Neutron Magic Number Research Articles

A New Mass Formula and New Neutron and Proton Magic Numbers in Light Nuclei

This mass formula explains gross features of the binding energy curves for all the elements from Li to Bi. It has no shell effects incorporated. Comparisons of separation energies computed from this formula and measured masses show extra-stability at N=6 (Z=3−8), Z=6 (N=6−9), N=14 (Z=7−10), Z=14 (N=14−19), N=16 (Z=7−8), Z=16 (N=24−26), loss of magicity at N=8 (Z=4), N=20 (Z=12−15) and quenching of N=50, 82, 126, Z=50 near driplines. Z=82 magicity rises at N=104 after strong quenching near N=107.

Clustering in Neutron-Rich Nuclei

Clustering in nuclei is discussed putting emphasis on the investigation of the role of nuclear clustering in neutron-rich nuclei. The subjects we discuss include clustering in neutron-rich Be, B and C isotopes, clustering in the island of inversion around N = 20, and clustering in the region with A ≈ 40. Be isotopes present us typical examples of clustering in neutron-rich nuclei not only in their ground band states but also in their excited band states, for which we show the analyses based on antisymmetrized molecular dynamics (AMD). Clustering in Be isotopes near neutron dripline is intimately related to the breaking of the neutron magic number N = 8. In this connection we report our study about the possible relation of the clustering with the breaking of the neutron magic number N = 20 in the island of inversion including 32Mg and 30Ne. Our discussion is not only about the positive parity states but also about negative parity states. Recently in the latter half of sd shell and in the pf shell many excited rotational bands with large deformation have been found to exist. Since the first excited K π = 0+ and K π = 0- bands in 40Ca have been regarded as constituting inversion doublet bands having the 36Ar + α structure, and since the first excited K π = 0- band in 44Ti has been concluded to have 40Ca + α structure through the α transfer reaction and by using the unique α optical potential on 40Ca, it is important to investigate the role of α clustering in these newly-found rotational bands with large deformation. We will report the AMD study about this problem.

Alpha Decay, Shell Structure, and New Elements**The project supported by National Natural Science Foundation of China (10125521), the 973 Project of China (G2000077400), the CAS Knowledge Innovation Project KJCX2-SW-N02, and the Fund of Education Ministry of China under Contract No. 20010284036

We systematically analyze the experimental data of alpha decay in even-even heavy nuclei far from stability and find that the Geiger–Nuttall law breaks for an isotopic chain when its neutron number is across a magic number or there is a deformed subshell. This break can be used to identify new magic numbers of superheavy nuclei. It is also discovered that there is a new linear relation between the logarithm of half-life and the reciprocal of the square root of decay energy for N=126 and N=152 isotones. It could be a new law of alpha decay for nuclei with magic neutron numbers but the physics behind it is to be explored. The significance of these researches for the search of new elements is discussed.

New neutron magic number N = 16 for neutron-rich nuclei

New neutron magic number N = 16 for neutron-rich nuclei

Further evidence for magic neutron number N=16 in neutron-rich nuclei

The properties of light exotic neutron-rich nuclei far from beta-stability have been found to differ substantially from the well known properties of nuclei near the valley of beta-stability. A very principal drastic changes can be seen in the locations of magic nucleon numbers that are of fundamental importance in nuclear physics as they form the basis of nuclear shell structure. We present an analysis of experiments performed at GANIL resulting in an appearance of a neutron magic number, N=16, in very neutron-rich nuclei. The behavior of the two neutron separation energies S 2 n gives a very clear evidence for the existence of the new shell closure N=16 for Z=9 and 10 appearing between 2 S 1 2 and 1 d 3 2 orbitals. Moreover, the analysis of its derivate, confirms the magic character of N=16 in the region 6⩽Z⩽10, while the shell closure at N=20 tends to disappear for Z⩽13.

New magic numbers and new islands of stability in drip-line regions in mass model

The systematics of the local energies and the two-neutron separation energies obtained in the mass predictions of infinite nuclear matter model of atomic nuclei and relativistic mean field study with NL3 interaction show strong evidence of new neutron magic numbers 100, 150, 164, new proton magic number 78 and new islands of stability around N = 100, Z ⋍ 62; N = 150, Z = 78; and N = 164, Z ⋍ 90 in the drip-line regions of nuclear chart. The usual magic numbers seen in the valley of stability are found to be no more valid in such regions.

Molecular structure of nuclei

Cluster structures of nuclei are discussed, with emphasis on nuclear clustering in unstable nuclei. The subjects we discuss are alpha condensed states, clustering in Be and B isotopes, and clustering in 32Mg and 30Ne. The subject of alpha cluster condensation comes from the clustering nature of dilute nuclear matter. We discuss that recent heavy-ion central collision experiments give us nice evidence of the clustering in dilute nuclear matter. We then present a new prediction of the existence of the “alpha cluster condensed states” in the self-conjugate 4n nuclei around the breakup threshold energy into n alpha-particles. As for the clustering in neutron-rich Be, we discuss the comparison between the antisymmetrized molecular-dynamics results and the recent experimental data, which shows that the clustering feature manifests itself very clearly in neutron-rich Be isotopes both in the ground and excited states. Clustering in Be isotopes near neutron dripline is intimately related to the breaking of the neutron magic number N = 8. We report our recent study about the possible relationship between the clustering and the breaking of the neutron magic number N = 20 in 32Mg and 30Ne.

Correlations beyond the mean field in magnesium isotopes: angular momentum projection and configuration mixing

The quadrupole deformation properties of the ground and low-lying excited states of the even–even magnesium isotopes with N ranging from 8 to 28 have been studied in the framework of the angular momentum projected generator coordinate method with the Gogny force. It is shown that the N=8 neutron magic number is preserved (in a dynamical sense) in 20Mg leading to a spherical ground state. For the magic numbers N=20 and N=28 this is not the case and prolate deformed ground states are obtained. The method yields values of the two neutron separation energies which are in much better agreement with experiment than those obtained at the mean field level. It is also obtained that 40Mg is at the neutron dripline. Concerning the results for the excitation energies of the 2+ excited states and their transition probabilities to the ground state we observe a good agreement with the available experimental data. On the theoretical side, we also present a detailed justification of the prescription used for the density dependent part of the interaction in our beyond-mean-field calculations.

Systematics of cluster decay modes

Spontaneous emission of some C, O, F, Ne, Mg, and Si isotopes, from heavy parent nuclei, have been experimentally observed since 1984, confirming earlier predictions. Experimental difficulties are mainly related to the low yield in the presence of a strong background of $\ensuremath{\alpha}$ particles. Until now, only some of the most favorable cases were investigated, leading to magic or almost magic proton and neutron numbers of daughter nuclei. We present a systematics of experimental results compared to calculations, clearly showing other possible candidates for future experiments. Universal curves may be used to estimate the expected half-lives.

Molecular-orbital structure in neutron-rich Be and C isotopes

The structure of Be and C isotopes are investigated based on the molecular-orbit (MO) model. The low-lying states are characterized by several configurations of valence neutrons, which are constructed as combinations of basic orbits. In 10Be, all of the observed positive-parity bands and the negative-parity bands are described within the model. The second 0+ state of 10Be has a large α-α cluster structure, and this is characterized by a (1/2+ σ)2 configuration. An enlargement of the α-α distance due to two-valence neutrons along the α-α axis makes their wave function smooth and reduces the kinetic energy drastically. Furthermore, the contribution of the spin-orbit interaction due to coupling between the S z = 0 and the S z = 1 configurations, is important. In the ground state of 12Be, the calculated energy exhibits similar characteristics, that the remarkable α clustering and the contribution of the spin-orbit interaction make the binding of the state with (3/2- π)2(1/2+ σ)2 configuration properly stronger in comparison with the closed p-shell (3/2- π)2(1/2- π)2 configuration. This is related to the breaking of the N = 8 (closed p-shell) neutron magic number. Also, the molecule-like structure of the C isotopes is investigated using a microscopic α+α+α+n+n+ . . . model. The combination of the valence neutrons in the π- and the σ-orbit is promising to stabilize the linear-chain state against the breathing and bending modes, and it is found that the excited states of 16C are the most promising candidates for such structure.

Nuclear Shells in the Superheavy Region within Meson Field Theory

There are fundamental questions in science, like e.g. “how did life emerge” or “how does our brain work” and others. However, the most fundamental of those questions is “how did the world originate?”. The material world has to exist before life and thinking can develop. Of particular importance are the substances themselves, i.e. the particles the elements are made of (baryons, mesons, quarks, gluons), i.e. elementary matter. The vacuum and its structure is closely related to that. On this I want to report today. I begin with the discussion of modern issues in nuclear physics. The elements existing in nature are ordered according to their atomic (chemical) properties in the periodic system which was developed by Mendeleev and Lothar Meyer. The heaviest element of natural origin is Uranium. Its nucleus is composed of Z = 92 protons and a certain number of neutrons (N = 128–150). They are called the different Uranium isotopes. The transuranium elements reach from Neptunium (Z = 93) via Californium (Z = 98) and Fermium (Z = 100) up to Lawrencium (Z = 103). The heavier the elements are, the larger are their radii and their number of protons. Thus, the Coulomb repulsion in their interior increases, and they undergo fission. In other words: the transuranium elements become more instable as they get bigger. In the late sixties the dream of the superheavy elements arose. Theoretical nuclear physicists around S. G. Nilsson (Lund) and from the Frankfurt school predicted that so-called closed proton and neutron shells should counteract the repelling Coulomb forces. Atomic nuclei with these special “magic” proton and neutron numbers and their neighbours could again be rather stable. These magic proton (Z) and neutron (N) numbers were thought to be Z = 114 and N = 184 or 196. Typical predictions of their lifetimes varied between seconds and many thousand years. Figure 1 summarizes the expectations at the time. One can see the islands of superheavy elements around Z = 114, N = 184 and 196, respectively, and the one around Z = 164, N = 318. The important question was how to produce these superheavy nuclei. There were many attempts, but only little progress was made. It was not until the middle of the seventies that the Frankfurt school of theoretical physics together with foreign guests (R. K. Gupta (India), A. Sandulescu (Romania)) theoretically understood and substantiated the concept of bombarding of double magic lead nuclei with suitable projectiles, which had been proposed intuitively by the Russian nuclear physicist Y. Oganessian. The two-center shell model, which is essential for the description of fission, fusion, and nuclear molecules, was developed in 1969–1972 together with my then students U. Mosel and J. Maruhn. It showed that the shell structure of the two final fragments was visible far beyond the barrier into the fusioning nucleus. The collective potential energy surfaces of heavy nuclei, as they were calculated in the framework of the two-center shell model, exhibit pronounced valleys, such that these valleys

Breaking of the Neutron Magic Number N=20 in 32Mg and 30Ne and Its Possible Relation to the Cluster Structure

The neutron-rich nuclei 32Mg and 30Ne are studied using a new version of antisymmetrized molecular dynamics (AMD). Due to the use of a deformed Gaussian as the single particle wave packet, the description of the deformation properties of Mg isotopes is drastically improved, and l·s force is accounted for more largely and sufficiently. After the angular momentum projection, 32Mg and 30Ne have deformed ground states, and the experimental data for 32Mg are reproduced with reasonable accuracy. The deformation of the mean field is found to be essential to describe these nuclei. The ground states of these nuclei have neutron 2p2h structure. In 30Ne, the cluster-like and the mean-field-like structures coexist. The ground state of 32Mg has a mean-field-like structure. In the vicinity of the ground state, this nucleus is expected to have excited states with cluster-like structure that possesses the neutron 4p4h configuration.

Nuclear structure studies for the astrophysical r-process

The production of the heaviest elements in nature occurs via the r-process, i.e. a combination of rapid neutron captures, the inverse photodisintegrations, and slower β −-decays, β-delayed processes as well as fission and possibly interactions with intense neutrino fluxes. A correct understanding and modeling requires the knowledge of nuclear properties far from stability and a detailed prescription of the astrophysical environment. Experiments at radioactive ion beam facilities have played a pioneering role in exploring the characteristics of nuclear structure in terms of masses and β-decay properties. Initial examinations paid attention to highly unstable nuclei with magic neutron numbers and their β-decay properties, related to the location and height of r-process peaks, while recent activities focus on the evolution of shell effects at large distances from the valley of stability. We show in site-independent applications the effect of both types of nuclear properties on r-process abundances. Next, we explore also additional nuclear properties in calculations related to possibly “realistic” astrophysical sites representative for the two options of neutron-rich high entropy or low entropy environments like (i) the supernova neutrino wind and (ii) neutron star mergers or possibly also axial jets from proto-neutron stars in supernova explosions. We close with a list of remaining theoretical, experimental and observational challenges needed to overcome for a full understanding of the nature of the r-process.

LOCAL ENERGY-DENSITY FUNCTIONAL APPROACH TO MANY-BODY NUCLEAR SYSTEMS WITH S-WAVE PAIRING

The ground-state properties of superfluid nuclear systems with 1S0 pairing are studied within a local energy-density functional (LEDF) approach. A new form of the LEDF is proposed with a volume part which fits the Friedman-Pandharipande and Wiringa-Fiks-Fabrocini equation of state at low and moderate densities and allows an extrapolation to higher densities which preserves causality. For inhomogeneous systems, a surface term is added, with two free parameters, which has a fractional form like a Padé approximant containing the square of the density gradient in both the numerator and denominator. In addition to the direct and exchange Coulomb interaction energy, an effective density-dependent Coulomb-nuclear correlation term is included with one more free parameter. A three-parameter fit to the masses and radii of about 100 spherical nuclei has shown that the latter term gives a contribution of the same order of magnitude as the Nolen-Schiffer anomaly in the Coulomb displacement energy. The root-mean-square deviations from experimental masses and radii with the proposed LEDF come out about a factor of two smaller than those obtained with the conventional functionals based on the Skyrme or finite-range Gogny force, or on relativistic mean-field theory. The generalized variational principle is formulated leading to the self-consistent Gor'kov equations which are sovled exactly, with physical boundary conditions both for the bound and scattering states. The method is used to calculate the differential observables such as odd-even mass differences and staggering in charge radii. With a zero-range density-dependent cutoff pairing interaction incorporating a density-gradient term, the evolution of these observables is reproduced reasonably well, including the kinks at magic neutron numbers and the sizes of the associated staggering. An extrapolation from the pairing properties of finite nuclei to pairing in infinite nuclear matter is discussed. A "reference" value of the pairing gap ΔF≈ 3.3 MeV is found for subsaturated nuclear matter at about 0.65 of the equilibrium density. With the formulated LEDF approach, we study also the dilute limit in both the weak and strong coupling regimes. Within the sum rules approach it is shown that the density-dependent pairing may also induce sizeable staggering and kinks in the evolution of the mean energies of multipole excitations.

Pairing properties in relativistic mean field models obtained from effective field theory

We apply recently developed effective field theory nuclear models in mean field approximation (parameter sets G1 and G2) to describe ground-state properties of nuclei from the valley of $\beta$-stability up to the drip lines. For faster calculations of open-shell nuclei we employ a modified BCS approach which takes into account quasi-bound levels owing to their centrifugal barrier, with a constant pairing strength. We test this simple prescription by comparing with available Hartree-plus-Bogoliubov results. Using the new effective parameter sets we then compute separation energies, density distributions and spin--orbit potentials in isotopic (isotonic) chains of nuclei with magic neutron (proton) numbers. The new forces describe the experimental systematics similarly to conventional non-linear $\sigma-\omega$ relativistic force parameters like NL3.

Isotope effects of samarium and ytterbium in the acetate/amalgam separation system

The data on the fractionation of all stable isotopes of ytterbium and samarium in a Me(III) acetate/Me amalgam system were obtained. The light isotopes were preferentially fractionated into the amalgam phase. The single separation gains per mass difference were found to be in the range (4.90÷17.8)·10−4 for ytterbium and (2.2÷3.8)·10−4 for samarium. The divergence from the linear dependence between separation gains and isotope mass differences was observed and explained as the effect of nuclear field shift and nuclear spin effects. The maximum value of eu.m. was observed to be 17.8·10−4 for the lightest, neutron deficit isotope of ytterbium, 168Yb. The same value for the lightest isotope of samarium, 144Sm, with a magic number of neutrons (82), was close to the mean for other isotopes.

Nuclear isotope shifts within the local energy-density functional approach

The foundation of the local energy-density functional method to describe the nuclear ground-state properties is given. The method is used to investigate differential observables such as the odd–even mass differences and odd–even effects in charge radii. For a few isotope chains of spherical nuclei, the calculations are performed with an exact treatment of the Gor'kov equations in the coordinate-space representation. A zero-range cutoff density-dependent pairing interaction with a density-gradient term is used. The evolution of charge radii and nucleon separation energies is reproduced reasonably well including kinks at magic neutron numbers and sizes of staggering. It is shown that the density-dependent pairing may also induce sizeable staggering and kinks in the evolution of the mean energies of multipole excitations. The results are compared with the conventional mean field Skyrme–HFB and relativistic Hartree–BCS calculations. With the formulated approach, an extrapolation from the pairing properties of finite nuclei to pairing in infinite matter is considered, and the dilute limit near the critical point, at which the regime changes from weak to strong pairing, is discussed.

Semiempirical formula for describing neutron shell effects in the resonance-level density for s-wave resonances

A simple formula for the resonance-level density for s-wave resonances is found by fitting the assumed function of the number of neutrons to numerous experimental data. To describe shell effects, demonstrated as strong decreases in experimental data at the magic numbers of neutrons, the dependence of the assumed function on the “complexity” of a compound nucleus is introduced. The resulting function describes quite well the character of changes in the resonance-level density with the number of neutrons, including the regions of the magic numbers.

Shell-Model Half-Lives forN=82Nuclei and Their Implications for therProcess

We have performed large-scale shell-model calculations of the half-lives and neutron-branching probabilities of the $r$-process waiting-point nuclei at the magic neutron number $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}82$. We find good agreement with the measured half-lives of ${}^{129}\mathrm{Ag}$ and ${}^{130}\mathrm{Cd}$. Our shell-model half-lives are noticeably shorter than those currently adopted in $r$ process simulations. Our calculation suggests that ${}^{130}\mathrm{Cd}$ is not produced in $\ensuremath{\beta}$-flow equilibrium with the other $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}82$ isotones on the $r$-process path.

Systematics of nuclear central densities

The nuclear central density is calculated from the nuclear charge-density parameters measured by elastic electron scattering and muonic atom spectroscopy. The nucleon number and asymmetry dependences of the obtained nuclear central density are discussed based on the macroscopic description of nuclei. It is shown that the nuclear central density decreases slowly as the nucleon number or the nuclear asymmetry increases. The proton number and neutron number dependences of the nuclear central density show some structure that seems like the shell effect, since density peaks are formed around the proton and neutron magic numbers. The data fit to the nuclear half density radii measured by muonic atom spectroscopy yields the nuclear radius constant ${r}_{0}=1.141 \mathrm{fm}$, and the data fit to the calculated nuclear central densities gives an estimation for the nuclear matter incompressibility ${K}_{0}$ in the range around 220--250 MeV.