Liquid Drop Mass Formula Research Articles

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.

Nuclear Fission Within the Lublin–Strasbourg Drop Model

The nuclear liquid drop (LD) model proposed first in 1935 by Weizsacker has explained quantitatively the systematics of nuclear binding energies known at that time [1]. This model was extended in 1939 by Meitner and Frisch by adding the deformation dependence to the surface and Coulomb terms in the LD mass formula [2]. This extension was needed to explain new phenomenon, nuclear fission, discovered in fall 1938 by Hahn and Strassman who have discovered barium in the natural uranium radiated with neutrons [3]. First quantitative calculation related to the fission process with the use of deformed liquid drop was performed by Bohr and Wheeler who have expanded the surface of a fissioning nucleus in the spherical harmonics series [4]. They have got a reasonable estimate of the fission barrier height and kinetic energies of the fission fragments. Next important step, which we would like to remind, was done by Myers and Świątecki who have added the shell and pairing corrections to the liquid drop binding energy [5]. This new macroscopic–microscopic model became

Mass and isospin dependence of symmetry energy coefficients of finite nuclei

The mass and isospin dependence of symmetry energy coefficients ${a}_{\mathrm{sym}}$ of finite nuclei are investigated with the measured nuclear masses incorporating the liquid drop mass formula. The enhanced ${a}_{\mathrm{sym}}$ for nearly symmetric nuclei are observed. To describe the mass and isospin dependence of ${a}_{\mathrm{sym}}$, a modified formula based on the conventional surface-symmetry term is proposed and the corresponding rms deviation of nuclear masses is checked.

Shell corrections to a liquid-drop description of nuclear masses and radii

It is shown that a consistent treatment of nuclear bulk and surface effects leads to an improved version of the liquid-drop mass formula with modified symmetry and Coulomb terms. If in addition shell effects are modelled through the counting of the number of valence nucleons, a very simple mass formula is obtained with an rms deviation from the 2003 database of atomic masses of about 800keV. As an application, the volume and surface symmetry energy for A $ \rightarrow$ ∞ are determined. A similar description of nuclear radii is suggested with specific reference to the neutron skin.

A Criterion to Obtain the Equation of State of Asymmetric Nuclear Matter

A criterion to obtain the equation of state of asymmetric nuclear matter experimentally is discussed from the viewpoint of surface and asymmetry energies in neutron-rich nuclei. A simple analysis with a liquid-drop mass formula suggests that the proton fraction of a nucleus, Z/A, should be as small as 0.3. In the case of Ni isotopes, this corresponds to the neutron number N = 52 and the asymmetry energy per nucleon is 2 MeV. A preliminary result of a systematic study using a macroscopic nuclear model concerning the asymmetry energy is also briefly discussed.

Ionic vibrational breathing mode of metallic clusters

The energy of the vibrational mode with spherical symmetry, in which the ionic cores oscillate in the radial direction around the equilibrium geometry (ionic breathing mode) is calculated for trivalent (AlN, 2≤N≤50) and monovalent (NaN, 2≤N≤73; CsN, 2≤N≤74) metallic clusters. The ground-state total energy is calculated using density functional theory, with a spherically averaged pseudopotential to describe the ion–electron interaction and optimizing the geometry by the simulated annealing technique. The energy of the ionic mode is calculated by diagonalization of the dynamical matrix including the electronic relaxation in the linear response approximation. The compressibility and bulk modulus of the metallic cluster are obtained from the energies of the monopole oscillations. These energies present a linear behavior on the inverse of the cluster radius, which is analyzed using a semiclassical liquid drop mass formula for the total energy of the clusters and a scaling model. The values of the vibrational frequencies present electronic shell closing effects for the three metals.©1997 John Wiley & Sons, Inc.

A modified backpropagation algorithm for training neural networks on data with error bars

A method is proposed for training multilayer feedforward neural networks on data contaminated with noise. Specifically, we consider the case that the artificial neural system is required to learn a physical mapping when the available target values for the output variable are subject to experimental uncertainties, but are characterized by error bars. The proposed method, based on a maximum-likelihood criterion for parameter estimation that allows for nonzero model error, introduces two simple modifications of the on-line backpropagation learning algorithm: 1. (i) The differential reliability of individual training patterns is taken into account by multiplying the learning-rate constant with a factor [1 + ( σ r σ ) 2] −1 , where σ r is the experimental uncertainty associated with data point r and σ is the current estimate of the model error parameter. 2. (ii) After each full pass through the training sample, the model error measure σ is updated by solution of a nonlinear algebraic equation, at the current values of the connection weights. The extended backpropagation algorithm is tested on two problems related to the modeling of atomic-mass systematics by neural networks, the most extensive study being performed for data generated by the liquid-drop mass formula and deliberately obscured by additive Gaussian noise. Provided the underlying mapping is reasonably smooth, neural nets trained with the new procedure are able to learn the true function to a good approximation even in the presence of high levels of noise.

Application of a fusion model using collective variables and microscopically related mass parameters

Abstract A calculation of nucleus-nucleus collisions is presented, using a model which starts from a TDHF equation and leads to classical equations of motion for a set of four collective variables. Restricting to axial symmetry and assuming the liquid drop mass formula to hold, a differential equation is derived, which describes nuclear deformations and energies and is used to construct a potential energy surface for the collective variables. The nuclear deformations are obtained without the need of shape parameters. The equations of motion for the collective variables are solved numerically.

On the use of a variable moment of pairing

The concept of a variable moment of pairing is introduced in analogy to the variable moment of inertia. It is shown that deviations from the limiting cases of a rigid pairing rotor and the harmonic pairing vibrator can be systematised and interpreted very conveniently within this semi-empirical model. Its bearing on the liquid-drop mass formula is also discussed.

Fission Barriers Deduced from the Analysis of Fission Isomer Results

Available experimental data on fission-isomer excitation functions have been reanalyzed using an improved statistical model to determine values for the energies relative to the ground state of the secondary minimum and second maximum in the fission barrier for a series of plutonium, americium, and curium isotopes. The statistical model incorporates realistic decay-width calculations for neutron emission, fission, and $\ensuremath{\gamma}$-ray emission which are based on nuclear level densities derived from appropriate single-particle-level spectra. Values for the curvature $\ensuremath{\hbar}{\ensuremath{\omega}}_{B}$ of the second barrier are also estimated from the observed fission-isomer half-lives. The fission-barrier shapes determined from the analysis of experimental results are compared to theoretical calculations of barriers performed by several groups. The experimental barrier parameters agree with a variety of theoretical calculations to an accuracy of about 1 MeV. Systematic deviations between the experimental and theoretical results suggest that the surface asymmetry constant $\ensuremath{\kappa}$ in the liquid-drop mass formula should be substantially larger than the value of 1.7826 used in the second mass formula of Myers and Swiatecki.

The binding energy and density distribution of finite nuclei

A single-particle nuclear model and an effective nucleon potential are used to investigate the binding energy and density distribution of finite nuclei. The results for the binding energy are analysed in the form of the liquid-drop mass formula. We find that the surface energy coefficient is smaller by 20% for a nucleus of ten nucleons than for large nuclei, the symmetry energy coefficient is constant for all real nuclei, and the surface symmetry energy coefficient is given by 70 [1-exp(−8 I)] MeV. The half-density radii of the neutron and proton distributions are equal for stable nuclei, but the neutron surface thickness increases with neutron excess, being some 50% larger than the proton surface thickness for large nuclei.