folded-Yukawa Single-particle Potential Research Articles

Essentials of the macroscopic-microscopic folded-Yukawa approach and examples of its record in providing nuclear-structure data for simulations

The macroscopic-microscopic model based on the folded-Yukawa singleparticle potential and a “finite-range” macroscopic model is probably the approach that has provided the most reliable predictions of a large number of nuclear-structure properties for all nuclei between the proton and neutron drip lines. I will describe some basic features of the model and the development philosophy that may be the reason for its success. Examples of quantities modeled within the same model framework are, nuclear masses, ground-state level structure, including spins, ground-state shapes, fission barriers, heavy-ion fusion barriers, sub-barrier fusion cross sections, β-decay half-lives and delayed neutron emission probabilities, shape coexistence, and α-decay Qα energies to name a few. I will show how well it predicted various properties measured after published results. Rather than giving an incomplete model description here I will give a timeline of model development and provide references to typical applications and references that are sufficiently complete that several individuals have written computer codes based on these references, codes whose results have excellent agreement with ours.

Study of Fission Barrier Heights of Uranium Isotopes by the Macroscopic-Microscopic Method

Potential energy surfaces of uranium nuclei in the range of mass numbers 229 through 244 are investigated in the framework of the macroscopic-microscopic model and the heights of static fission barriers are obtained in terms of a double-humped structure. The macroscopic part of the nuclear energy is calculated according to Lublin—Strasbourg-drop (LSD) model. Shell and pairing corrections as the microscopic part are calculated with a folded-Yukawa single-particle potential. The calculation is carried out in a five-dimensional parameter space of the generalized Lawrence shapes. In order to extract saddle points on the potential energy surface, a new algorithm which can effectively find an optimal fission path leading from the ground state to the scission point is developed. The comparison of our results with available experimental data and others' theoretical results confirms the reliability of our calculations.

Combinatorial nuclear level-density model

A microscopic nuclear level-density model is presented. The model is a completely combinatorial (micro-canonical) model based on the folded-Yukawa single-particle potential and includes explicit treatment of pairing, rotational and vibrational states. The microscopic character of all states enables extraction of level-distribution functions with respect to pairing gaps, parity and angular momentum. The results of the model are compared to available experimental data: level spacings at neutron separation energy, data on total level-density functions from the Oslo method, cumulative level densities from low-lying discrete states, and data on parity ratios. Spherical and deformed nuclei follow basically different coupling schemes, and we focus on deformed nuclei.

PAIRING CORRELATIONS AND FISSION BARRIER HEIGHTS

The influence of a pairing strength proportional to the surface of the deformed nucleus, on the fission barrier heights evaluated within the macroscopic-microscopic model which does not contain the average pairing energy in its macroscopic part is investigated. The calculations were carried out with the Lublin-Strasbourg drop model and the folded-Yukawa single-particle potential.

Dependence of direct neutron capture on nuclear-structure models

The prediction of cross sections for nuclei far off stability is crucial in the field of nuclear astrophysics. We calculate direct neutron capture on the even-even isotopes ${}^{124\ensuremath{-}145}$Sn and ${}^{208\ensuremath{-}238}$Pb with energy levels, masses, and nuclear density distributions taken from different nuclear-structure models. The utilized structure models are a Hartree-Fock-Bogoliubov model, a relativistic mean field theory, and a macroscopic-microscopic model based on the finite-range droplet model and a folded-Yukawa single-particle potential. Due to the differences in the resulting neutron separation and level energies, the investigated models yield capture cross sections sometimes differing by orders of magnitude. This may also lead to differences in the predicted astrophysical $r$-process paths.

Applications of a global nuclear structure model to studies of the heaviest elements

We present some new results on heavy element nuclear structure properties calculated on the basis of the finite-range droplet model and folded Yukawa single-particle potential. Specifically, we discuss calculations of nuclear ground state masses and microscopic corrections, α decay properties, β decay properties, fission potential energy surfaces, and spontaneous fission half-lives. These results, obtained with a global nuclear structure approach, are particularly reliable for describing the stability properties of the heaviest elements.

Global nuclear-structure calculations

The revival of interest in nuclear ground-state octupole deformations that occurred in the 1980's was stimulated by observations in 1980 of particularly large deviations between calculated and experimental masses in the Ra region, in a global calculation of nuclear ground-state masses. By minimizing the total potential energy with respect to octupole shape degrees of freedom in addition to ϵ 2 and ϵ 4 used originally, a vastly improved agreement between calculated and experimental masses was obtained. To study the global behaviour and interrelationships between other nuclear properties, we calculate nuclear ground-state masses, spins, pairing gaps and β-decay half-lives and compare the results to experimental quantities. The calculations are based on the macroscopic-microscopic approach, with the microscopic contributions calculated in a folded-Yukawa single-particle potential.

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 |ϵ2| and ϵ4 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 |ϵ2| 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.

Nuclear mass formula with a Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential

We use a Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential to systematically calculate the ground-state masses of 4023 nuclei ranging from 16O to {279}112. The method is also used to calculate the fission-barrier heights of 28 nuclei ranging from 109Cd to 252Cf. We introduce several previously neglected physical effects, including a smaller nuclear radius constant, a proton form factor, an exact diffuseness correction, an A 0 term, a chargeasymmetry term, and microscopic zero-point energies. The nuclear radius constant is determined from elastic electron scattering and microscopic calculations of nuclear density distributions, the range of the Yukawa-plus-exponential folding function is determined from heavy-ion elastic scattering, the surface-energy constant and surface-asymmetry constant are determined from the fission-barrier heights of the 28 nuclei that are considered, and the remaining constants are determined from the ground-state masses of 1323 nuclei ranging from 16O to 259No for which experimental values are known with experimental errors less than 1 MeV. For the final formula, the root-mean-square error in the ground-state masses is 0.835 MeV and the root-mean-square error in the fission-barrier heights is 1.331 MeV. Some of the remaining discrepancies in the groundstate masses can be understood in terms of instabilities with respect to ε 3 and ε 6 deformations.

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 Nilsson's perturbed-spheroid parametrization.