Quantum Phase Transition in the Shape of Zr isotopes

Demonstration of nuclear gamma-ray polarimetry based on a multi-layer CdTe Compton camera

To detect and track structural changes in atomic nuclei, the systematic study of nuclear levels with firm spin-parity assignments is important. While linear polarization measurements have been applied to determine the electromagnetic character of gamma-ray transitions, the applicable range is strongly limited due to the low efficiency of the detection system. The multi-layer Cadmium-Telluride (CdTe) Compton camera can be a state-of-the-art gamma-ray polarimeter for nuclear spectroscopy with the high position sensitivity and the detection efficiency. We demonstrated the capability to operate this detector as a reliable gamma-ray polarimeter by using polarized 847-keV gamma rays produced by the [Formula: see text]([Formula: see text]) reaction. By combining the experimental data and simulated calculations, the modulation curve for the gamma ray was successfully obtained. A remarkably high polarization sensitivity was achieved, compatible with a reasonable detection efficiency. Based on the obtained results, a possible future gamma-ray polarimetery is discussed.

α-Clustering in atomic nuclei from first principles with statistical learning and the Hoyle state character

A long-standing crucial question with atomic nuclei is whether or not α clustering occurs there. An α particle (helium-4 nucleus) comprises two protons and two neutrons, and may be the building block of some nuclei. This is a very beautiful and fascinating idea, and is indeed plausible because the α particle is particularly stable with a large binding energy. However, direct experimental evidence has never been provided. Here, we show whether and how α(-like) objects emerge in atomic nuclei, by means of state-of-the-art quantum many-body simulations formulated from first principles, utilizing supercomputers including K/Fugaku. The obtained physical quantities exhibit agreement with experimental data. The appearance and variation of the α clustering are shown by utilizing density profiles for the nuclei beryllium-8, -10 and carbon-12. With additional insight by statistical learning, an unexpected crossover picture is presented for the Hoyle state, a critical gateway to the birth of life.

Emerging Concepts in Nuclear Structure Based on the Shell Model

Some emerging concepts of nuclear structure are overviewed. (i) Background: the many-body quantum structure of atomic nucleus, a complex system comprising protons and neutrons (called nucleons collectively), has been studied largely based on the idea of the quantum liquid (à la Landau), where nucleons are quasiparticles moving in a (mean) potential well, with weak “residual” interactions between nucleons. The potential is rigid in general, although it can be anisotropic. While this view was a good starting point, it is time to look into kaleidoscopic aspects of the nuclear structure brought in by underlying dynamics and nuclear forces. (ii) Methods: exotic features as well as classical issues are investigated from fresh viewpoints based on the shell model and nucleon–nucleon interactions. The 70-year progress of the shell–model approach, including effective nucleon–nucleon interactions, enables us to do this. (iii) Results: we go beyond the picture of the solid potential well by activating the monopole interactions of the nuclear forces. This produces notable consequences in key features such as the shell/magic structure, the shape deformation, the dripline, etc. These consequences are understood with emerging concepts such as shell evolution (including type-II), T-plot, self-organization (for collective bands), triaxial-shape dominance, new dripline mechanism, etc. The resulting predictions and analyses agree with experiment. (iv) Conclusion: atomic nuclei are surprisingly richer objects than initially thought.

Single-Particle and Collective Structures in Neutron-Rich Sr Isotopes

Neutron-rich Sr nuclei around N=60 exhibit a sudden shape transition from a spherical ground state to strongly prolate-deformed. Recently, much new insight into the structure of Sr isotopes in this region has been gained through experimental studies of the excited levels, transition strengths, and spectroscopic factors. In this work, a “classic” shell model description of strontium isotopes from N=50 to N=58 is provided, using a natural valence space outside the 78Ni core. Both even–even and even–odd isotopes are addressed. In particular, spectroscopic factors are computed to shed more light on the structure of low-energy excitations and their evolution along the Sr chain. The origin of deformation at N=60 is mentioned in the context of the present and previous shell model and Monte Carlo shell model calculations.

Effects of the Tensor Force on the Ground Properties of Zr Isotopes

The effects of the tensor force on the ground properties of Zr isotopes are studied in the framework of the Skyrme–Hartree–Fock approach. It is found that the tensor force strongly affects the ground state energies and the geometric symmetry properties, in particular for those isotopes near N=60 region. The effects are attributed to the fact that the tensor force enlarges the spin and pseudospin symmetry breaking and therefore results in a ∼2 MeV sub-shell gap between d3/2 and s1/2 single-particle levels.

Shape Coexistence at Zero Spin in ^{64}Ni Driven by the Monopole Tensor Interaction

The low-spin structure of the semimagic ^{64}Ni nucleus has been considerably expanded: combining four experiments, several 0^{+} and 2^{+} excited states were identified below 4.5 MeV, and their properties established. The Monte Carlo shell model accounts for the results and unveils an unexpectedly complex landscape of coexisting shapes: a prolate 0^{+} excitation is located at a surprisingly high energy (3463 keV), with a collective 2^{+} state 286 keV above it, the first such observation in Ni isotopes. The evolution in excitation energy of the prolate minimum across the neutron N=40 subshell gap highlights the impact of the monopole interaction and its variation in strength with N.

Metastable States of ^{92,94}Se: Identification of an Oblate K Isomer of ^{94}Se and the Ground-State Shape Transition between N=58 and 60

Here we present new information on the shape evolution of the very neutron-rich ^{92,94}Se nuclei from an isomer-decay spectroscopy experiment at the Radioactive Isotope Beam Factory at RIKEN. High-resolution germanium detectors were used to identify delayed γ rays emitted following the decay of their isomers. New transitions are reported extending the previously known level schemes. The isomeric levels are interpreted as originating from high-K quasineutron states with an oblate deformation of β∼0.25, with the high-K state in ^{94}Se being metastable and K hindered. Following this, ^{94}Se is the lowest-mass neutron-rich nucleus known to date with such a substantial K hindrance. Furthermore, it is the first observation of an oblate K isomer in a deformed nucleus. This opens up the possibility for a new region of K isomers at low Z and at oblate deformation, involving the same neutron orbitals as the prolate orbitals within the classic Z∼72 deformed hafnium region. From an interpretation of the level scheme guided by theoretical calculations, an oblate deformation is also suggested for the ^{94}Se_{60} ground-state band.

g Factor of the ^{99}Zr (7/2^{+}) Isomer: Monopole Evolution in the Shape-Coexisting Region

The gyromagnetic factor of the low-lying E=251.96(9)  keV isomeric state of the nucleus ^{99}Zr was measured using the time-dependent perturbed angular distribution technique. This level is assigned a spin and parity of J^{π}=7/2^{+}, with a half-life of T_{1/2}=336(5)  ns. The isomer was produced and spin aligned via the abrasion-fission of a ^{238}U primary beam at RIKEN RIBF. A magnetic moment |μ|=2.31(14)μ_{N} was deduced showing that this isomer is not single particle in nature. A comparison of the experimental values with interacting boson-fermion model IBFM-1 results shows that this state is strongly mixed with a main νd_{5/2} composition. Furthermore, it was found that monopole single-particle evolution changes significantly with the appearance of collective modes, likely due to type-II shell evolution.

Underlying Structure of Collective Bands and Self-Organization in Quantum Systems

The underlying structure of low-lying collective bands of atomic nuclei is discussed from a novel perspective on the interplay between single-particle and collective degrees of freedom, by utilizing state-of-the-art configuration interaction calculations on heavy nuclei. Besides the multipole components of the nucleon-nucleon interaction that drive collective modes forming those bands, the monopole component is shown to control the resistance against such modes. The calculated structure of ^{154}Sm corresponds to the coexistence between prolate and triaxial shapes, while that of ^{166}Er exhibits a deformed shape with a strong triaxial instability. Both findings differ from traditional views based on β/γ vibrations. The formation of collective bands is shown to be facilitated from a self-organization mechanism.

Intertwined quantum phase transitions in the Zr chain

We introduce the notion of intertwined quantum phase transitions (IQPTs), for which a crossing of two configurations coexists with a pronounced shape-evolution of each configuration. A detailed analysis in the framework of the interacting boson model with configuration mixing, provides evidence for this scenario inthe Zr isotopes. The latter exhibit a normal configuration which remains spherical along the chain, but exchanges roles with an intruder configuration, which undergoes first a spherical to prolate-deformed [U(5)→SU(3)] QPT and then a crossover to γ-unstable [SU(3)→SO(6)].

Shape transitions between and within Zr isotopes

The Zirconium isotopes across the N=56,58 neutron sub­shell closures have been of special interest since years, sparked by the near doubly-magic features of 96Zr and the subsequent rapid onset of collectivity with a deformed ground-state structure already in 100Zr. Recent state-of-the-art shell model approaches did not only correctly describe this shape-phase transition in the Zr isotopic chain, but alsothe coexistence of non-collective structures and pronounced collectivity especially in 96,98Zr. Theisotope 98Zr is located on the transition from spherical to deformed ground state structures. We summarize recent experimental work to obtain the B(E2) excitation strengths of the first 2+ state of98Zr, including a new experiment employing the recoil-distance Doppler-shift method following a two-neutron transfer reaction.

Shape coexistence in 94Zr studied via Coulomb excitation

In recent years, a number of both theoretical and experimental investigations have been performed focusing on the zirconium isotopic chain. In particular, state-of-the-art Monte Carlo shell-model calculations predict shape coexistence in these isotopes. In this context, the 94Zr nucleus, which is believed to possess a nearly spherical ground state, is particularly interesting since the purported deformed structure is basedon the low-lying 02+ state, making it amenable for detailed study. In order to provide definitive conclusionson the shapes of the low-lying states, two complementary experiments to study 94Zr by means of low-energy Coulomb excitation were performed. This data will allow the quadrupole moments of the 21,2+ levels to be extracted as well as for the deformation parameters of the 01,2+ states to be determined and, thus, definitive conclusions to be drawn on the role of shape coexistence in this nucleus for the first time.

The first experiment was performed at the INFN Legnaro National Laboratory with the GALILEO-SPIDER setup, which, for the first time, was coupled with 6 lanthanum bromide scintillators (LaBr3:Ce) in order to maximize the γ-ray detection effciency. The second experiment was performed at the Maier-Leibnitz Laboratory (MLL) in Munich and used a Q3D magnetic spectrograph to detect the scattered 12C ions following Coulomb excitation of 94Zr targets.

Enhanced Quadrupole and Octupole Strength in Doubly Magic ^{132}Sn

The first 2^{+} and 3^{-} states of the doubly magic nucleus ^{132}Sn are populated via safe Coulomb excitation employing the recently commissioned HIE-ISOLDE accelerator at CERN in conjunction with the highly efficient MINIBALL array. The ^{132}Sn ions are accelerated to an energy of 5.49  MeV/nucleon and impinged on a ^{206}Pb target. Deexciting γ rays from the low-lying excited states of the target and the projectile are recorded in coincidence with scattered particles. The reduced transition strengths are determined for the transitions 0_{g.s.}^{+}→2_{1}^{+}, 0_{g.s.}^{+}→3_{1}^{-}, and 2_{1}^{+}→3_{1}^{-} in ^{132}Sn. The results on these states provide crucial information on cross-shell configurations which are determined within large-scale shell-model and Monte Carlo shell-model calculations as well as from random-phase approximation and relativistic random-phase approximation. The locally enhanced B(E2;0_{g.s.}^{+}→2_{1}^{+}) strength is consistent with the microscopic description of the structure of the respective states within all theoretical approaches. The presented results of experiment and theory can be considered to be the first direct verification of the sphericity and double magicity of ^{132}Sn.

Evidence for Coexisting Shapes through Lifetime Measurements in ^{98}Zr

The lifetimes of the first excited 2^{+}, 4^{+}, and 6^{+} states in ^{98}Zr were measured with the recoil-distance Doppler shift method in an experiment performed at GANIL. Excited states in ^{98}Zr were populated using the fission reaction between a 6.2  MeV/u ^{238}U beam and a ^{9}Be target. The γ rays were detected with the EXOGAM array in correlation with the fission fragments identified by mass and atomic number in the VAMOS++ spectrometer. Our result shows a very small B(E2;2_{1}^{+}→0_{1}^{+}) value in ^{98}Zr, thereby confirming the very sudden onset of collectivity at N=60. The experimental results are compared to large-scale Monte Carlo shell model and beyond-mean-field calculations. The present results indicate the coexistence of two additional deformed shapes in this nucleus along with the spherical ground state.

Novel Shape Evolution in Sn Isotopes from Magic Numbers 50 to 82

A novel shape evolution in the Sn isotopes by the state-of-the-art application of the Monte Carlo shell model calculations is presented in a unified way for the ^{100-138}Sn isotopes. A large model space consisting of eight single-particle orbits for protons and neutrons is taken with the fixed Hamiltonian and effective charges, where protons in the 1g_{9/2} orbital are fully activated. While the significant increase of the B(E2;0_{1}^{+}→2_{1}^{+}) value, seen around ^{110}Sn as a function of neutron number (N), has remained a major puzzle over decades, it is explained as a consequence of the shape evolution driven by proton excitations from the 1g_{9/2} orbital. A second-order quantum phase transition is found around N=66, connecting the phase of such deformed shapes to the spherical pairing phase. The shape and shell evolutions are thus described, covering topics from the Gamow-Teller decay of ^{100}Sn to the enhanced double magicity of ^{132}Sn.

Nuclear shapes: Quest for triaxiality in 86Ge and the shape of 98Zr

The region of neutron-rich nuclei above the N = 50 magic neutron shell closure encompasses a rich variety of nuclear structure, especially shapeevolutionary phenomena. This can be attributed to the complexity of sub-shell closures, their appearance and disappearance in the region, such as the N = 56 sub shell or Z = 40 for protons. Structural effects reach from a shape phase transition in the Zr isotopes, over shape coexistence between spherical, prolate, and oblate shapes, to possibly rigid triaxial deformation. Recent experiments in this region and their main physics viewpoints are summarized.

Shell Evolution and E2 Collectivity: New Spectroscopic Information

New spectroscopic information on electric quadrupole collectivity has recently been obtained. We report on our identification of a coexisting deformed structure in the quasi-doubly closed-shell nucleus 96Zr from our measurement of a corresponding absolute E2 intraband transition strength in electron scattering reactions and its small mixing matrix element with the spherical ground state structure. The even-mass zirconium isotopes exhibit a first order quantum shape phase transition between neutron numbers 58 and 60. We have used photon scattering reactions for a first measurement of the E2 decay strength of the nuclear 1+ scissors mode. Evidence for its 2+ rotational excitation and its extraordinarily large rotational moment of inertia are presented.

Lifetime measurement in neutron-rich A~100 nuclei

Lifetimes of excited states of the 98;100;102Zr nuclei were measured by using the Generalized Centroid Difference Method. The nuclei of interest were populated via neutron-induced fission of 241Pu and 235U during the EXILL-FATIMA campaign. The obtained lifetimes were used to calculate the B(E2) transition strengths and β deformation parameters which were then compared with the recent theoretical predictions obtained with Monte Carlo Shell Model.

Single-particle states vs. collective modes: friends or enemies ?

The quantum self-organization is introduced as one of the major underlying mechanisms of the quantum many-body systems. In the case of atomic nuclei as an example, two types of the motion of nucleons, single-particle states and collective modes, dominate the structure of the nucleus. The collective mode arises as the balance between the effect of the mode-driving force (e.g., quadrupole force for the ellipsoidal deformation) and the resistance power against it. The single-particle energies are one of the sources to produce such resistance power: a coherent collective motion is more hindered by larger spacings between relevant single particle states. Thus, the single-particle state and the collective mode are “enemies” against each other. However, the nuclear forces are rich enough so as to enhance relevant collective mode by reducing the resistance power by changing single-particle energies for each eigenstate through monopole interactions. This will be verified with the concrete example taken from Zr isotopes. Thus, the quantum self-organization occurs: single-particle energies can be self-organized by (i) two quantum liquids, e.g., protons and neutrons, (ii) monopole interaction (to control resistance). In other words, atomic nuclei are not necessarily like simple rigid vases containing almost free nucleons, in contrast to the naïve Fermi liquid picture. Type II shell evolution is considered to be a simple visible case involving excitations across a (sub)magic gap. The quantum self-organization becomes more important in heavier nuclei where the number of active orbits and the number of active nucleons are larger.

Shape Evolution in Neutron-Rich Krypton Isotopes Beyond N=60: First Spectroscopy of ^{98,100}Kr

We report on the first γ-ray spectroscopy of low-lying states in neutron-rich ^{98,100}Kr isotopes obtained from ^{99,101}Rb(p,2p) reactions at ∼220  MeV/nucleon. A reduction of the 2_{1}^{+} state energies beyond N=60 demonstrates a significant increase of deformation, shifted in neutron number compared to the sharper transition observed in strontium and zirconium isotopes. State-of-the-art beyond-mean-field calculations using the Gogny D1S interaction predict level energies in good agreement with experimental results. The identification of a low-lying (0_{2}^{+}, 2_{2}^{+}) state in ^{98}Kr provides the first experimental evidence of a competing configuration at low energy in neutron-rich krypton isotopes consistent with the oblate-prolate shape coexistence picture suggested by theory.

_{36}^{96}Kr_{60}-Low-Z Boundary of the Island of Deformation at N=60

Prompt γ-ray spectroscopy of the neutron-rich ^{96}Kr, produced in transfer- and fusion-induced fission reactions, has been performed using the combination of the Advanced Gamma Tracking Array and the VAMOS++ spectrometer. A second excited state, assigned to J^{π}=4^{+}, is observed for the first time, and a previously reported level energy of the first 2^{+} excited state is confirmed. The measured energy ratio R_{4/2}=E(4^{+})/E(2^{+})=2.12(1) indicates that this nucleus does not show a well-developed collectivity contrary to that seen in heavier N=60 isotones. This new measurement highlights an abrupt transition of the degree of collectivity as a function of the proton number at Z=36, of similar amplitude to that observed at N=60 at higher Z values. A possible reason for this abrupt transition could be related to the insufficient proton excitations in the g_{9/2}, d_{5/2}, and s_{1/2} orbitals to generate strong quadrupole correlations or to the coexistence of competing different shapes. An unexpected continuous decrease of R_{4/2} as a function of the neutron number up to N=60 is also evidenced. This measurement establishes the Kr isotopic chain as the low-Z boundary of the island of deformation for N=60 isotones. A comparison with available theoretical predictions using different beyond mean-field approaches shows that these models fail to reproduce the abrupt transitions at N=60 and Z=36.

Multifaceted Quadruplet of Low-Lying Spin-Zero States in ^{66}Ni: Emergence of Shape Isomerism in Light Nuclei

A search for shape isomers in the ^{66}Ni nucleus was performed, following old suggestions of various mean-field models and recent ones, based on state-of-the-art Monte Carlo shell model (MCSM), all considering ^{66}Ni as the lightest nuclear system with shape isomerism. By employing the two-neutron transfer reaction induced by an ^{18}O beam on a ^{64}Ni target, at the sub-Coulomb barrier energy of 39 MeV, all three lowest-excited 0^{+} states in ^{66}Ni were populated and their γ decay was observed by γ-coincidence technique. The 0^{+} states lifetimes were assessed with the plunger method, yielding for the 0_{2}^{+}, 0_{3}^{+}, and 0_{4}^{+} decay to the 2_{1}^{+} state the B(E2) values of 4.3, 0.1, and 0.2 Weisskopf units (W.u.), respectively. MCSM calculations correctly predict the existence of all three excited 0^{+} states, pointing to the oblate, spherical, and prolate nature of the consecutive excitations. In addition, they account for the hindrance of the E2 decay from the prolate 0_{4}^{+} to the spherical 2_{1}^{+} state, although overestimating its value. This result makes ^{66}Ni a unique nuclear system, apart from ^{236,238}U, in which a retarded γ transition from a 0^{+} deformed state to a spherical configuration is observed, resembling a shape-isomerlike behavior.

Status and future of nuclear matrix elements for neutrinoless double-beta decay: a review

The nuclear matrix elements that govern the rate of neutrinoless double beta decay must be accurately calculated if experiments are to reach their full potential. Theorists have been working on the problem for a long time but have recently stepped up their efforts as ton-scale experiments have begun to look feasible. Here we review past and recent work on the matrix elements in a wide variety of nuclear models and discuss work that will be done in the near future. Ab initio nuclear-structure theory, which is developing rapidly, holds out hope of more accurate matrix elements with quantifiable error bars.

Are There Signatures of Harmonic Oscillator Shells Far from Stability? First Spectroscopy of ^{110}Zr

The first measurement of the low-lying states of the neutron-rich ^{110}Zr and ^{112}Mo was performed via in-beam γ-ray spectroscopy after one proton removal on hydrogen at ∼200  MeV/nucleon. The 2_{1}^{+} excitation energies were found at 185(11) keV in ^{110}Zr, and 235(7) keV in ^{112}Mo, while the R_{42}=E(4_{1}^{+})/E(2_{1}^{+}) ratios are 3.1(2), close to the rigid rotor value, and 2.7(1), respectively. These results are compared to modern energy density functional based configuration mixing models using Gogny and Skyrme effective interactions. We conclude that first levels of ^{110}Zr exhibit a rotational behavior, in agreement with previous observations of lighter zirconium isotopes as well as with the most advanced Monte Carlo shell model predictions. The data, therefore, do not support a harmonic oscillator shell stabilization scenario at Z=40 and N=70. The present data also invalidate predictions for a tetrahedral ground state symmetry in ^{110}Zr.

First Measurement of Collectivity of Coexisting Shapes Based on Type II Shell Evolution: The Case of ^{96}Zr

BACKGROUND

Type II shell evolution has recently been identified as a microscopic cause for nuclear shape coexistence.

PURPOSE

Establish a low-lying rotational band in ^{96}Zr.

METHODS

High-resolution inelastic electron scattering and a relative analysis of transition strengths are used.

RESULTS

The B(E2;0_{1}^{+}→2_{2}^{+}) value is measured and electromagnetic decay strengths of the 2_{2}^{+} state are deduced.

CONCLUSIONS

Shape coexistence is established for ^{96}Zr. Type II shell evolution provides a systematic and quantitative mechanism to understand deformation at low excitation energies.