Liquid Drop Model Research Articles

Roles of tensor force and pairing correlation in two-proton radioactivity of halo nuclei* *Supported by the National Natural Science Foundation of China (U1832120, 11675265), the Natural Science Foundation for Outstanding Young Scholars of Hebei Province, China (A2020210012), the Natural Science Foundation of Hebei Province, China (A2021210010), the Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy

The tensor force and pairing correlation effects on the two-proton radioactivity of Mg and Si with a pronounced two-proton halo are explored in the framework of spherical Skyrme-Hartree-Fock-Bogoliubov theory. It is shown that the halo sizes are enhanced with the increase in the strength of the tensor force and pairing correlation. Furthermore, the increasing halo sizes lead to the enhancement of diproton emission. Then, the tensor force is found to have a small influence on the two-proton decay energies, and the two-proton decay energies calculated with strong surface pairing are smaller than those with weak mixed pairing. Because the two-proton decay energies are relatively large, the predicted order of magnitude of half-lives within the effective liquid drop model is not sensitive to the decay energy variation caused by the tensor force and pairing correlation, which has a value of approximately 10 s.

A systematic analysis for one proton radioactivity of ground state nuclei

We systematically studied proton radioactivity half-lives in the atomic number range 53≤Z≤83 using macroscopic models such as Coulomb and Proximity potential model, effective liquid drop model (ELDM), Generalized liquid drop model (GLDM), Universal decay law for proton emission (UDLP), Gammow-like model (GLM) and Unified fission model (UFM). The proton decay half-lives produced by the macroscopic models are compared with those of semi-microscopic approach such as DDM3Y and using effective interaction potentials such as JLM, M3Y+EX, R3Y+EX, SLy4, SRG, SKC and SKD. After detail analysis, it is found that among macroscopic models ELDM and semi-empirical formula UDLP produce proton decay half-lives close to experiments. Furthermore, among semi-microscopic approach, DDM3Y produces proton decay half-lives close to the experiments. Thus, Both ELDM and DDM3Y model can be effectively used in the prediction of half-lives of unexplored proton emitters.

Derivation of a new formula for calculating the two neutrons' separation energy ([formula omitted]) based on the liquid-drop model

In this paper, the separation energy of two neutrons (S2n) for light, medium, and heavy nuclei was determined (even-odd, odd-even), which ranges its atomic numbers between (10≤Z≤97). An equation was derived from calculating the two neutrons' separation energy (S2n) in nuclear binding energies based on the liquid drop model (LDM). Statistical relationships such as the standard deviation (σ) in addition to the root mean square deviation (rmsd) were used to determine the extent to which this model can be relied upon in determining the values of the two neutrons' separation energy (S2n). The results showed a very acceptable agreement between the experimental and theoretical values, and the proposed model was called the modified liquid drop model (MLDM).

Strong magnetic fields and pasta phases reexamined

In this work, we compute the structure and composition of the inner crust of a neutron star in the presence of a strong magnetic field, such as it can be found in magnetars. To determine the geometry and characteristics of the crust inhomogeneities, we consider the compressible liquid drop model, where surface and Coulomb terms are included in the variational equations, and we compare our results with previous calculations based on more approximate treatments. For the equation of state (EoS), we consider two non-linear relativistic mean-field models with different slopes of the symmetry energy, and we show that the extension of the inhomogeneous region inside the star core due to the magnetic field strongly depends on the behavior of the symmetry energy in the crustal EoS. Finally, we argue that the extended spinodal instability observed in previous calculations can be related to the presence of small amplitude density fluctuations in the magnetar outer core, rather than to a thicker solid crust. The compressible liquid drop model formalism, while in overall agreement with the previous calculations, leads to a systematic suppression of the metastable solutions, thus allowing a more precise estimation of the crust-core transition density and pressure, and therefore a better estimation of the crustal radius.

Adding a correction terms for the shell of the Liquid Drop Model and the Quark-Like Model

Adding a correction terms for the shell of the Liquid Drop Model and the Quark-Like Model

New semi-empirical systematic of (p,n) reaction cross section at 7.5 MeV

Abstract It is advised to use a new formula to calculate the (p, n) reaction cross section at 7.5 MeV. We propose six new parameters for the formula proposed by Broeders, C. and Konobeyev, A.Y. (2008. Systematics of (p,n) reaction crosssection. Radiochim. Acta 96: 387–397) to fit experimental data. It should be noted that our systematics is only applicable to isotopes with a ratio of 7.5 MeV to the reaction threshold above 1.3 (7.5/E th > 1.3). This is based on analytical calculations generated from the semi-empirical mass formula, the evaporation model, and the pre-equilibrium exciton model. We were able to find new parameters for the Broders et al. formula through this inquiry that guarantee a good fit with the revised experimental data (EXFOR2022) and provide a minimum value for the statistical parameters ∑ and χ 2.

Clustering effects in Ar36 nuclei produced via the Mg24 + C12 reaction

The nuclear properties of $^{36}\mathrm{Ar}$ composite $\ensuremath{\alpha}$-like nuclei produced at 46.72 MeV of excitation energy via the $^{24}\mathrm{Mg}+^{12}\mathrm{C}$ reaction were investigated. In the past, at this excitation energy, resonant structures with peak-to-valley variation of 50--100 mb have been observed for this system and associated with fusion cross sections. To reveal the nature of this phenomenon the fusion channel observable was investigated. Exclusive measurements of $\ensuremath{\alpha}$ particles and evaporation residues were carried out at Laboratori Nazionali di Legnaro using the $8\ensuremath{\pi}\mathrm{LP}$ apparatus coupled to an evaporation residue detector. Then the experimental data were interpreted through the comparison with statistical model predictions. The energy spectra in coincidence with evaporation residues evidence the limitation of the statistical model assuming nuclear shape of the compound nucleus according to the rotating liquid drop model. To reproduce the experimental data very elongated nuclear shapes and reduced barriers for light particle emission have to be considered, with a major to minor axis ratio up to 3 at higher angular momenta. This large value for the axis ratio is selected in agreement with the predictions of the cranked cluster model and is consistent with previous findings for $\ensuremath{\alpha}$-like nuclei.

α and 2α decay of nuclei in the region 94≤Z≤101 using the modified generalized liquid drop model

The \ensuremath{\alpha} decay and $2\ensuremath{\alpha}$ decay of various isotopes of Pu, Am, Cm, Bk, Cf, Es, Fm, and Md in the mass region $A=218$ to 273 are investigated using the modified generalized liquid drop model (MGLDM) and universal decay law. The comparison of experimental alpha decay half-lives with the predicted half-lives proves the consistency of our calculations. As a result we broadened our investigation to also include a study of double alpha decay of these isotopes. There is a considerable interest in performing research on double alpha decay, as seen by the proposal submitted at CERN to investigate the double alpha radioactivity of $^{224}\mathrm{Ra}$. In view of this, the decay half-lives of most probable $2\ensuremath{\alpha}$ emitters, calculated using MGLDM by employing different preformation factors, are presented which can be beneficial for upcoming experimental investigations on this topic. A peak or dip in \ensuremath{\alpha} and $2\ensuremath{\alpha}$ decay half-life is witnessed which indicates the stability of parent or daughter isotopes correspondingly. Minimum and maximum half-lives for decay are observed when the daughter and parent nuclei, respectively, contain a magic number of neutrons. From our study, 126 and 162 are found as magic/semimagic numbers. A linear plot is obtained while plotting ${\mathrm{log}}_{10}{T}_{1/2}$ of all the isotopes against $Z{Q}^{\ensuremath{-}1/2}$, which emphasizes that our estimations are reliable.

Magnetized neutron star crust within effective relativistic mean-field model

Even though the crystallize nature of the neutron star crust plays a pivotal role in describing various fascinating astrophysical observations, its microscopic structure is not fully understood in the presence of a colossal magnetic field. In the present work, we study the crustal properties of a neutron star within an effective relativistic mean field framework in the presence of magnetic field strength $\sim 10^{17}$G. We calculate the equilibrium composition of the outer crust by minimizing the Gibbs free energy using the most recent atomic mass evaluations. The magnetic field significantly affects the equation of state (EoS) and the properties of the outer crust, such as neutron drip density, pressure, and melting temperature. For the inner crust, we use the compressible liquid drop model for the first time to study the crustal properties in a magnetic environment. The inner crust properties, such as mass and charge number distribution, isospin asymmetry, cluster density, etc., show typical quantum oscillations (De Haas-van Alphen effect) sensitive to the magnetic field's strength. The density-dependent symmetry energy influences the magnetic inner crust like the field-free case. We study the probable modifications in the pasta structures and it is observed that their mass and thickness changes by $\sim 10-15 \%$ depending upon the magnetic field strength. The fundamental torsional oscillation mode frequency is investigated for the magnetized crust in the context of quasiperiodic oscillations (QPO) in soft gamma repeaters. The magnetic field strengths considered in this work influences only the EoS of outer and shallow regions of the inner crust, which results in no significant change in global neutron star properties. However, the outer crust mass and its moment of inertia increase considerably with increase in magnetic field strength.

Decay Times of Compound Nuclei in Heavy Ion Fusion Reactions

The decay time of the compound nuclei has been studied using different statistical models. In the present work, we have improved the back-shifted microscopic level density model by including the pairing effect, an angular momentum-dependent moment of inertia, fission barriers, and a dynamical effect. The decay time of the compound nuclei is evaluated with and without dynamical effect and also compared with various statistical models and available experiments. It is also observed that the decay times of the compound nuclei formed during inverse kinematics is found to be larger when compared to direct kinematics at same excitation energy. Furthermore, the possible fission fragments have been identified using modified generalised liquid drop model. The identified fission fragments are compared with available experiments. The failure of the experiments to synthesize the superheavy element Z = 119 and 120 may be confirmed by the identification of fission fragments in these reactions. The predicted decay times of the compound nuclei and possible fission fragments gives a complete over view of failure of reactions to synthesize the superheavy element Z = 119 and 120.

Elastic properties of nuclear pasta in a fully three-dimensional geometry

Realistic estimations on the elastic properties of neutron star matter are carried out with a large strain (ε≲0.5) in the framework of relativistic-mean-field model with Thomas-Fermi approximation, where various crystalline configurations are considered in a fully three-dimensional geometry with reflection symmetry. Our calculation confirms the validity of assuming Coulomb crystals for the droplet phase above neutron drip density, which nonetheless does not work at large densities since the elastic constants are found to be decreasing after reaching their peaks. Similarly, the analytic formulae derived in the incompressible liquid-drop model give excellent description for the rod phase at small densities, which overestimates the elastic constants at larger densities. For slabs, due to the negligence on the variations of their thicknesses, the analytic formulae from liquid-drop model agree qualitatively but not quantitatively with our numerical estimations. By fitting to the numerical results, these analytic formulae are improved by introducing dampening factors. The impacts of nuclear symmetry energy are examined adopting two parameter sets, corresponding to the slope of symmetry energy L=41.34 and 89.39 MeV. Even with the uncertainties caused by the anisotropy in polycrystallines, the elastic properties of neutron star matter obtained with L=41.34 and 89.39 MeV are distinctively different, results in detectable differences in various neutron star activities.

An improved α-decay energy formula for heavy and superheavy nuclei * *Supported by National Natural Science Foundation of China (Grant No. 12175100), the construct program of the key discipline in Hunan province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 18A237), the Innovation Group of Nuclear and Particle Physics in USC, the Shandong Province Natural Science

Based on the liquid-drop model and using the first derivative of the normalized Gaussian function to consider the shell correction, a simple α-decay energy formula is proposed for heavy and superheavy nuclei. The values of corresponding adjustable parameters are obtained by fitting α-decay energies of 209 nuclei ranging from Z = 90 to Z = 118 with N ≥ 140. The calculated results are in good agreement with the experimental data. The average and standard deviations between the experimental data and theoretical results are 0.141 and 0.190 MeV, respectively. For comparison, the reliable formulae proposed by Dong T K et al (2010, Phys. Rev. C 82, 034 320), Dong J M et al (2010, Phys. Rev. C 81, 064 309) and the WS3+ nuclear mass model proposed by Wang N et al (2011, Phys. Rev. C 84, 051 303) are also used. The results indicate that our improved 7-parameter formula is superior to these empirical formulae and is largely consistent with the WS3+ nuclear mass model. In addition, we extend this formula to predict the α-decay energies for nuclei with Z = 117, 118, 119 and 120. The predicted results of these formulae are basically consistent.

Trends of Neutron Skins and Radii of Mirror Nuclei from First Principles.

The neutron skin of atomic nuclei impacts the structure of neutron-rich nuclei, the equation of state of nucleonic matter, and the size of neutron stars. Here we predict the neutron skin of selected light- and medium-mass nuclei using coupled-cluster theory and the auxiliary field diffusion MonteCarlo method with two- and three-nucleon forces from chiral effective field theory. We find a linear correlation between the neutron skin and the isospin asymmetry in agreement with the liquid-drop model and compare with data. We also extract the linear relationship that describes the difference between neutron and proton radii of mirror nuclei and quantify the effect of charge symmetry breaking terms in the nuclear Hamiltonian. Our results for the mirror-difference charge radii and binding energies per nucleon agree with existing data.

Gravitational Term in Semi Empirical Mass Formula

Gravitational Term in Semi Empirical Mass Formula

Canonical ensemble of many fermions in harmonic oscillator potentials: deformation and the liquid drop model

Abstract The thermodynamic quantities of a many-fermion system at low excitation energy are studied in terms of the temperature-dependent antisymmetrized density matrices in spherical and spheroidally deformed harmonic oscillator potentials at quite low temperature. The internal energies of the canonical ensemble of many fermions coincide numerically with those of microcanonical ensemble in the same spherical HO potentials. The saturation properties of binding energy per fermion are found partly as effects of deformation. The relationship between this treatment and the liquid drop model is studied. The characteristic feature of the liquid drop model is found to be not inconsistent with this treatment.

Variational inference with vine copulas: an efficient approach for Bayesian computer model calibration

With the advancements of computer architectures, the use of computational models proliferates to solve complex problems in many scientific applications such as nuclear physics and climate research. However, the potential of such models is often hindered because they tend to be computationally expensive and consequently ill-fitting for uncertainty quantification. Furthermore, they are usually not calibrated with real-time observations. We develop a computationally efficient algorithm based on variational Bayes inference (VBI) for calibration of computer models with Gaussian processes. Unfortunately, the standard fast-to-compute gradient estimates based on subsampling are biased under the calibration framework due to the conditionally dependent data which diminishes the efficiency of VBI. In this work, we adopt a pairwise decomposition of the data likelihood using vine copulas that separate the information on dependence structure in data from their marginal distributions and leads to computationally efficient gradient estimates that are unbiased and thus scalable calibration. We provide an empirical evidence for the computational scalability of our methodology together with average case analysis and describe all the necessary details for an efficient implementation of the proposed algorithm. We also demonstrate the opportunities given by our method for practitioners on a real data example through calibration of the Liquid Drop Model of nuclear binding energies.

Neutron star inner crust: reduction of shear modulus by nuclei finite size effect

ABSTRACT The elasticity of neutron star crust is important for adequate interpretation of observations. To describe elastic properties one should rely on theoretical models. The most widely used is Coulomb crystal model (system of point-like charges on neutralizing uniform background), in some works it is corrected for electron screening. These models neglect finite size of nuclei. This approximation is well justified except for the innermost crustal layers, where nuclei size becomes comparable with the inter-nuclear spacing. Still, even in those dense layers it seems reasonable to apply the Coulomb crystal result, if one assumes that nuclei are spherically symmetric: Coulomb interaction between them should be the same as interaction between point-like charges. This argument is indeed correct; however, as we point here, shear of crustal lattice generates (microscopic) quadrupole electrostatic potential in a vicinity of lattice cites, which induces deformation on the nuclei. We analyse this problem analytically within compressible liquid drop model. In particular, for ground state crust composition the effective shear modulus is reduced for a factor of $1-u^{5/3}/(2+3\, u-4\, u^{1/3})$, where u is the ratio of the nuclei volume to the volume of the cell. This result is universal, i.e. it does not depend on the applied nucleon interaction model within applied approach. For the innermost layers of inner crust u ∼ 0.2 leading to reduction of the shear modulus by $\sim 25{{\ \rm per\ cent}}$, which can be important for correct interpretation of quasi-periodic oscillations in the tails of magnetar flares.

Pasta Phases in Neutron Star Mantle: Extended Thomas–Fermi vs. Compressible Liquid Drop Approaches

Nuclear pasta phases in the neutron stars mantle can affect the mechanical and transport properties of superdense matter, thus playing an important role in the dynamics and evolution of neutron stars. In this paper, we compare results obtained by the Extended Thomas–Fermi (ETF) method with the compressible liquid drop model (CLDM), based on the thermodynamically consistent description of the surface properties calculated for the two-phase plane interface and the same energy-density functional (for numerical illustration, we applied the Skyrme-type functional SLy4). Our ETF calculations found that pasta phases in cylindrical form cover a significant crustal region (both normal and inverse phases, aka spaghetti and bucatini are presented). Meanwhile, within the applied CLDM framework, which includes the thermodynamically required effect of neutron adsorption on the cluster’s surface but neglects curvature corrections, only the spaghetti phase was found to be energetically favorable in the small density range prior to crust–core transition. On the other hand, the recent CLDM of Dinh Thi et al., 2021, which, on the contrary, accounts for curvature term but neglects neutron adsorption, predicts pasta phase onset in better agreement with the ETF. This fact highlights the importance of the curvature effects and allows counting on the potential validity of the CLDMs as a convenient, transparent and accurate tool for investigation of the pasta-phase properties.

The Electromagnetic Considerations of the Nuclear Force—PART II: The Determination of the Lowest Energy Configurations for Nuclei

This paper explores how the electromagnetic energies of the quarks within the atomic nucleus affect the behavior of the Nuclear Force. By examining the electromagnetic energies and forces, many questions about nuclear behavior can be answered, and many insights into the atomic nucleus can be gained. Previous theoretical models for the Nuclear Force include only the Coulomb electric force of the protons, but with little or no consideration of the electromagnetic characteristics of the quarks. By incorporating the electromagnetic energies and forces into nuclear theory, this model has been able to achieve predictions of binding energy better than any previous model, doing so by using only one parameter instead of the five parameters used in the semi-empirical Weizsäcker formula of the Liquid Drop Model. The Electromagnetic Model unifies the Nuclear Force to the Electromagnetic Force. The Electromagnetic Model of the Nuclear Force includes the calculation of electromagnetics of the quarks, and by doing such, it is shown that the Nuclear Force is significantly influenced by the Electromagnetic Forces of the quarks. This paper, Part II of this series, illustrates the ground state configurations of the atomic nuclei, showing the basic segments of how the protons and neutrons cluster together, and how these segments bond to form larger atomic nuclei. A pattern emerges for the ground state configurations due to the uniformity of the electromagnetic laws. Diagrams are shown for this basic pattern, for both stable and radioactive atomic nuclei. By incorporating the electromagnetic energies and forces into nuclear theory, this model has been able to achieve not only excellent predictions of binding energy, but the ability to answer many other questions regarding the various behaviors of atomic nuclei.

Theoretical calculation of alpha decay half-lives of Neptunium nuclei using modified generalized liquid drop model

We considered the α-decay half-lives of neptunium (ZNp = 93) nuclei in the case of deformed nuclei in the range 219 ≤ A ≤ 239 using the generalized liquid-drop model (GLDM) with 1977 nuclear proximity potential suggested by Blocki et al. (1977) [29]. Also, we analyzed the α-decay half-lives of Np isotopes within the WKB method and the Deformed Proximity Potential Model (DPPM). We consider the quadrupole and hexadecapole deformations at θ=0, θ=30, θ=45, and θ=90 degrees. The calculated half-lives and the alpha decay branching ratios are compared with experimental data and are in good agreement. We compared the results obtained with the corresponding experimental data and with the models UNIV, UDL, SemFIS, the Coulomb and Proximity Potential Model (CPPM), SLB and Royer and it is found that they match well over a wide range. The calculated results are in reasonable agreement with the experimental data. Therefore, DPPM gives reliable and accurate results for favored α-decays in the neptunium mass region.