Liquid Drop Model Research Articles

Surface properties for Ne, Na, Mg, Al, and Si isotopes in the coherent density fluctuation model using the relativistic mean-field densities

We have systematically studied the surface properties, such as symmetry energy, neutron pressure, and symmetry energy curvature coefficient for Ne, Na, Mg, Al, and Si nuclei from the proton to neutron drip lines. The coherent density fluctuation model (CDFM) is used to estimate these quantities taking the relativistic mean-field densities as inputs. The Brückner energy density functional is taken for the nuclear matter binding energy and local density approximation is applied for its conversion to coordinate space. The symmetry energy again decomposed to the volume and surface components within the liquid drop model formalism to the volume and surface parts separately. Before calculating the surface properties of finite nuclei, the calculated bulk properties are compared with the experimental data, whenever available. The NL3* parameter set with the Bardeen–Cooper–Schrieffer (BCS) pairing approach in an axially deformed framework is used to take care of the pairing correlation when needed. The deformed density is converted to its spherical equivalent with a two-Gaussian fitting, which is used as an input for the calculation of weight function in the CDFM approximation. With the help of the symmetry energy, the isotopes 29F, 28Ne, 29,30Na, and 31,35,36Mg are considered to be within the island of inversion (Han et al. Phys. Lett. B, 772, 529 (2017). doi:10.1016/j.physletb.2017.07.007). Although we get large symmetry energies corresponding to a few neutron numbers for this isotopic chain as expected, an irregular trend appears for all these considered nuclei. The possible reason behind this abnormal behavior of symmetry energy for these lighter mass nuclei is also included in the discussion, which gives a direction for future analysis.

Generalizing the Liquid Drop Model (LDM) to Assess (Q_β^+-value) for Nuclei in the Range of 10≤Z≤98.

In this research, a new formula of positive beta decay energy (Q_β^+-value) was obtained for light, medium and heavy nuclei (even – even, even – odd, odd – even and odd – odd) for the range of nuclei (10≤Z≤98). Instead of mass differences values, this formula is expressed in the form of nuclear binding energies. After multiple mathematical derivations, the decay energy is defined using the liquid drop model (LDM) to arrive at a final equation for the positive beta decay energy. The (Q_β^+-value) was calculated using four terms: the first term represents the symmetry energy with the opposite sign as in the liquid drop model, the second term represents the Coulomb repulsion energy, and the other two terms represent the pairing and constant (C) terms, where (C) is provided by the mass difference between the neutron and proton. In the investigation of our novel formula of (Q_β^+-value).The findings showed a good match between experimental and theoretical values, especially for heavy and medium nuclei, but a less match for light nuclei due to the presence of a magic numbers. In the investigation of our novel Q_β^+-value formula, the standard deviation (σ) was used to calculate the dependency of (LDM).

Large mass rigidity for a liquid drop model in2D with kernels of finite moments

Motivated by Gamow’s liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter P(E) is replaced by P(E)-γP ε (E), with 0<γ<1 and P ε a nonlocal energy such that P ε (E)→P(E) as ε vanishes. We prove that unit area minimizers are disks for ε>0 small enough. More precisely, we first show that in dimension 2, minimizers are necessarily convex, provided that ε is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension n≥2) the unit ball in ℝ n is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass m 0 such that for any m>m 0 , the disk is the unique minimizer of area m up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass.

Symmetry Energy in the Semi-empirical Mass Formula

Symmetry Energy in the Semi-empirical Mass Formula

Non-spherical nucleon clusters in the mantle of a neutron star: CLDM based on Skyrme-type forces

Neutron stars are superdense compact astrophysical objects. The central region of the neuron star (the core) consists of locally homogeneous nuclear matter, while in the outer region (the crust) nucleons are clustered. In the outer crust these nuclear clusters represent neutron-rich atomic nuclei and all nucleons are bound within them. Whereas in the inner crust some neutrons are unbound, but nuclear clusters still keeps generally spherical shape. Here we consider the region between the crust and the core of the star, so-called mantle, where non-spherical nuclear clusters may exist. We apply compressible liquid drop model to calculate the energy density for several shape types of nuclear clusters. It allows us to identify the most energetically favorable configuration as function of baryon number density. Employing four Skyrme-type forces (SLy4 and BSk24, BSk25, BSk26), which are widely used in the neutron star physics, we faced with strong model dependence of the ground state composition. In particular, in agreement with previous works within liquid drop model, mantle is absent for SLy4 (nuclear spheres directly transit into homogeneous nuclear matter; exotic nuclear shapes do not appear).

Size and charge-state dependence of detachment energies of polyanionic silver clusters.

The electronic properties of silver clusters (N up to 800) charged by attachment of up to z = 7 excess electrons are investigated. As an essential preparation step, the technique of in-trap electron attachment to size-selected monoanions within a linear Paul trap is applied. By taking advantage of tunable laser pulses, the photoelectron spectra allow us to evaluate details of the electronic structure of polyanionic metal clusters, giving a multidimensional dataset. The subsequent analysis based on the liquid drop model provides information about the atomic structure and the bulk work function at a hitherto unknown quality.

Improved effective liquid drop model for α-decay half-lives

The effective liquid drop model (ELDM) is improved by introducing an accurate nuclear charge radius formula and an analytic expression of the effective assaulting frequency. Within the improved effective liquid drop model (IMELDM), the α-decay half-lives of 421 nuclei which have been measured in experiment are calculated. It is shown that the accuracy of the IMELDM is improved evidently compared to its predecessor. Meanwhile, by focusing on the superheavy nuclei (SHN), the α-decay half-lives of SHN are investigated within the IMELDM. It is found that the experimental α-decay half-lives of 80 SHN are reproduced well. Then, within the IMELDM the α-decay half-lives of the Z=118–121 isotopes are predicted by inputting the α-decay energies (Qα) extracted from the Weizsäcker-Skyrme-4 (WS4) mass table. At last, the competition between the α-decay and spontaneous fission (SF) of the SHN of the N=178 α-decay chains is studied in detail, which may be useful for searching for the new SHN in future.

Systematic study of two-proton radioactivity half-lives using the two-potential and Skyrme-Hartree-Fock approaches * *Supported in part by the National Natural Science Foundation of China (11205083, 11505100, 11705055), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (18A237), the Natural Science Foundation of Hunan Province, China (2018JJ2321), the Innovation Group

In this work, we systematically study the two-proton ( ) radioactivity half-lives using the two-potential approach, and the nuclear potential is obtained using the Skyrme-Hartree-Fock approach and the Skyrme effective interaction of SLy8. For true radioactivity ( 0 and 0, where and are the released energies of the one-proton and two-proton radioactivity, respectively), the standard deviation between the experimental half-lives and our theoretical calculations is 0.701. In addition, we extend this model to predict the half-lives of 15 possible radioactivity candidates with 0 obtained from the evaluated atomic mass table AME2016. The calculated results indicate a clear linear relationship between the logarithmic radioactivity half-lives ( ) and coulomb parameters [( + ) ] considering the effect of orbital angular momentum proposed by Liu [Chin. Phys. C 45, 024108 (2021)]. For comparison, the generalized liquid drop model (GLDM), effective liquid drop model (ELDM), and Gamow-like model are also used. Our predicted results are consistent with those obtained using other relevant models.

Cluster pre-formation probabilities and decay half-lives for trans-lead nuclei using modified generalised liquid drop model (MGLDM)

Cluster pre-formation probabilities and decay half-lives for trans-lead nuclei using modified generalised liquid drop model (MGLDM)

Signature of magic numbers in light exotic nuclei

The advent of technology widened the study of shell closure for nuclei lying far away from the stability line. The classic shell closures are observed to be valid for nuclei near the stability line. The light exotic nuclei exhibit unique behavior compared to those of stable ones. The objective of this work is to study the shell closure in light nuclei near the neutron drip-line to the proton drip-line region. The systematics of two nucleon separation energy difference ([Formula: see text]) between calculated and experimental data are taken as a tool to study the shell closure of light nuclei. Nucleon separation energy is determined using the liquid drop model (LDM) expression with modified asymmetry and pairing energy terms. The light mass numbered isotopes and isotones as a function of even neutron and even proton numbers were considered for this work. The shell structure predicted by [Formula: see text] systematics is compared with the first excitation energy [Formula: see text]. The modified LDM is successful in identifying the shell structure properties of light nuclei in exotic region. The study supports the emergence of new shell closure and the disappearance of existing classic shell closure for both neutron and proton numbers. The shell structure predicted by [Formula: see text] systematics for isotopes (isotones) of magic nucleon number [Formula: see text] was in agreement with the results of [Formula: see text] analysis.

Machine learning the nuclear mass

The masses of $$\sim$$ 2500 nuclei have been measured experimentally; however, >7000 isotopes are predicted to exist in the nuclear landscape from H ( $$Z=1$$ ) to Og ( $$Z=118$$ ) based on various theoretical calculations. Exploring the mass of the remaining isotopes is a popular topic in nuclear physics. Machine learning has served as a powerful tool for learning complex representations of big data in many fields. We use Light Gradient Boosting Machine (LightGBM), which is a highly efficient machine learning algorithm, to predict the masses of unknown nuclei and to explore the nuclear landscape on the neutron-rich side from learning the measured nuclear masses. Several characteristic quantities (e.g., mass number and proton number) are fed into the LightGBM algorithm to mimic the patterns of the residual $$\delta (Z,A)$$ between the experimental binding energy and the theoretical one given by the liquid-drop model (LDM), Duflo–Zucker (DZ, also dubbed DZ28) mass model, finite-range droplet model (FRDM, also dubbed FRDM2012), as well as the Weizsäcker–Skyrme (WS4) model to refine these mass models. By using the experimental data of 80 $$\%$$ of known nuclei as the training dataset, the root mean square deviations (RMSDs) between the predicted and the experimental binding energy of the remaining 20% are approximately $$0.234\pm 0.022$$ , $$0.213\pm 0.018$$ , $$0.170\pm 0.011$$ , and $$0.222\pm 0.016$$ MeV for the LightGBM-refined LDM, DZ model, WS4 model, and FRDM, respectively. These values are approximately 90%, 65%, 40%, and 60% smaller than those of the corresponding origin mass models. The RMSD for 66 newly measured nuclei that appeared in AME2020 was also significantly improved. The one-neutron and two-neutron separation energies predicted by these refined models are consistent with several theoretical predictions based on various physical models. In addition, the two-neutron separation energies of several newly measured nuclei (e.g., some isotopes of Ca, Ti, Pm, and Sm) predicted with LightGBM-refined mass models are also in good agreement with the latest experimental data. LightGBM can be used to refine theoretical nuclear mass models and predict the binding energy of unknown nuclei. Moreover, the correlation between the input characteristic quantities and the output can be interpreted by SHapley additive exPlanations (a popular explainable artificial intelligence tool), which may provide new insights for developing theoretical nuclear mass models.

The effect of the energy functional on the pasta-phase properties of catalysed neutron stars

Nuclear pasta, that is an inhomogeneous distribution of nuclear matter characterised by non-spherical clustered structures, is expected to occur in a narrow spatial region at the bottom of the inner crust of neutron stars, but the width of the pasta layer is strongly model dependent. In the framework of a compressible liquid-drop model, we use Bayesian inference to analyse the constraints on the sub-saturation energy functional and surface tension imposed by both ab-initio chiral perturbation theory calculations and experimental measurements of nuclear masses. The posterior models are used to obtain general predictions for the crust-pasta and pasta-core transition with controlled uncertainties. A correlation study allows us to extract the most influential parameters for the calculation of the pasta phases. The important role of high-order empirical parameters and of the surface tension is underlined.

Uncertainties in the pasta-phase properties of catalysed neutron stars

Context. The interior of a neutron star is expected to exhibit different states of matter. In particular, complex non-spherical configurations known as ‘pasta’ phases may exist at the highest densities in the inner crust, potentially having an impact on different neutron-star phenomena. Aims. We study the properties of the pasta phase and the uncertainties in the pasta observables which are due to our incomplete knowledge of the nuclear energy functional. Methods. To this aim, we employed a compressible liquid-drop model approach with surface parameters optimised either on experimental nuclear masses or theoretical calculations. To assess the model uncertainties, we performed a Bayesian analysis by largely varying the model parameters using uniform priors, and generating posterior distributions with filters accounting for both our present low-density nuclear physics knowledge and high-density neutron-star physics constraints. Results. Our results show that the nuclear physics constraints, such as the neutron-matter equation of state at very low density and the experimental mass measurements, are crucial in determining the crustal and pasta observables. Accounting for all constraints, we demonstrate that the presence of pasta phases is robustly predicted in an important fraction of the inner crust. We estimate the relative crustal thickness associated with pasta phases as Rpasta/Rcrust = 0.128 ± 0.047 and the relative moment of inertia as Ipasta/Icrust = 0.480 ± 0.137. Conclusions. Our findings indicate that the surface and curvature parameters are more influential than the bulk parameters for the description of the pasta observables. We also show that using a surface tension that is inconsistent with the bulk functional leads to an underestimation of both the average values and the uncertainties in the pasta properties, thus highlighting the importance of a consistent calculation of the nuclear functional.

Evaluation of pre-neutron-emission mass distributions of neutron-induced typical actinide fission using scission point model * *Supported by the NSFC-Nuclear Technology Innovation Joint Fund (U1867213), the NSAF (U1830102), the National Natural Science Foundations of China (12075105, 11705071, 11875155) and the Fundamental Research Funds for the Central Universities (lzujbky-2021-kb09)

The scission point model is improved by considering the excitation-dependent liquid drop model to calculate mass distributions for neutron-induced actinide nuclei fission. Excitation energy effects influence the deformations of light and heavy fragments. The improved scission point model shows a significant advance with regard to accuracy for calculating pre-neutron-emission mass distributions of neutron-induced typical actinide fission with incident-neutron-energies up to 99.5 MeV. The theoretical frame assures that the improved scission point model is suitable for evaluating the fission fragment mass distributions, which will provide guidance for studying fission physics and designing nuclear fission engineering and nuclear transmutation systems.

Observation of a fission mode with very short elongation for the neutron-rich Md257 nucleus at high excitation energy

Mass and total kinetic-energy (TKE) distributions of fission fragments from $^{257}\mathrm{Md}$ nucleus populated by complete fusion of $^{19}\mathrm{F}$ with $^{238}\mathrm{U}$ have been measured in the range of excitation energy ${E}^{*}\ensuremath{\approx}37\text{\ensuremath{-}}48\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. Due to the skewness of the measured TKE distributions, two Gaussian distributions were required to explain the data. The events with high TKE must originate from the supershort mode of fission, the presence of which has been confirmed by measuring two additional reactions: $^{19}\mathrm{F}+^{232}\mathrm{Th}$ and $^{18}\mathrm{O}+^{238}\mathrm{U}$ involving first a different target and then a different projectile leading to compound nuclei $^{251}\mathrm{Es}$ and $^{256}\mathrm{Fm}$, respectively, at similar excitation energies and comparing their mass-TKE correlation plots. The existence of considerable supershort mode of fission for the $^{257}\mathrm{Md}$ nucleus up to ${E}^{*}\ensuremath{\approx}48$ MeV, observed for the first time, reveals that such rare fission modes for neutron-rich exotic nuclei are far off from the liquid drop model predictions, even at high excitation energies.

Existence and nonexistence in the liquid drop model

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

Systematic study of two-proton radioactivity with a screened electrostatic barrier * *Supported in part by the National Natural Science Foundation of China (11205083, 11505100, 11705055), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (18A237), the Natural Science Foundation of Hunan Province, China (2018JJ3324), the Innovation Group of Nuclear

In this study, a phenomenological model is proposed based on Wentzel-Kramers-Brillouin (WKB) theory and applied to investigate the two-proton ( ) radioactive half-lives of nuclei near or beyond the proton drip line. The total diproton-daughter nucleus interaction potential is composed of the Hulthen-type electrostatic term and the centrifugal term. The calculated radioactive half-lives can accurately reproduce the existing 10 experimental datasets of five true radioactive nuclei with σ = 0.736. In addition, we extend this model to predict the half-lives of possible radioactive nuclei whose radioactivity is energetically allowed or observed but not yet quantified in NUBASE2016. The predicted results are in agreement with those obtained using the Gamow-like model, generalized liquid drop model, Sreeja formula, and Liu formula.

Prior Probability Distributions of Neutron Star Crust Models

Abstract To make best use of multifaceted astronomical and nuclear data sets, probability distributions of neutron star models that can be used to propagate errors consistently from one domain to another are required. We take steps toward a consistent model for this purpose, highlight where model inconsistencies occur, and assess the resulting model uncertainty. Using two distributions of nuclear symmetry energy parameters—one uniform, the other based on pure neutron matter theory—we prepare two ensembles of neutron star inner crust models. We use an extended Skyrme energy density functional within a compressible liquid drop model (CLDM). We fit the surface parameters of the CLDM to quantum 3D Hartree–Fock calculations of crustal nuclei. All models predict that more than 50% of the crust by mass and 15% of the crust by thickness comprises pasta with medians of around 62% and 30%, respectively. We also present 68% and 95% ranges for the crust composition as a function of density. We examine the relationships between crust–core boundary and pasta transition properties, the thickness of the pasta layers, the symmetry energy at saturation and subsaturation densities, and the neutron skins of 208Pb and 48Ca. We quantify the correlations using the maximal information coefficient, which can effectively characterize nonlinear relationships. Future measurements of neutron skins, information from nuclear masses and giant resonances, and theoretical constraints on PNM will be able to place constraints on the location of the pasta and crust–core boundaries and the amount of pasta in the crust.

Heavy particle radioactivity of superheavy element Z = 126

The concept of heavy particle radioactivity is studied using modified generalised liquid drop model (MGLDM) in the superheavy element Z = 126. The eight different proximity functions and different mass excess values were used to evaluate cluster/HPR. The logarithmic half-lives using different proximity functions and mass excess values are compared with that of experiments. The HPR of 60Ni to 102Ru have been studied in the superheavy region 126306 to 126326. The HPR half-lives has been compared with the different decay modes such as α-decay, β-decay and spontaneous fission. 9 HPR emitters, 4 α emitters, 1 β+ emitter and 7 spontaneous fission nuclei were identified in the superheavy nuclei 126306−314, 126315−318, 126319 and 126320−326 respectively.

Erratum: Two-proton radioactivity within a generalized liquid drop model [Phys. Rev. C 101 , 014301 (2020)

Erratum: Two-proton radioactivity within a generalized liquid drop model [Phys. Rev. C <b>101</b> , 014301 (2020)