Calculation of microscopic nuclear level densities based on covariant density functional theory

In this study,a microscopic method for calculating the nuclear level density(NLD)based on the covariant density functional theory(CDFT)is developed.The particle-hole state density is calculated by a combinatorial method using single-particle level schemes obtained from the CDFT,and the level densities are then obtained by considering collective effects such as vibration and rotation.Our results are compared with those of other NLD models,including phenomenological,microstatisti-cal and nonrelativistic Hartree–Fock–Bogoliubov combinatorial models.This comparison suggests that the general trends among these models are essentially the same,except for some deviations among the different NLD models.In addition,the NLDs obtained using the CDFT combinatorial method with normalization are compared with experimental data,including the observed cumulative number of levels at low excitation energies and the measured NLDs.The CDFT combinatorial method yields results that are in reasonable agreement with the existing experimental data.

A Cautionary Note on the Application of GIS in Spatial Optimization Modeling

Spatial optimization as part of spatial modeling has been facilitated significantly by integration with GIS techniques. However, for certain research topics, applying standard GIS techniques may create problems which require attention. This paper serves as a cautionary note to demonstrate two problems associated with applying GIS in spatial optimization, using a capacitated p-median facility location optimization problem as an example. The first problem involves errors in interpolating spatial variations of travel costs from using kriging, a common set of techniques for raster files. The second problem is inaccuracy in routing performed on a graph directly created from polyline shapefiles, a common vector file type. While revealing these problems, the paper also suggests remedies. Specifically, interpolation errors can be eliminated by using agent-based spatial modeling while the inaccuracy in routing can be improved through altering the graph topology by splitting the long edges of the shapefile. These issues suggest the need for caution in applying GIS in spatial optimization study.

Dynamic Spatio-Temporal Modeling in Disease Mapping

Spatio-temporal models are valuable tools for disease mapping and understanding the geographical distribution of diseases and temporal dynamics. Spatio-temporal models have been proven empirically to be very complex and this complexity has led many to oversimply and model the spatial and temporal dependencies independently. Unlike common practice, this study formulated a new spatio-temporal model in a Bayesian hierarchical framework that accounts for spatial and temporal dependencies jointly. The spatial and temporal dependencies were dynamically modelled via the matern exponential covariance function. The temporal aspect was captured by the parameters of the exponential with a first-order autoregressive structure. Inferences about the parameters were obtained via Markov Chain Monte Carlo (MCMC) techniques and the spatio-temporal maps were obtained by mapping stable posterior means from the specific location and time from the best model that includes the significant risk factors. The model formulated was fitted to both simulation data and Kenya meningitis incidence data from 2013 to 2019 along with two covariates;Gross County Product (GCP) and average rainfall. The study found that both average rainfall and GCP had a significant positive association with meningitis occurrence. Also, regarding geographical distribution, the spatio-temporal maps showed that meningitis is not evenly distributed across the country as some counties reported a high number of cases compared with other counties.

Level density of odd-A nuclei at saddle point

Based on the covariant density functional theory,by employing the core–quasiparticle coupling(CQC)model,the nuclear level density of odd-A nuclei at the saddle point is achieved.The total level density is calculated via the convolution of the intrinsic level density and the collective level density.The intrinsic level densities are obtained in the finite-temperature covariant density functional theory,which takes into account the nuclear deformation and pairing self-consistently.For saddle points on the free energy surface in the(β_(2),γ)plane,the entropy and the associated intrinsic level density are compared with those of the global minima.By introducing a quasiparticle to the two neighboring even–even core nuclei,whose properties are determined by the five-dimensional collective Hamiltonian model,the collective levels of the odd-A nuclei are obtained via the CQC model.The total level densities of the^(234-240)U agree well with the available experimental data and Hilaire’s result.Furthermore,the ratio of the total level densities at the saddle points to those at the global minima and the ratio of the total level densities to the intrinsic level densities are discussed separately.

Three-dimensional potential energy surface for fission of^(236)U within covariant density functional theory

We calculate the three-dimensional potential energy surface(PES)for the fission of the compound nucleus^(236)U using covariant density functional theory with constraints on the axial quadrupole and octupole deformations(β_(2),β_(3))coexistence of the elongated and compact fission modes is predicted for comes shallow across a large range of quadrupole and octupole deformations for small scission line in the(β_(2),β_(3))plane extends to a shallow band,leading to fluctuations of several to ten MeV in the estimated total kinetic energies and of several to approximately ten nucleons in the fragment masses.

Evolution of N=20,28,50 shell closures in the 20≤Z≤30 region in deformed relativistic Hartree-Bogoliubov theory in continuum

Magicity,or shell closure,plays an important role in our understanding of complex nuclear phenomena.In this work,we employ one of the state-of-the-art density functional theories,the deformed relativistic Hartree-Bogoliubov theory in continuum(DRHBc)with the density functional PC-PK1,to investigate the evolution of the N=20,28,50 shell closures in the 20≤Z≤30 region.We show how these three conventional shell closures evolve from the proton drip line to the neutron drip line by studying the charge radii,two-neutron separation energies,two-neutron gaps,quadrupole deformations,and single-particle levels.In particular,we find that in the 21≤Z≤27 region,the N=50 shell closure disappears or becomes quenched,mainly due to the deformation effects.Similarly,both experimental data and theoretical predictions indicate that the N=28 shell closure disappears in the Mn isotopic chain,mainly due to the deformation effects.The DRHBc theory predicts the existence of the N=20 shell closure in the Ca,Sc,and Ti isotopic chains,but the existing data for the Ti isotopes suggest the contrary,and therefore further research is needed.

Magnetic moments of odd-A aluminum isotopes in covariant density functional theory

The ground-state properties,especially the magnetic moments,of odd-A aluminum isotopes have been studied and well reproduced in covariant density functional theory after considering the rotational coupling.The present calculations support the rotational structure in the ground state of odd-A aluminum isotopes,i.e.the ground state 5/2^+is built on the intrinsic state 5/2[202].In addition,the contribution from the time-odd fields is also discussed.

Nuclear chart in covariant density functional theory with dynamic correlations: From oxygen to tin

Nuclear masses of even-even nuclei with the proton number 8≤Z≤50(O to Sn isotopes)from the proton drip line to neutron drip line are investigated using the triaxial relativistic Hartree-Bogoliubov theory with the relativistic density functional PC-PK1.Further,the dynamical correlation energies(DCEs)associated with the rotational motion and quadrupole-shaped vibrational motion are taken into account by the five-dimensional collective Hamiltonian(5DCH)method.The root-mean-square deviation with respect to the experimental masses reduces from 2.50 to 1.59 MeV after the consideration of DCEs.The inclusion of DCEs has little influence on the position of drip lines,and the predicted numbers of bound even-even nuclei between proton and neutron drip lines from O to Sn isotopes are 569 and 564 with and without DCEs,respectively.

The interplay of single-particle and collective motions in the low-lying states of ^(21)_(A)Ne with quadrupole-octupole correlations

The beyond mean-field approach for low-lying hypernuclear states is extended by mixing the configurations associated with both single-particle and quadrupole-octupole collective excitations within the generator coordinate method based on a covariant den-sity functional theory.The method is demonstrated in the application to the low-lying states of 21ΛNe,where the configurations with theΛhyperon occupying the first(Λ_(s))and second(Λ_(p))lowest-energy states are considered.The results indicate that the positive-parity states are dominated by theα+^(12)C+α+Λ_(s) structure.In contrast,the low-lying negative-parity states are domi-nated by a strong admixture ofα+16O+Λ_(s) structure andα+12C+α+Λ_(p) structure due to the inclusion of octupole correlations.As a result,the low-lying negative-parity states become much lower than what is expected from the previous studies without the mixing,and the electric multipole transition strengths are significantly quenched.

Low-lying state investigations of odd-A Mn isotopes around N=28

Based on the systematic studies for low-lying states of the odd-A^(49-57)Mn isotopes,the groundstates inversion and the rotational properties of a ground-state-based sequence are revealed and discussed.The energy levels of low-lying states and electromagnetic moments in odd-A^(49-57)Mn isotopes have been well reproduced in shell-model calculations,and the above phenomena could be understood with obviously different occupation numbers in proton orbitals such asπf_(7/2)andπp_(3/2),which changes similarly with the obtained quadrupole deformation in covariant density functional theory(CDFT).After considering the coupling of collective rotation and intrinsic single-particle motion,the available experimental magnetic moments in53Mn and adjacent nuclei can be well explained with CDFT.The present calculations suggest that the 5/2^(-)and 7/2^(-)states in53Mn are formed byπ5/2^(-)[312]andπ7/2^(-)[303]respectively.Together with the behavior of levels,this provides proofs for the level sequences of low-lying states in53Mn distinct from the K^(π)=5/2^(-)rotational band in^(49)Cr and other odd-A Mn isotopes.

Triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis

A triaxially deformed relativistic Hartree–Bogoliubov theory in the Woods–Saxon basis is developed with the aim of treating the triaxial deformation,pairing correlations and continuum in a unified way.In order to consider the triaxial deformation,the deformed potentials are expanded in terms of spherical harmonic functions in the coordinate space.In order to take the pairing correlations into account and treat the continuum properly,by using the Dirac Woods–Saxon basis,which has correct asymptotic behavior,the relativistic Hartree–Bogoliubov equation with triaxial deformation is solved.The formalism of triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis is presented.Taking an axially deformed nucleus24Ne and a triaxially deformed nucleus76Ge as examples,the numerical checks are performed.A weakly bound nucleus112Ge is taken as an example to carry out the necessary converge checks for the numerical parameters.In addition,the ground-state properties of even–even germanium isotopes are investigated.The evolutions of two-neutron separation energy,deformation,root-mean-square radii and density distribution with mass number are analyzed.The comparison between the calculations from the relativistic Hartree–Bogoliubov theory based on harmonic-oscillator basis and the triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis is performed.It is found that the neutron drip line is extended from114Ge to118Ge in the triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis.

Density-dependent relativistic mean field approach and its application to single-Λhypernuclei in oxygen hyperisotopes

The in-medium feature of nuclear force, which includes both nucleon-nucleon( NN) and hyperon-nucleon( ΛN) interactions, impacts the description of single-Λ hypernuclei. With the alternated mass number or isospin of hypernuclei, such effects may be unveiled by analyzing the systematic evolution of the bulk and single-particle properties. From a density-dependent meson-nucleon/hyperon coupling perspective, a new ΛN effective interaction in the covariant density functional(CDF) theory, namely, DD-LZ1-Λ1, is obtained by fitting the experimental data ofΛ separation energies for several single-Λ hypernuclei. It is then used to study the structure and transition properties of single-Λ hypernuclei in oxygen hyperisotopes, in comparison with those determined using several selected CDF Lagrangians. A discrepancy is explicitly observed in the isospin evolution of Λ1p spin-orbit splitting with various effective interactions, which is attributed to the divergence of the meson-hyperon coupling strengths with increasing density. In particular, the density-dependent CDFs introduce an extra contribution to reduce the value but enhance the isospin dependence of the splitting, which originates from the rearrangement terms of Λ self-energies. In addition, the characteristics of hypernuclear radii are studied along the isotopic chain. Owing to the impurity effect of theΛ hyperon, a size shrinkage is observed in the matter radii of hypernuclei compared with the cores of normal nuclei,and its magnitude is further elucidated to correlate with the incompressibility of nuclear matter. Moreover, there is a sizable model-dependent trend in which the Λ hyperon radii evolve with neutron number, which is decided partly by the in-medium NN interactions and core polarization effects.

Local variations of charge radii for nuclei with even Z from 84 to 120

Pronounced changes of nuclear charge radii provide a stringent benchmark on the theoretical models and play a vital role in recognizing various nuclear phenomena.In this work,the systematic evolutions of nuclear charge radii along even Z=84-120 isotopic chains are first investigated by the recently developed new ansatz under the covariant density functional.The calculated results show that the shell closure effects of nuclear charge radii appear remarkably at the neutron numbers N=126 and 184.Interestingly,the arch-like shapes of charge radii between these two strong neutron-closed shells are naturally observed.Across the N=184 shell closure,the abrupt increase in charge radii is still evidently emerged.In addition,the rapid raise of nuclear charge radii from the neutron numbers N=138 to N=144 is disclosed clearly in superheavy regions due to the enhanced shape deformation.

Unified neutron star EOSs and neutron star structures in RMF models

In the framework of the Thomas-Fermi approximation,we systematically study the EOSs and microscopic structures of neutron star matter in a vast density range with n_(b)≈10^(-10)-2 fm^(-3),where various covariant density functionals are adopted,i.e.,those with nonlinear self couplings(NL3,PK1,TM1,GM1,MTVTC)and density-dependent couplings(DD-LZ1,DDME-X,PKDD,DDME2,DD2,TW99).It is found that the EOSs generally coincide with each other at nb■10^(-4)fm^(-3)and 0.1 fm^(-3)■n_(b)■0.3 fm^(-3),while in other density regions they are sensitive to the effective interactions between nucleons.By adopting functionals with a larger slope of symmetry energy L,the curvature parameter K_(sym)and neutron drip density generally increases,while the droplet size,proton number of nucleus,core-crust transition density,and onset density of non-spherical nuclei,decrease.All functionals predict neutron stars with maximum masses exceeding the two-solar-mass limit,while those of DD2,DD-LZ1,DD-ME2,and DDME-X predict optimum neutron star radii according to the observational constraints.Nevertheless,the corresponding skewness coefficients J are much larger than expected,while only the functionals MTVTC and TW99 meet the start-of-art constraints on J.More accurate measurements on the radius of PSR J0740+6620 and the maximum mass of neutron stars are thus essential to identify the functional that satisfies all constraints from nuclear physics and astrophysical observations.Approximate linear correlations between neutron stars’radii at M=1.4M⊙and 2M⊙,the slope L and curvature parameter K_(sym)of symmetry energy are observed as well,which are mainly attributed to the curvature-slope correlations in the functionals adopted here.The results presented here are applicable for investigations of the structures and evolutions of compact stars in a unified manner.

Beyond-mean-field dynamical correlations for nuclear mass table in deformed relativistic Hartree-Bogoliubov theory in continuum

We extend the deformed relativistic Hartree-Bogoliubov theory in continuum(DRHBc)to go beyondmean-field framework by performing a two-dimensional collective Hamiltonian.The influences of dynamical correlations on the ground-state properties are examined in different mass regions,picking Se,Nd,and Th isotopic chains as representatives.It is found that the dynamical correlation energies(DCEs)and the rotational correction energies E_(rot) in the cranking approximation have an almost equivalent effect on the description of binding energies for most deformed nuclei,and the DCEs can provide a significant improvement for the(near)spherical nuclei close to the neutron shells and thus reduce the rms deviations of S_(2n) by≈17%.Furthermore,it is found that the DCEs are quite sensitive to the pairing correlations;taking ^(148)Nd as an example,a 10%enhancement of pairing strength can raise the DCE by≈37%.