Macroscopic-microscopic Model Research Articles

Exploring the production of new superheavy nuclei with proton and α -particle evaporation channels

Existing experimental facilities limit the possibilities for discovery of new nuclides to those synthesized with cross sections above 100 fb, but the perspectives for future high current accelerators could lower this limit by two orders of magnitude. Therefore, in the present work excitation functions for fusion-$xn$ evaporation reaction channels induced not only by $^{48}Ca$ but also by heavier projectiles (usually leading to smaller cross sections) on actinide targets were calculated in the framework of the fusion-by-diffusion (FBD) model. For the first time, in this approach, channels in which a proton ($pxn$) or alpha particle ($\alpha$$xn$) is evaporated have been included in the first step of the deexcitation cascade. To calculate the synthesis cross sections entry data such as fission barriers, ground-state masses, deformations and shell effects of the superheavy nuclei calculated in a consistent way within the Warsaw macroscopic-microscopic model were used. The only adjustable parameter of the FBD model is the injection point distance $s_{inj}$ and the value determined in our previous analysis of experimental cross sections for the synthesis of superheavy nuclei of Z=114-118 has been used. Excitation functions for the synthesis of selected (cross section above a few fb) new superheavies in the range of atomic numbers 112-120 are presented. Observation of 21 new heaviest isotopes is predicted. A realistic discussion of the FBD model uncertainties is presented for the first time.

Study of fission dynamics with a three-dimensional Langevin approach

The three-dimensional Langevin model plus a constraint on the heavy fragment deformation is used to study the fission dynamics for uranium and plutonium isotopes at low excitation energies. The potential energy surface is calculated with the macroscopic-microscopic model based on the two-center shell model. The Werner-Wheeler approximation is used to calculate the inertia tensor and the wall-and-window model is applied to calculate the friction tensor. In this work, the influence of the model parameters on the fission fragment mass distribution is investigated. The fission fragment mass distributions for $^{234,236,239}\mathrm{U}$ and $^{240}\mathrm{Pu}$ at low excitation energies are calculated and compared with the results of gef code as well as the evaluated data of ENDF/B-VIII.0. A nice agreement is found in the comparison, in which the incorporation of the constraint on the heavy fragment deformation plays an important role. In addition, the dependence of the mass distribution on the excitation energies for $n+^{235}\mathrm{U}$ fission is also studied within the model. Furthermore, the correlation between the elongation and mass asymmetry at the scission point and the correlations of the fission time with both the elongation and mass asymmetry are studied. This study may shed light on understanding the dynamics of the superlong channel for symmetric fission and the standard channels for asymmetric fission in the gef model and other phenomenological fission models.

Influence of nuclear physics inputs and astrophysical conditions on r-process

The rapid neutron-capture process ($r$-process) is one of the main mechanisms to explain the origin of heavy elements in the universe. Although the past decades have seen great progress in understanding this process, the related nuclear physics inputs to $r$-process models include significant uncertainty. In this study, ten nuclear mass models, including macroscopic, macroscopic-microscopic, and microscopic models, are used to calculate the $\\beta$-decay rates and neutron-capture rates of the neutron-rich isotopes for the $r$-process simulations occurring in three classes of astrophysical conditions. The final $r$-process abundances include uncertainties introduced by the nuclear mass model mainly through the variation of neutron-capture rates, whereas the uncertainties of $\\beta$-decay rates make a relatively small contribution. The uncertainties in different astrophysical scenarios are also investigated, and are found to be connected to the diverse groups of nuclei produced during nucleosynthesis.

Effects of Various Deformation on the First Fission Barrier in Even-A N = 152 Isotones**Supported by the National Natural Science Foundation of China under Grant No. 11675148, the Project of Youth Backbone Teachers of Colleges and Universities of Henan Province under Grant No. 2017GGJS008, the Foundation and Advanced Technology Research Program of Henan Province under Grant No. 162300410222, the Outstanding

The first (namely, inner) fission barriers for even-A N = 152 nuclei have been studied systematically in the framework of macroscopic-microscopic model by means of potential energy surface (PES) calculations in the three-dimensional () deformation space. Their collective properties, such as ground-state deformations, are compared with previous calculations and available observations, showing a consistent trend. In addition, it has been found that the microscopic shell correction energy plays an important role on surviving fission in these N = 152 deformed shell nuclei. The inclusion of non-axial symmetric degree of freedom γ will pull the fission barrier down more significantly with respect to the calculation involving in hexadecapole deformation β4. Furthermore, the calculated Woods-Saxon (WS) single particle levels indicate that the large microscopic shell correction energies due to low level densities may be responsible for such a reduction on the inner fission barrier.

Strutinsky shell correction energies in relativistic Hartree-Fock theory

The single-particle spectra in relativistic Hartree-Fock (RHF) theory are employed to extract shell correction energies with the Strutinsky method. It is found that proper box sizes ${R}_{\mathrm{box}}$ used in solving the RHF equations are important to extract reliable Strutinsky shell correction energies. An empirical formula to determine ${R}_{\mathrm{box}}$ is proposed by systematic studies on nuclei over a wide range of nuclear charts, including light, heavy, neutron-deficient, and neutron-rich nuclei. Furthermore, the shell correction energies of Ca, Pb, and Og isotopes and $N=50$ isotones are calculated with proper ${R}_{\mathrm{box}}$ and their uncertainties due to the uncertainty of ${R}_{\mathrm{box}}$ are investigated as well. It is found that the results extracted by the Strutinsky method provide a reliable evaluation of the nuclear shell structure information, which can be employed to construct a macroscopic-microscopic mass model in the future.

Fine structure of 211Bi α-decay from Coriolis coupling

The α-decay is considered as a superasymmetric fission process. A fragmentation path is obtained by mean of the least action principle in a configuration space spanned by five degrees of freedom. The potential barrier is obtained within the macroscopic-microscopic model while the effective mass and the momentum of inertia within the cranking approach. The single-particle wave functions and the single-particle energies are supplied by the Woods-Saxon two-center shell model. The fine structure of 211Bi α-decay is treated within a set of generalized time-dependent pairing equations that takes into account the Landau-Zener effect and the Coriolis coupling. A low value of the internuclear collective velocity was used. The theoretical results showed a good agreement with the experimental data. Essentially, in the framework of the formalism, the fine structure for the 211Bi is due to the occurrence of the Coriolis coupling.

Pairing corrections within the density-dependent delta interaction

The validity of the Strutinsky prescriptions is tested within a state-dependent pairing interaction instead of a constant pairing strength. The amplitude of the pairing field is determined to reproduce the experimental values of the pairing gaps in the ground state of the parent [Formula: see text]Th. The density-dependent delta interaction was used to compute the quantities related to the pairing corrections along a fission trajectory, such as Fermi energies, pairing gaps, and pairing effects. An enhancement of the pairing gaps during the penetration of the second barrier was evidenced in the case of the state-dependent pairing formalism. Moreover, the outer barrier calculated in the framework of the macroscopic–microscopic model is strongly suppressed.

SIMULATE5-HEX extension for VVER analyses

Abstract Studsvik's 3D, steady-state, nodal code SIMULATE5, which is widely used for square lattice geometries, has been recently extended to hexagonal geometry for the analysis of VVER 1000 and VVER 1200s. The lattice physics code CASMO5 is used for generating nodal cross sections data from single assembly calculations. SIMULATE5's HEX extension solves the multi-group diffusion equation, or optionally SP3 equations, for the hexagonal-z geometry by dividing each hexagonal into six triangles. In each triangle, the Fourier solution of the wave equation is approximated by eight plane waves to describe the intra-nodal flux accurately. The cross sections are described by a hybrid microscopic-macroscopic model that includes approximately 60 heavy nuclides and fission products. Heterogeneities, due to spacer/grid and control rod, in the axial direction of an assembly are treated systematically. This paper summarizes the physics models employed in SIMULATE5-HEX as well as the initial benchmarking activities.

Sensitivity of the nuclear deformability and fission barriers to the equation of state

The model-dependent analysis of the fission data impacts the extracted fission-related quantities, which are not directly observables, such as the super- and hyperdeformed isomeric states and their energies. We investigated the model dependence of the deformability of a nucleus and its fission barriers on the nuclear equation of state. Within the microscopic–macroscopic model based on a large number of Skyrme nucleon–nucleon interactions, the total energy surfaces and the double-humped fission barrier of 230Th are calculated in a multidimensional deformation space. In addition to the ground-state (GS) and the superdeformed (SD) minima, all the investigated forces yielded a hyperdeformed (HD) minimum. The contour map of the shell-plus-pairing energy clearly displayed the three minima. We found that the GS binding energy and the deformation energy of the different deformation modes along the fission path increase with the incompressibility coefficient K0, while the fission barrier heights and the excitation energies of the SD and HD modes decrease with it. Conversely, the surface-energy coefficient asurf, the symmetry-energy, and its density-slope parameter decrease the GS energy and the deformation energies, but increase the fission barrier heights and the excitation energies. The obtained deformation parameters of the different deformation modes exhibit almost independence on K0, and on the symmetry-energy and its density-slope. The principle deformation parameters of the SD and HD isomeric states tend to decrease with asurf.

First Direct Mass Measurements of Nuclides around Z=100 with a Multireflection Time-of-Flight Mass Spectrograph.

The masses of ^{246}Es, ^{251}Fm, and the transfermium nuclei ^{249-252}Md and ^{254}No, produced by hot- and cold-fusion reactions, in the vicinity of the deformed N=152 neutron shell closure, have been directly measured using a multireflection time-of-flight mass spectrograph. The masses of ^{246}Es and ^{249,250,252}Md were measured for the first time. Using the masses of ^{249,250}Md as anchor points for α decay chains, the masses of heavier nuclei, up to ^{261}Bh and ^{266}Mt, were determined. These new masses were compared with theoretical global mass models and demonstrated to be in good agreement with macroscopic-microscopic models in this region. The empirical shell gap parameter δ_{2n} derived from three isotopic masses was updated with the new masses and corroborates the existence of the deformed N=152 neutron shell closure for Md and Lr.

Hauser-Feshbach fission fragment de-excitation with calculated macroscopic-microscopic mass yields

The Hauser-Feshbach statistical model is applied to the de-excitation of primary fission fragments using input mass yields calculated with macroscopic-microscopic models of the potential energy surface. We test the sensitivity of the prompt fission observables to the input mass yields for two important reactions, $^{235}$U$(n_\mathrm{th},f)$ and $^{239}$Pu$(n_\mathrm{th},f)$, for which good experimental data exist. General traits of the mass yields, such as the location of the peaks and their widths, can impact both the prompt neutron and $\gamma$-ray multiplicities, as well as their spectra. Specifically, we use several mass yields to determine a linear correlation between the calculated prompt neutron multiplicity $\bar{\nu}$ and the average heavy-fragment mass $\langle A_h\rangle$ of the input mass yields $\partial\bar{\nu}/\partial\langle A_h\rangle = \pm 0.1\,n/f/\mathrm{u}$. The mass peak width influences the correlation between the total kinetic energy of the fission fragments and the total number of prompt neutrons emitted $\bar{\nu}_T(\mathrm{TKE})$. Typical biases on prompt particle observables from using calculated mass yields instead of experimental ones are: $\delta \bar{\nu} = 4\%$ for the average prompt neutron multiplicity, $\delta \bar{M}_\gamma = 1\%$ for the average prompt $\gamma$-ray multiplicity, $\delta \bar{\epsilon}_n^\mathrm{LAB} = 1\%$ for the average outgoing neutron energy, $\delta \bar{\epsilon}_\gamma = 1\%$ for the average $\gamma$-ray energy, and $\delta \langle\mathrm{TKE}\rangle = 0.4\%$ for the average total kinetic energy of the fission fragments.

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.

Fission fragment mass and total kinetic energy distributions of spontaneously fissioning plutonium isotopes

The fission-fragment mass and total kinetic energy (TKE) distributions are evaluated in a quantum mechanical framework using elongation, mass asymmetry, neck degree of freedom as the relevant collective parameters in the Fourier shape parametrization recently developed by us. The potential energy surfaces (PES) are calculated within the macroscopic-microscopic model based on the Lublin-Strasbourg Drop (LSD), the Yukawa-folded (YF) single-particle potential and a monopole pairing force. The PES are presented and analysed in detail for even-even Plutonium isotopes with A = 236–246. They reveal deep asymmetric valleys. The fission-fragment mass and TKE distributions are obtained from the ground state of a collective Hamiltonian computed within the Born-Oppenheimer approximation, in the WKB approach by introducing a neck-dependent fission probability. The calculated mass and total kinetic energy distributions are found in good agreement with the data.

Detailed illustration of the accuracy of currently used nuclear-mass models

The ability of nuclear mass models to describe experimentally known nuclear masses is studied. Twelve frequently used models of various kinds are considered. From the selected models, eleven are of global character and one is a local model specially adapted to describe heavy nuclei. To decrease the number of nuclei over which the accuracy is averaged, the global region (Z,N≥8) is divided into four subregions, in which the accuracy is studied separately. Finally, to reach the best precision, the accuracy is investigated without any averaging for each nucleus separately. The results are presented in the form of colored maps, large enough to be easily and accurately read.Besides the accuracy of the studied models, their predictive power is also considered.It is found that the accuracy of the description of the mass strongly depends on the nuclear-mass model and on the region of nuclei to which the model is applied. The best accuracy is obtained by the two recent Chinese models WS3+ and WS4+. Generally, no clear, strong correlation between the accuracy of the description of already known masses by a given model and its predictive power for new masses is observed. However, such a correlation is found for separate models and in separate regions of nuclei, more often for the macroscopic–microscopic models than for the other approaches.

On possible shape isomers in the Pt-Ra region of nuclei

.A certain number of yet unknown super- and hyper-deformed shape isomeric states are predicted in even-even nuclei of the region between Pt and Ra, using the macroscopic-microscopic model based on the Lublin Strasbourg Drop for the macroscopic energy and shell plus pairing corrections evaluated through the Yukawa-folded mean-field potential for the microscopic part. A new, rapidly converging shape parametrization is used to describe the vast range of nuclear deformations between the ground and the fission isomeric states up to near the scission configuration. Quadrupole moments are evaluated in the local minima and turn out to be in good agreement with the experimental data wherever available.

Spontaneous fission properties of superheavy elements

Spontaneous fission properties of transuranium isotopes are reviewed. Specific emphasis was laid on brief historical overviews of theoretical descriptions and experimental determination of basic properties as spontaneous fission half-lives, fission barriers, or total kinetic energy release in fission. Experimental spontaneous fission half-lives are compared with the results of recent theoretical predictions. Hindrance factors for spontaneous fission of odd-mass nuclei are discussed in context with the configuration (spin, parity) of the fissioning states and the change in energy of single particle levels at deformation. Kinetic energy release and mass distributions are discussed in the context of different fission modes, as symmetric and asymmetric or fission from elongated or compact shapes of the nascent fission fragments. An overview of recent fission barrier calculations of superheavy elements on the basis of macroscopic-microscopic models or self-consistent calculations is given, and the results are compared for selected examples.

Effect of non-axial octupole shapes in heavy and superheavy nuclei

We search for effects of tetrahedral deformation $\beta_{32}$ over a range of $\sim 3000$ heavy and superheavy nuclei, $82\leq Z \leq 126$, using a microscopic-macroscopic model based on the deformed Woods-Saxon potential, well tested in the region. We look for the energy minima with a non-zero tetrahedral distortion, both absolute and conditional - with the quadrupole distortion constrained to zero. In order to assure reliability of our results we include 10 most important deformation parameters in the energy minimization. We could not find any cases of stable tetrahedral shapes. The only sizable - up to 0.7 MeV - lowering of the ground state occurs in superheavy nuclei $Z\geq 120$ for $N=173-188$, as a result of a {\it combined} action of two octupole deformations: $\beta_{32}$ and $\beta_{30}$, in the ratio $\beta_{32}/\beta_{30}\approx \sqrt{3/5}$. The resulting shapes are moderately oblate, with the superimposed distortion $\beta_{33}$ {\it with respect to the oblate axis}, which makes the equator of the oblate spheroid slightly triangular. Almost all found conditional minima are excited and not protected by any barrier, a handful of them are degenerate with the axial minima.

Adiabatic fission barriers in superheavy nuclei

Using the microscopic-macroscopic model based on the deformed Woods-Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy we calculated static fission barriers $B_{f}$ for 1305 heavy and superheavy nuclei $98\leq Z \leq 126$, including even - even, odd - even, even - odd and odd - odd systems. For odd and odd-odd nuclei, adiabatic potential energy surfaces were calculated by a minimization over configurations with one blocked neutron or/and proton on a level from the 10-th below to the 10-th above the Fermi level. The parameters of the model that have been fixed previously by a fit to masses of even-even heavy nuclei were kept unchanged. A search for saddle points has been performed by the "Imaginary Water Flow" method on a basic five-dimensional deformation grid, including triaxiality. Two auxiliary grids were used for checking the effects of the mass asymmetry and hexadecapole non-axiallity. The ground states were found by energy minimization over configurations and deformations. We find that the non-axiallity significantly changes first and second fission barrier in many nuclei. The effect of the mass - asymmetry, known to lower the second, very deformed barriers in actinides, in the heaviest nuclei appears at the less deformed saddles in more than 100 nuclei. It happens for those saddles in which the triaxiallity does not play any role, what suggests a decoupling between effects of the mass-asymmetry and triaxiality. We studied also the influence of the pairing interaction strength on the staggering of $B_f$ for odd- and even-particle numbers. Finally, we provide a comparison of our results with other theoretical fission barrier evaluations and with available experimental estimates.

Macroscopic-microscopic model of nuclear potential energy

To improve the evaluation of nuclear observables, refined models are to be used more and more as underlying analysis tools. Fission is a complex process and is the less accurately described with current models. Standard evaluation models rely on the Hill-Wheeler formalism for the fission transmission coefficient, which in turns is based on phenomenological parameters “reflecting” the fission barrier heights and widths. To reduce the weight of phenomenology in the evaluation process, nuclear structure models are expected to embed more and more microscopic descriptions. As models are rarely exact, evaluators are often compelled to “tune” model parameters so that observables can be properly reproduced. Related computation time can thus be a major hindrance to the use of advanced models in evaluation as final adjustments are expected to remain necessary. For this reason, a macroscopic-microscopic model has been selected to replace the current phenomenological description of fission barriers. The Finite-Range Liquid-Drop Model (FRLDM) has been implemented in the CONRAD evaluation code and its present implementation shows remarkable consistency with experimental and published benchmark data. The CONRAD code can be used to provide expectation values but also related uncertainties and covariance data. Sensitivity of FRLDM parameters and the correlation matrix between these parameters have been obtained so that further uncertainty propagation on barrier heights can be carried out in the near future.

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.