Shell Correction Research Articles

Binding energies of proton-rich nuclei determined from their mirror pairs* *Supported by the National Natural Science Foundation of China (12075121 and 11605089), and by the Natural Science Foundation of Jiangsu Province (BK20190067 and BK20150762)

In addition to the Coulomb displacement energy, the residual differences between the binding energies of mirror nuclei (a pair of nuclei with the same mass number plus interchanged proton and neutron numbers) contribute to the shell effect via the valence scheme in this study. To this end, one linear combining type of valence nucleon number, namely, , is chosen to tackle this shell correction, in which and are the valence proton and neutron numbers with respect to the nearest shell closure, respectively. The mass differences of mirror nuclei, as the sum of the empirical Coulomb displacement energy and shell effect correction, are then used to obtain the binding energies of proton-rich nuclei through the available data of their mirror partners to explore the proton dripline of the nuclear chart.

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.

Bethe stopping-power formula and its corrections

The classical and quantum theories leading to the asymptotic Bethe formula of the stopping power of matter for charged particles heavier than the electron are briefly reviewed. Models and approximations for the practical calculation of various corrections that extend the validity range of the formula are described. The asymptotic formula and the associated shell correction were determined previously from an extensive database of atomic generalized oscillator strengths, calculated for an independent-electron model with the Dirac-Hartree-Fock-Slater (DHFS) self-consistent potential, with due account for relativistic departures from the Bethe sum rule. The nonrelativistic Bloch correction is extended to the relativistic domain by means of the Lindhard-S\o{}rensen formulation, and an accurate parametrization for point projectiles with small charges is proposed. The density-effect correction and the Barkas correction are obtained from a semiempirical model of the optical oscillator strength (OOS), built from the calculated DHFS contributions of inner electron subshells plus the OOS of outer-shell electrons represented by an analytical expression, which is determined by the composition, mass density, and empirical mean excitation energy, or $I$ value of the material. Inclusion of the shell, density-effect, Lindhard-S\o{}rensen, and Barkas corrections into the asymptotic formula leads to the corrected Bethe formula. A general strategy is proposed to determine the stopping power in terms of only the $I$ value of the material. It is shown that, with the empirical $I$ values recommended in Report 37 of the International Commission on Radiation Units and Measurements, the stopping powers calculated numerically from the corrected formula are in close agreement with available measurements of the stopping power of elemental materials for protons and alpha particles with energies higher than 0.75 and 5 MeV, respectively.

Investigation of empirical heat capacity in hot-rotating A ∼ 200 nuclei

The empirical heat capacities of some hot-rotating A ∼ 200 nuclei (184Re, 200Tl, 211Po, and 212At) have been investigated by combining the angular-momentum dependent back-shifted Fermi gas (BSFG) model of nuclear level density (NLD) with the experimental NLD data extracted from the neutron-evaporation spectra at the average total angular momentum ⟨J⟩ = 12 ℏ. The parameters of the BSFG are obtained by fitting its NLD to the corresponding measured data using an advanced package of program modeling (CPM) provided by Python feature of IBM decision optimization CPLEX. The results obtained show that the shell correction plays an important role in the formation of empirical S-shaped heat capacity, which serves as a fingerprint for the pairing phase transition in finite nuclear systems. The 184Re nucleus, which is deformed and has small shell correction, exhibits a weaker S-shaped heat capacity than the remaining three spherical 200Tl, 211Po, and 212At nuclei that have large shell effects. This result contrasts with that recently predicted by the microscopic exact pairing plus independent-particle model at finite temperature (EP + IPM), in which the S-shaped heat capacity was predicted in 184Re only. This discrepancy between the heat capacities obtained within the BSFG and EP + IPM models suggests that an NLD model capable of well describing the experimental data while also having intrinsic and as complete as possible physical interpretations is still required in order to provide the exact description of nuclear thermodynamic quantities. In addition, more experimental NLD data in other mass and higher energy regions are also demanded.

A Global Weizsäcker mass model with relativistic mean field shell correction* *Supported by the Open Project of Guangxi Key Laboratory of Nuclear Physics and Nuclear Technology (NLK2021-01), the National Natural Science Foundation of China (U2032141), Natural Science Foundation of Henan Province (202300410480, 202300410479), the Foundation of Fundamental Research for Young Teachers of Zhengzhou University (JC202041041), and the Physics Research and

A relativistic Weizsäcker mass model is proposed based on the single-particle levels and ground state deformations obtained in axial deformed relativistic mean field theory. The density functional of relativistic mean field theory is chosen as DD-LZ1, which can partially remove spurious shell closures. Compared with the fourth Weizsäcker-Skyrme mass model, the proposed model provides shell correction energies that exhibit wide spreading, and the root-mean-square mass deviation is 1.353 MeV. Further improvement is in progress.

Statistical model calculation of the compound nuclear fission timescale in fusion-fission reactions

In the decay of an excited heavy nucleus, both the fission or evaporation residue (ER) cross sections and the fission time are determined by the competition between the various evaporation channels and fission. Statistical model (SM) calculations for several heavy-ion-induced fusion-fission reactions are presented which reproduce the fission or ER excitation functions. The corresponding calculated fission times are found to depend upon the compound nuclei and their excitation energies. The multichance fission probabilities and the corresponding fission times are also obtained, and they are shown to depend sensitively on the shell correction energies and the neutron separation energies of the compound nuclei involved. The experimental neutron multiplicities are reproduced by adjusting the saddle-to-scission transition time interval. An additional delay time for compound nuclear formation is found necessary to fit the experimental neutron multiplicities for a system with higher entrance channel mass symmetry.

Empirical formula for pre-formation probability in actinide region within unified fission model

Abstract An improved semi-empirical formulae for pre-formation probability is proposed within unified fission model. The role of pairing effect and shell correction term were taken in to account. Around 39 even–even (E–E) nuclei, 24 even–odd (E–O), 29 odd–even (O–E) and 14 odd–odd (O–O) nuclei were considered in the atomic and mass number range 89 ≤ Z ≤ 103 and 206 ≤ A ≤ 257 respectively. The standard deviation are evaluated for Zhang et al. [Phys. Rev. C 95, 014311 (2017)], Ismail and Adel [Phys. Rev. C 88.054604 (2013)], Seif et al. [Phys. Rev. C 92, 044302 (2015)] are evaluated. We have also compared the standard deviation of present work with that of available semi-empirical formulae. The standard deviation using Royer [J. Phys. G: Nucl. Part. Phys. 26 1149 (2000)] in case of E–E is smaller when compared to present work. In all other cases the standard deviation obtained from the present work is smaller when compared to other studied semi-empirical relations. Hence, involvement of pairing effect and shell correction term of parent nuclei in the pre-formation probability reproduces experimental α-decay half-lives in the actinide nuclei. As a result, the current work’s detailed examination of the unified fission model and application of pre-formation probability may be utilised to anticipate unexplored isotopes in actinide nuclei.

Semi-bubble structure of superheavy nuclei

The nuclear structure properties of heavy and superheavy nuclei lying at the extremes of charge, especially [Formula: see text], and their density profiles have been explored using the relativistic Hartree–Bogoliubov in axial symmetry and the Skyrme–Hartree–Fock–Bogoliubov framework in transformed harmonic oscillator basis. In this study, we found that the semi-bubble (reduced central density) structure is mainly driven by Coulomb repulsion which stabilizes the nuclei either by pushing the nuclear matter in the surface region or by shape deformation. The shell effect is found maximum in spherical symmetric nuclei as the vanishing fission barrier in Liquid Drop formula indicates that the shell correction energy is responsible for the stabilization of these nuclei.

Comparative Evaluation on Nuclear Shell Correction Calculation using Regulation Bayesian and Moving Average Methods

Since Strutinsky had proposed a shell correction for nuclear mass calculations in 1967, there has been much further research on this topic. In addition to the development carried out on the Strutinsky method, some studies try to apply the shell correction. This paper is included in the development of the shell correction method. There are two techniques used in this work the first is the average method, while the second is the Bayesian regulation method. The two techniques are very different from those Strutinsky originally developed. He used the hermit polynomial as an extension of delta Dirac. This paper will show a comparative study of the two alternative techniques for calculating shell correction.

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.

Inelastic collisions of fast charged particles with atoms: Bethe asymptotic formulas and shell corrections

The relativistic plane-wave Born approximation is applied to the study of inelastic collisions of charged particles with atoms, by considering atomic wave functions calculated from the independent-electron approximation with the self-consistent Dirac-Hartree-Fock-Slater potential. A database of longitudinal and transverse generalized oscillator strengths (GOSs) has been computed by using accurate numerical methods for all the subshells of the ground-state configurations of the elements with atomic numbers from 1 (hydrogen) to 99 (einsteinium). The calculated GOS do not satisfy the Bethe sum rule; departures from the sum rule are in accordance with previous theoretical estimates. Asymptotic high-energy formulas for the total cross section, the stopping cross section, and the energy-straggling cross section are derived with proper account of the relativistic departure from the Bethe sum rule. The shell correction is calculated as the energy-dependent term that, when added to the asymptotic formula, reproduces the value of the atomic cross section calculated by integrating the energy-loss differential cross section. Shell corrections to the stopping cross section obtained from the present approach are presented and compared with previous estimates.

Accurate estimation of the neutron and fission decay widths for hot fusion reactions

Yanez et al. [Phys. Rev. Lett. 112, 152702 (2014)] measured unusually high neutron decay width for the hot fusion reaction $^{25,26}\mathrm{Mg}+^{248}\mathrm{Cm}$ compared to the theoretical estimations available till that date. Various theoretical models have been developed since then to upgrade our understanding of the phenomenon but good agreement is not yet achieved. We have made an attempt for the same, which is based on a modified back-shifted Fermi gas model that also includes the shell and pairing energy correction. Though it works well for the neutron decay, it needs a dissipative effect also to evaluate the fission decay width correctly. It predicts the neutron to total decay width ratio 0.82--0.89 for dissipation coefficients 15--22 and thus closed to the measured value $0.89\ifmmode\pm\else\textpm\fi{}0.13$.

Research on α-decay for the superheavy nuclei with Z= 118–120 * *Partly supported by the National Natural Science Foundation of China (11935001,11575002), the Key Research Foundation of Education Ministry of Anhui Province (KJ2018A0028), and Anhui project (Z010118169)

The generalized liquid-drop model (GLDM) with the microscopic shell correction from relativistic Hartree-Fock (RHF) calculations is used to explore the α-decay of superheavy nuclei. The known nuclei with are chosen as examples for testing. The calculated half-lives of α-decay agree with the experimental data better than those from the GLDM with the shell correction in the Weizs cker-Skyrme model. Moreover, the influence of the decay energy on α-decay is investigated. It is determined that the values obtained from the WS4 model with radial basis function (RBF) correction match the experimental data optimally. Owing to these advantages, the GLDM with the RHF shell correction and WS4+RBF values is adopted to predict the α-decay lifetime for the unknown superheavy nuclei with . The trend of the available α-decay half-lives according to the neutron number is similar to the trends of the values from the GLDM calculation without shell correction as well as the universal decay law (UDL) formula. Comparably, the RHF shell correction depresses (raises) the α-decay lifetime for most nuclei with ( ). In comparison with the half-lives of spontaneous fission, it can be concluded that the α-decay is dominant in the superheavy nuclei , , and . These results are beneficial to the exploration of superheavy nuclei in experiments.

New Coulomb barrier scaling law with reference to the synthesis of superheavy elements

The synthesis of superheavy elements based on fusion reactions and multinucleon transfer reactions is sensitive to the reaction energies, where the Coulomb barrier plays a crucial role as it must be overcome for the projectile and target to contact each other. The Coulomb barrier cannot be measured directly, and the synthesis of superheavy elements is sensitive to it. In this study, we systematically extract the barrier information from the experimental fusion excitation functions by the empirical cross section formula, which is based on a single-Gaussian distribution of fusion barrier heights. A total of 403 sets of experimental data are fitted, among which 243 sets have good results with ${\ensuremath{\chi}}^{2}/pt$ less than one. The extracted fusion barrier is a dynamical barrier, which includes the overall effect of coupled channels. Different from the prediction of the WKJ formula, the new scaling law proposed in this work is almost identical to the CW76 Coulomb barrier at the $z=170--300$ region and reproduces well the centroid barrier extracted from quasielastic scattering. The comparison to the other 14 bare potential models and exploration of the very large $z$ region is also discussed. It is also remarkable to find that the predictions of the DP2015 potential, including of the contributions of the macroscopic and shell correction terms, are highly consistent with the extracted results in this work and the CW76 potential. The numerical results demonstrated that the dynamical fusion barrier and the centroid barrier extracted from the quasielastic reaction (the reaction threshold) could be the same physical quantity for superheavy reactions. This study could provide important references for the synthesis of superheavy nuclei based on fusion reactions and very heavy multinucleon transfer reactions.

Level density within a micro-macroscopic approach

Statistical level density ρ(E,A) is derived for nucleonic system with a given energy E, particle number A and other integrals of motion in the micro-macroscopic approximation beyond the standard saddle-point method of the Fermi gas model. This level density reaches the two limits; the well-known Fermi gas grand-canonical ensemble limit for a large entropy S related to large excitation energies, and the finite micro-canonical limit for a small combinatorical entropy S at low excitation energies. The inverse level density parameter K as function of the particle number A in the semiclassical periodic orbit theory, taking into account the extended Thomas-Fermi and Strutinsky shell corrections, is calculated and compared with experimental data.

Nuclear level densities away from line of β-stability

The variation of total nuclear level densities (NLDs) and level density parameters with proton number (Z) are studied around the β-stable isotope, Z 0, for a given mass number. We perform our analysis for a mass range A = 40 to 180 using the NLDs from popularly used databases obtained with the single-particle energies from two different microscopic mass-models. These NLDs which include microscopic structural effects such as collective enhancement, pairing and shell corrections, do not exhibit inverted parabolic trend with a strong peak at Z 0 as predicted earlier. We also compute the NLDs using the single-particle energies from macroscopic–microscopic mass-model. Once the collective and pairing effects are ignored, the inverted parabolic trends of NLDs and the corresponding level density parameters become somewhat visible. Nevertheless, the factor that governs the (Z − Z 0) dependence of the level density parameter, leading to the inverted parabolic trend, is found to be smaller by an order of magnitude. We further find that the (Z − Z 0) dependence of NLDs is quite sensitive to the shell effects.

Measurements of evaporation residue cross-sections for 48Ti + 140,142Ce reactions

Evaporation residue (ER) cross-sections have been measured for the 48Ti + 140,142Ce reactions to understand the influence of neutron shell closure on compound nucleus (CN) formation. The measured ER cross-sections for 48Ti + 140Ce (NT = 82) system are found to be very close to those of 48Ti + 142Ce (NT = 84) near the Coulomb barrier. The results show that the effect of shell closure in the target nucleus on fusion cross-sections is negligible for the present systems which can be attributed to the small values of the shell correction energies in the entrance channel. Statistical model calculations are performed to interpret the experimental results. Further, comparison of the present results with those from a reaction with higher mass asymmetry in the entrance channel shows no evidence of quasifission in the present systems.

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 $^{295--297,299}120$.