Macroscopic-microscopic Approach Research Articles

Macroscopic-Microscopic Fission Yields

We utilize the macroscopic-microscopic approach to fission to calculate nascent fragment distributions. Assuming strongly damped shape motion, we run many iterations of a Metropolis random walk across nuclear potential-energy surfaces to obtain sufficient scission statistics. Our nuclear potential surfaces consist of a macroscopic energy from the Finite-Range Liquid-Drop Model (FRLDM) and microscopic terms that arise from the single-particle spectra. We compare our predictions for two major actinides to experimental data. We present global trends from this modeling that manifest as a function of mass number of the fissioning species. We discuss the impact of fission yield modeling on nucleosynthetic outcomes.

Cassini-oval description of the multidimensional potential energy surface for U236 : Role of octupole deformation and calculation of the most probable fission path

Multidimensional potential energy surfaces (PES) are calculated using the microscopic-macroscopic approach. The nuclear shapes are described by Cassinian ovals generalized by the inclusion of ${\ensuremath{\alpha}}_{1}$, ${\ensuremath{\alpha}}_{3}$, and ${\ensuremath{\alpha}}_{4}$ shape parameters in addition to the main fission coordinate $\ensuremath{\alpha}$. The influence of the octupole deformation $({\ensuremath{\alpha}}_{3})$ on the PES is studied in the case of the nuclear fission of $^{236}\mathrm{U}$. It is found that ${\ensuremath{\alpha}}_{3}$ plays an important role in the last stage of the fission process; for instance, it lowers the third minimum and the third barrier. Two methods to calculate the static fission path are investigated. They are found to be consistent in the sense that they lead to the same fission barrier. In certain subspaces, the least energy paths from the ground state to scission present discontinuities around one of the saddles. They are caused by sharp changes in the nuclear shapes involved occurring without a change in energy. Such transitions are smoothed out by the principle of stationary action, which transforms a discontinuous path into a continuous one. Finally, various macroscopic models have been employed in order to study their influence on the energies and positions of the saddles and the minima.

Giant dipole resonance and its fine structures in 144−152Nd,152Sm studied within the linear response theory

We present a microscopic approach for giant dipole resonance (GDR) where the linear response by the nuclear density to the dipole radiation is represented through the single-particle wavefunctions calculated with a triaxial Woods-Saxon potential. We follow a microscopic-macroscopic approach to estimate the nuclear deformation with the same potential. We explain the recent experimental data for even–even nuclei 144−152Nd and 152Sm and present a comparison with the macroscopic approach for GDR. We highlight the cases where the results from the microscopic approach are sensitive to the change in single-particle configuration despite no change in shape and mass but with a change of two protons in a mid-shell region. We also present the fine structure analysis of the GDR cross-section using the continuous wavelet transform (CWT) framework and elucidate the origin of such fine structures.

Study of shape change in T = 1 triplet 70Se, 70Br and 70Kr in microscopic–macroscopic approach

The shapes of the A = 70, T = 1 triplet nuclei [Formula: see text]Se, [Formula: see text]Br and [Formula: see text]Kr have been investigated by Total Routhian Surface (TRS) calculations in the microscopic–macroscopic formalism. A Woods–Saxon potential with global set of parameters was used for obtaining the shell correction energy and the Strutinsky method has been applied to calculate the TRS. The [Formula: see text] and [Formula: see text] shape parameters, determined from the minimum of the TRS, show a structural change between mirror pairs [Formula: see text]Se and [Formula: see text]Kr. This is in support of the recent measurements performed at RIKEN.

Role of spin-orbit strength in the prediction of closed shells in superheavy nuclei

Using the microscopic-macroscopic approach based on the modified two-center shell model, the ground-state shell corrections are analyzed as a function of spin-orbit strength for even-$Z$ superheavy nuclei in the $\ensuremath{\alpha}$-decay chains containing $^{295--300,302,304}120$ nuclei. The influence of the spin-orbit strength on the positions of shell closures and the description of the low-lying one-quasiparticle spectra for $^{251}\mathrm{Cf}, ^{243}\mathrm{Cm}, ^{243}\mathrm{Bk}$, and $^{251}\mathrm{Es}$ nuclei are studied in detail. The importance of studying the nuclear structure of actinides for predicting the next double magic nucleus beyond $^{208}\mathrm{Pb}$ is demonstrated.

Improved macroscopic microscopic mass formula

A nuclear mass formula based on the macroscopic microscopic approach is proposed, in which the number of model parameters is reduced compared with other macroscopic microscopic models. The root mean square (RMS) deviation with respect to 2314 training sets (measured nuclear masses) is reduced to 0.447 MeV, and the calculated value of each nucleus is no more than 0.8% different from the experimental value. The single and two nucleon separation energies and the shell gaps are calculated to test the model. The shell corrections and double magic number of superheavy nuclei are also analyzed.

Rate of decline of the production cross section of superheavy nuclei with Z=114–117 at high excitation energies

The production cross sections of superheavy nuclei with charge numbers $114-117$ are predicted in the $(5-9)n$-evaporation channels of the $^{48}$Ca-induced complete fusion reactions for future experiments. The estimates of synthesis capabilities are based on a uniform and consistent set of input nuclear data provided by the multidimensional macroscopic-microscopic approach. The contributions of various factors to the final production cross section are discussed. As shown, the specific interplay between survival and fusion probabilities unexpectedly leads to a relatively slow decline of the total cross-sections with increasing excitation energy. This effect is supported by a favorable arrangement of fission barriers protecting the compound nucleus against splitting concerning energetic thresholds for the emission of successive neutrons. In particular, the probabilities of the formation of superheavy nuclei in the $5n$-, $6n$-, and in some cases even in $7n$-evaporation channels are still promising. This may offer a new opportunity for the future synthesis of unknown neutron-deficient superheavy isotopes.

Improvements to the macroscopic-microscopic approach of nuclear fission

The well established macroscopic-microscopic (mac-mic) description of nuclear fission enables the prediction of fission fragment yields for a broad range of fissioning systems. In this work, we present several key enhancements to this approach. We improve upon the microscopic sector of nuclear potential energy surfaces by magnifying the resolution of the Lipkin-Nogami equations and strengthening the Strutinsky procedure, thus reducing spurious effects from the continuum. We further present a novel deterministic method for calculating fission dynamics under the assumption of strongly damped nucleonic motion. Our technique directly determines the evolution of the scissioned shape distribution according to the number of random walk steps rather than the statistical accumulation of fission events. We show that our new technique is equivalent to the Metropolis random walk pioneered over the past decade by Randrup and colleagues. It further improves upon it, as we remove the need for altering the nuclear landscape via a biased potential. With our final improvement, we calculate fission fragments mass and charge distributions using particle number projection, which affords the simultaneous calculation of both mass and charge yield distributions. Fission fragments are thus calculated from the quantum mechanical $A$-body states of the potential energy surface rather than the collective mass asymmetry variable ($\alpha_{\rm g}$) of the Finite-Range Liquid-Drop Model (FRLDM) used in past work. We highlight the success of our enhancements by predicting the odd-even staggering and the charge polarization for the neutron-induced fission of ${}^{233}$U and ${}^{235}$U.

Microscopic-macroscopic approach for ground-state energies based on the Gogny force with the Wigner-Kirkwood averaging scheme

In the previous paper I \cite{bhagwat20} we have shown that self-consistent Extended Thomas-Fermi (ETF) potentials and densities associated with a given finite-range interaction can be parametrized by generalized Fermi distributions. As a next step, a comprehensive calculation of ground-state properties of a large number of spherical and deformed even-even nuclei is carried out in the present work using the Gogny D1S force within the ETF scheme. The parametrized ETF potentials and densities of paper I are used to calculate the smooth part of the energy and the shell corrections within the Wigner-Kirkwood semiclassical averaging scheme. It is shown that the shell corrections thus obtained, along with a simple liquid drop prescription, yield a good description of ground-state masses and potential energy surfaces for nuclei spanning the entire periodic table.

Isobaric-multiplet mass equation in a macroscopic-microscopic approach

We study the a, b and c coefficients of the isobaric-multiplet mass equation using a macroscopic-microscopic approach developed by P. Moeller and his collaborators in ADNDT 59, 185 (1995) and ADNDT 109-110, 1 (2016). We show that already the macroscopic part of the finite-range liquid-drop model (FRLDM) describes the general trend of the a and b coefficients relatively well, while the staggering behavior of b coefficients for doublets and quartets can be understood in terms of the difference of average proton and neutron pairing energies. The sets of isobaric masses, predicted by the full macroscopic-microscopic approaches, are used to explore the general trends of IMME coefficients up to A=100. We conclude that while the agreement for a coefficients is quite satisfactory, the global approaches have less sensitivity to predict the staggering pattern observed for b coefficients of doublets and quartets. The best set of theoretical b coefficients for multiplets up to about A=100 is used to predict masses of proton-rich nuclei based on the known experimental masses of neutron-rich mirror partners, and subsequently to investigate their one- and two-proton separation energies. The estimated position of the proton-drip line is in fair agreement with known experimental data. These masses are important for simulations of the astrophysical rp-process.

Possibilities of direct production of superheavy nuclei with Z=112–118 in different evaporation channels

The production cross sections of heaviest isotopes of superheavy nuclei with charge numbers 112–118 are predicted in the xn–, pxn–, and αxn–evaporation channels of the 48Ca-induced complete fusion reactions for future experiments. The estimates of synthesis capabilities are based on a uniform and consistent set of input nuclear data. Nuclear masses, deformations, shell corrections, fission barriers and decay energies are calculated within the macroscopic-microscopic approach for even-even, odd-Z and odd-N nuclei. For odd systems the blocking procedure is used. To find saddle points, the Imaginary Water Flow technique is used and non-axiallity is taken into account. As shown, our calculations, based on a new set of mass and barriers, agree very well with the experimentally known cross-sections, especially in the 3n–evaporation channel. The dependencies of these predictions on the mass/fission barriers tables, the ratio af/a, and fusion models are discussed. A way is shown to produce directly unknown superheavy isotopes in the 1n– or 2n–evaporation channels. The synthesis of new superheavy isotopes unattainable in reactions with emission of neutrons is proposed in the promising channels with emission of protons (σpxn≃10−200 fb) and alphas (σαxn≃50−500 fb).

Mass yields of fission fragments of Pt to Ra isotopes

An effective Fourier nuclear shape parametrization which describes well the most relevant degrees of freedom on the way to fission is used to construct a 3D collective model. The potential energy surface is evaluated within the macroscopic-microscopic approach based on the Lublin-Strasbourg Drop (LSD) macroscopic energy and Yukawa-folded single particle potential. A phenomenological inertia parameter is used to describe the kinetic properties of the fissioning system. The fission fragment mass yields are obtained by using an approximate solution of the underlying Hamiltonian. The predicted mass fragmentations for even-even Pt to Ra isotopes are compared with available experimental data. Their main characteristics are well reproduced when the neck rupture probability dependent on the neck radius is introduced.

Fine structure of $$\alpha $$ decay from the time-dependent pairing equations

The $$\alpha $$ decay half-lives and the fine structure phenomenon are investigated with fission-like models. A superasymmetric fission path in a configuration space spanned by five degrees of freedom is determined in accordance with the least action principle. The deformation energy is evaluated within the macroscopic–microscopic approach while the inertia is obtained in the framework of the cranking model. The single particle levels schemes are calculated connecting the ground state of the parent nucleus and the asymptotic configuration of two separated nuclei. The probabilities to find different seniority configurations are obtained by solving time-dependent pairing equations generalized by including the Landau–Zener effect and the Coriolis coupling. The microscopic equations of motion for even numbers of particles are deduced, those for odd-nuclear systems being obtained in previous works. The models used in the calculations are reviewed within a detailed description. The microscopic equations of motion are solved by starting from the ground state configuration and arriving at the scission point. A description of all the possible configurations at scission together with their realization probabilities is given. By fitting the inter-nuclear velocity, the best agreement between experimental and theoretical hindrance factors is retained. The theoretical results for the $$\alpha $$ decay half-lives for $$^{211,212}$$ Po and $$^{211}$$ Bi are compared with experimental data showing discrepancies ranging over three orders of magnitude. The accuracy of the model concerning the calculations of the half-lives for different channels is discussed. The connections between the classical theories concerning the preformation of the $$\alpha $$ particle and the fission-like descriptions are highlighted.

Number of particles in fission fragments

In current simulations of fission, the number of protons and neutrons in a given fission fragment is almost always obtained by integrating the total density of particles in the sector of space that contains the fragment. Because of the antisymmetry of the many-body wave function of the whole nucleus, this procedure systematically gives noninteger numbers of particles in the fragments. We introduce a novel sampling method to estimate rigorously the probability of finding $Z$ protons and $N$ neutrons in a fission fragment without resorting to projectors, which can sometimes give unwieldy results. When applied on standard Hartree-Fock-Bogoliubov many-body states, we show that our approach reproduces indeed the results of full particle-number projection. We then estimate the charge and mass number dispersion of several scission configurations in $^{240}$Pu with and without pairing correlations included. We show that odd-even effects in the charge probability naturally occur within our approach, which could explain the well-known odd-even staggering of charge distributions. Our method is applicable either in static calculations of scission configurations such as, e.g., in the macroscopic-microscopic approach or energy density functional theory, but also in explicitly time-dependent density functional theory simulations of fission.

Lifetime of the recently identified 10+ isomeric state at 3279 keV in the Nd136 nucleus

Background: The γ softness of 136Nd makes it possible to study the shape changes induced by two-proton or two-neutron excitation. Purpose: We measure the lifetimes of two-quasiparticle states of the bands based on the 10+ states at 3296 and 3279 keV to investigate the shape change induced by the alignment of two protons or two neutrons in the h11/2 orbital. Methods: The recoil-distance Doppler shift method was used for the study of 136Nd studies, which was formed by the fusion reaction 120Sn(20Ne, 4n)136Nd, at Ebeam = 85 MeV. Calculations were performed within the microscopic-macroscopic approach, based on the deformed Woods-Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy. Results: The lifetime of the 10+ state at 3279 keV of 136Nd was measured to be T 10+1/2 = 1.63(9) ns. The lifetimes of the 2+ state at 374 keV and of the 12+ state at 3686 keV of the ground band were also measured to be T 2+1/2 = 26.5(14) ps and T 12+1/2 = 22.5(14) ps. Conclusions: The measured lifetime of 10+ the state at 3279 keV together with other observables confirm the structure change in 136Nd. A rather small reduced hindrance of the electromagnetic decay of the 10+ state at 3279 keV would be consistent with its K-mixed character.

Description of the mass-asymmetric fission of the Pt isotopes, obtained in the reaction Ar36+Nd142 within the two-stage fusion-fission model

The two stages dynamical stochastic model developed earlier for description of fusion-fission reactions is applied to the calculation of mass- and energy-distributions of fission fragments of platinum isotopes in reaction ${\rm ^{36}Ar + ^{142}Nd \to ^{178-x}Pt + xn}$. The first stage of this model is the calculation of the approaching of projectile nucleus to the target nucleus. On the second stage of the model, the evolution of the system formed after the touching of the projectile and target nuclei is considered. The evolution of the system on both stages is described by three-dimensional Langevin equations for the shape parameters of the system. The mutual orientation of the colliding ions and tunneling through the Coulomb barrier in the entrance channel are also taken into account. The potential energy of the system is calculated within the macroscopic-microscopic approach. The calculated mass-energy distributions of fission fragments are compared with the available experimental data. The impact of shell effects, rotation of the system and neutron evaporation on the calculated results is discussed.

Incorporating self-consistent single-particle potentials into the microscopic-macroscopic method

Effective single-particle potentials obtained by self-consistent HFB calculations from the established non-relativistic and relativistic nuclear EDF approaches are incorporated into the microscopic-macroscopic method, a widely and successfully used approach for superheavy nuclei. We determine the Schrodinger-equivalent central and spin-orbit potentials incorporating effective mass effects. The method can be applied to non-relativistic and relativistic mean-fields. A parametrization in terms of the Wood-Saxon form is introduced to derive the proton and neutron potentials, appropriate for the microscopic-macroscopic method. As the first application, the extended microscopic-macroscopic approach is used to calculate the shell corrections in the heaviest nuclei. Constraints on parameters sets for central and spin-orbit potentials are derived for which the shell effects are amplified towards $ Z=120$ .

Distinct ground state features and the decay chains of Z = 121 superheavy nuclei

A fully systematic study of even and odd isotopes [Formula: see text] of [Formula: see text] superheavy nuclei is presented in theoretical frameworks of Relativistic mean-field plus state dependent BCS approach and macroscopic–microscopic approach with triaxially deformed Nilsson-Strutinsky prescription. The ground state properties namely shell correction, binding energy, two- and one-proton and neutron separation energy, shape, deformation, density profile and the radius are estimated that show strong evidences for magicity in [Formula: see text], 228. Central depletion in the charge density due to large repulsive Coulomb field indicating bubble like structure is reported. A comprehensive analysis for the possible decay modes specifically [Formula: see text]-decay and spontaneous fission (SF) is presented and the probable [Formula: see text]-decay chains are evaluated. Results are compared with Finite Range Droplet Model (FRDM) calculations and the available experimental data which show excellent agreement.

Hindered α decays of heaviest high- K isomers

To find candidates for long-lived high-$K$ isomers in even-even $Z=106\text{--}112$ superheavy nuclei, we study dominant $\ensuremath{\alpha}$-decay channels of two- and four-quasiparticle configurations at a low excitation. Energies are calculated within the microscopic-macroscopic approach with the deformed Woods-Saxon potential. Configurations are fixed by a standard blocking procedure and their energy found by a subsequent minimization over deformations. Different excitation energies of a high-$K$ configuration in parent and daughter nucleus seem particularly important for hindrance of the $\ensuremath{\alpha}$ decay. A strong hindrance is found for some four-quasiparticle states, particularly ${K}^{\ensuremath{\pi}}={20}^{+}$ and/or ${19}^{+}$ states in $^{264\text{--}270}\mathrm{Ds}$. Contrary to what was suggested in experimental papers, it is rather a proton configuration that leads to this strong hindrance. If not shortened by the electromagnetic decay, $\ensuremath{\alpha}$ half-lives of $\ensuremath{\sim}1\phantom{\rule{0.28em}{0ex}}\mathrm{s}$ could open new possibilities for studies of chemical/atomic properties of related elements.

Stability of superheavy nuclei

The potential-energy surfaces of an extended set of heavy and super-heavy even-even nuclei with $92 \le Z \le 126$ and isospins $40 \le N-Z \le 74$ are evaluated within the recently developed Fourier shape parametrization. Ground-state and decay properties are studied for 324 different even-even isotopes in a four-dimensional deformation space, defined by non-axiality, quadrupole, octupole, and hexadecapole degrees of freedom. Nuclear deformation energies are evaluated in the framework of the macroscopic-microscopic approach, with the Lublin-Strasbourg-Drop model and a Yukawa-folded mean-field potential. The evolution of the ground-state equilibrium shape (and possible isomeric, metastable states) is studied as a function of $Z$ and $N$. Alpha-decay $Q$-values and half-lives, as well as fission-barrier heights, are deduced. In order to understand the transition from asymmetric to symmetric fission along the Fm isotopic chain, the properties of all identified fission paths are investigated. Good agreement is found with experimental data wherever available. New interesting features about the population of different fission modes for nuclei beyond Fm are predicted.