Liquid Drop Model Research Articles

Novel soliton solutions of liquid drop model appear in fluid dynamics and modulation instability of dynamical system

Novel soliton solutions of liquid drop model appear in fluid dynamics and modulation instability of dynamical system

On minimizers in the liquid drop model

AbstractWe prove that round balls of volume uniquely minimize in Gamow's liquid drop model.

Prediction of binding energy using machine learning approach

The liquid drop model is an empirical hypothesis established on the idea that nuclei can be thought of as incompressible liquid droplets. The AME2020 dataset was used in this work to determine binding energy using a semi-empirical mass formula and compare it with binding energies predicted by a machine learning algorithm. Random forest regressor, MLPRegressor, and XGBoost models were employed. In terms of accuracy, root mean square error, and mean absolute error, machine learning models performed better than the semi-empirical mass formula. Compared to RFR, XGBoost, and SEMF, MLPRegressor performed better in predicting binding energies for lighter nuclei. Using estimated binding energies, nuclear masses were computed, and it was shown that all three models adequately predicted nuclear masses with minimal error. This finding highlights how machine learning can be applied to nuclear physics to predict various nuclei’s properties.

Theoretical predictions of nuclear binding energy for the observed nuclei: the influence of coefficients and terms in a semi-empirical mass formula

The aim of this study is to calculate the nuclear binding energy for the 2457 nuclei which experimentally observed, as tabulated in the atomic mass evaluation AME2020, by using the semi-empirical mass formula based on the renowned liquid drop model (LDM). The Bethe-Weizsäcker (BW) mass formula-four terms (BWMF-4T) is extended to incorporate an additional six terms, resulting in a formula of ten terms. Our findings demonstrate commendable accuracy and reliability, as substantiated by the acceptable level of uncertainty observed when compared them to experimental data and existing models. The influence of coefficients and terms in the semi-empirical mass formula has been determined. All possible combinations of the terms are fitted in turn to the measured nuclear masses, and the outcomes are analyzed in order to reveal correlations and mutual influences between the various terms in all observed mass regions of the periodic table. The fitted surface energy and symmetry coefficients in the semi-empirical mass formula-ten terms (SEMF-10T) are remaining 26.4 and 33.3 MeV respectively. In contrast, coefficients fitted in the classic BW formula with and without one or two additional terms are around 17 and 23 MeV, respectively. The interplay between different terms in this mass formula is found to be significant and interesting.

A simple model for two-proton radioactivity* *Supported in part by the National Natural Science Foundation of China (12175100, 11975132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (21B0402, 18A237), the Natural Science Foundation of Hunan Province, China (2018JJ2321), the Innovation Group of Nuclear and Particle Physics in USC,

In this work, considering the preformation factor of the emitted two protons in parent nucleus and the effect of the parent nucleus deformation, based on the Wentzel-Kramers-Brillouin approximation and Bohr-Sommerfeld quantization condition, we improve a simple phenomenological model proposed by Bayrak [J. Phys. G: 47, 025102 (2020)] to systematically study radioactivity half-lives. This model contains two adjustable parameters and , which are related to the depth of nuclear potential and effect of deformation. The calculated results show that this model can effectively reproduce the experimental data with a corresponding root-mean-square (RMS) standard deviation of σ = 0.683. For comparison, we include the Gamow-like model (GLM) proposed by Liu et al. [Chin. Phys. C 45, 044110 (2021)], generalized liquid drop model (GLDM) proposed by Cui et al. [Phys. Rev. C 101, 014301 (2020)], effective liquid drop model (ELDM) proposed by M. Gonalves et al. [Phys. Lett. B 774, 14 (2017)], two-potential approach with Skyrme-Hartree-Fock (TPASHF) proposed by Pan et al. [Chin. Phys. C 45, 124104 (2021)], phenomenological model with a screened electrostatic barrier (SEB) propoesed by Zou et al. [Chin. Phys. C 45, 104101 (2021)], unified fission model (UFM) proposed by Xing et al. [Chin. Phys. C 45, 124105 (2021)], Coulomb and proximity potential model for deformed nuclei (CPPMDN) proposed by Santhosh [Phys. Rev. C 104, 064613 (2021)], two-parameter empirical formula proposed by Liu et al. [Chin. Phys. C 45, 024108 (2021)], and four-parameter empirical formula proposed by Sreeja et al. [Eur. Phys. J. A 55, 33 (2019)]. In addition, we use this model to predict the radioactive half-lives of some possible potential nuclei whose radioactivity are energetically allowed or observed but not yet quantified in NUBASE2020.

New Approach to Calculate Nuclear Binding Energies of Nickel Isotopes 57-78Ni28 by Using Shell Model

Depending on the basic assumptions of the Shell Model and by using suitable potentials, we found a new formula to calculate nuclear binding energies of nickel isotopes 57-78Ni28. This formula is related only to the mass number of these isotopes and the number of valence nucleons outside the closed core 56Ni28. When calculating the standard deviation of the experimental data from the theory, we found that our value is better than the value calculated by the Semi-Empirical Mass Formula. It is also better than those calculated by the Integrated Model and calculated by the Modified Integrated Model. This indicates that our inferred formula is better than the most important formulas, which were previously used to calculate the nuclear binding energies for our studied nuclei.

Inference of neutron-star properties with unified crust-core equations of state for parameter estimation

Context. Relating different global neutron-star (NS) properties, such as tidal deformability and radius, or mass and radius, requires an equation of state (EoS). Determining the NS EoS is therefore not only the science goal of a variety of observational projects, but it also enters in the analysis process; for example, to predict a NS radius from a measured tidal deformability via gravitational waves (GW) during the inspiral of a binary NS merger. To this aim, it is important to estimate the theoretical uncertainties on the EoS, one of which is the possible bias coming from an inconsistent treatment of the low-density region; that is, the use of a so called non-unified NS crust. Aims. We propose a numerical tool allowing the user to consistently match a nuclear-physics informed crust to an arbitrary high-density EoS describing the core of the star. Methods. We introduce an inversion procedure of the EoS close to saturation density that allows users to extract nuclear-matter parameters and extend the EoS to lower densities in a consistent way. For the treatment of inhomogeneous matter in the crust, a standard approach based on the compressible liquid-drop (CLD) model approach was used in our work. A Bayesian analysis using a parametric agnostic EoS representation in the high-density region is also presented in order to quantify the uncertainties induced by an inconsistent treatment of the crust. Results. We show that the use of a fixed, realistic-but-inconsistent model for the crust causes small but avoidable errors in the estimation of global NS properties and leads to an underestimation of the uncertainties in the inference of NS properties. Conclusions. Our results highlight the importance of employing a consistent EoS in inference schemes. The numerical tool that we developed to reconstruct such a thermodynamically consistent EoS, CUTER, has been tested and validated for use by the astrophysical community.

Revision of the semi-empirical mass formula coefficients by using the ame2020 database

This work aims to revise the coefficients of the Bethe-Weizsäcker’s Semi-empirical mass formula by using the latest version release of the Atomic Mass Evaluation Database (AME2020). The updated values of five energy term coefficients are determined based on the binding energy of a total of 2548 nuclei from AME2020 database. Least-squares fitting is used to obtain a new set of mass formula coefficients. The results obtained are in good agreement with the experimental AME2020 data in a wide range of the nuclei binding energy. The relative error, with respect to the experimental data does not exceed 1%, which is less than the oscillations obtained in the other works using earlier database version AME2016 due to the more accurate measurements and the larger data sample. The semi-empirical mass formula provides a good approximation of the atomic masses and thereby, with the refined coefficient values given here, other effects in the nuclear fission engineering can be calculated with higher accuracy.

Skyrmion crystals stabilized by ω-mesons

We investigate the ground state crystalline structure of nuclear matter in the ω-meson variant of the Skyrme model. After minimizing energy with respect to variations of both the Skyrme field and the period lattice, we find four distinct periodic solutions which are similar to those found in the standard Skyrme model. We use these crystals to calculate coefficients in the Bethe-Weizsäcker semi-empirical mass formula and the compression modulus of infinite nuclear matter, and find a significant improvement as compared with other variants of the Skyrme model.

Modeling in Nuclear Physics of Visual Approach to the Semi-empirical Mass Formula SEMF

An empirical framework that accurately describes radioactive binding energies is the Somewhat Empirical Mass Equation (SEMF). They showcase many implementations and uses of the idea that rely on graphics and printed objects. A key new addition is a contrast with real experiments, as well as a visualization of the energy environment as supplied by the SEMF. The shortcomings of the empirical theory are shown by our visualization of this differential energy scenery, which also highlights the significance of what is known as magic numbers—an explanation provided by the outermost approach, which was developed much more recently than the water drop theory. This provides a great chance to talk about the advantages and limitations of simulations everywhere within the framework of science teaching.

Systematic study of α-decays from the ground and isomeric states of the nuclei with 67 ≤ Z ≤ 106

The Modified Generalised Liquid Drop Model (MGLDM) is employed to analyze both the favored and unfavored α-decays in the ground states (g.s.) and isomeric states (i.s.) of some nuclei having proton numbers, 67 ≤ Z ≤ 106. The calculated half-lives closely match the experimental results, with a least standard deviation, 0.3598, which shows that the g.s. model ‘MGLDM’ with some improvements can effectively be applied to the isomeric state decay. Royer’s formula and Qian’s formula are employed in our calculations and the reported results of the Two Potential Approach (TPA) and the GLDM are included, to compare the half-lives reproduced by MGLDM. The straight-line behaviour of New Geiger-Nuttall (New G-N) plots for various isotopes for different kinds of transitions shows the New G-N law can be applied successfully in the isomeric state α-decays also.

Abstract 6224: Drop-like deformation of nuclei in cancer patient tissue

Abstract In this study, we investigated nuclear deformations in cancer tissues and their clinical relevance to diagnostics. Introduction: Cancers are graded by pathologists examining biopsy slides stained with hematoxylin and eosin for factors including nuclear atypia, which can include features like lobules, invaginations, large nuclear size, and prominent nucleoli. While the importance of nuclear atypia has been known for centuries, its cause remains unclear. Nuclear shaping has been studied extensively in vitro, leading to the liquid drop model of nuclear shaping, which states that nuclear shape changes can occur due to excess area in the form of wrinkles or folds in the lamina. When the lamina becomes fully taut, the nucleus reaches a limiting steady shape. Wrinkles allow the nuclear shape to mirror the cell shape such as in invading cancer cells. The applicability of this model to nuclei in cancer tissues from human patients is untested. Here, tissues were stained for nuclear lamina proteins to test for the presence of excess area in the lamina. Diagnostic relevance of the drop model was assessed by quantitative Fourier analysis of high-resolution images of lamin-stained nuclei. Methods: Deidentified patient tissues from breast, colon, head and neck, and skin were immunostained for lamin B1 and imaged at high resolution (60X, 1.5 NA). Nuclei were then roughly segmented by the deep learning program Cellpose, followed by segmentation with a custom approach that sensitively detects micron-scale variations in the laminar contour. An Elliptical Fourier Analysis was performed to quantify nuclear irregularity by describing nuclear shape with a series of elliptical harmonics. More irregular nuclei are described by larger Fourier coefficients at higher frequencies. Results: Folds and wrinkles in the nuclear lamina were observed in healthy tissue from all four tissue types studied, indicating the presence of excess area in the lamina. The presence of irregular nuclear shape including wrinkling/folds in the lamina was quantified with the Elliptical Fourier Coefficient (EFC) ratio. Posterior probability analysis related EFC ratio to tumor grade to assess the clinical value of the excess area found in the nuclear lamina. This revealed cancer-type dependent trends between EFC ratio and tumor grade. Conclusions: Observations of wrinkling/folds in the nuclear lamina in healthy and cancerous tissue are consistent with the liquid drop model of nuclear deformation. Quantification of nuclear lamina wrinkling with the EFC ratio, along with other nuclear shape parameters, can be a powerful approach for computer-based, explainable, clinical grading. Our findings reveal a fundamental principle of nuclear deformation in cancer tissues and suggest a new approach for improving the accuracy in cancer grading. Citation Format: Christina R. Dubell, Ting-Ching Wang, Ishita Singh, Daniel G. Rosen, Saptarshi Chakraborty, Tanmay P. Lele. Drop-like deformation of nuclei in cancer patient tissue [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 6224.

Systematic study of cluster radioactivity within the generalized liquid drop model* *Supported by the National Natural Science Foundation of China (12175170), Hubei Provincial Natural Science Foundation of China (2023AFB035), Hunan Outstanding Youth Science Foundation (2022JJ10031), and Natural Science Research Project of Yichang City (A23-2-028)

Cluster radioactivity is studied within the generalized liquid drop model (GLDM), in which the shell correction energy, pairing energy, and cluster preformation factor are considered. The calculations show significant improvements and can reproduce the experimental data within a factor of 8.04 after considering these physical effects. In addition, the systematic trend of the cluster preformation factors is discussed in terms of the scheme to study the influence of the valence proton-neutron interaction and shell effect on cluster radioactivity. It is found that is linearly related to . This is in agreement with a recent study [L. Qi et al., Phys. Rev. C 108, 014325 (2023)], in which , obtained using different theoretical models and treatment methods than those used in this study, also had a linear relationship with . Combined with the work by Qi et al., this study suggests that the linear relationship between and is model-independent and both the shell effect and valence proton-neutron interaction play essential roles in cluster radioactivity. An analytical formula is proposed to calculate the cluster preformation factor based on the scheme. In addition, the cluster preformation factors and the cluster radioactivity half-lives of some heavy nuclei are predicted, which can provide a reference for future experiments.

Surface Vibrations of Bubble-like Superheavy Nuclei

The shape vibrations of a superheavy nucleus with a complete (bubble) or a partially (semi-bubble) depleted density in its central region and sharp-edge inner and outer surfaces are investigated in the frame of the Liquid-Drop Model. The quadrupole oscillations of the two existing surfaces are coupled in both velocity and coordinate and, upon decoupling, a low-energy and a high-energy component are predicted. The electric transition probabilities are estimated for the decay of the low-lying mode first 2+ state to the ground state for the entire range of the radius and density of the depleted core.

An improved effective liquid drop model for cluster radioactivity

The effective liquid drop model (ELDM) is improved by introducing an accurate nuclear charge radius formula and an analytic expression for assaulting frequency. Within the improved effective liquid drop model (IMELDM), the experimental cluster radioactivity half-lives of the trans-lead region are calculated. It is shown that the accuracy of the IMELDM is improved compared with that of the ELDM. At last, the cluster radioactivity half-lives that are experimentally unavailable for the trans-lead nuclei are predicted by the IMELDM. These predictions may be useful for searching for new candidates for cluster radioactivity in future experiments.

Systematic study of two proton radioactivity within the effective liquid drop model

In this study, we investigated two-proton radioactivity using the effective liquid drop model with varying mass asymmetry shapes and effective inertia coefficients (VMAS-EFF). Gaudin’s expression term for Coulomb potential energy is replaced in the current study’s calculation of potential energy. The VMAS-EFF combination predicts the half-life with the lowest standard deviation ( σ =1.03) compared to other models and previous constant mass asymmetry with Werner-inertial Wheeler’s coefficient combinations. The lowest standard deviation value indicates that the VMAS-EFF combination is suitable for reproducing the experimental data. We extended the half-life predictions to another 54 nuclei and compared them with six theoretical model predictions and two empirical formula predictions. The present work also investigated the decay possibilities using quantum tunnelling by computing the driving potential (V-Q) for various states of angular momentum l = 0, 2, 3, and 4. Found that 6Be, 6B, 8C, 12O,15,16Ne, 19Mg, 22Si, 26S, 30Ar 38,39Ti, 40V, 42Cr, 45Fe, 48,49Ni, 54Zn, 58,59,60Ge and 64Se nuclei have a high chance of emission for two protons, while 7B, 10N, 14F, 17Na, 24P, 28Cl,32K, 34Ca, 36,37Sc, 44Mn, 47Co, 52Cu, 56,57,58Ga, 61,62As nuclei have a lower chance for two proton than one proton decay. The computation of separation energy findings agree well with the driving potential findings.

Macroscopic-Microscopic Fission Yields

We utilize the macroscopic-microscopic approach to fission to calculate nascent fragment distributions. Assuming strongly damped shape motion, we run many iterations of a Metropolis random walk across nuclear potential-energy surfaces to obtain sufficient scission statistics. Our nuclear potential surfaces consist of a macroscopic energy from the Finite-Range Liquid-Drop Model (FRLDM) and microscopic terms that arise from the single-particle spectra. We compare our predictions for two major actinides to experimental data. We present global trends from this modeling that manifest as a function of mass number of the fissioning species. We discuss the impact of fission yield modeling on nucleosynthetic outcomes.

Heavy particle radioactivity of superheavy nuclei.

We investigated different decay modes such as heavy particle radioactivity (HPR), spontaneous fission (SF), alpha decay and beta-decay in superheavy (SH) region 104≤Z≤126. In HPR, different emissions from Zmin=28 to maximum heavy particle (HP) emission up to Zmax=Z-82 were considered. In the evaluation of Q-values, mass excess Weizsäcker-Skyrme 4+radial basis function (WS4+RBF) values were taken into account. The half-lives of cluster decay determined using modified generalised liquid drop model (MGLDM) and Coulomb and proximity potential model (CPPM) were in close agreement with the experimental results. The MGLDM produce less deviation compared with CPPM. The role of asymmetry effect, relative neutron excess, pairing effect and Coulomb effect on half-lives were studied. A band of neutron numbers from 193 to 200 shows extra stability against HPR. The HPR of 86Kr, 94Zr, 91Y and 96Mo is observed in the superheavy elements Z=118, 122-124 and 126. The dominant decay mode identified is compared with SF, alpha and beta± decay, the majority of decay chains end with the lead nuclei, which is also evident in supernova and galaxy spectrum. This study could be useful in nuclear astrophysics.

Energy transfer and interaction between liquid metal with water

The interactions and energy transfer between liquid metal and water are of great research value in developing and utilizing clean energy (solar and nuclear). Understanding how much energy transfer exists in interactions and the consequences of these interactions is crucial for ensuring the stability of energy development and response measures in case of accidents, especially in third-generation ultra-high temperature concentrated solar power and fourth-generation nuclear energy systems. In this paper, a jet visualization test was performed to investigate the interaction characteristics of liquid metal with water. We found that the energy transfer between liquid metals and water is constrained, distinguishing three distinct pressurization modes due to mass transfer, while also developing an energy transfer model for liquid metals. First, the visualization data show that liquid metal-water interaction consists of four stages: coarse mixing, large steam bubble expansion and fragmentation, direct contact and bubble migration. This process is accompanied by the splashing of metal droplets and shaking of the free liquid surface. Furthermore, the water escapes from the melt pool under some conditions, and modal boiling occurs at the surface of the melt pool, hindering heat exchange. Subsequently, three cover gas pressurization modes were distinguished according to the different interactions of liquid metal with water. Finally, based on the perspective of restricted energy transfer, the liquid metal temperature drop model was established, and there is an appropriate matching between the model and test data with ±20 % uncertainty.

Estimation of Nuclear Mass Formulas Coefficients Using Least-Squares Method Based on Gauss-Seidel Scheme: A Comparative Study Between Three Models

This paper presents the analysis and implementation of the least-squares method based on the Gauss-Seidel scheme for solving nuclear mass formulas. The least-squares method leads to the solution of the system by iterations. The main advantages of the discussed method are simplicity and high accuracy. Moreover, the method enables us to process large data quickly in practice. To demonstrate the effectiveness of the method, implementation using the FORTRAN language is carried out. The steps of the algorithm are detailed. Using 2331 nuclear masses with Z ≥ 8 and N ≥ 8, it was shown that the performance of the liquid drop mass formula with six parameters improved in terms of root mean square (r.m.s. deviation equals 1.28 MeV), compared to the formula of liquid drop mass with six parameters without microscopic energy, deformation energy and congruence energy (r.m.s. deviation equals 2.65 MeV). The nuclear liquid drop model is revisited to make explicit the role of the microscopic corrections (shell and pairing). Deformation energy and the congruence energy estimate have been used to obtain the best fit. It is shown that the performance of the new approach is improved by a model of eight parameters, compared to the previous model of six parameters. The obtained r.m.s. result for the new liquid drop model in terms of masses is equal to 1.05 MeV.