Monte Carlo Shell Model Calculations Research Articles

First spectroscopic study of odd-odd Cu78

Nuclei in the vicinity of Ni78 are important benchmarks for nuclear structure, which can reveal changes in the shell structure far from stability. Spectroscopy of the odd-odd isotope Cu78 was performed for the first time in an experiment with the EURICA setup at the Radioactive Isotope Beam Factory at RIKEN Nishina Center. Excited states in the neutron-rich isotope were populated following the β decay of Ni78 produced by in-flight fission and separated by the BigRIPS separator. A level scheme based on the analysis of γ−γ coincidences is presented. Tentative spin and parity assignments were made when possible based on the β-decay feeding intensities and γ-decay properties of the excited states. Time correlations between β and γ decay show clear indications of an isomeric state with a half-life of 3.8(4) ms. Large-scale Monte Carlo shell-model calculations were performed using the A3DA-m interaction and a valence space comprising the full fp shell and the 1g9/2 and 2d5/2 orbitals for both protons and neutrons. The comparison of the experimental results with the shell-model calculations allows interpreting the excited states in terms of spin multiplets arising from the proton-neutron interaction. The results provide further insight into the evolution of the proton single-particle orbitals as a function of neutron number, and quantitative information about the proton-neutron interaction outside the doubly magic Ni78 core.Received 6 October 2022Revised 3 January 2023Accepted 9 January 2023DOI:https://doi.org/10.1103/PhysRevC.107.044301©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBeta decayEnergy levels & level densitiesIsomer decaysNuclear spin & parityNuclear structure & decaysNucleon-nucleon interactionsShell modelProperties59 ≤ A ≤ 89Nuclear Physics

Two-particle transfer processes as a signature of shape phase transition in Zirconium isotopes

We explore two-particle transfer reactions as a crucial probe of the occurrence of shape coexistence in shape phase transitions. The (t,p) reactions to the ground state and to excited 0+ states are calculated for the isotope chain of even-even Zirconium isotopes starting from stable nuclei up to beyond current experimental limits. Two-particle spectroscopic factors derived from Monte Carlo Shell Model calculations are used, together with the sequential description of the two-particle transfer reaction mechanism. The calculation shows a clear signature for a shape phase transition between 98Zr and 100Zr, which displays coexistence of a deformed ground state with an excited spherical 0+ state. Furthermore, we show that there is a qualitative difference with respect to the case of a normal shape phase transition that can be discriminated with two-neutron transfer reactions.

Study of the N=32 and N=34 Shell Gap for Ti and V by the First High-Precision Multireflection Time-of-Flight Mass Measurements at BigRIPS-SLOWRI.

The atomic masses of ^{55}Sc, ^{56,58}Ti, and ^{56-59}V have been determined using the high-precision multireflection time-of-flight technique. The radioisotopes have been produced at RIKEN's Radioactive Isotope Beam Factory (RIBF) and delivered to the novel designed gas cell and multireflection system, which has been recently commissioned downstream of the ZeroDegree spectrometer following the BigRIPS separator. For ^{56,58}Ti and ^{56-59}V, the mass uncertainties have been reduced down to the order of 10keV, shedding new light on the N=34 shell effect in Ti and V isotopes by the first high-precision mass measurements of the critical species ^{58}Ti and ^{59}V. With the new precision achieved, we reveal the nonexistence of the N=34 empirical two-neutron shell gaps for Ti and V, and the enhanced energy gap above the occupied νp_{3/2} orbit is identified as a feature unique to Ca. We perform new MonteCarlo shell model calculations including the νd_{5/2} and νg_{9/2} orbits and compare the results with conventional shell model calculations, which exclude the νg_{9/2} and the νd_{5/2} orbits. The comparison indicates that the shell gap reduction in Ti is related to a partial occupation of the higher orbitals for the outer two valence neutrons at N=34.

Multistep Coulomb excitation of Ni64 : Shape coexistence and nature of low-spin excitations

The structure of $^{64}\mathrm{Ni}$, the heaviest stable Ni isotope, has been investigated via high-statistics, multistep safe Coulomb excitation to search for shape coexistence, a phenomenon recently observed in neutron-rich $^{66}\mathrm{Ni}$ and $^{70}\mathrm{Ni}$ as well as in doubly magic, $N=40, ^{68}\mathrm{Ni}$. The study was motivated by recent, state-of-the-art Monte Carlo shell-model calculations (MCSM), where a Hamiltonian with effective interactions incorporating the monopole tensor force predicts the existence of shape coexistence, also in the lower-mass $^{62,64}\mathrm{Ni}$ isotopes. A set of transition and static $E2$ matrix elements for both yrast and near-yrast structures was extracted from the differential Coulomb excitation cross sections. From comparisons between the new results and MCSM as well as other shell-model calculations, a clearer picture of the structure of $^{64}\mathrm{Ni}$ emerges. Specifically, the low-spin states are shown to be dominated by proton and neutron excitations mainly within the $fp$ shell, with minimal contribution from the ${g}_{9/2}$ shape-driving neutron orbital. The agreement between experimental data and MCSM results indicates a small oblate deformation for the ${0}_{2}^{+}$ level and a spherical shape for the ${0}_{3}^{+}$ state. In addition, the small upper limit determined for the $B(E2)$ probability of a transition associated with the decay of the recently observed 3463-keV, ${0}_{4}^{+}$ state agrees with its proposed assignment to a prolate shape, herewith providing first evidence for triple shape coexistence in a stable Ni isotope.

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.

Detailed low-spin spectroscopy of Ni65 via neutron capture reaction

An extended investigation of the low-spin structure of the $^{65}$Ni nucleus was performed at the Institut Laue-Langevin, Grenoble, via the neutron capture reaction $^{64}$Ni(n,$\gamma$)$^{65}$Ni, using the FIPPS HPGe array. The level scheme of $^{65}$Ni was significantly expanded, with 2 new levels and 87 newly found transitions. Angular correlation analyses were also performed, allowing us to assign spins and parities for a number of states, and to determine multipolarity mixing ratios for selected $\gamma$ transitions. The low-energy part of the experimental level scheme (up to about 1.4 MeV) was compared with Monte Carlo Shell Model calculations, which predict spherical shapes for all states, apart from the 9/2$^+$ and the second excited 1/2$^-$ states of oblate deformation.

β decay of Ni75 and the systematics of the low-lying level structure of neutron-rich odd- A Cu isotopes

Background: Detailed spectroscopy of neutron-rich odd-A Cu isotopes is of great importance for studying the shell evolution in the region of Ni78. While there is experimental information on excited states in Cu69−73,77,79 isotopes, the information concerning Cu75 is very limited. Purpose: Experimentally observed single-particle, core-coupling, and proton-hole intruder states in Cu75, will complete the systematics of these states in the chain of isotopes. Method: Excited states in Cu75 were populated in the β decay of Ni75 isotopes. The Ni nuclei were produced by the in-flight fission of U238 projectiles, and were separated, identified, and implanted in a highly segmented Si detector array for the detection of the β-decay electrons. The β-delayed γ rays were detected in a HPGe cluster array. Monte Carlo shell model calculations were performed using the A3DA interaction built on the pfg9/2d5/2 model space for both neutrons and protons. Results: A level scheme of Cu75 was built up to ≈4 MeV by performing a γ-γ coincidence analysis. The excited states below 2 MeV were interpreted based on the systematics of neutron-rich odd-A Cu isotopes and the results of the shell model calculations. Conclusions: The evolution of the single-particle, core-coupling, and proton-hole intruder states in the chain of neutron-rich odd-A Cu isotopes is discussed in the present work, in connection with the newly observed level structure of Cu75.

Shape-coexistence Studies in the Ni Isotopic Chain by Using the Selectivity of Different Reaction Mechanisms

We report on the investigation of the shape coexistence phenomenon in the Ni isotopic chain, from A = 62 to A = 66, by using γ-ray spectroscopy techniques and different reaction mechanisms, such as sub-Coulomb barrier transfer reactions and thermal-neutron capture. Our aim is to understand, from the microscopic point of view, the appearance of nuclear deformation in Ni isotopes at low excitation energy. A series of experiments was performed at the Tandem Accelerator Laboratory in Bucharest, at ALTO IPN-Orsay and at the ILL reactor in Grenoble. Various mean-field theoretical approaches, as well as recent state-of-the-art Monte Carlo Shell Model (MCSM) calculations, predict in 66Ni a deep secondary prolate minimum in the nuclear potential energy surface at spin zero, resulting in a hindered electromagnetic decay towards the spherical ground state (i.e., with an E2 transition probability less than 1 W.u.). This has been confirmed in the first experiment performed in Bucharest. Less pronounced prolate minima, at higher excitation energies, are also expected in lighter neutron-rich Ni isotopes, from state-of-the-art Monte Carlo Shell Model (MCSM) calculations. Preliminary results are discussed for 62Ni and 64Ni, in comparison with theory predictions.

Revealing microscopic origins of shape coexistence in the Ni isotopic chain

In a two–neutron transfer experiment, performed in Bucharest in July 2016 at sub–Coulomb barrier energy,a photon decay hindered – solely – by a nuclear shape change was identified in the 66Ni nucleus. Such a rare process, at spin zero, was clearly observed before only in actinide nuclei in the 1970’s,where fission isomers were found. The experimental findings on 66Ni have been well reproduced by the Monte Carlo Shell Model Calculations of the Tokyo group, which predict a multifaceted scenario of coexistence of spherical, oblate and prolate shapes in neutron–rich Ni isotopes. The results on 66Ni encouraged a comprehensive gamma–spectroscopy investigation of neutron–rich Ni isotopes, in particular 62Ni and 64–Ni, at IFIN–HH (Bucharest), IPN Orsay and ILL (Grenoble), employing different reaction mechanisms to pin down the wave function composition of selected excited states. The aim is to shed light on the microscopic origin of deformation in neutron–rich Ni nuclei, possibly locating other examples of “shape–isomer–like” structures inthis region.

Shape staggering of midshell mercury isotopes from in-source laser spectroscopy compared with density-functional-theory and Monte Carlo shell-model calculations

Neutron-deficient $^{177-185}$Hg isotopes were studied using in-source laser resonance-ionization spectroscopy at the CERN-ISOLDE radioactive ion-beam facility, in an experiment combining different detection methods tailored to the studied isotopes. These include either alpha-decay tagging or Multi-reflection Time-of-Flight gating to identify the isotopes of interest. The endpoint of the odd-even nuclear shape staggering in mercury was observed directly by measuring for the first time the isotope shifts and hyperfine structures of $^{177-180}$Hg. Changes in the mean-square charge radii for all mentioned isotopes, magnetic dipole and electric quadrupole moments of the odd-A isotopes and arguments in favor of $I = 7/2$ spin assignment for $^{177,179}$Hg were deduced. Experimental results are compared with Density Functional Theory (DFT) and Monte-Carlo Shell Model (MCSM) calculations. DFT calculations with several Skyrme parameterizations predict a large jump in the charge radius around the neutron $N = 104$ mid shell, with an odd-even staggering pattern related to the coexistence of nearly-degenerate oblate and prolate minima. This near-degeneracy is highly sensitive to many aspects of the effective interaction, a fact that renders perfect agreement with experiment out of reach for current functionals. Despite this inherent diffculty, the SLy5s1 and a modified UNEDF1^{SO} parameterization predict a qualitatively correct staggering that is off by two neutron numbers. MCSM calculations of states with the experimental spins and parities show good agreement for both electromagnetic moments and the observed charge radii. A clear mechanism for the origin of shape staggering within this context is identified: a substantial change in occupancy of the proton $\pi h_{9/2}$ and neutron $\nu i_{13/2}$ orbitals.

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.

β− decay study of the Mn66−Fe66−Co66−Ni66 decay chain

Background: Shell evolution can impact the structure of the nuclei and lead to effects such as shape coexistence. The nuclei around $^{68}$Ni represent an excellent study case, however, spectroscopic information of the neutron-rich, $Z<28$ nuclei is limited. Purpose: The goal is to measure $\gamma$-ray transitions in $^{66}$Fe, $^{66}$Co and $^{66}$Ni populated in the $\beta^-$ decay of $^{66}$Mn, to determine absolute $\beta$-feedings and relative $\gamma$-decay probabilities and to compare the results with Monte Carlo Shell Model calculations in order to study the influence of the relevant single neutron and proton orbitals occupancies around $Z=28$ and $N=40$. Method: The low-energy structures of $^{65,66}$Fe, $^{66}$Co and $^{66}$Ni were studied in the $\beta^-$ decay of $^{66}$Mn produced at ISOLDE, CERN. The beam was purified by means of laser resonance ionization and mass separation. The $\beta$ and $\gamma$ events detected by three plastic scintillators and two MiniBall cluster germanium detectors, respectively, were correlated in time to build the low-energy excitation schemes and to determine the $\beta$-decay half-lives of the nuclei. Results: The relative small $\beta$-decay ground state feeding of $^{66}$Fe obtained in this work is at variant to the earlier studies. Spin and parity $1^+$ was assigned to the $^{66}$Co ground state based on the strong ground state feeding in the decay of $^{66}$Fe as well as in the decay of $^{66}$Co. Experimental log(ft) values, $\gamma$-ray deexcitation patterns and energies of excited states were compared to Monte Carlo Shell Model calculations. Based on this comparison, spin and parity assignments for the selected number of low-lying states in the $^{66}$Mn to $^{66}$Ni chain were proposed. Conclusions: The $\beta$-decay chain starting $^{66}$Mn towards $^{66}$Ni, crossing $N=40$, evolves from deformed nuclei to sphericity...

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 MonteCarlo 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.

Revised B(E3) transition rate and structure of the 3− level in 96Zr

The B(E3) transition strength from the 1897-keV, 3− level in 96Zr has been reevaluated from six high-statistics, independent data sets. The measured value of 42(3) W.u. is significantly lower than that adopted in recent compilations. It is, however, in line with the global systematics of collective B(E3) rates found throughout the periodic table. Thus, the “exceptional” character of this transition, that challenged theory, no longer applies. Monte Carlo shell-model calculations indicate that the collectivity of the octupole vibration arises from both proton and neutron excitations involving a large number of orbitals.

Characterization of the shape-staggering effect in mercury nuclei

In rare cases, the removal of a single proton (Z) or neutron (N) from an atomic nucleus leads to a dramatic shape change. These instances are crucial for understanding the components of the nuclear interactions that drive deformation. The mercury isotopes (Z = 80) are a striking example1,2: their close neighbours, the lead isotopes (Z = 82), are spherical and steadily shrink with decreasing N. The even-mass (A = N + Z) mercury isotopes follow this trend. The odd-mass mercury isotopes 181,183,185Hg, however, exhibit noticeably larger charge radii. Due to the experimental difficulties of probing extremely neutron-deficient systems, and the computational complexity of modelling such heavy nuclides, the microscopic origin of this unique shape staggering has remained unclear. Here, by applying resonance ionization spectroscopy, mass spectrometry and nuclear spectroscopy as far as 177Hg, we determine 181Hg as the shape-staggering endpoint. By combining our experimental measurements with Monte Carlo shell model calculations, we conclude that this phenomenon results from the interplay between monopole and quadrupole interactions driving a quantum phase transition, for which we identify the participating orbitals. Although shape staggering in the mercury isotopes is a unique and localized feature in the nuclear chart, it nicely illustrates the concurrence of single-particle and collective degrees of freedom at play in atomic nuclei.

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 MonteCarlo 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.

Skyrme density functional description of the double magic Ni78 nucleus

We calculate the single particle spectrum of the double magic nucleus $^{78}$Ni in a Hartree-Fock approach using the Skyrme density-dependent effective interaction containing central, spin-orbit and tensor parts. We show that the tensor part has an important effect on the spin-orbit splitting of the proton $1f$ orbit which may explain the survival of magicity so far from the stability valley. We confirm the inversion of the $1f5/2$ and $2p3/2$ levels at the neutron number 48 in the Ni isotopic chain expected from previous Monte Carlo shell model calculations and supported by experimental observation.

Persistence of the Z=28 Shell Gap Around ^{78}Ni: First Spectroscopy of ^{79}Cu.

In-beam γ-ray spectroscopy of ^{79}Cu is performed at the Radioactive Isotope Beam Factory of RIKEN. The nucleus of interest is produced through proton knockout from a ^{80}Zn beam at 270  MeV/nucleon. The level scheme up to 4.6MeV is established for the first time and the results are compared to MonteCarlo shell-model calculations. We do not observe significant knockout feeding to the excited states below 2.2MeV, which indicates that the Z=28 gap at N=50 remains large. The results show that the ^{79}Cu nucleus can be described in terms of a valence proton outside a ^{78}Ni core, implying the magic character of the latter.

Nucleon correlations and the structure of [formula omitted]

The structure of 71Zn was investigated by one-neutron transfer and heavy-ion induced complex (deep-inelastic) reactions using the GRETINA-CHICO2 and the Gammasphere setups, respectively. The observed inversion between the 9/2+ and 1/2− states is explained in terms of the role of neutron pairing correlations. Non-collective sequences of levels were delineated above the 9/2+ isomeric state. These are interpreted as being associated with a modest oblate deformation in the framework of Monte-Carlo shell-model calculations carried out with the A3DA-m Hamiltonian in the pfg9/2d5/2 valence space. Similarities with the structure of Ni402868 were observed and the shape-coexistence mechanism in the N=40 region of neutron-rich nuclei is discussed in terms of the so-called Type-II shell evolution, with an emphasis on proton–neutron correlations between valence nucleons, especially those involving the shape-driving g9/2 neutron orbital.

Particle-number projection in the finite-temperature mean-field approximation

Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical deformed nucleus $^{162}$Dy, a typical spherical nucleus $^{148}$Sm, and a transitional nucleus $^{150}$Sm in which the pairing condensate is deformed. We compare the results of this projection with results from the saddle-point approximation and exact shell model Monte Carlo calculations. We find that the approximate canonical HF entropy in the particle-number projection decreases monotonically to zero in the limit when the temperature goes to zero. However, in a nucleus with a strong pairing condensate, the approximate canonical HFB entropy in the particle-number projection decreases monotonically to a negative value, reflecting the violation of particle-number conservation. Computationally, the exact particle-number projection is more efficient than calculating the derivatives required in the saddle-point approximation.