Shell-model States Research Articles

Multi-Layered Shell Model and Seismic Limit States of a Containment Building in Nuclear Power Plant Considering Deterioration and Voids

Multi-Layered Shell Model and Seismic Limit States of a Containment Building in Nuclear Power Plant Considering Deterioration and Voids

Octupole vibrational excitations built on shell-model states in Xe, Cs, and Ba isotopes up to mass 142

Octupole vibrational excitations built on shell-model states in Xe, Cs, and Ba isotopes up to mass 142

Generalized molecule-like structure in atomic nuclei

Starting from a |ϕ|4 field theory with universal pn interacting angular and distance coordinates (θ,r) to depict nuclear long-range correlations, we have derived a cubic Schrödinger equation for NpNnpn interacting virtual bosons, which are believed to dominate the generic evolution of nuclear structure from spherical multi-nucleon shell-model states to deformed collective rotational states. It is shown that two-soliton solution in θ space manifests two opposite pn-type Ising-like chains existing in ground states and their gradual alignment responsible for nuclear rotational excitations. Of particular prominence is two-soliton solution with respect to r: the mutual relative motion of valence protons and neutrons around valence-nucleon mass centers considered as the dynamic equilibrium positions of a non-linear Schrödinger breather. And the resulting non-vanishing root mean square 〈r2〉 divides half the valence nucleons belonging to a chain into four local mass centers and proposes two chain-corresponding molecule-like tetrads outside inert cores pairwise hidden in ground states. Such solitons turn out to be an analogue of opposite axion fluids around a local mass center in the galactic dark-matter halo outside the soliton core (Garnier et al. (2021) [67]) and the present study facilitates making the latter more clear. The generalized molecule-like structure arouses a requisite to the addition of cluster-interior and cluster-to-core electric dipole moments in particular to pear-shaped nuclei, whose variants Schiff moments play an important role in measuring diamagnetic atomic electric dipole moments and therefore understanding CP-violating new physics beyond the standard model. We put additional parts in evidence by befitting the NpNn scheme verified already by electric quadrupole moments in same nuclei.

Cluster-shell competition and effect of adding hyperons

Background: The fundamental question is how the hyperon plays a role in the nuclear structure. It is of particular importance, especially in the light mass region, to verify the structure change when $\mathrm{\ensuremath{\Lambda}}$ particle(s) is added to normal nuclei.Purpose: The ground state of $^{8}\mathrm{Be}$ has been known to have a well-developed $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\alpha}$ cluster structure, whereas $^{12}\mathrm{C}$ has a mixed structure of three $\ensuremath{\alpha}$ clusters and $jj$-coupling shell model, where $\ensuremath{\alpha}$ clusters are partially broken. Adding $\mathrm{\ensuremath{\Lambda}}$ particle(s) could induce the structure change. We compare the Be and C cases.Methods: Using the antisymmetrized quasicluster model (AQCM), the $\ensuremath{\alpha}$-cluster states and $jj$-coupling shell-model states of $^{8}\mathrm{Be}$ and $^{12}\mathrm{C}$ are prepared on the same footing, and we add $\mathrm{\ensuremath{\Lambda}}$ particles. The cluster-shell competition in the ground state can be well described with this model. Using AQCM, we calculate $^{8}\mathrm{Be}, _{\mathrm{\ensuremath{\Lambda}}}^{9}\mathrm{Be}, _{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}^{10}\mathrm{Be}, ^{12}\mathrm{C}, _{\mathrm{\ensuremath{\Lambda}}}^{13}\mathrm{C}$, and $_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}^{14}\mathrm{C}$.Results: By adding one or two $\mathrm{\ensuremath{\Lambda}}$ particle(s), the ground state of $^{12}\mathrm{C}$ approaches the $jj$-coupling shell model side. On the other hand, in the Be case, although the $\mathrm{\ensuremath{\Lambda}}$ particle(s) shrinks the $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\alpha}$ distance, the breaking effect of the cluster structure is rather limited.Conclusions: The spin-orbit interaction is the driving force of breaking the $\ensuremath{\alpha}$ clusters, and whether the glue-like effect of $\mathrm{\ensuremath{\Lambda}}$ particle(s) attracts the cluster inside the range of this interaction is crucial. In $_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}^{14}\mathrm{C}$, the breaking of $\ensuremath{\alpha}$ clusters in $^{12}\mathrm{C}$ is much enhanced by the addition of the $\mathrm{\ensuremath{\Lambda}}$ particles than the case of free $^{12}\mathrm{C}$. We also found that breaking $\ensuremath{\alpha}$ clusters in the ground state of $_{\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}}^{14}\mathrm{C}$ affects the excited state with the pure cluster structure.

High-spin states of $$^{204}$$At: isomeric states and shears band structure

High-spin states of neutron-deficient trans-lead nucleus \(^{204}\hbox {At}\) were populated up to \(\sim 8\,\mathrm{MeV}\) excitation through the \(^{12}\hbox {C} + ^{197}\hbox {Au}\) fusion evaporation reaction. Decay of the associated levels through prompt and delayed \(\gamma \)-ray emissions were studied to evaluate the underlying nuclear structure. The level scheme, which was partly known, was extended further. An isomeric \(16^+\) level with observed mean lifetime \(\tau =52 \pm 5\, \mathrm{ns}\), was established from our measurements. Attempts were made to interpret the excited states based on multi quasiparticle and hole structures involving \(2f_{5/2}\), \(1h_{9/2}\), and \(1i_{13/2}\) shell model states, along with moderate core excitation. Magnetic dipole band structure over the spin parity range: \(16^+\)–\(23^+\) was confirmed and evaluated in more detail, including the missing cross-over E2 transitions. Band-crossing along the shears band was observed and compared with the evidence of similar phenomena in the neighbouring \(^{202}\hbox {Bi}\), \(^{205}\hbox {Rn}\) isotones and the \(^{203}\hbox {At}\) isotope. Based on comparison of the measured B(M1)/B(E2) values for transitions along the band with the semiclassical model based estimates, the shears band of \(^{204}\hbox {At}\) was established along with the level scheme.

Role of tensor interaction as salvation of cluster structure in Ti44

Background: The $^{44}$Ti nucleus has been known to have a $^{40}$Ca+$\alpha$ cluster structure, and inversion doublet structure has been observed; however, $\alpha$ cluster structure tends to be washed out when the breaking of the $\alpha$ cluster is allowed due to the spin-orbit interaction. Nevertheless, $\alpha$ clustering in medium-heavy nuclei is quite a hot subject recently. Purpose: The tensor interaction has been known to play an essential role in the strong binding of the $^4$He nucleus, which induces the two-particle-two-hole (2p2h) excitation. Since this excitation is blocked when another nucleus approaches, it is worthwhile to show whether the tensor effect works to keep the distance between $^4$He and $^{40}$Ca and becomes the salvation of the clustering in $^{44}$Ti. Methods: The spin-orbit effect is included in the cluster model by using the antisymmetrized quasi cluster model (AQCM) developed by the authors. We have also developed an improved version of the simplified method to include the tensor contribution ($i$SMT), which allows us to estimate the tensor effect within the cluster model. The competition of these two is investigated in the medium-heavy mass region for the first time. Results: According to AQCM, the spin-orbit interaction completely breaks the $\alpha$ cluster and restores the symmetry of $jj$-coupling shell model when the $\alpha$ cluster approaches the $^{40}$Ca core. On the other hand, $i$SMT gives a large distance between $\alpha$ and $^{40}$Ca due to the tensor effect. Conclusions: In $^{44}$Ti, because of the strong spin-orbit and tensor contributions, two completely different configurations ($jj$-coupling shell model and cluster states) almost degenerate, and their mixing becomes important.

New β -decaying state in Bi214

A new β-decaying state in Bi214 has been identified at the ISOLDE Decay Station at the CERN-ISOLDE facility. A preferred Iπ=(8−) assignment was suggested for this state based on the β-decay feeding pattern to levels in Po214 and shell-model calculations. The half-life of the Iπ=(8−) state was deduced to be T1/2=9.39(10) min. The deexcitation of the levels populated in Po214 by the β decay of this state was investigated via γ−γ coincidences and a number of new levels and transitions was identified. Shell-model calculations for excited states in Bi214 and Po214 were performed using two different effective interactions: the H208 and the modified Kuo-Herling particle interaction. Both calculations agree on the interpretation of the new β-decaying state as an Iπ=8− isomer and allow for tentative assignment of shell-model states to several high-spin states in Po214.Received 12 July 2021Accepted 21 October 2021DOI:https://doi.org/10.1103/PhysRevC.104.054301Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBeta decayElectromagnetic transitionsLifetimes & widthsNuclear structure & decaysShell modelProperties190 ≤ A ≤ 219Nuclear Physics

Neutron-unbound states in Ne31

Background: The island of inversion near the $N=20$ shell gap is home to nuclei with reordered single-particle energy levels compared with the spherical shell model. Studies of $^{31}\mathrm{Ne}$ have revealed that its ground state has a halo component characterized by a valence neutron orbiting a deformed $^{30}\mathrm{Ne}$ core. This lightly bound nucleus with a separation energy of only ${S}_{n}=170$ keV is expected to have excited states that are neutron unbound.Purpose: The purpose of this experiment was to investigate the low-lying excited states in $^{31}\mathrm{Ne}$ that decay by the emission of a single neutron.Methods: An 89 MeV/nucleon $^{33}\mathrm{Mg}$ beam impinged on a segmented Be reaction target. Neutron-unbound states in $^{31}\mathrm{Ne}$ were populated via a two-proton knockout reaction. The $^{30}\mathrm{Ne}$ fragment and associated neutron from the decay of $^{31}\mathrm{Ne}$ were detected by the MoNA-LISA-Sweeper experimental setup at the National Superconducting Cyclotron Laboratory. Invariant-mass spectroscopy was used to reconstruct the two-body decay energy ($^{30}\mathrm{Ne}+n$).Results: The two-body decay energy spectrum exhibits two features: a low-lying peak at $0.30\ifmmode\pm\else\textpm\fi{}0.17$ MeV and a broad enhancement at $1.50\ifmmode\pm\else\textpm\fi{}0.33$ MeV, each fit with an energy-dependent asymmetric Breit-Wigner lineshape representing a resonance in the continuum. Accompanying shell-model calculations using the FSU interaction within NuShellX, combined with cross-section calculations using the eikonal reaction theory, indicate that these peaks in the decay energy spectrum are caused by multiple resonant states in $^{31}\mathrm{Ne}$.Conclusions: Excited states in $^{31}\mathrm{Ne}$ were observed for the first time. Transitions from calculated shell-model final states in $^{31}\mathrm{Ne}$ to bound states in $^{30}\mathrm{Ne}$ are in good agreement with the measured decay energy spectrum. Cross-section calculations for the two-proton knockout populating $^{31}\mathrm{Ne}$ states as well as spectroscopic factors pertaining to the decay of $^{31}\mathrm{Ne}$ into $^{30}\mathrm{Ne}$ are used to examine the results within the context of the shell-model expectations.

α -cluster formation and decay: The role of shell structure

Shell structure effects on $\ensuremath{\alpha}$-cluster formation and decay are studied by using the quartetting wave function approach (QWFA). Both the intrinsic and center-of-mass (c.o.m.) motions of an $\ensuremath{\alpha}$ cluster inside a core nucleus are investigated with different contributing shell model wave functions. The overlap between intrinsic wave functions of four nucleons in the $\ensuremath{\alpha}$-like quartet state and in a free $\ensuremath{\alpha}$ particle is analyzed in detail. The change of the effective potential describing the c.o.m. motion of the quartet is explicitly shown for the major shell closures $Z=82$ and $N=126$. It is found that both the $\ensuremath{\alpha}$-cluster formation probability and the half-life are sensitive to the quartet shell model states. By extending the QWFA calculations from $^{212}\mathrm{Po}$ to $\ensuremath{\alpha}$ emitters with one or two extra nucleons, we show that the bound state and scattering wave functions of the $\ensuremath{\alpha}$ cluster are changed accordingly. The calculated $\ensuremath{\alpha}$-decay half-lives agree nicely with the experimental data.

Microscopic structure and mathematical background of the multiconfigurational dynamical symmetry

Background: The multiconfigurational dynamical symmetry (MUSY) is the common intersection of the shell, collective, and cluster models for the multi-major-shell problem. It is able to describe the spectra of different configurations in different energy windows in a unified way. It is based on some heuristic arguments, related to the connection of the wave functions, and energy eigenvalues. The detailed mathematical background has been worked out so far only for the simplest case of the two binary cluster configurations.Purpose: I intend to construct the exact algebraic framework for the general case of the MUSY, i.e., for any number of configurations and any number of clusters or nucleons. As an illustrative example, the spectrum of different configurations of the $^{44}\mathrm{Ti}$ nucleus is described by a simple Hamiltonian.Methods: Classification schemes defined by different algebra chains need to be combined; in particular, those of the major shell scheme and of the particle index scheme.Results: A class of Hamiltonians, which is invariant under the transformations from one configuration to the other, is determined. In case of the $^{44}\mathrm{Ti}$ low-lying shell-model spectrum, as well as $^{40}\mathrm{Ca}+^{4}\mathrm{He}$ and $^{28}\mathrm{Si}+^{16}\mathrm{O}$ cluster states are obtained in a unified way.Conclusions: The MUSY is based on two pillars: (i) a unified multiplet structure for shell, collective, and cluster model states and (ii) a Hamiltonian which is invariant with respect to the transformations from one configuration to the other.

Updated systematics of intermediate-energy single-nucleon removal cross sections

The body of experimental measurements of intermediate-energy reactions that remove a single nucleon from a secondary beam of neutron- or proton-rich nuclei continues to grow. These data have been analyzed consistently using an approximate, eikonal-model treatment of the reaction dynamics combined with appropriate shell-model descriptions of the projectile initial state, the bound final states spectrum of the reaction residue, and single-particle removal strengths computed from their wave-function overlaps. The systematics of the ratio ${R}_{s}$ of the measured inclusive cross section to all bound final states and the calculated cross section to bound shell-model states---in different regions of the nuclear chart and involving both very weakly bound and strongly bound valence nucleons---is important in relating the empirically deduced orbital occupancies to those from the best available shell-model predictions. Importantly, several new higher-energy measurements, for which the sudden-approximation aspect of the dynamical description is placed on an even stronger footing, now supplement the previously analyzed measurements. These additional data sets are discussed. Their ${R}_{s}$ values are shown to conform to and reinforce the earlier-observed systematics, with no indication that the approximately linear reduction in ${R}_{s}$ with increasing nucleon separation energy is a consequence of a breakdown of the sudden approximation.

Large-scale nuclear shell-model calculations of isotopes in the southwest region of 208Pb

Abstract Large-scale nuclear shell-model calculations are performed in the neutron- and proton-deficient Pt, Au, Hg, and Tl isotopes ($Z < 82$ and $N \le 126$) near $^{208}$Pb. All the single-particle levels in the one-major shells, six neutron ($2p_{1/2}$, $1f_{5/2}$, $2p_{3/2}$, $0i_{13/2}$, $1f_{7/2}$, and $0h_{9/2}$) orbitals and five proton ($2s_{1/2}$, $1d_{3/2}$, $0h_{11/2}$, $1d_{5/2}$, and $0g_{7/2}$) orbitals are considered. For an effective two-body interaction, one set of the multipole pairing, quadrupole–quadrupole interactions is employed for all the nuclei considered. These phenomenological interactions are determined to reproduce the experimental energy spectra. Some of the isomeric states are analyzed in terms of the shell-model configurations. Octupole correlated states are discussed in terms of a collective octupole excitation on top of each shell model state.

Collectivity of the electromagnetic transitions in near-threshold resonances

Mixing of the shell model (SM) eigenstates due to the coupling via the common decay channel influences leads in many cases to the formation of a collective eigenstate which carries many features of the nearby decay channel. This generic mechanism in open quantum systems explains the phenomenological Ikeda diagram and generalizes it for various clusters/correlations in the vicinity of the respective cluster decay thresholds. The near-threshold collectivization of the SM states may also influence their electromagnetic decays. We discuss this phenomenon on the example of B(Eλ) decays of near-threshold 2+ states in 14C.

He8 and Li9 cluster structures in light nuclei

The possibility of the $^8$He and $^{9}$Li clusters in atomic nuclei is discussed. Until now most of the clusters in the conventional models have been limited to the closures of the three-dimensional harmonic oscillators, such as $^4$He, $^{16}$O, and $^{40}$Ca. In the neutron-rich nuclei, however, the neutron to proton ratio is not unity, and it is worthwhile to think about more neutron-rich objects with $N>Z$ as the building blocks of cluster structures. Here the nuclei with the neutron number six, which is the subclosure of the $p_{3/2}$ subshell of the $jj$-coupling shell model, are assumed to be clusters, and thus we study the $^8$He and $^9$Li cluster structures in $^{16}$Be ($^8$He+$^8$He), $^{17}$B ($^8$He+$^9$Li), $^{18}$C ($^9$Li+$^9$Li), and $^{24}$C ($^8$He+$^8$He+$^8$He). Recent progress of the antisymmetrized quasi cluster model (AQCM) enables us to utilize $jj$-coupling shell model wave functions as the clusters rather easily. It is shown that the $^8$He+$^9$Li and $^9$Li+$^9$Li cluster configurations cover the lowest shell-model states of $^{17}$B and $^{18}$C, respectively. To predict the cluster states with large relative distances, we increase the expectation value of the principal quantum numbers by adding the nodes to the lowest states under the condition that the total angular momentum is unchanged (equal to $J^\pi =0$). As a result, developed cluster states are obtained around the corresponding threshold energies. The rotational band structure of $^{24}$C, which reflect the symmetry of equilateral triangular configuration ($D_{3h}$ symmetry) of three $^8$He clusters, also appears around the threshold energy.

Calculating reactions with use of no-core shell-model states

A scheme to compute reactions is described that uses the Slater determinants constructed of oscillator orbitals. Simple linear equations are suggested for this purpose and shown to be efficient in model examples. A universal method to evaluate the required matrix elements is given.

Reflections on the origin of the EMC effect

In the 35 years since the European Muon Collaboration announced the astonishing result that the valence structure of a nucleus was very different from that of a free nucleon, many explanations have been suggested. The first of the two most promising explanations is based upon the different effects of the strong Lorentz scalar and vector mean fields known to exist in a nucleus on the internal structure of the nucleon-like clusters which occupy shell model states. The second links the effect to the modification of the structure of nucleons involved in short-range correlations, which are far off their mass shell. We explore some of the methods which have been proposed to give complementary information on this puzzle, especially the spin-dependent EMC effect and the isovector EMC effect, both proposed by Cloët, Bentz and Thomas. It is shown that the predictions for the spin-dependent EMC effect, in particular, differ substantially within the mean-field and short-range correlation approaches. Hence, the measurement of the spin-dependent EMC effect at Jefferson Lab should give us a deeper understanding of the origin of the EMC effect and, indeed, of the structure of atomic nuclei.

Octupole phonon model based on the shell model for octupole vibrational states

A model is proposed for a description of the octupole vibrational states based on the nuclear shell model. In this model, a one-octupole-phonon representing the collective octupole vibration across the magic core is introduced onto the shell-model states that are microscopically calculated within the valence nucleon space. With this model, a unified description is now possible for the pure shell-model states made by valence nucleons, and the octupole-phonon states excited on the shell-model states. The model is applied to various nuclei around the $^{208}\mathrm{Pb}$ nucleus. Energy levels and electric octupole transition probabilities are well reproduced for both the pure shell-model states and the octupole vibrational states.

Classification of F20 states below 4.4 MeV

Angle-integrated cross sections and forward/backward ratios for the reaction $^{18}\mathrm{O}(^{3}\mathrm{He},p)^{20}\mathrm{F}$ have been combined with existing information in order to classify $^{20}\mathrm{F}$ states up to 4.4 MeV excitation. Experimental counterparts are now known for all 13 ${(sd)}^{4}$ shell-model states in this region.

Effects of cluster–shell competition and BCS-like pairing in 12C

The antisymmetrized quasi-cluster model (AQCM) was proposed to describe {\alpha}-cluster and $jj$-coupling shell models on the same footing. In this model, the cluster-shell transition is characterized by two parameters; $R$ representing the distance between {\alpha} clusters and {\alpha} describing the breaking of {\alpha} clusters, and the contribution of the spin-orbit interaction, very important in the $jj$-coupling shell model, can be taken into account starting with the {\alpha} cluster model wave function. Not only the closure configurations of the major shells, but also the subclosure configurations of the $jj$-coupling shell model can be described starting with the {\alpha}-cluster model wave functions; however, the particle hole excitations of single particles have not been fully established yet. In this study we show that the framework of AQCM can be extended even to the states with the character of single particle excitations. For $^{12}$C, two particle two hole (2p2h) excitations from the subclosure configuration of $0p_{3/2}$ corresponding to BCS-like pairing are described, and these shell model states are coupled with the three {\alpha} cluster model wave functions. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part. We try to pave the way to establish a generalized description of the nuclear structure.

Alpha Decay Width of212Po from a quartetting wave function approach

An effective α particle equation is derived for cases where an α particle is bound to a doubly magic nucleus. As an example, we consider212Po with the α on top of the208Pb core. The fully quantal solution of the problem is inspired by the THSR (Tohsaki-Horiuchi-Schuck-Röpke) wave function concept that has been successfully applied to light nuclei. Shell model calculations are improved by including four-particle (α-like) correlations that are of relevance when the matter density becomes low. In the region where the α-like cluster penetrates the core nucleus, the intrinsic bound state wave function transforms at a critical density into an unbound four-nucleon shell model state. We present a microscopic calculation of both α-cluster preformation probability and decay width in typical α-emitter212Po. Using actually measured density distribution of the208 Pb core, the calculated alpha decay width of212Po agrees fairly well with the measured one. Applications to other nuclei (20Ne) are discussed.