Odd-even Nuclei Research Articles

A study of U-isotope ground state properties with Covariant Energy Density Functional

Abstract In this paper we have systematically analyzed the ground state of uranium isotopes from 225 to 240. In our calculations, we used the covariant energy density functional of density dependent meson exchange interaction (DD-ME2) with separable pairing interaction (TMR). Using the multiple deformation constraint, we calculate the potential energy surface (PES) of the uranium isotopes for
both even-even and even-odd nuclei with quadrupole and octupole deformation. By comparison with experimental data and Hartree-Fock-Bogoliubov calculations with Gogny D1S, the ground state of the uranium isotopes is always preferred to reflection-asymmetric deformation with our calculation.

Exploration of the ground state properties of neutron-rich sodium isotopes using the deformed relativistic mean field theory in complex momentumrepresentations with BCS pairings* *Partly supported by the National Natural Science Foundation of China (11935001, 11575001), the Natural Science Foundation of Anhui Province (2008085MA26), Anhui project (Z010118169), the Heavy Ion Research Facility in Lanzhou (HIRFL) (HIR2021PY007), and the project of Key

This study explores the ground-state characteristics of neutron-rich sodium isotopes, encompassing two-neutron separation energies, root-mean-square radii, quadrupole moments of proton and neutron distributions, single-particle levels of bound and resonant states, and neutron density distributions and shapes. Simultaneously, special attention is paid to the distinctive physical phenomena associated with these isotopes. The deformed relativistic mean field theory in complex momentum representations with BCS pairings (DRMF-CMR-BCS) employed in our research provides resonant states with real physics, offering insights into deformed halo nuclei. Four effective interactions (NL3, NL3*, PK1, and NLSH) were considered to assess the influence of continuum and deformation effects on halo structures. Calculations for odd-even nuclei 35–43Na revealed the dependence on the chosen effective interaction and number of considered resonant states. Neutron occupation patterns near the Fermi surface, particularly in orbitals and , were determined to be crucial in halo formation. The study provided detailed insights into the density distributions, shape evolution, and structure of neutron-rich sodium isotopes, contributing valuably to the field of nuclear physics.

Superheavy nuclei and other exotics – opportunities at SPIRAL2 and S3

The structure of very heavy and superheavy nuclei (SHN) as well as the location of the next proton and neutron shell closures beyond 208Pb is still one of the most intriguing topics in modern nuclear physics [1]. Worldwide competitive, high beam intensities provided by the accelerator facility SPIRAL2 at GANIL which started operation recently, will cover in future all ions up to uranium thanks to the new injector project NEWGAIN. Combined with the separator-spectrometer installation S3 [3], it will provide the instrumental prerequisites for an ambitious science program. Apart from SHN/SHE research, the envisaged physics case at S3 covers, among other, the structure of N=Z nuclei, low energy physics (fundamental properties of the atomic nucleus etc.), interdisciplinary research, atomic physics and reaction studies (fission, deep inelastic reactions etc.). The state of the art of the field is discussed in this paper with an emphasis on the role of the odd particle(s) in odd-even, even-odd and odd-odd nuclei and the consequences for nuclear structure features like K-isomers, trends of single-particle energies as a function of deformation, and the competition of spontaneous fission (SF) and α decay. As an alternative approach to produce heavy and in particular more neutron-rich nuclear species multi-nucleon transfer reactions are briefly discussed as well.

Alpha decay around shell closures and correlation of half-life times with the neutron energy levels of the emitting nuclei

The α-decay half-lives (T1/2) for 16 isotopic chains of even-even and even-odd nuclei, from 52Te to 82Pb were investigated by employing the density dependent cluster model. The α-nucleus potential was calculated by the double-folding model (DFM) using the M3Y-Reid NN interaction with the zero-range exchange part assuming spherical and deformed density distributions of the daughter nuclei. In the framework of the WKB approximation, the half-lives of the α-emitters were calculated and compared using six analytic formulas (UDL, Horoi, BKAG, UNIV, NRDX and TM formulas). We found a clear similarity in the behavior of log10⁡T1/2 versus the neutron daughter number (Nd) when the calculations performed using both DFM and these mentioned analytical formulas for each isotopic chain. In addition, the theoretical results of log10⁡T1/2 obtained with the UDL and UNIV formulas slightly better agree with the DFM calculations than those calculated from the other analytical methods. Moreover, the present work pointed out some magic or semi-magic neutron numbers of the daughter nuclei corresponding to dips in the behavior of log10⁡T1/2 with variation in Nd. The interplay of these magic (semi-magic) numbers is inspected as the total number of neutrons filling the upper neutron level in the parent nucleus. We test the predicted results of the above mentioned models by comparing them with the available experimental α-decay half-lives. Starting from the number of neutrons needed for level closure and the small available values of the confirmed spin of parent and daughter nuclei, we predicted the level schemes around the known magicities N=50, 82 and 126. We found changes in the neutron levels for isotopes of the same element which become too large for heavy nuclei. Example for these changes is the repetition of the spin value 1/2− for the three isotopes 195−199Hg and their daughters, we need five different neutron level arrangements with 3p1/2 level at the top.

Spontaneous fission hindrance in even-odd nuclei within a cluster approach

For even-odd isotopes of actinides and superheavy nuclei, spontaneous fission and $\ensuremath{\alpha}$-decay half-lives are calculated within the dinuclear system model and compared with the existing experimental data. All these processes are considered as the evolution of a nucleus in the charge (mass) asymmetry coordinate and in the relative distance between the centers of clusters formed. The important role of the spin dependence of total potential energy in the charge asymmetry coordinate is shown and the origin of the fission hindrance in even-odd nuclei is explained.

Ground state magnetic dipole moment of Sc40

The hyperfine coupling constants of the proton dripline odd-odd $^{40}\mathrm{Sc}$ nucleus were deduced from the hyperfine spectrum of the $3d4s{\phantom{\rule{0.16em}{0ex}}}^{3}{\text{D}}_{2}\ensuremath{\leftrightarrow}3d4p{\phantom{\rule{0.16em}{0ex}}}^{3}{\text{F}}_{3}^{\ensuremath{\circ}}$ transition in Sc II, measured by the bunched beam collinear laser spectroscopy technique. The ground-state magnetic-dipole and electric-quadrupole moments were determined as $\ensuremath{\mu}=+5.57(4)(2){\ensuremath{\mu}}_{N}$ and $Q=+42(38)(28)\phantom{\rule{4pt}{0ex}}{e}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{fm}}^{2}$, respectively. The magnetic moment is well reproduced by the additivity rule with magnetic moments of neighboring odd-even nuclei in the vicinity of the doubly magic $^{40}\mathrm{Ca}$ nucleus. An ab initio multishell valence-space Hamiltonian was also employed to calculate the magnetic moment of $^{40}\mathrm{Sc}$, which spans across the $sd$ and $fp$ nuclear shells, where we obtained good agreements.

Positive parity low-spin states of even–odd 129−133Ba isotopes

The positive parity low-spin states of even-odd 129−133Ba isotopes were studied in this study using the Interacting Boson–Fermion Model (IBFM-1). The single fermion is predicted to be in one of three single-particle orbits: 2d5/2, 2d3/2, and 3s1/2. The Interacting Boson Model (IBM-1) was used to investigate the energy levels, electric quadrupole transition probabilities, and potential energy surface of even-even Barium isotopes (a core for even-odd nuclei).The measured positive parity low-state energy spectra and predicted energy levels, as well as the B(E2) transition probabilities, are reasonably consistent with the experimental data and previous research for Ba isotopes. The potential energy surface contour plot reveals that all interesting nuclei are deformed and have γ-unstable-like properties.

Nuclear ground states in a consistent implementation of the time-dependent density matrix approach

Background: Time-dependent techniques in nuclear theory often rely on mean-field or Hartree-Fock descriptions. Beyond mean-field dynamical calculations within the time-dependent density matrix (TDDM) theory have often invoked symmetry restrictions and ignored the connection between the mean-field and the induced interaction. Purpose: We study the ground states obtained in a TDDM approach for nuclei from $A=12$ to $A=24$, including examples of even and odd-even nuclei with and without intrinsic deformation. We overcome previous limitations using three-dimensional simulations and employ density-independent Skyrme interactions self-consistently. Methods: The correlated ground states are found starting from the Hartree-Fock solution, by adiabatically including the beyond-mean-field terms in real time. Results: We find that, within this approach, correlations are responsible for $\approx 4-5 \%$ of the total energy. Radii are generally unaffected by the introduction of beyond mean-field correlations. Large nuclear correlation entropies are associated to large correlation energies. By all measures, $^{12}$C is the most correlated isotope in the mass region considered. Conclusions: Our work is the starting point of a consistent implementation of the TDDM technique for applications into nuclear reactions. Our results indicate that correlation effects in structure are small, but beyond-mean-field dynamical simulations could provide new insight into several issues of interest.

Symmetry-projected variational calculations with the numerical suite TAURUS

We present the numerical code \(\textsf {TAURUS}_{\textsf {vap}}\) that solves the variation after particle-number projection equations for real general Bogoliubov quasiparticle states represented in a spherical harmonic oscillator basis. The model space considered is invariant under spatial and isospin rotations but no specific set of orbits is assumed such that the code can carry out both valence-space and no-core calculations. In addition, no number parity is assumed for the Bogoliubov quasiparticle states such that the code can be used to describe even-even, odd-even and odd-odd nuclei. The variational procedure can be performed under several simultaneous constraints on the expectation values of a variety of operators such as the multipole deformations, the pairing field or the components of the angular momentum. To demonstrate the potential and versatility of the code, we perform several example calculations using an empirical shell-model interaction as well as a chiral interaction. The ability to perform advanced variational Bogoliubov calculations offered by this code will, we hope, be beneficial to the shell model and ab initio communities.

Unexpected transitional paths in the prolate to oblate shape phase transitions for Bose–Fermi systems

The quantum shape phase transitions in odd-even nuclei are investigated within the intrinsic frame approach to the interacting boson-fermion model. In this work, the case of a single-j fermion coupled to a bosonic core that performs a transition from prolate to oblate shapes is considered. The focus is on the effect of the coupled fermion on the whole system along the transitional path from prolate to oblate shapes, passing through the $${\gamma }$$ -unstable shape. One could expect that all the magnetic substates of the coupled single-fermion with $$j=9/2$$ would be driven by the shape of the bosonic core. However, the present work shows that the odd-fermion follows some unexpected and unique paths. The five-components of the $$j=9/2$$ orbital do show quite interesting and diverse behaviour. Two of them move slowly from the prolate to the oblate shape by venturing into the triaxial region and also show $$\gamma $$ -softness around their slow shape-changing region. The other three odd-states show sudden jumps from prolate side to oblate side and shape coexistence appears, although one of them is fairly close to $$\gamma $$ -instability. These unexpected situations make this shape phase transition worth of investigation and discussion.

E1 moments from a coherent set of measured photoneutron cross sections

Recently, the $E1$ photoneutron cross sections were measured consistently using quasimonochromatic laser Compton-scattering $\ensuremath{\gamma}$-ray beams at NewSUBARU (Japan) for a set of ten odd-even nuclei, spherical as well as deformed, covering relatively widely light to heavy species and at energies ranging between the neutron threshold and 40 MeV. This consistent set of experimental photoneutron cross sections allow us to estimate the total $E1$ photoabsorption cross sections, through validated theoretical models, by excluding the quasideuteron component and by including the missing low-energy and charged-particle emission contributions. In turn, the total $E1$ photoabsorption cross sections are used to derive the three main moments of the $E1$ distributions, namely the integrated cross section, the centroid energy, and the polarizability. These so-called model-dependent experimental moments follow a rather smooth trend with atomic mass, as theoretically expected, except for the specific case of $^{209}\mathrm{Bi}$ which presents a surprising 10% increase of both the integrated strength and the polarizability with respect to its even-even spherical neighbor $^{208}\mathrm{Pb}$. A rather consistent value for the enhancement factor of the $E1$ energy-weighted sum rule could be extracted from the present experimental data. The experimentally derived moments are compared for the ten odd-even systems with calculations based on the mean-field plus quasiparticle random phase approximation (QRPA). Spherical as well as axially deformed predictions based on the Skyrme and Gogny interactions are considered to discuss the results. Special attention is also paid to the approximate treatment of odd systems of nucleons in this context. It is found that in general theoretical predictions can rather well describe the data for the full set of nuclei, except for $^{209}\mathrm{Bi}$ presenting a kink in all the three main $E1$ moments with respect to the well-studied $^{208}\mathrm{Pb}$ case. Such an experimental determination cannot be explained by mean-field plus QRPA calculations. New measurements of photoabsorption cross section on $^{208}\mathrm{Pb}$ and $^{209}\mathrm{Bi}$ in the 14--19 MeV range as well as future theoretical calculations, in particular of odd-even systems, may help to solve such a mystery. The present data are also used to reinvestigate the correlation between the nuclear matter symmetry energy and its slope at saturation density. The analysis based on both Skyrme and Gogny Hartree-Fock-Bogolyubov (HFB) plus QRPA calculations confirm previous results, though quantitatively larger variations of the correlation parameters are found.

Alpha and heavy cluster radioactivity of superheavy nuclei 100 ⩽ Z ⩽ 120

Using Cubic plus Yukawa plus Exponential (CYE) model, half lives of alpha decay and heavy cluster radioactivity of Superheavy nuclei have been systematically investigated for even-even, even-odd, odd-even and odd-odd nuclei with 100 ⩽ Z ⩽ 120. We have done our calculations by considering the Coulomb, centrifugal and Yukawa plus exponential potentials as an interacting barrier for separated fragments and the cubic potential for the overlapping region.The predicted half life time values by including deformation effects on parent and parent cluster have been compared with theAnalytical Super Asymmetric Fission (ASAF) model of D.N.Poenaru et.al. In this work, we have compared the half life time values of alpha and heavy cluster radioactivity of Super heavy nuclei leading to 208Pb. This study suggests that heavy-cluster radioactivity may be comparable to or even dominant over a decay for some of the isotopes with Z ⩾ 118. Furthermore, branching ratio calculations have been performed to find out the probable cluster emitters. The predictions for cluster emitters are in agreement with CPPM model by Santhosh et. al.,and Wentzel-Kramers-Brillouin (WKB) method by A.Soylu et. al. We hope that this study will help in future measurements on α-decay and cluster radioactivity half-lives of SHN.

Some Electromagnetic Transition Properties of Odd-A Palladium Isotopes

In this study, the Palladium nuclei have been studied by interacting boson fermion model for even-odd nuclei. The isotopes under study have proton number 46 and neutron numbers 63-67. The calculations involved the eigenstates and reduce electric probability. The calculations show that the energy levels are reasonable with experimental data. The B(E2) values compared with those of experimental and are in good agreements. All calculations compared with experimental and give more data for reader.

Neutron Skin, Central Depletion, Neutron–Proton Interaction Effects and 2pF Parameterization for Zr in Neutron Stars

A study of the neutron skin thicknesses, central depletions and neutron–proton interactions for the isotopic chain of Zr (32 ≤ N ≤ 98) with even-even and even-odd nuclei is carried out within the framework of the Skyrme–Hartree–Fock approach. The neutron skin thicknesses are calculated by three different methods, one of which is the hard spheres, and the neutron thicknesses including surface and bulk parts are investigated for Zr isotopes as a function of the relative neutron excess I = (N − Z)/A. For that purpose, the level density distributions and 2pF parameters are calculated, and theoretical methods are used to obtained the residual and average neutron–proton interactions (δVnp, δnp), in addition to the single-particle energy levels for $$ j=l\mp \frac{1}{2} $$ states, the charge form factors Fch(k), and the pairing, Fermi and binding energies. The importance of the obtained results is discussed in terms of the quantum effects, the Coulomb repulsion and neutron–neutron interaction.

Quantum phase transition phenomena in odd-A nuclei

Quantum phase transitions in even-even nuclei been extensively studied both in theory and experiment in the recent years. Odd-even nuclei can be considered as examples of mixed Bose-Fermi systems with a single fermion coupled to an even-even core. The eÆect of a fermion with angular momentum j on quantum phase transi- tions of a bosonic system comprised of s (L=0) and d (L=2) bosons is investigated. The analysis is based on an Interacting Boson Fermion Model (IBFM) Hamiltonian and its classical limit and it demostrates the changes in the energy level structure as well as the classical and quantum order parameters involved. Some experimental evidence is also presented and compared with theoretical calculations

Evidence for α -cluster structure in Ne21 in the first measurement of resonant O17+α elastic scattering

The first study of resonances in $^{17}$O+$\alpha$ elastic scattering was carried out using the Thick Target Inverse Kinematics (TTIK) method. The data were analyzed in the framework of an $\textit{R}$-matrix approach. Many $\alpha$-cluster states were found in the $^{21}$Ne excitation region of the 9-13 MeV excitation energy including the first observation of a broad $\textit{l}$=0 state in an odd-even nucleus, which is likely the analog of the broad 0$^+$ at 8 MeV in $^{20}$Ne. The observed structure in $^{21}$Ne appeared to be strikingly similar to that in $^{20}$Ne populated in the resonance $^{16}$O+$\alpha$ scattering. The results are also useful for refinement of data on an $^{17}$O($\alpha$,$\textit{n}$) reaction important for astrophysics.

Investigation of γ-unstable odd-even nuclei in the Collective model framework with the Morse potential

An odd-mass nucleus has been thought as coupling of a single nucleon in the j=3/2 single-particle orbit with the γ-unstable even-core. To get the properties of such a system, the Collective model Hamiltonian with the Morse potential has been used. Two cases, in which the interaction strength between the nucleon and the core are fixed or deformation dependent, are examined. The excitation energy spectrum and the electric quadrupole transition ratios have been obtained. Then the results have been used to predict the experimental data of the some selected odd-mass nuclei.

Consistency of two different approaches to determine the strength of a pairing residual interaction in the rare-earth region

Two fits of the pairing residual interaction in the rare-earth region are independently performed. One is made on the odd-even staggering of masses by comparing measured and explicitly calculated three-point binding-energy differences centered on odd-even nuclei. Another deals with the moments of inertia of the first 2+ states of well deformed even-even nuclei upon comparing experimental data with the results of Inglis-Belyaev moments (supplemented by a crude estimate of the so-called Thouless-Valatin corrections). The sample includes 24 even-even and 31 odd-mass nuclei selected according to two criteria: they should have good rotor properties and should not correspond to low pairing-correlation regimes in their ground states. Calculations are performed in the self-consistent Hartree-Fock plus BCS framework (implementing a self-consistent blocking in the case of odd-mass nuclei). The Skyrme SIII parametrization is used in the particle-hole channel and the fitted quantities are the strengths of |Tz|=1 proton and neutron seniority residual interactions. As a result, the two fits yield sets of strengths in excellent agreement: about 0.1% for the neutron parameters and 0.2% for protons. In contrast when one performs such a fit on odd-even staggering from quantities deduced from BCS gaps or minimal quasiparticle energies in even-even nuclei, as is traditional, one obtains results significantly different from those obtained in the same nuclei by a fit of moments of inertia. As a conclusion, beyond providing a phenomenological tool for microscopic calculations in this region, we have illustrated the proposition made in the seminal paper of Bohr, Mottelson and Pines that moments of inertia and odd-even staggering in selected nuclei were excellent measuring sticks of nuclear pairing correlations.

Shell-model investigation of odd-mass nuclei in the 132Sn region

New shell-model calculations are performed in the mass region around 132Sn robust core, where the low-lying states energies and the magnetic dipole moments of even-odd (135,137Te, 137,l39Xe, 139,141Ba, 141,143Ce), and odd-even (135Sb, 135,137I, 135,139Cs, 139,141La) nuclei are obtained with the N3LOP effective interaction. The position of 5/2+ state in 135Sb is discussed with the variation of the proton gap between 0g7/2 and 1d5/2 orbits.

Effects of single particle on shape phase transitions and phase coexistence in odd-even nuclei**Supported by National Natural Science Foundation of China (11375005)

A classical analysis of shape phase transitions and phase coexistence in odd-even nuclei has been performed in the framework of the interacting boson-fermion model. The results indicate that the effects of a single particle may influence different types of transitions in different ways. Especially, it is revealed that phase coexistence can clearly emerge in the critical region and thus be taken as a indicator of the shape phase transitions in odd-even nuclei.