Nucleon Separation Energies Research Articles

Nuclear transformations and radioactive emissions: Part I—primary transformation pathways of unstable nuclei

To understand the origin and character of individual radioactive emissions accompanying nuclear transformation processes, it is essential to understand what an unstable nucleus is, what its motivation is to transform, and how the nucleus selects the best way to transform. The following chapter represents part I of a series of two articles. It introduces the composition of stable atomic nuclei, the meaning of “isotopes”, the discrepancy between the mass of the nucleus and the sum of the mass of the nucleons it is composed of (the “mass defect”), and the parameter of “mean nucleon binding energy”. It is the fate of those unstable nuclei, that they transform to more stable ones. Part I will focus on the “primary” pathways of those transformations, i.e., those, where the nucleon composition of an unstable nucleus is modified to gain mean nucleon binding energy. Those transformation processes are the origin of certain radioactive emissions, and part I will discuss those originating from the primary transformation pathways, i.e., mainly β- and α-emissions.

Nuclear structure and decay properties of Nd isotopes

Using the Hartree–Fock–Bogoliubov mean-field theory, the ground-state structural and decay properties of Nd isotopes are investigated from the proton-rich side up to the neutron drip-line. Quantities such as binding energies per nucleon, one and two-neutron separation energies, rms charge radii, and quadrupole deformation parameters have been calculated. Compared with the relativistic mean-field results, the present calculations are in better agreement with the available experimental data. The results show clearly the signature of a shape transition at [Formula: see text] and an abrupt increase in the deformation near the neutron drip-line. Further, the possible decay modes like alpha, cluster and [Formula: see text]-decay are analyzed in a unified fission model and phenomenological formulas. Overall, a good agreement is achieved between the calculated and experimental [Formula: see text]-values and half-lives wherever available. The most likely decay modes are thus identified throughout the isotopic chain.

Single-particle strength from nucleon transfer in oxygen isotopes: Sensitivity to model parameters

In the analysis of transfer reaction data to extract nuclear structure information the choice of input parameters to the reaction model such as distorting potentials and overlap functions has a significant impact. In this paper we consider a set of data for the ($d,t$) and ($d,^{3}\mathrm{He}$) reactions on $^{14,16,18}\mathrm{O}$ as a well-delimited subject for a study of the sensitivity of such analyses to different choices of distorting potentials and overlap functions with particular reference to a previous investigation of the variation of valence nucleon correlations as a function of the difference in nucleon separation energy $\mathrm{\ensuremath{\Delta}}S=|{S}_{p}\ensuremath{-}{S}_{n}|$ [Phys. Rev. Lett. 110, 122503 (2013)].

Radial overlap correction to superallowed 0+→0+ β decay reexamined

Within the nuclear shell model, we investigate the correction $\delta_{RO}$ to the Fermi matrix element due to a mismatch between proton and neutron single-particle radial wave functions. Eight superallowed $0^+ \to 0^+$ $\beta$ decays in the $sd$-shell, comprised of $^{22}$Mg, $^{26m}$Al, $^{26}$Si, $^{30}$S, $^{34}$Cl, $^{34}$Ar, $^{38m}$K and $^{38}$Ca are re-examined. The radial wave functions are obtained from a spherical Woods-Saxon potential whose parametrizations are optimized in a consistent adjustment of the depth and the length parameter to relevant experimental observables, such as nucleon separation energies and charge radii, respectively. The chosen fit strategy eliminates the strong dependence of the radial mismatch correction to a specific parametrization, except for calculations with an additional surface-peaked term. As an improvement, our model proposes a new way to calculate the charge radii, based on a parentage expansion which accounts for correlations beyond the extreme independent-particle model. Apart from the calculations with a surface-peak term and the cases where we used a different model space, the new sets of $\delta_{RO}$ are in general agreement with the earlier result of Towner and Hardy [1]. Small differences of the corrected average $\overline{\mathcal{F}t}$ value are observed.

The Potential Energy of Nucleon and Nuclear Structure

This article proposes the potential energy function of nucleon in nucleus, derives the expression equation of nuclear force, shows that nucleus has the shell structure by the solving the Schr?dinger equation of nucleon, obtains the magic numbers, and interprets the past experimental results in theory; for example the radius of nucleus is proportional to the cubic root of nucleon number, the nuclear force is repulsive in the depths of nucleus and attractive in the surface layer, and the variation of average binding energy of nucleons with the nucleon number.

The highest limiting Z in the extended periodic table

The problem of finding the highest limiting Z in the extended periodic table is discussed. The upper limit suggested by the atomic many body theory at Z = 172 may be reached much earlier due to nuclear instabilities. Therefore, an extensive set of calculations based on the relativistic mean field formulation are carried out for the ground state properties of nuclei with Z = 100 to 180 and N/Z ratio ranging from 1.19 to 2.70. This choice of Z and N extends far beyond the corresponding values of all the known heavy to superheavy elements. To facilitate the analysis of the huge quantity of calculated results, various filters depending upon the pairing energies, one and two nucleon separation energies, binding energy per particle (BE/A) and α-decay plus fission half lives, are introduced. The limiting value of Z is found to be 146. For the specific filter with 5.5 MeV a few nuclei with Z = 180 also appear. No evidence for the limiting Z value 172 is found. We stress the need to bridge the atomic and nuclear findings and to arrive at an acceptable limiting value of highest Z (or rather combination of Z and N) of the extended periodic table.

Direct mass measurements of the heaviest elements with Penning traps

Penning-trap mass spectrometry (PTMS) is a mature technique to provide atomic masses with highest precision. Applied to radionuclides it enables us to investigate their nuclear structure via binding energies and derived quantities such as nucleon separation energies. Recent progress in slowing down radioactive ion beams in buffer gas cells in combination with advanced ion-manipulation techniques has opened the door to access even the elements above fermium by PTMS. Such elements are produced in complete fusion–evaporation reactions of heavy ions with lead, bismuth, and actinide targets at very low rates. Pioneering high-precision mass measurements of nobelium and lawrencium isotopes have been performed with SHIPTRAP at the GSI Darmstadt, Germany. These have illustrated that direct mass measurements provide reliable anchor points to pin down decay chains and that they allow mapping nuclear shell effects, the reason for the very existence of the heaviest elements. Thus, accurate masses contribute to our understanding of these exotic nuclei with extreme proton numbers. In this article experimental challenges in mass measurements of the heaviest elements with Penning traps are discussed. Some illustrative examples of the nuclear structure features displayed based on the presently known masses are given.

Asymmetry of the parallel momentum distribution of (p,pN) reaction residues

The parallel momentum distribution (PMD) of the residual nuclei of the 14O(p,pn)13O and 14O(p,2p)13N reactions at 100 and 200 MeV/nucleon in inverse kinematics is investigated with the framework of the distorted wave impulse approximation. The PMD shows an asymmetric shape characterized by a steep fall-off on the high momentum side and a long-ranged tail on the low momentum side. The former is found to be due to the phase volume effect reflecting the energy and momentum conservation, and the latter is to the momentum shift of the outgoing two nucleons inside an attractive potential caused by the residual nucleus. Dependence of these effects on the nucleon separation energy of the projectile and the incident energy is also discussed.

Structural evolution in transitional nuclei of mass82≤A≤132

In this theoretical study, we report an investigation on the behavior of two-neutron separation energy, a differential variation of the nucleon separation energy, the nuclear charge radii, and the single-particle energy levels along the isotopic chains of transitional nuclei. We have used the relativistic mean-field formalism with NL3 and NL3* forces for this present analysis. The study refers to the even-even nuclei such as Zr, Mo, Ru, and Pd for $N=42\ensuremath{-}86$, where a rich collective phenomena such as proton radioactivity, cluster or nucleus radioactivity, exotic shapes, island of inversion, etc. are observed. We found that there are few nonmonotonic aspects over the isotopic chain, which are correlated with the structural properties such as shell/subshell closures, the shape transition, clustering, magicity, etc. In addition to these, we have shown the internal configuration of these nuclei to get a further insight into the reason for these discrepancies.

Application of robust estimation methods to simple models of nucleon separation energies

Some works have recently shown the usefulness of simple models of nucleon separation energies in terms of neutron and proton numbers. However, the customary use of least squares in the process of parameter estimation turns out to be extremely sensible to the accuracy of the model and the extent and quality of data (e.g. highly vulnerable to the sample size or the possible existence of undesired errors in the experimental values). We will show how robust estimation by global optimization instead of least squares estimation improves on both the stability of the estimated parameters and the extrapolation to unknown energies. Comparison against recently determined experimental data will show a level of agreement comparable to the predictions made by the best and much more complex models.

Systematic study of shell gaps in nuclei

The nucleon separation energies and shell gaps in nuclei over the whole nuclear chart are systematically studied with eight global nuclear mass models. For unmeasured neutron-rich and super-heavy regions, the uncertainty of the predictions from these different mass models is still large. The latest version (WS4) of the Weizs\"acker-Skyrme mass formula, in which the isospin dependence of model parameters is introduced into the macroscopic-microscopic approach inspired by the Skyrme energy-density functional, is found to be the most accurate one in the descriptions of nuclear masses, separation energies and shell gaps. Based on the predicted shell gaps in nuclei, the possible magic numbers in super-heavy nuclei region are investigated. In addition to the shell closures at $N=184, Z=114$, the sub-shell closures at around $N=178, Z=120$ could also play a role for the stability of super-heavy nuclei.

Quasifree (p,2p) and (p,pn) reactions with unstable nuclei

We study ($p,2p$) and ($p,pn$) reactions at proton energies in the range of 100 MeV--1 GeV. Our purpose is to explore the most sensitive observables in unpolarized reactions with inverse kinematics involving radioactive nuclei. We formulate a model based on the eikonal theory to describe total cross sections and momentum distributions of the recoiled residual nucleus. The model is similar to the one adopted for knockout reactions with heavy ions. We show that momentum distributions are sensitive to the angular momentum of the ejected nucleon which can be used as an spectroscopic tool. The total cross sections are sensitive to the nucleon separation energies and to multiple scattering effects. Our calculations also indicate that a beam energy around 500 MeV/nucleon has a smaller dependence on the anisotropy of the nucleon-nucleon elastic scattering.

Nuclear symmetry energy from the Fermi-energy difference in nuclei

The neutron-proton Fermi-energy difference and the correlation to nucleon separation energies for some magic nuclei are investigated with the Skyrme energy density functionals and nuclear masses, with which the nuclear symmetry energy at sub-saturation densities is constrained from 54 Skyrme parameter sets. The extracted nuclear symmetry energy at sub-saturation density of 0.11 fm$^{-3}$ is 26.2 $\pm$ 1.0 MeV with 1.5 $\sigma$ uncertainty. By further combining the neutron-skin thickness of 208Pb, ten Skyrme forces with slope parameter of 28<L<65 are selected for the description of the symmetry energy around saturation densities.

Limited asymmetry dependence of correlations from single nucleon transfer.

Single nucleon pickup reactions were performed with a 18.1 MeV/nucleon (14)O beam on a deuterium target. Within the coupled reaction channel framework, the measured cross sections were compared to theoretical predictions and analyzed using both phenomenological and microscopic overlap functions. The missing strength due to correlations does not show significant dependence on the nucleon separation energy asymmetry over a wide range of 37 MeV, in contrast with nucleon removal data analyzed within the sudden-eikonal formalism.

The effect of discontinuous wave functions on optimizations of VMC calculations of quantum systems

Here, a relatively good comparison has been done between the performance results of discontinuous and continuous forms of wave function on quantum systems as well. In this project, we’ve estimated the energy of ground and first state as the expectation value of Hamiltonian of harmonic oscillator and binding energy of nucleons confined in a Yukawa potential by making use of following three wave function: exponential, linear and Gaussian form which are convenient to optimize and may be evaluated rapidly are devised and tested and we stress the advantages of using expressions which are Gaussian in the variable parameters. Exponential function is quite different from a Gaussian ones as it has a cusp at x=0, hence a discontinuity in its derivatives. We then have performed to test inaccuracy of the discontinuous wave function and energy’s error of its exact value because using of exponential and linear function by mathematical statements of calculation of delta and Heaviside function and either VMC calculation of wave function optimization, as both of them will have the same result of error value. The result of our study is very promising for future applications of quantum Monte Carlo methods especially in many body systems

Systematic study ofp-shell nuclei via single-nucleon knockout reactions

A systematic investigation of the inclusive cross sections for single-nucleon knockout reactions from $p$-shell nuclei has been performed. A total of seven reactions were studied for projectiles with masses between $A=7$ and 10, having a wide range of nucleon separation energies. Results were obtained for a range of incident beam energies and targets. These differences were found to have a minimal impact on the deduced cross sections. Experimental results were compared to theoretical predictions based on variational Monte Carlo (VMC) nuclear structure calculations, whose radial overlap functions and neutron and proton densities were included in the reaction description. These results are compared with the conventional model, developed for heavier nuclei, that uses shell-model and Hartree-Fock structure inputs. The VMC-based calculations agreed with the experimental data for several reactions where deeply bound nucleons are removed but does not describe some of the more weakly bound nucleon removal cases with comparable accuracy.

Nonsudden Limits of Heavy-Ion Induced Knockout Reactions

We report on the single neutron and proton removal reactions from unstable nuclei with large asymmetry ΔS = S(n)-S(p) at incident energies below 80 MeV/nucleon. Strong nonsudden effects are observed in the case of deeply-bound-nucleon removal. The corresponding parallel momentum distributions exhibit an abrupt cutoff at high momentum that corresponds to an energy threshold occurring when the incident energy per particle is of comparable magnitude to the nucleon separation energy. A large low-momentum tail is related to both dissipative processes and the dynamics of the nucleon removal process. New limits for the applicability of the sudden and eikonal approximations in nucleon knockout are given.

Radioactive decays at limits of nuclear stability

The last decades brought impressive progress in synthesizing and studying properties of nuclides located very far from the beta stability line. Among the most fundamental properties of such exotic nuclides, the ones usually established first are the half-life, possible radioactive decay modes, and their relative probabilities. When approaching limits of nuclear stability, new decay modes set in. First, beta decays are accompanied by emission of nucleons from highly excited states of daughter nuclei. Second, when the nucleon separation energy becomes negative, nucleons start being emitted from the ground state. A review of the decay modes occurring close to the limits of stability is presented. The experimental methods used to produce, identify, and detect new species and their radiation are discussed. The current theoretical understanding of these decay processes is reviewed. The theoretical description of the most recently discovered and most complex radioactive process---the two-proton radioactivity---is discussed in more detail.

The Gamma Decay of the Pygmy Resonance and the Neutron Skin of Nuclei

The Coulomb excitation of the neutron rich 68Ni was measured using it as as secondary radioactive beam at 600 MeV/nucleon impinging on a 197Au target. The dipole strength in the region at around the nucleon binding energy was extracted from the gamma decay measurement. A sizable fraction of "pygmy" dipole strength, energetically located below the giant dipole resonance, is observed in this nucleus. The experimental data is discussed in connection with predictions obtained using different Random Phase Approximation (RPA) models for the dipole response, based on a representative set of Skyrme effective forces plus meson-exchange effective Lagrangians. A comparison with the experimental data for both 68Ni and 132Sn has allowed us to constrain the value of the derivative of the symmetry energy at saturation. The neutron skin radius is deduced under this constraint.

Possibility of implementing a nondirect experiment aimed at determining the cross section for nucleon interaction with an excited nucleus

With the aid of experimental data on the level densities in the even-odd tungsten isotopes 183,185,187W as determined in the respective (n, 2ρ) reactions, the ratio of the cross section for the interaction of an excited nucleus with a neutron to its counterpart calculated on the basis of the optical model of the nucleus was obtained from the spectra of evaporated neutrons in the reaction 181Ta(p, n)181W. A significant local increase in this ratio is qualitatively interpreted as that which is due to the possible increase in the penetrability of the nuclear surface for nucleon reaction products because of the effect of phonon-type excitations. Sources of possible nonstandard systematic errors in experimentally determining the level density in an arbitrary nucleus at energies below the nucleon binding energy Bn are analyzed. The extraction of information from the distribution of the intensities of cascades belonging to the nucleon and photon type is simulated. The resulting information may be highly reliable if use is made of the entire body of experimental data on the nucleus under analysis.