Reflection Asymmetric Shapes Research Articles

Examination of static fission properties of 236\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{236}$$\\end{document}U and 233\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{233}$$\\end{document}Th using Cassinian oval parametrization within the macroscopic–microscopic approach

We have conducted a thorough analysis of the potential energy surfaces (PES) in 236\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{236}$$\\end{document}U and 233\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{233}$$\\end{document}Th using the Cassini-ovals parameterization within the macro–micro approach. We employed the state-of-the-art immersion water flow (IWF) method to study the saddles on four-dimensional energy grids encompassing reflection-asymmetric shapes. For 233\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{233}$$\\end{document}Th, we computed the adiabatic potential energy surfaces by minimizing configurations with one blocked neutron within ten levels below and above the Fermi level. Our results show satisfactory agreement with empirical and experimental estimates for both nuclei, specifically regarding the first and second fission barriers. This suggests that our method holds promise in efficiently describing non-compact shapes while reducing the dimensionality of the space without sacrificing accuracy. Interestingly, employing Cassinian oval parameterization fails to reveal a pronounced, hyper-deformed third minimum in the potential energy landscape. Instead, only a shallow third minimum is observed for 233\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{233}$$\\end{document}Th, while in 236\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{236}$$\\end{document}U, this minimum ultimately vanishes. This finding holds significant importance when considering the modeling of fission cross-sections.

Evolution of quadrupole-octupole collectivity in the even–even 54 ≤ Z ≤ 58 nuclei

We explore the quadrupole-octupole collectivity in the mass region of even–even 54 ≤ Z ≤ 58 (Xe, Ba and Ce) nuclei by analysing the fine structure of energy spectra. As a basic criterion for the presence of reflection-asymmetric degree of freedom we consider the formation of alternating-parity level-sequences. As a clear-cut criterion for the emergence of pronounced octupole deformation we consider the reduction of the parity-shift at certain angular momentum and the formation of a single energy sequence which can be interpreted as an octupole band. Applying these criteria we select out alternating-parity bands (APBs) in the region of Xe, Ba and Ce isotopes for which a manifestation of pronounced octupole collectivity can be considered. Their fine structure is probed in a collective quadrupole-octupole rotation model (QORM) providing a detailed analysis in terms of odd–even staggering diagrams. The obtained model descriptions of the energy levels and staggering patterns reveal the most characteristic features of the quadrupole-octupole deformations in this mass region and outline a clearly determined area of manifestation of nuclear reflection-asymmetric shapes.

VADER: A novel decay station for actinide spectroscopy

A research programme focused on the study of the nuclear structure of actinide isotopes has recently been implemented at the IGISOL facility, University of Jyväskylä. Within this scope, a new decay station named VADER (Versatile Actinides DEcay spectRoscopy setup) has been developed and commissioned. The system consists of a compact array of silicon detectors, a liquid-nitrogen-cooled silicon lithium (Si(Li)) detector and three broad energy germanium detectors (BEGe), placed around a thin implantation carbon foil. The combined use of different detectors allows the measurement of α particles, conversion electrons and de-excitation γ rays in coincidence, enabling a full reconstruction of nuclear decay schemes. The measurement of basic nuclear decay observables provides a picture of the nuclear shell evolution in neutron-deficient actinides, and highlights the possible emergence of reflection-asymmetric shapes in the region.

Skyrme pseudopotentials at next-to-next-to-leading order: Construction of local densities and first symmetry-breaking calculations

There is an ongoing quest to improve on the spectroscopic quality of nuclear energy density functionals (EDFs) of the Skyrme type through extensions of its traditional form. One direction for such activities is the inclusion of terms of higher order in gradients in the EDF. We report on exploratory symmetry-breaking calculations performed for an extension of the Skyrme EDF that includes central terms with four gradients at next-to-next-to-leading order (N2LO) and for which the high-quality parametrization SN2LO1 has been constructed recently [P. Becker et al, Phys. Rev. C 96, 044330 (2017)]. Up to now, the investigation of such functionals with higher-order terms was limited to infinite matter and spherically symmetric configurations of singly- and doubly-magic nuclei. We address here nuclei and phenomena that require us to consider axial and non-axial deformation, both for reflection-symmetric and also reflection-asymmetric shapes, as well as the breaking of time-reversal invariance. Achieving these calculations demanded a number of formal developments. These all resulted from the formulation of the N2LO EDF requiring the introduction of new local densities with additional gradients that are not present in the EDF at NLO. Their choice is not unique, but can differ in the way the gradients are coupled. While designing a numerical implementation of N2LO EDFs in Cartesian 3d coordinate-space representation, we have developed a novel definition and a new unifying notation for normal and pair densities that contain gradients at arbitrary order. The resulting scheme resolves several issues with some of the choices that have been made for local densities in the past, in particular when breaking time-reversal symmetry. Guided by general practical considerations, we propose an alternative form of the N2LO contribution to the Skyrme EDF that is built from a different set of densities.

Pear-shaped nuclei

This brief review presents current experimental evidence on the existence of deformed axially symmetric but reflection asymmetric or pear shapes in nuclei. A summary of the main findings on the properties of nuclear energy levels and electric transition probabilities is given. These experimental investigations have been possible due to the development of high efficiency gamma-detector arrays and production and availability of exotic radioactive ion beams which facilitated in-beam Coulomb excitation. The nuclear structure of odd-mass pear-shaped nuclei has assumed importance as they can be vital in solving some of the fundamental problems in physics. This paper is sequel to the publication of the book by the author on Pear-shaped Nuclei (World Scientific, 2020).

Microscopic origin of reflection-asymmetric nuclear shapes

Background: The presence of nuclear ground states with stable reflection-asymmetric shapes is supported by rich experimental evidence. Theoretical surveys of odd-multipolarity deformations predict the existence of pear-shaped isotopes in several fairly localized regions of the nuclear landscape in the vicinity of near-lying single-particle shells with $\Delta\ell=\Delta j=3$. Purpose: We analyze the role of isoscalar, isovector, neutron-proton, neutron-neutron, and proton-proton multipole interaction energies in inducing the onset of reflection-asymmetric ground-state deformations. Methods: The calculations are performed in the framework of axial reflection-asymmetric Hartree-Fock-Bogoliubov theory using two Skyrme energy density functionals and density-dependent pairing force. Results: We show that reflection-asymmetric ground-state shapes of atomic nuclei are driven by the odd-multipolarity neutron-proton (or isoscalar) part of the nuclear interaction energy. This result is consistent with the particle-vibration picture, in which the main driver of octupole instability is the isoscalar octupole-octupole interaction giving rise to large $E3$ polarizability.

Properties of heaviest nuclei with [formula omitted] and [formula omitted

We systematically determine ground-state and saddle-point shapes and masses for 1305 heavy and superheavy nuclei with Z=98–126 and N=134–192, including odd-A and odd–odd systems. From these we derive static fission barrier heights, one- and two-nucleon separation energies, and Qα values for g.s. to g.s. transitions. Our study is performed within the microscopic–macroscopic method with the deformed Woods–Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy taken as the smooth part. We use parameters of the model that were fitted previously to masses of even–even heavy nuclei. For systems with odd numbers of protons, neutrons, or both, we use a standard BCS method with blocking. Ground-state shapes and energies are found by the minimization over seven axially-symmetric deformations. A search for saddle-points was performed by using the ”imaginary water flow” method in three consecutive stages, using five- (for nonaxial shapes) and seven-dimensional (for reflection-asymmetric shapes) deformation spaces. Calculated ground-state mass excess, nucleon separation- and Qα energies, total, macroscopic (normalized to the macroscopic energy at the spherical shape) and shell corrections energies, and deformations are given for each nucleus in Table 1. Table 2 contains calculated properties of the saddle-point configurations and the fission barrier heights. In Tables 3-7, are given calculated ground-state, inner and outer saddle-point and superdeformed secondary minima characteristics for 75 actinide nuclei, from Ac to Cf, for which experimental estimates of fission barrier heights are known. These results are an additional test of our model.

Robustness of chiral symmetry in atomic nuclei with reflection-asymmetric shapes

Abstract Nearly degenerate quantum states arise from fundamental symmetries, which are often broken in finite many-body systems like atomic nuclei. Chirality can be static, like in molecules composed of more than four atoms, or dynamic, like in particle physics distinguishing between parallel and antiparallel orientation of the spin with respect to the momentum of massless fermions. Frauendorf and Meng.

Landscape of pear-shaped even-even nuclei

The phenomenon of reflection-asymmetric nuclear shapes is relevant to nuclear stability, nuclear spectroscopy, nuclear decays and fission, and the search for new physics beyond the standard model. Global surveys of ground-state octupole deformation, performed with a limited number of models, suggest that the number of pear-shaped isotopes is fairly limited across the nuclear landscape. We carry out global analysis of ground-state octupole deformations for particle-bound even-even nuclei with $Z \leq 110$ and $N \leq 210$ using nuclear density functional theory (DFT) with several non-relativistic and covariant energy density functionals. In this way, we can identify the best candidates for reflection-asymmetric shapes. The calculations are performed in the frameworks of axial reflection-asymmetric Hartree-Fock-Bogoliubov theory and relativistic Hartree-Bogoliubov theory using DFT solvers employing harmonic oscillator basis expansion. We consider five Skyrme and four covariant energy density functionals. We predict several regions of ground-state octupole deformation. In addition to the "traditional" regions of neutron-deficient actinide nuclei around $^{224}$Ra and neutron-rich lanthanides around $^{146}$Ba, we identified vast regions of reflecion-asymmetric shapes in very neutron-rich nuclei around $^{200}$Gd and $^{288}$Pu, as well as in several nuclei around $^{112}$Ba. Our analysis suggests several promising candidates with stable ground-state octupole deformation, primarily in the neutron-deficient actinide region, that can be reached experimentally. Detailed comparison between Skyrme and covariant models is performed.

Coexistence of Reflection Asymmetric and Symmetric Shapes in ^{144}Ba.

Level structures in the neutron-rich ^{144}Ba nucleus have been reinvestigated by measuring prompt γ rays in the spontaneous fission of ^{252}Cf. The previous s=+1 octupole band structure with reflection asymmetric shape has been expanded, and a side quadrupole band structure based on a 3^{+} state with reflection symmetric shape is identified. Thus, the results show the coexistence of reflection asymmetric and symmetric shapes in ^{144}Ba. This is a first identification of such a shape coexistence structure in a nuclear structure. The other structural characteristics are discussed.

Static fission properties of actinide nuclei

Fission barriers heights and excitation energies of superdeformed isomeric minima are calculated within the microscopic - macroscopic Woods - Saxon model for 75 actinide nuclei for which the experimental data are known. State - of - the - art methods were used: minimization over many deformation parameters for minima and the imaginary water flow on many - deformation energy grid for saddles, including nonaxial and reflection-asymmetric shapes. We obtain 0.82 - 0.94 MeV rms deviation between the calculated and experimental barriers and 0.53 MeV rms error in the excitation of superdeformed minima (SD). Experimental vs theory discrepancies seem to be of various nature and not easy to eliminate, especially if one cares for more than one or two observables. As an example, we show that by strengthening pairing in odd systems one can partially improve agreement in barriers, while spoiling it for masses. We also discuss the "thorium anomaly" and suggest its possible relation to a different way in which the Ac and Th barriers are derived from experimental data.

Search for octupole-deformed actinium isotopes using resonance ionization spectroscopy

In-source resonance ionization spectroscopy of the neutron-rich actinium isotopes $^{225\ensuremath{-}229}\mathrm{Ac}$ has been performed at the ISAC facility in TRIUMF, probing a $^{2}D_{3/2}\ensuremath{\rightarrow}^{4}P_{5/2}^{\ensuremath{\circ}}$ atomic transition. New data on the magnetic dipole moments and changes in mean-square charge radii $\ensuremath{\delta}\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}$ of $^{226,228,229}\mathrm{Ac}$ have been obtained. The comparison of the measured isotope shifts and magnetic dipole coupling constants $a(^{4}P_{5/2}^{\ensuremath{\circ}})$ of $^{225,227}\mathrm{Ac}$ with a high-resolution data set is used to identify systematic uncertainties on the deduced $\ensuremath{\delta}{\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}}^{A,215}$ and magnetic dipole moment values. The charge radii odd-even staggering is evaluated for the odd-$N$ isotopes, showing that $^{226}\mathrm{Ac}$ has an inverted odd-even staggering that might be linked with a reflection-asymmetric shape. Comparison of the magnetic dipole moments of $^{225,227,229}\mathrm{Ac}$ with Nilsson-model estimates supports the assumption of octupole deformation in isotopes $^{225,227}\mathrm{Ac}$ and its gradual decrease toward isotope $^{229}\mathrm{Ac}$. The changes in mean-square charge radii are compared to self-consistent calculations employing multiple modern energy density functionals: SLy5s1, BSk31, and DD-MEB1. For SLy5s1 in particular, self-consistent time-reversal breaking calculations of odd-odd nuclei incorporating finite octupole deformation are reported for the first time. For these calculations, the overall best agreement is obtained when the octupole degree of freedom is taken into account.

Deformed band structures in neutron-rich Pm152–158 isotopes

High spin band structures of neutron-rich $^{152-158}$Pm isotopes have been obtained from the measurement of prompt $\gamma$-rays of isotopically identified fragments produced in fission of $^{238}$U+$^{9}$Be and detected using the VAMOS++ magnetic spectrometer and EXOGAM segmented Clover array at GANIL and also from the high statistics $\gamma$-$\gamma$-$\gamma$ and $\gamma$-$\gamma$-$\gamma$-$\gamma$ data from the spontaneous fission of $^{252}$Cf using Gammasphere. The excited states in $^{157}$Pm and those above the isomers in even-A Pm isotopes $^{152,154,156,158}$Pm have been identified for the first time. The spectroscopic information on the rotational band structures in odd-A Pm isotopes has been extended considerably to higher spins and the possibility of the presence of reflection asymmetric shapes is explored. The configuration assignments are based on the results of Cranked Relativistic Hartree-Bogoliubov calculations. From the systematics of bands in odd-A Pm isotopes and weak population of opposite parity bands, octupole deformed shapes in neutron rich Pm isotopes beyond $N=90$ seem unlikely to be present.

Studies of the Shapes of Heavy Nuclei at ISOLDE

In this talk I will describe how recent experiments carried out at REX-ISOLDE have furthered our understanding of shape co-existence and reflection-asymmetric shapes in atomic nuclei. I will discuss how the experimental results have constrained nuclear theory and how the latter measurements might contribute to tests of extensions of the Standard Model. I will also discuss future prospects for measuring nuclear shapes from inelastic scattering.

Laser spectroscopy of francium isotopes at the borders of the region of reflection asymmetry

The magnetic dipole moments and changes in mean-square charge radii of the neutron-rich $^{218m,219,229,231}\text{Fr}$ isotopes were measured with the newly-installed Collinear Resonance Ionization Spectroscopy (CRIS) beam line at ISOLDE, CERN, probing the $7s~^{2}S_{1/2}$ to $8p~^{2}P_{3/2}$ atomic transition. The $\delta\langle r^{2}\rangle^{A,221}$ values for $^{218m,219}\text{Fr}$ and $^{229,231}\text{Fr}$ follow the observed increasing slope of the charge radii beyond $N~=~126$. The charge radii odd-even staggering in this neutron-rich region is discussed, showing that $^{220}\text{Fr}$ has a weakly inverted odd-even staggering while $^{228}\text{Fr}$ has normal staggering. This suggests that both isotopes reside at the borders of a region of inverted staggering, which has been associated with reflection-asymmetric shapes. The $g(^{219}\text{Fr}) = +0.69(1)$ value supports a $\pi 1h_{9/2}$ shell model configuration for the ground state. The $g(^{229,231}\text{Fr})$ values support the tentative $I^{\pi}(^{229,231}\text{Fr}) = (1/2^{+})$ spin, and point to a $\pi s_{1/2}^{-1}$ intruder ground state configuration.

Multidimensionally constrained relativistic mean field model and applications in actinide and transfermium nuclei

We present some results of potential energy surfaces of actinide and transfermium nuclei from multidimensionally constrained relativistic mean field (MDC-RMF) models. Recently we developed multidimensionally constrained covariant density functional theories (MDC-CDFT), in which all shape degrees of freedom with even μ are allowed, and the functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the nonlinear or density-dependent couplings. In MDC-RMF models, the pairing correlations are treated with the BCS method. With MDC-RMF models, the potential energy surfaces of even−even actinide nuclei were investigated and the effect of triaxiality on the fission barriers in these nuclei was discussed. The nonaxial reflection-asymmetric shape in some transfermium nuclei with N = 150, namely Cm, Cf, Fm and No, were also studied.

Reflection asymmetric shapes in covariant density functional theory

Reflection asymmetric shapes in covariant density functional theory

Multidimensionally-constrained relativistic mean-field models and potential-energy surfaces of actinide nuclei

Background: Many different shape degrees of freedom play crucial roles in determining the nuclear ground state and saddle point properties and the fission path. For the study of nuclear potential energy surfaces, it is desirable to have microscopic and self-consistent models in which all known important shape degrees of freedom are included.Purpose: By breaking both the axial and the spatial reflection symmetries simultaneously, we develop multidimensionally-constrained relativistic mean field (MDC-RMF) models.Methods: The nuclear shape is assumed to be invariant under the reversion of $x$ and $y$ axes, i.e., the intrinsic symmetry group is ${V}_{4}$ and all shape degrees of freedom ${\ensuremath{\beta}}_{\ensuremath{\lambda}\ensuremath{\mu}}$ with even $\ensuremath{\mu}$, such as ${\ensuremath{\beta}}_{20}$, ${\ensuremath{\beta}}_{22}$, ${\ensuremath{\beta}}_{30}$, ${\ensuremath{\beta}}_{32}$, ${\ensuremath{\beta}}_{40}$, $...,$ are included self-consistently. The single-particle wave functions are expanded in an axially deformed harmonic oscillator (ADHO) basis. The RMF functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the nonlinear or density-dependent couplings. The pairing effects are taken into account with the BCS approach.Results: The one-, two, and three-dimensional potential energy surfaces of ${}^{240}$Pu are illustrated for numerical checks and for the study of the effect of the triaxiality on the fission barriers. Potential energy curves of even-even actinide nuclei around the first and second fission barriers are studied systematically. Besides the first ones, the second fission barriers in these nuclei are also lowered considerably by the triaxial deformation. This lowering effect is independent of the effective interactions used in the RMF functionals. Further discussions are made about different predictions on the effect of the triaxiality between the macroscopic-microscopic and MDC-RMF models, possible discontinuities on PES's from self-consistent approaches, and the restoration of broken symmetries.Conclusions: MDC-RMF models give a reasonably good description of fission barriers of even-even actinide nuclei. It is important to include both the nonaxial and the reflection asymmetric shapes simultaneously for the study of potential energy surfaces and fission barriers of actinide nuclei and of those in unknown mass regions such as, e.g., superheavy nuclei.

Third minima in thorium and uranium isotopes in a self-consistent theory

Background: Well-developed third minima, corresponding to strongly elongated and reflection-asymmetric shapes associated with dimolecular configurations, have been predicted in some non-self-consistent models to impact fission pathways of thorium and uranium isotopes. These predictions have guided the interpretation of resonances seen experimentally. On the other hand, self-consistent calculations consistently predict very shallow potential-energy surfaces in the third minimum region.Purpose: We investigate the interpretation of third-minimum configurations in terms of dimolecular (cluster) states. We study the isentropic potential-energy surfaces of selected even-even thorium and uranium isotopes at several excitation energies. In order to understand the driving effects behind the presence of third minima, we study the interplay between pairing and shell effects.Methods: We use the finite-temperature superfluid nuclear density functional theory. We consider two Skyrme energy density functionals: a traditional functional SkM${}^{*}$ and a recent functional UNEDF1 optimized for fission studies.Results: We predict very shallow or no third minima in the potential-energy surfaces of ${}^{232}$Th and ${}^{232}$U. In the lighter Th and U isotopes with $N=136$ and 138, the third minima are better developed. We show that the reflection-asymmetric configurations around the third minimum can be associated with dimolecular states involving the spherical doubly magic ${}^{132}$Sn and a lighter deformed Zr or Mo fragment. The potential-energy surfaces for ${}^{228,232}$Th and ${}^{232}$U at several excitation energies are presented. We also study isotopic chains to demonstrate the evolution of the depth of the third minimum with neutron number.Conclusions: We show that the neutron shell effect that governs the existence of the dimolecular states around the third minimum is consistent with the spherical-to-deformed shape transition in the Zr and Mo isotopes around $N=58$. We demonstrate that the depth of the third minimum is sensitive to the excitation energy of the nucleus. In particular, the thermal reduction of pairing, and related enhancement of shell effects, at small excitation energies help to develop deeper third minima. At large excitation energies, shell effects are washed out and third minima disappear altogether.

Reflection-asymmetric nuclear deformations within the Density Functional Theory

Within the nuclear density functional theory (DFT) we study the effect of reflection-asymmetric shapes on ground-state binding energies and binding energy differences. To this end, we developed the new DFT solver AXIALHFB that uses an approximate second-order gradient to solve the Hartree-Fock-Bogoliubov equations of superconducting DFT with the quasi-local Skyrme energy density functionals. Illustrative calculations are carried out for even-even isotopes of radium and thorium.