J.R. Nix

Nuclear pairing models

Abstract We calculate nuclear pairing gaps for nuclei throughout the periodic system in both the BCS and Lipkin-Nogami pairing models. The energy levels required for the calculations are obtained from the folded-Yukawa single-particle model for ground-state shapes obtained in the macroscopic-microscopic approach by minimizing the total potential energy with respect to ϵ 2 and ϵ 4 shape degrees of freedom. For both pairing models we study two proposed forms for the effective-interaction pairing gap that is used to determine the pairing-gap parameter G that enters directly into the pairing equations. By comparing the calculated pairing gaps to experimental odd-even mass differences we determine parameter values for the proposed forms of the effective-interaction pairing gap by least-squares minimization. These comparisons to data lead to a preferred form for the effective-interaction pairing gap and to values of its parameters for both the BCS and Lipkin-Nogami models. From this microscopic study we conclude that no explicit isospin dependence is required for the effective-interaction pairing gap that is used to determine the pairing-gap parameter G .

Nuclear Shapes and Shape Transitions

We study nuclear potential-energy surfaces, ground-state masses and shapes calculated by use of the Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential for 4023 nuclei ranging from 16O to 279112. We present an overview of the results in the form of four colour contour diagrams vs. proton number Z and neutron number N. The four diagrams show calculated values of |2| and 4 at the ground state, ground-state microscopic shell-plus-pairing corrections and the deviations between experimental and calculated masses. The diagrams vividly display the regions of magic and deformed nuclei. In particular, the plot of |2| vs. Z and N clearly shows the well-known deformed actinide and rare-earth regions and the two new deformed regions around A = 80 and A = 100. The plots indicate differences between the various deformed regions. For instance, there are differences in the magnitude of the deformation and in the character of the transition from spherical to deformed shapes. We discuss extensively the transition from spherical to deformed shapes and study the relation between shape changes and the mass corresponding to the ground-state minimum, and the significance of additional minima in the nuclear potential-energy surface. For a few illustrative cases we discuss the effect of angular momentum on the nuclear shape. The calculated values for the ground-state mass and shape show good agreement with experimental data throughout the periodic system, but some discrepancies remain that deserve further study.

CALCULATION OF HALF-LIVES FOR SUPERHEAVY NUCLEI.

Abstract We have performed a new calculation of the half-lives of superheavy nuclei with respect to spontaneous fission, α-decay and β-decay (including electron capture). The half-lives are calculated from fission barriers and decay energies that have been obtained by means of the macroscopic-microscopic method applied to realistic diffuse-surface single-particle potentials. The results indicate that the longest total half-life is 10 9.4 y for the nucleus 294 110, which decays predominantly by the emission of α-particles. As a general rule, the predominant decay mode is α-emission for nuclei containing more than 110 protons or a few more neutrons than 184, β-emission for nuclei containing less than 110 protons, and spontaneous fission for nuclei containing either less than 184 neutrons or substantially more. Therefore, if nuclei in the vicinity of 304 122 could be formed successfully in their ground states, they should decay primarily by the rapid emission of a series of high-energy α-particles, which provides a simple identification method. However, once the closed proton shell at Z = 114 is reached, electron capture becomes the predominant decay mode. This leads to nuclei whose calculated total half-lives are sufficiently long that they could be studied by conventional chemical methods. Some fission barriers along the r-process path have been calculated for two sets of liquid-drop-model constants that differ primarily in their values of the surface-asymmetry constant K . For K = 2.84, which is representative of the values suggested by a variety of new evidence, the potential barrier against fission has practically disappeared for nuclei in the vicinity of mass number 290. The neutron-induced fission of such nuclei should be extremely rapid, thus terminating the r-process somewhat before the island of superheavy nuclei is reached.

Calculated ground-state properties of heavy nuclei

Abstract Ground-state distortions and single-particle corrections are calculated fornuclei with Z ≧ 68 and N ≧ 106 by use of the macroscopic-microscopic method as developed by Strutinsky. The microscopic part is calculated primarily by use of the folded Yukawa single-particle potential. Its parameters are redetermined to fit a actinide data. The modified oscillator potential is also used in some of the studies. Two methods for calculating the macroscopic energy are investigated. One is the droplet model of Myers and Swiatecki, and other is a modified liquid-drop model in which the surface-energy term is modified to take into account the finite range of the nuclear force. Single-particle level diagrams for the folded Yukawa potential are also presented. They are plotted as functions of the distortion parameters ϵ, ϵ 4 and ϵ 6 . Theoretical and experimental single-particle levels at the ground state for actinide nuclei are also compared.

Calculated fission properties of the heaviest elements

A quantitative calculation is presented that shows where high-kinetic-energy symmetric fission occurs and why it is associated with a sudden and large decrease in fission half-lives. The study is based on calculations of potential-energy surfaces in the macroscopic-microscopic model and a semi-empirical model for the nuclear inertia. For the macroscopic part a Yukawa-plus-exponential model is used and for the microscopic part a folded-Yukawa single-particle potential is used. The three-quadratic-surface parameterization generates shapes for which the potential-energy surfaces are calculated. The use of this parameterization and the use of the finite-range macroscopic model allows for the study of two touching spheres and similar shapes. The results of the calculations in terms of potential-energy surfaces and fission half-lives are presented for heavy even nuclei. The surfaces are displayed in the form of contour diagrams as functions of two moments of the shape. 53 refs., 15 figs., 1 tab.

Calculation of fission barriers with the droplet model and folded Yukawa single-particle potential

Abstract Single-particle levels in a folded Yukawa potential are studied as functions of various types of distortions. Some level diagrams are plotted versus an elongation coordinate in the fission direction. The levels are also plotted as functions of the asymmetric coordinate ϵ 35 . The levels are labeled by their asymptotic quantum numbers.These diagrams thus show the calculated single-particle level structure at the second minimum and second peak in the fission barrier for nuclei in the actinide region. Fission barriers are calculated by the use of the macroscopic-microscopic method; the macroscopic part of the energy is obtained from the droplet model of Myers and Swiatecki. The relevant equilibrium points in the fission barrier are tabulated for the actinide elements and also for some lighter elements in the region 76 ≦ Z ≦ 84. The calculated energies of the equilibrium points reproduce experimental values to within an accuracy of 1 or 2 MeV. A detailed comparison is made between the three-quadratic-surface parametrization and the ϵ-coordinates in Nilssons perturbed-spheroid parametrization.

New developments in the calculation of heavy-element fission barriers

Abstract We present calculated potential-energy surfaces and fission half-lives for heavy even and odd nuclei between Pu and Z = 110. We base our study on the macroscopic-microscopic model. For the macroscopic part we use the Yukawa-plus-exponential (finite-range) model and for the microscopic part a folded-Yukawa (diffuse-surface) single-particle potential. To remove some deficiences of the model that were associated with describing the transition from a single nuclear system to two different nuclear systems that occurs in the scission region, we have included the following new features. We have increased the smoothing range in the Strutinsky method from 1.0 × ħω 0 to 1.4 × ħω 0 and we use shape-dependent Wigner and A 0 terms. The effects of these two improvements are large, up to a magnitude of about 10 MeV close to scission. However, since the changes due to the two improvements have different signs, the results obtained in our previous study are approximately retained. To allow studies of the fission properties of odd nuclei we have also added the possibility of calculating the specialization energies associated with the odd particles. This allows us to calculate fission half-lives also for odd systems for fission along both the new and old fission paths. In most cases we find good agreement between calculated and experimental half-lives. Our results show that we have obtained a good understanding of the fission properties also of odd heavy nuclei.

How much of a bubble is there in 184Hg

Abstract Recent suggestions of the existence of bubble-shape nuclei are examined for a few selected nuclei in terms of a Strutinsky shell-correction method type of calculation, based on the folded-Yukawa model. The inner surface is treated by a modified version of the liquid-drop model, allowing for the finite range of the nuclear diffuseness and nucleon-nucleon interaction. It appears safe to conclude that 184 Hg is not bubble-like. The observed large 〈 r 2 〉 in this region of Hg nuclei is explained as being associated with a change in distortion. This change is shown to be largely an effect of the introduction of quadrupole pairing.

Two-dimensional calculation of sub-barrier heavy-ion fusion cross sections

Abstract Using a two-dimensional potential-energy surface and a diagonal kinetic energy we calculate the sub-barrier heavy-ion fusion cross section for the reaction 58 Ni + 58 Ni → 116 Ba . We represent the effective potential energy by an expression based on separated nuclei that includes monopole and quadrupole Coulomb interaction energies, a Yukawa-plus-exponential nuclear interaction energy, a parabolic deformation energy and a centrifugal energy. To approximate the penetrability through the resulting two-dimensional barrier surface in the separation and deformation coordinates, we take into account the spheroidal zero-point motion and average dynamical deformation of the colliding nuclei by using a gaussian superposition of straight-line paths centered about a nuclear deformation which is determined by solving classical equations of motion up to the turning point. Relative to the result calculated for a one-dimensional barrier corresponding to spherical nuclei, the sub-barrier fusion cross section is enhanced by the zeropoint motion but is suppressed by the dynamically induced oblate deformations. The combined effect for 58Ni+58Ni gives only a slight enhancement which does not explain the experimental measurements.

Potential-Energy Surfaces for Heavy-Ion Collisions

We calculate the nuclear potential energy of deformation for the collision of two heavy nuclei by means of a macroscopic-microscopic method. The nuclear macroscopic energy is calculated in terms of a double volume integral of a Yukawa function, and the microscopic shell and pairing corrections are calculated by use of Strutinskys method from the single-particle levels of a realistic diffuse-surface single-particle potential. The time evolution of the system after the point of first contact is determined by solving the classical equations of motion for incompressible, irrotational hydrodynamical flow. The effect of nuclear viscosity on the fusion path is to slow down the formation of the neck and to inhibit the excitation of collective shape vibrations. For nuclear systems in which the fission saddle point lies well outside the contact point it is possible to interpret experimental fusion cross sections at relatively low bombarding energies in terms of a one-dimensional interaction barrier, as is customarily done. For heavier nuclear systems and higher bombarding energies, where the larger Coulomb and centrifugal forces tend to deform the fusing nuclei and lead to immediate fission, only those dynamical paths that pass inside the fission saddle point contribute significantly to fusion.