Macroscopic-microscopic Method Research Articles

Fission valleys and favored isotopes emitted from 310\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{310}$$\\end{document}126

The decay channels for the superheavy nucleus 310\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{310}$$\\end{document}126 are studied under the binary macroscopic–microscopic method. The macroscopic part is calculated as the heavy-light interacting charged liquid drops, linked through the finite range Yukawa-plus-exponential force. The microscopic energy is calculated based on the spheroidally deformed two-center shell model. The proton and neutron single-particle levels are introduced in the binary Strutinsky procedure. The resulting shell corrections are added to the macroscopic part to obtain the fission barriers. The dynamical part follows by introducing the Werner–Wheeler inertia. The final lifetimes are calculated for all possible isotopic channels around the most favored heavy-light fragment pairs.

Fission barriers of superheavy nuclei for emitted fragment isotopes near proton magic numbers

Fission barriers for supposed doubly magic $^{298}\mathrm{Fl}$ are calculated using a specialized binary macroscopic-microscopic method. The microscopic effects are obtained from the developed binary Strutinsky shell correction method. The main input data are the proton and neutron energy levels calculated with the deformed two-center shell model. One is able in this way to provide the microscopic transition via the level schemes, from the parent to two overlapped and finally separated level schemes. The macroscopic part is acquired by the deformed charged liquid drop model with the double Yukawa-plus-exponential finite range potential. The method is applied to the reactions accompanied by isotopes of Sn and Pb within the fission channels, as being favored by strongly negative shell corrections.

Fission channels for fragment isotopes from Fl298 with magic nucleon numbers

Short-lived systems such as superheavy nuclei decay mainly by $\ensuremath{\alpha}$ emission. However, fission fragment emission is also probable, as theoretical calculations prove it. $^{298}\mathrm{Fl}$ dynamics decay is studied with a specialized binary macroscopic-microscopic method combined with the mass inertia tensor. The multidimensional WKB tunneling method is used to obtain the penetrabilities and half-lives. The calculations are applied to fission channels around the isotopes of two presumably favored fragments, Pb and Sn. The choice is due to their proton magicity shell correction favorable fission valleys.

Exotic shape symmetries around the fourfold octupole magic number N=136 : Formulation of experimental identification criteria

We employ a realistic nuclear mean-field theory using the phenomenological, Woods-Saxon Hamiltonian with newly adjusted parameters containing no parametric correlations; originally present correlations are removed employing the Monte Carlo approach. We find very large neutron shell gaps at $N=136$ for all the four octupole deformations ${\ensuremath{\alpha}}_{3\ensuremath{\mu}=0,1,2,3}$. These shell gaps generate well-pronounced double potential-energy minima in the standard multipole $({\ensuremath{\alpha}}_{20},{\ensuremath{\alpha}}_{22},{\ensuremath{\alpha}}_{3\ensuremath{\mu}},{\ensuremath{\alpha}}_{40})$ representation, often at ${\ensuremath{\alpha}}_{20}=0$, which in turn generate exotic symmetries ${C}_{2v}$, ${D}_{2d}$, ${T}_{d}$, and ${D}_{3h}$, discussed in detail. The main goal of the article is to formulate spectroscopic criteria for experimental identification. Calculations employing macroscopic-microscopic method are performed for nuclei with $Z\ensuremath{\ge}82$ and $N\ensuremath{\ge}126$ in multidimensional deformation spaces to analyze the expected exotic symmetries and octupole shape instabilities in the mass table ``northeast'' of the doubly magic $^{208}\mathrm{Pb}$ nucleus. Whereas the proton-unperturbed properties of neutron-generated octupole shell effects are illustrated in detail for exotic ${}_{Z=82}{\mathrm{Pb}}_{N>126}$ nuclei, our discussion is extended into even-even $Z>82$ nuclei approaching the less exotic $Z/N$ ratios, to encourage experiments which could identify the predicted exotic symmetries. In addition to the tetrahedral point group symmetry, ${T}_{d}$, of which experimental evidence has recently been published, we present ${D}_{2d}$ symmetry resulting from a superposition of axially symmetric quadrupole and tetrahedral symmetries and two new point group symmetries, ${D}_{3h}$ and ${C}_{2v}$, associated with the octupole ${\ensuremath{\alpha}}_{33}$ and ${\ensuremath{\alpha}}_{31}$ energy minima, respectively. The multidimensional $n>2$ deformation spaces are treated as usual by projecting the total potential energies onto the $n=2$ subspace. Using the representation theory of point groups we formulate quantum mechanical criteria for experimental identification of exotic symmetries through analysis of the specific properties of the collective rotational bands generated by the symmetries. The resulting band structures happen to be markedly distinct from the structure of the bands generated by ellipsoidal symmetry quantum rotors; those various rotational properties are discussed in detail.

Landscape of the island of stability with self-consistent mean-field potentials

Incorporating effective nucleon mass from the noncovariant energy-density functional, the Schr\"odinger-equivalent central and spin-orbit mean-field potentials are determined and used in the microscopic-macroscopic method to calculate the ground-state shell corrections in superheavy nuclei with the charge numbers $Z=112--126$. The island of stability of superheavy nuclei is found to be rather flat and looks like one of coral reef origin due to the interplay between the proton shells at $Z=114$ and 120, and neutron shells at $N=174$ and 184, respectively. The shape coexistence in superheavy nuclei depends on spin-orbit potentials and can affect the spectrum of $\ensuremath{\alpha}$ decay. The one-quasiparticle spectra, isomeric states, and possible $\ensuremath{\alpha}$-decay energies are predicted in the nuclei of $\ensuremath{\alpha}$-decay chains of $^{295}119$ and $^{295--297,299}120$.

Shaping the archipelago of stability by the competition of proton and neutron shell closures

The evolution of superheavy nuclei in proton and neutron number is explored by investigating the ground-state shell correction energy in the $Z=112\text{--}126$ and $N=170\text{--}190$ region of the $(Z,N)$ plane. Cuts through the $(Z,N)$-dependent distribution of shell corrections are chosen for constant neutron-proton differences as accessible in $\ensuremath{\alpha}$-decay chains and varying the neutron number at fixed charge number. A new approach is used, integrating elements of energy-density functional theory into the microscopic-macroscopic method. The topology of the shell correction implies for binding energies a distribution resembling in shape a ridge on a coral reef, formed by the competition of proton and neutron shell closures at $Z=114$ and $Z=120$, and $N=174$ and $N=184$, respectively. $^{288}\mathrm{Fl}$ is predicted as the next double magic nucleus after $^{208}\mathrm{Pb}$, and $^{304}120$ is identified as the most likely candidate for next-to-next double magic nucleus.

Correlation studies of fission-fragment neutron multiplicities

We calculate neutron multiplicities from fission fragments with specified mass numbers for events having a specified total fragment kinetic energy. The shape evolution from the initial compound nucleus to the scission configurations is obtained with the Metropolis walk method on the five-dimensional potential-energy landscape, calculated with the macroscopic-microscopic method for the three-quadratic-surface shape family. Shape-dependent microscopic level densities are used to guide the random walk, to partition the intrinsic excitation energy between the two proto-fragments at scission, and to determine the spectrum of the neutrons evaporated from the fragments. The contributions to the total excitation energy of the resulting fragments from statistical excitation and shape distortion at scission is studied. Good agreement is obtained with available experimental data on neutron multiplicities in correlation with fission fragments from $^{235}$U(n$_{\rm th}$,f). At higher neutron energies a superlong fission mode appears which affects the dependence of the observables on the total fragment kinetic energy.

Properties of heaviest nuclei with [formula omitted] and [formula omitted

We systematically determine ground-state and saddle-point shapes and masses for 1305 heavy and superheavy nuclei with Z=98–126 and N=134–192, including odd-A and odd–odd systems. From these we derive static fission barrier heights, one- and two-nucleon separation energies, and Qα values for g.s. to g.s. transitions. Our study is performed within the microscopic–macroscopic method with the deformed Woods–Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy taken as the smooth part. We use parameters of the model that were fitted previously to masses of even–even heavy nuclei. For systems with odd numbers of protons, neutrons, or both, we use a standard BCS method with blocking. Ground-state shapes and energies are found by the minimization over seven axially-symmetric deformations. A search for saddle-points was performed by using the ”imaginary water flow” method in three consecutive stages, using five- (for nonaxial shapes) and seven-dimensional (for reflection-asymmetric shapes) deformation spaces. Calculated ground-state mass excess, nucleon separation- and Qα energies, total, macroscopic (normalized to the macroscopic energy at the spherical shape) and shell corrections energies, and deformations are given for each nucleus in Table 1. Table 2 contains calculated properties of the saddle-point configurations and the fission barrier heights. In Tables 3-7, are given calculated ground-state, inner and outer saddle-point and superdeformed secondary minima characteristics for 75 actinide nuclei, from Ac to Cf, for which experimental estimates of fission barrier heights are known. These results are an additional test of our model.

Predictions of identification and production of new superheavy nuclei with Z=119 and 120

Effective single-particle potentials obtained from the established nonrelativistic nuclear energy density functional approach are incorporated into the microscopic-macroscopic method to calculate the ground-state shell corrections, mass excesses, and ${Q}_{\ensuremath{\alpha}}$ values for heaviest nuclei. The low-lying one-quasiparticle states are studied in the isotonic chains with neutron numbers $N=175$, 176, and 177. The isomeric states are discussed in the odd isotopes. The possible $\ensuremath{\alpha}$-decay chains of nuclei $^{295}119$ and $^{295,297}120$ are analyzed and compared with the available experimental data. The termination of the $\ensuremath{\alpha}$-decay chains by spontaneous fission is analyzed. The production cross sections are calculated for superheavy nuclei with $112\ensuremath{\le}Z\ensuremath{\le}120$ in the actinide-based complete fusion reactions with projectiles $^{48}\mathrm{Ca}, ^{48,50}\mathrm{Ti}, ^{54}\mathrm{Cr}, ^{58}\mathrm{Fe}$, and $^{64}\mathrm{Ni}$.

Phenomenological Estimates of the Profiles of Giant Dipole Resonances Built on Ground- and Low-lying Exited States

Thermal Shape Fluctuation (TSF) Model describes the properties of the Giant Dipole Resonance (GDR) radiation emitted by hot rotating nu-clei. This approach turned out to be a very efficient method of extracting the information about the shapes of nuclei and their evolution with increasing angular momentum. The GDR can be built also on the ground-state band but this process needs advanced microscopic methods to be described adequately. We propose to investigate the GDR strength function with the developed for this purpose TSF model. The potential energy surface, which allows to calculate the shape probability using the so-called Boltz-mann factor, is obtained within macroscopic-microscopic method, where the Folded-Yukawa model is used for macroscopic part and Folded-Yukawa plus exponential mean-field potential to generate the single-particle energies. The Strutinsky shell-, and the particle number projected BCS pairing-correction energies give the microscopic contribution to the energy of a nucleus at each given deformation. The method has been tested for Se-lenium and Neodymium isotopes for which the photo-absorption spectra were measured and preliminary results will be presented.

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$ .

Possible candidate nuclei for chirality-parity quartet bands**Supported by Natural Science Foundation of China (11705102, 11622540, 11675094), the Shandong Natural Science Foundation (ZR2017PA005, JQ201701), the China Postdoctoral Science Foundation (2017M612254), and the Young Scholars Program of Shandong University, Weihai (2015WHWLJH01)

The potential energy surfaces of the even-even 68–92Se, 112–150Ba, and 208–230Ra isotopes are calculated using the macroscopic-microscopic method in a multidimensional space {αλ,μ} including quadrupole (λ = 2, μ = 0, 2) and octupole (λ = 3, μ = 0, 1, 2, 3) degrees of freedom. The calculated results show that the even-even isotopes 92Se, 112,114,144–150Ba and 220–228Ra can exhibit the coexistence of triaxial and octupole deformations, thereby leading to simultaneous chiral and reflected symmetry breaking. Therefore, chirality-parity quartet bands are expected in these and their neighboring odd-A/odd-odd nuclei.

Rotational bands in the quadrupole-octupole collective model

A collective bands of positive and negative parity could be composed of the vibrations and rotations. The rotations of the octupole configurations can be based either on the axial or the non-axial octupole vibrations. A consistent approach to the quadrupole-octupole collective vibrations coupled with the rotational motion enables to distinguish between various scenarios of disappearance of the E2 transitions in negative-parity bands. The here presented theoretical estimates are compared with the recent experimental energies and transition probabilities in and between the ground-state and low-energy negaive-parity bands in $^{156}$Dy. A realistic collective Hamiltonian contains the potential energy term obtained through the macroscopic-microscopic Strutinsky-like method with particle-number-projected BCS approach and deformation-dependent mass tensor defined in vibrational-rotational, nine-dimensional collective space. The symmetrization procedure ensures the uniqueness of the Hamiltonian eigensolutions with respect to the laboratory coordinate system. This quadrupole-octupole collective approach may also allow to find and/or verify some fingerprints of possible high-order symmetries (e.g. tetrahedral, octahedral,...) in nuclear collective bands.

Spectroscopic criteria for identification of nuclear tetrahedral and octahedral symmetries: Illustration on a rare earth nucleus

We formulate criteria for identification of the nuclear tetrahedral and octahedral symmetries and illustrate for the first time their possible realization in a Rare Earth nucleus 152Sm. We use realistic nuclear mean-field theory calculations with the phenomenological macroscopic-microscopic method, the Gogny-Hartree-Fock-Bogoliubov approach and the general point-group theory considerations to guide the experimental identification method as illustrated on published experimental data. Following group-theory the examined symmetries imply the existence of exotic rotational bands on whose properties the spectroscopic identification criteria are based. These bands may contain simultaneously states of even and odd spins, of both parities and parity doublets at well defined spins. In the exact-symmetry limit those bands involve no E2-transitions. We show that coexistence of tetrahedral and octahedral deformations is essential when calculating the corresponding energy minima and surrounding barriers and that it has a characteristic impact on the rotational bands. The symmetries in question imply the existence of long-lived shape-isomers and, possibly, new waiting point nuclei -- impacting the nucleosynthesis processes in astrophysics -- and an existence of 16-fold degenerate particle-hole excitations. Specifically designed experiments which aim at strengthening the identification arguments are briefly discussed.

Cluster preformation at the nuclear surface in cold fission

Microscopic theories of alpha decay and cluster radioactivity explain these decay modes as a quantum tunnelling of a preformed cluster at the nuclear surface. In the present work we show that in a spontaneous cold-fission process the shell plus pairing corrections, calculated with Strutinsky's procedure based on the two-center shell model, may give a strong argument for preformation of a light fission fragment near the nuclear surface. It is obtained when the radius of the light fragment, R2, is increased linearly with the separation distance, R, of the two fragments, while for R2 = const one gets the well-known two-hump potential barrier for heavy and superheavy nuclei. Nuclear-physics community also contributed to nanocluster physics by applying the macroscopic-microscopic method to explain the shell effects experimentally observed since 1984. Applications are shown for two nuclei, 260Rf and 264Sg, whose half-life against spontaneous fission is very well known. We stress a new aspect of the cold spontaneous fission, unifying its theory with that of α- and cluster decays, all having in common a preformed light cluster which will penetrate the potential barrier by quantum tunelling.

Mass distribution of fission fragments within the Born-Oppenheimer approximation

.The fission fragments mass-yield of 236 U is obtained by an approximate solution of the eigenvalue problem of the collective Hamiltonian that describes the dynamics of the fission process whose degrees of freedom are: the fission (elongation), the neck and mass-asymmetry modes. The macroscopic-microscopic method is used to evaluate the potential energy surface. The macroscopic energy part is calculated using the liquid drop model and the microscopic corrections are obtained using a Woods-Saxon single-particle levels. The four-dimensional modified Cassini ovals shape parametrization is used to describe the shape of the fissioning nucleus. The mass tensor is taken within a cranking-type approximation. The final fragment mass distribution is obtained by weighting the adiabatic density distribution in the collective space with the neck-dependent fission probability. The neck degree of freedom is found to play a significant role in determining the final fragment mass distribution.

Fragment Mass Distributions in Low-energy Fission of $^{236}$Pu

The potential energy landscapes of Hg–U nuclei were calculated within the macroscopic–microscopic method with a Fourier nuclear shape parametrisation and the Lublin–Strasbourg Drop for the macroscopic energy. Microscopic corrections based on the Yukawa folded single-particle potential were obtained with the Strutinsky shell-correction method and the BCS approximation. The energy landscapes of even–even isotopes of Hg, Po, Ra and U were analysed in a 4-dimensional deformation space and projected onto the quadrupole–octupole plane. Extrema, ridges and valleys were localized and the electric quadrupole moments in these minima were evaluated. A comparison with the experimental data for the ground state is shown for the Hg isotopes.

Potential Energy Surfaces of Mercury up to Uranium Isotopes in the 4D Fourier Shape Parametrisation

The potential energy landscapes of Hg–U nuclei were calculated within the macroscopic–microscopic method with a Fourier nuclear shape parametrisation and the Lublin–Strasbourg Drop for the macroscopic energy. Microscopic corrections based on the Yukawa folded single-particle potential were obtained with the Strutinsky shell-correction method and the BCS approximation. The energy landscapes of even–even isotopes of Hg, Po, Ra and U were analysed in a 4-dimensional deformation space and projected onto the quadrupole–octupole plane. Extrema, ridges and valleys were localized and the electric quadrupole moments in these minima were evaluated. A comparison with the experimental data for the ground state is shown for the Hg isotopes.

Study of five-dimensional potential-energy surfaces for actinide isotopes by the macroscopic-microscopic method

In this work, the nuclear potential-energy of the deformed nuclei as a function of shape coordinates is calculated in a five-dimensional (5D) parameter space of the axially symmetric generalized Lawrence shapes, on the basis of the macroscopic-microscopic method. The liquid-drop part of the nuclear energy is calculated according to the Myers-Swiatecki model and the Lublin-Strasbourg-drop (LSD) formula. The Woods-Saxon and the folded-Yukawa potentials for deformed nuclei are used for the shell and pairing corrections of the Strutinsky-type. The pairing corrections are calculated at zero temperature, T, related to the excitation energy. The eigenvalues of Hamiltonians for protons and neutrons are found by expanding the eigen-functions in terms of harmonic-oscillator wave functions of a spheroid. Then the BCS pair is applied on the smeared-out single-particle spectrum. By comparing the results obtained by different models, the most favorable combination of the macroscopic-microscopic model is known as the LSD formula with the folded-Yukawa potential. Potential-energy landscapes for actinide isotopes are investigated based on a grid of more than 4,000,000 deformation points and the heights of static fission barriers are obtained in terms of a double-humped structure on the full 5D parameter space. In order to locate the ground state shapes, saddle points, scission points and optimal fission path on the calculated 5D potential-energy surface, the falling rain algorithm and immersion method are designed and implemented. The comparison of our results with available experimental data and others' theoretical results confirms the reliability of our calculations.

Fission approach to cluster radioactivity

Fission theory is used to explain α decay. Also, the analytical superasymmetric fission (ASAF) model is successfully employed to make a systematic search and to predict, with other models, cluster radioactivity. The macroscopic–microscopic method is illustrated for the superheavy nucleus 286Fl. Then a few results of the theoretical approach of α decay (ASAF, UNIV and semFIS models), cluster decay (ASAF and UNIV) and spontaneous fission dynamics are described with Werner–Wheeler and cranking inertia. UNIV denotes universal curve and semFIS the fission-based semiempirical formula.