the lowest-energy configuration in both the Nilsson perturbed-spheroid (ǫ) and in the spherical-harmonic (β) parameterizations, including axially asymmetric deformations. The lower of the two results (correcting for zero-point motion) is defined as the ground-state energy. The effect of axial asymmetry on the inner barrier peak is calculated in the (ǫ, γ) parameterization. We have earlier benchmarked our calculated barrier heights to experimentally extracted barrier parameters and found average agreement to about one MeV for known data across the nuclear chart. Here we do additional benchmarks and investigate the qualitative, and when possible, quantitative agreement and/or consistency with data on β-delayed fission, isotope generation along prompt-neutron-capture chains in nuclear-weapons tests, and superheavy-element stability. These studies all indicate that the model is realistic at considerable distances in Z and N from the region of nuclei where its parameters were determined. PACS numbers: 24.75.+i,25.85.Ec,26.30.Hj,21.60.n I. INTRODUCTION degrees of freedom corresponding to: elongation, neck ra- dius, left fragment shape, right fragment shape and the In actinide fission a nucleus typically evolves from a asymmetry of the mass division [1, 3–6]. It is also neces- single nucleus in its deformed ground state to two sep- sary to space the deformation points fairly densely so that arated fragments: one large roughly spherical fragment shell-structure features appear accurately in the calcu- with nucleon number A ≈ 140 and a smaller deformed lated potential-energy surfaces. As an example, consider fragment with the remainder of the nucleons. In Ref. [1] the shell-structure effects in the tin-like fragment with we presented calculations of fission potential-energy sur- A ≈ 140. In ground-state mass calculations in which the faces and associated barrier heights for 1585 nuclei, and associated microscopic corrections are also tabulated, for some conclusions based on these studies. We now extend example in the FRDM(1992) mass calculation [7] we find 132 the study and tabulate barriers for 5239 nuclides in the that the ground-state microscopic corrections for 50Sn82, 130 130 region 171 ≤ A ≤ 330 for all nuclei between the proton 50Sn80, and 48Cd82 are −11.55 MeV, −9.14 MeV, and and neutron drip lines. All aspects of the calculations −9.85 MeV, respectively. Thus, near magic numbers a are identical to those presented in Ref. [1] where full de- change by one nucleon changes the microscopic correc- tails are given. However, we review both some important tion by about 1 MeV. See Ref. [7] for precise definitions ingredients and features of our approach and how it is po- of the shell-plus-pairing correction, and the associated sitioned with respect to other fission-barrier calculations. microscopic correction. The latter quantity is a sum of An overview of the results is presented in Figs. 1 and 2. the shell-plus-pairing correction, the deformation change We have previously made the case that to study the of the macroscopic energy relative to the sphere and a nuclear potential energy during the shape evolution in fis- zero-point energy. We therefore space our asymmetry sion and to allow the nucleus freedom to evolve into frag- coordinate so that one step in the asymmetry coordinate ments with different shapes and into different fragment corresponds to moving about one proton and one neu- mass divisions, it is necessary to calculate the nuclear po- tron from one fragment to the other. For example, for tential energy as a function of, at a minimum, five shape an A = 200 nucleus the symmetric configuration obvi- ously corresponds to a mass division MH/ML = 100/100 and our next grid point in the asymmetry coordinate cor- ∗ [email protected] responds to MH/ML = 102/98 where MH is the heavy- 2 Calculated Fission-Barrier Height 130 Bf(Z,N) (MeV) 1 2 3 4 5 6 7 8 Z 120 110 100 Proton Number 90 80 130 140 150 160 170 180 190 200 210 220 230 Neutron Number N FIG. 1. (Color online) Calculated fission-barrier heights for 3282 nuclei. The highly variable structure is mostly due to ground- 252 270 state shell effects. Ground-state shell effects are particularly strong in the deformed regions around 100Fm152 and 108Hs162 and 298 in the nearly spherical region near the next doubly-magic nuclide postulated to be at 114Fl184. Our strongest shell effects are slightly offset to the left with respect to this isotope. fragment mass and ML is the light-fragment mass. To al- mations) and “minimize” the potential energy with re- 208 low studies of cold-fusion heavy-ion reactions on 82Pb126 spect to additional multipoles; typical examples are Refs. targets leading to superheavy elements, we use 35 grid [8, 9]. Although such approaches intuitively seem promis- points in the asymmetry coordinate [1]. In the elonga- ing, there are significant concerns about the uniqueness tion coordinate Q2 we use 45 grid points; in the remaining and stability of such results. First, when minimizations three deformation coordinates, 15 in each. This leads to are carried out at a specific location (β2,β3), what are more than 5000000 shapes. It should be obvious, but is the starting values of the additional shape variables over perhaps not immediately intuitively clear that the con- which the minimization is carried out? A trivial sugges- sequence is large data storage needs. If the energies for tion is that the values obtained for a previous point be each shape for each of the about 5000 nuclei is stored as used, but which is the “previous point” will depend upon a 10 digit number this means that the total data storage the sequence in which the grid points are considered. It space needed is 5000000 × 10 × 5000= 2.5 × 1011 bytes, is easy to visualize a surface, even in two dimensions, that is 250 Gb of storage. When we started this type of for which a different result may be found by approach- calculation based on millions of shapes in 1999, [3], this ing a particular point from opposite directions. Another was indeed a problem; now it is not. strategy could be that the minimizations are started at the value zero of the additional variables at each point (β2,β3), but these approaches would miss possible mul- II. OTHER FISSION POTENTIAL-ENERGY tiple deformed minima. And, even if found, it would be CALCULATIONS impossible to display multiple minima versus the “hid- den” shape variables in a two-dimensional contour plot. In most previous fission studies various schemes were Furthermore, none of these methods, which only access a employed to avoid calculating a complete “hypercube” limited part of the higher dimensional space, are guaran- in all the deformation variables considered. Such com- teed to find the true saddle points with reasonable accu- plete calculations were impractical until computer perfor- racy. In some cases, the saddle solutions will be correct, mance had evolved sufficiently, roughly achieved around but there is no way to mathematically evaluate the pos- 1995–2000. In macroscopic-microscopic calculations it sible errors inside the model framework itself. In many was the norm to plot energies versus two shape variables, of these minimization studies points that seem near each for example β2 and β3 (quadrupole and octupole defor- other in the two-dimensional (β2,β3) plots are actually 3 Calculated Fission-Barrier Height (MeV) 3 6 9 12 15 18 21 24 27 30 33 36 100 Z 90 80 Proton Number 70 60 90 100 110 120 130 140 150 160 Neutron Number N FIG. 2. (Color online) Calculated fission-barrier heights for 2113 nuclei with generally lower proton and neutron numbers than those in Fig. 1. Because the macroscopic energy contributes the major part of the fission-barrier height for most nuclei in this region, and because of the different energy scale compared to Fig. 1, the only shell effects clearly visible come from the N = 126 spherical neutron shell. quite distant in the higher-dimensional space. This is potential-energy surfaces and saddle points is the often manifested as strong discontinuities appearing in constrained Hartree-Fock method introduced in 1973 published potential-energy contour diagrams or plots of [12]. Those authors state “One of the advantages of this energy surfaces. Despite these known deficiencies, these type of calculation is that deformation energy curves methods are still in routine use today [10].
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