Single-particle Energy Levels Research Articles

Shape evolution of thorium isotopes in between the drip-lines

Commonly encountered shapes in nuclei are spherical, prolate and oblate. The shape of a nucleus is decided by the residual interactions among its nucleons. In our current investigation, we incorporate both axial and triaxial degrees of freedom to study how the shape of thorium nuclei changes within the boundaries of the drip lines. We employed the relativistic mean-field theory and utilized density-dependent meson-exchange as the functional throughout our calculations. We calculated the total binding energy as it varies with the deformation parameter [Formula: see text] and subsequently depicted the resulting binding energy curves. Furthermore, we examined the self-consistent energy surface for thorium isotopes with triaxial quadrupole deformations. The emergence of potential energy minima can be attributed to the presence of gaps or regions with reduced density of single-particle energy levels around the Fermi surface, as the nucleus undergoes deformation. The nuclear shape evolution of thorium isotopes ranging from [Formula: see text] to [Formula: see text] was investigated, [Formula: see text] and [Formula: see text] are spherical in shape and the nearby isotopes are less deformed and nearly spherical. Isotopes ranging from [Formula: see text] to [Formula: see text] exhibit a higher level of deformation. The most deformed nucleus is [Formula: see text] and is prolate. The weak triaxial behavior is observed within the region associated with shape phase transition.

Midgap state requirements for optically active quantum defects

Optically active quantum defects play an important role in quantum sensing, computing and communication. The electronic structure and the single-particle energy levels of these quantum defects in the semiconducting host have been used to understand their optoelectronic properties. Optical excitations that are central for their initialization and readout are linked to transitions between occupied and unoccupied single-particle states. It is commonly assumed that only quantum defects introducing levels well within the band gap and far from the band edges are of interest for quantum technologies as they mimic an isolated atom embedded in the host. In this perspective, we contradict this common assumption and show that optically active defects with energy levels close to the band edges can display similar properties. We highlight quantum defects that are excited through transitions to or from a band-like level (bound exciton) such as the T center and Se Si+ in silicon. We also present how defects such as the silicon split-vacancy in diamond can involve transitions between localized levels that are above the conduction band or below the valence band. Loosening the commonly assumed requirement on the electronic structure of quantum defects offers opportunities in quantum defects design and discovery especially in smaller band gap hosts such as silicon. We discuss the challenges in terms of operating temperature for photoluminescence or radiative lifetime in this regime. We also highlight how these alternative type of defects bring their own needs in terms of theoretical developments and fundamental understanding. This perspective clarifies the electronic structure requirement for quantum defects and will facilitate the identification and design of new color centers for quantum applications especially driven by first principles computations.

Influence of single-particle energy on inclusive electron scattering

Extensive studies have been carried out over the past decades to explore the impacts of nucleon momentum distributions (NMDs) on the inclusive electron scattering process. Nevertheless, there remains limited research about the effects of the single-particle energy level distributions E s.p.(β) on reaction. For a given nucleus, the E s.p.(β) of different deformations exhibits distinct changes. In this paper, we investigate the influences of the E s.p.(β) of distinct deformations on process. Using the plane-wave impulse approximation method, we calculate the cross sections of different deformed configurations. The corresponding E s.p.(β) and NMDs are obtained from the deformed relativistic mean-field model. It is found that the theoretical cross sections show moderate dependence on the E s.p.(β) of distinct deformations. This is due to the conservation of energy, which restricts the number of nucleons participating in the process of diverse deformed configurations. The studies in this paper can provide useful guides for future lepton-nucleus scattering experiments.

Many body density of states of a system of non interacting spinless fermions

The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.

Hartree–Fock–Bogoliubov study of quantum shell effects on the path to fission in ^{180}Hg, ^{236}U and ^{256}Fm

Quantum shell effects stabilising fission fragments with various shapes have been invoked as a factor determining the distribution of nucleons between the fragments at scission. Shell effects also induce asymmetric shapes in the nucleus on its way to fission well before the final fragments are (pre)formed. These shell effects are studied in fission of ^{180}Hg, ^{236}U and ^{256}Fm with constrained Hartree-Fock-Bogoliubov calculations using the D1S parametrisation of the Gogny interaction. Strutinsky shell energy correction and single-particle energy level density near the Fermi surface are computed. Several neutron and proton shell effects are identified as drivers towards asymmetric fission. Shell effects are also used to identify the preformation of the fragments in the later stage of fission.

Excitation of the delta-resonance in compressed super heavy neutron-rich doubly magic [formula omitted] nucleus

By using constrained spherical Hartree-Fock approximation (CSHF), nuclear system of the super-heavy spherical U126310bh184 nucleus is investigated. CSHF is applied to no-core shell model, with an effective baryon-baryon interaction. Physical properties are affected by delta excitation under compression. It is shown when the △ resonances have occurred, the nucleus becomes more bound. Also, it can be seen that as the compression is continuous, the formation of △ increases. Under compression, the most of increment in energy of the system is gone to △-mass formation. Therefore, this appears to play a role in nuclear compression in subthreshold pion-production in heavy ions collisions. When the △ degree of freedom is created, reduced compressibility has occurred. In a ground state without nuclear compression, it is seen that the U126310bh184 nucleus is formed from the △ particles as the basic component besides nucleons. Also, there is a radial density distribution of the △ particles at the ground state for the U126310bh184 nucleus. If the nucleus was compressed to twenty-four of the ordinary density, then the △ particle would have been sharply increased to about 14.2% of all particles in the nucleus. The single particle spectra are studied under nuclear compression too. An excellent agreement was happened between the calculations and the shell model for the low levels of the single particle energies. Finally, there is a excellent consistency between the effective Hamiltonian and phenomenological shell model results for the lower single particle energy levels.

Transitional patterns in the spherical mean-field plus quadrupole-quadrupole and pairing model within two-j shells

Patterns of transition between several terms in the spherical mean-field plus quadrupole-quadrupole and pairing model within two-j shells are studied. By analyzing the variational features of the quantities in the quadrupole-quadrupole term to pairing term, mean-field term to quadrupole-quadrupole term, and mean-field term to pairing term, we find that with the even-n cases, some observables such as low-lying energy level ratio E(41+)/E(21+), the overlap |〈41+;x0|41+;x〉| with x0=0 and x0=1, B(E2;41+→21+) and electric quadrupole moment value Q(41+), are sensitive with respect to a control parameter or the number of valence nucleons. The investigation also indicates that increasing the spacing of single-particle energy level can produce the significant transitional behavior.

ΛΛ pairing effects in spherical and deformed multi- Λ hyperisotopes

The $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}$ pairing effects in spherical and deformed multi-$\mathrm{\ensuremath{\Lambda}}$ hyperisotopes are investigated in the framework of the Skyrme-Hartree-Fock approach employing a $\ensuremath{\delta}$ pairing force with the pairing strength of $\mathrm{\ensuremath{\Lambda}}$ hyperons being $4/9$ of that for nucleons. For spherical hyperisotopes, the occurrences of magic numbers $\ensuremath{-}S=2$, 8, 18, 20, 34, 58, 68, and 70, which are attributed to a Woods-Saxon-like $\mathrm{\ensuremath{\Lambda}}$ hyperon potential, are evinced by the sudden drop of $2\mathrm{\ensuremath{\Lambda}}$ separation energies and the vanishing pairing gaps and pairing energies. The results are compared with equivalent ones in recent Hartree-Fock-Bogoliubov and relativistic-Hartree-Bogoliubov calculations. For the deformed hyperisotopes, more possible $\mathrm{\ensuremath{\Lambda}}$ hyperon magic numbers $\ensuremath{-}S=4$, 6, 10, 14, 26, 30, and 32 are found based on the analysis of the single-particle energy levels, and are all sensitive to the quadrupole deformation ${\ensuremath{\beta}}_{2}$. The steps of the $2\mathrm{\ensuremath{\Lambda}}$ separation energies are accordingly smaller than in spherical hyperisotopes, and the possibilities for pairing are consistently reduced.

Bubble nuclei with shape coexistence in even-even isotopes of Hf to Hg

The shape of a nucleus is one of the fundamental nuclear properties. We perform a systematic investigation of bubble nuclei that also exhibit shape coexistence in Hf, W, Os, Pt, and Hg even-even isotopes using the deformed relativistic Hartree-Bogoliubov theory in continuum. For a systematic study, we first consider nuclear bubble structures and shape coexistence separately. We confirm that deformations and pairing correlations hinder bubble structures by comparing our results with those from relativistic continuum Hartree-Bogoliubov theory that assumes spherical symmetry in nuclei. We then predict candidate isotopes with both bubble structure and shape coexistence. We observe that the depletion fraction factor that characterizes bubble structure is mostly smaller in oblate deformation than in prolate, while some isotopes such as $^{206}\mathrm{Os}$ have bubble structures both in oblate and prolate deformations. We compare the proton single-particle energy levels for the candidates of shape coexistence both with only prolate bubble structure and with prolate and oblate bubble structures.

ΛΛ pairing effects in multi-Λ hyperisotopes within a Skyrme-Hartree-Fock model

The ΛΛ pairing effects are investigated in multi-Λ hyperisotopes with the Skyrme-Hartree-Fock approach. The shell closure of hyperon occurrs at −S = 2, 8, 18, 20, 34, 58, 68, as well as 70 in spherical hypernuclei, which are attributed from the Λ hyperon potential being close to theWoods-Saxon potential. The more possible Λ hyperon magic numbers −S = 4, 6, 10, 14, 26, 30, and 32 in deformed hypernuclei are found based on the analysis of the single-particle energy levels, and are all sensitive to the quadrupole deformation parameter β2.

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.

Neutron Matter and Interactions for Ni nuclei in a Supernova 1987A

Structure of neutron matter and neutron-interactions in a Supernova (SN) are investigated by the theoretical model approach. For that purpose, we calculate the neutron skin thicknesses, neutron-proton interactions, density distributions, central depletions in nuclei using Skyrme Hartree-Fock (SHF) model. The neutron skin thickness (NST) properties are obtained by bulk and surface contributions, based on two-parameter Fermi (2pF) distributions, as a function of the relative neutron excess I = (N− Z)/A. For the central depletion in the nucleus, the depletion fraction (DF) of nuclei is calculated by bubble parameters b and b′. To demonstrate the accuracy of the mass model in the theoretical model, calculated binding energies are compared with the theoretical and experimental data in the literature, namely the relativistic mean-field theory (RMFT), atomic mass evaluation (AME), and finite-range droplet model (FRDM). The fluctuations in the mass model are shown for average neutron-proton interactions δ,V1n−1p and the residual neutron-proton interaction δnp for both odd-A and even-A nuclei. Additionally, the single-particle energy levels and charge form factors for some nuclei are presented for the contribution experimental processes. The based on the theoretical and experimental data, the importance and relevance of results of the calculated results are discussed.

Microscopic study of shape evolution and some important ground state properties of 190–210Au isotopes

In this paper, we have examined the nuclear structural properties of 190–210Au isotopes within the framework of the Hartree–Fock–Bogoliubov Model. For the theoretical calculations, we have used the UNEDF2 and SKP interactions. The studies include the shape evolution, quadrupole deformation parameter [Formula: see text], nuclear electric quadrupole moment Q(b), single-particle energy levels, the binding energy per nucleon, nuclear charge radius, neutron rms radius, proton rms radius and neutron skin thickness. To analyze the accuracy of our theoretical results, we have presented the comparative analysis with experimental data and finite-range droplet model calculations. The adequate resemblance with experimental data is supporting our studies. We observe the oblate-prolate shape coexistence for 190Au, 193Au and 195Au using the UNEDF2 interaction.

Exploring the halo phenomena of medium-mass nuclei having approximately Z= 40 with point-coupled parameters in complex momentum representations * *Supported by the National Natural Science Foundation of China (11875070) and the Natural Science Foundation of Anhui Province (1908085MA16)

To explore the properties of neutron-rich nuclei with approximately 40 protons, the density-dependent point coupling (DD-PC1) effective interaction parameter is adopted in the relativistic mean-field theory with the complex momentum representation (RMF-CMR). The calculated two-neutron separation energy (S 2n ) and root-mean-square (rms) radii support the halo structure that appear in Mo and Ru isotopic chains. Besides, the neutron skin structures appear in Kr and Sr isotopes. The conclusions drawn are also supported by the single-particle energy levels and their occupancy probability and density distribution. Particularly, the energy levels, which reduce to bound states or are approximately 0 MeV with a small orbital angular momentum, are suggested to provide the primary contribution to increasing the neutron radius. Moreover, the single-particle energy levels significantly reflect the shell structure. In addition, the neutron drip line nuclei for Kr, Sr, Mo, and Ru elements are proposed via the changes in S 2n .

Electronic and optical properties of tapered tetrahedral semiconductor nanocrystals

The quantum confinement effect resulting from size reduction drastically alters the electronic structure and optical properties of optoelectronic materials. Quantum confinement in nanomaterials can be efficiently controlled by morphology variation combined characteristics of nanomaterials, such as their size, shape, and spatial organization. In this study, considering indium arsenide (InAs) in tetrahedral semiconductors as an example, we demonstrated the controllable morphology evolution of InAs nanostructures by tuning the growth conditions. We used the atomistic pseudopotential method to investigate the morphology-dependent electronic and optical properties of InAs nanostructures: tapered and uniform nanostructures, including the absorption spectra, single-particle energy levels, distribution and overlap integral of band-edge states, and exciton binding energies. Compared with uniform nanomaterials, a weaker quantum confinement effect was observed in the tapered nanomaterials, because of which tapered InAs nanostructures have a smaller bandgap, larger separation of photoinduced carriers, and smaller exciton binding energy. The absorption spectra of InAs nanostructures also exhibit strong morphology dependence. Our results indicate that morphology engineering can be exploited as a potential approach for modulating the electronic and optoelectronic properties of nanomaterials.

Resolution of the spin paradox in the Nilsson model

In the well-known Nilsson diagrams, depicting the dependence of the nuclear single-particle energy levels on quadrupole deformation, a spin paradox appears as the deformation sets in, leading from spherical shapes to prolate deformed shapes with cylindrical symmetry. Bunches of levels corresponding to a spherical shell model orbital, sharing the same orbital angular momentum and the same total angular momentum, appear to correspond to Nilsson energy levels, labeled by asymptotic quantum numbers in cylindrical coordinates, some of which have spin-up, while some others have spin-down. Furthermore, for some orbitals the correspondence between spherical shell model quantum numbers and Nilsson asymptotic quantum numbers is not the same for protons and for neutrons. Introducing a new rule of correspondence between the two sets of quantum numbers, we show that the spin paradox is resolved and full agreement between the proton and neutron Nilsson diagrams is established. The form of the Nilsson diagrams as a function of the quadrupole deformation remains unchanged, the only difference between the new diagrams and the traditional ones being the mutual exchange of the Nilsson labels for certain pairs of single-particle energy levels.

Anharmonicity in light nuclei near drip lines

Single-particle energy levels of nucleons moving in an anisotropic harmonic oscillator potential along with a spin-orbit and an orbit-orbit interaction have been calculated. The model proposed herein incorporates shapes generated by higher multipolar anisotropies other than the quadruple one. The gaps among the energy levels are strongly dependent on the degree of anisotropy and hence, affect the nucleonic separation energies. For large anisotropies, this characteristic disturbs the occurrences of the usual magic numbers and the sequential ordering of the shells characterized by principal quantum number, N, associated with the elementary shell model. For large asymmetries, the states characterized by higher N, may come down and lie below the energies of the states with lower N and vice versa. This feature of the model provides a natural explanation of intruder states or the island of inversion observed experimentally. For certain choices of the parameters, the large gaps in single nucleonic separation energies could occur at neutron numbers, 10,18,30, and 32 as observed, respectively, in 20Ne, 28Ne, 52Ti, and 54Ti. For neutron as well as proton number 14, the magnitude of this energy gap depends strongly on the choice of the strength of the spin-orbit and orbit-orbit terms. The model provides a simple understanding of the observed fact that some nuclei with proton or neutron number 14 sometimes exhibit this gap but not others. It also provides an interesting insight of having a low-laying (1/2 − ) state in 7He, and confirms the ground state spin and parity of 11Be to be (1/2+) and those of 9He to be (1/2+), both of which have been contentious. This investigation indicated the density distribution, i.e., the shapes of many of nuclei near drip line to be anisotropic having a symmetry about their body-fixed z-axis.

Nuclear shape evolution and shape coexistence in Zr and Mo isotopes

The phenomena of shape evolution and shape coexistence in even–even $$^{88-114}$$ Zr and $$^{90-116}$$ Mo isotopes are studied by employing covariant density functional theory (CDFT) with density-dependent point-coupling parameter set, DD-PCX, and with separable pairing interaction. The results for the rms deviation in binding energies, two-neutron separation energy, the differential variation of two-neutron separation energy, and rms charge radii, as a function of neutron number, are presented and compared with available experimental data. In addition to the oblate–prolate shape coexistence in $$^{96-110}$$ Zr isotopes, the correlations between shape transition and discontinuity in the observables are also examined. A smooth trend of charge radii in Mo isotopes is found to be due to the manifestation of triaxiality softness. The observed oblate and prolate minima are related to the low single-particle energy level density around the Fermi level of neutron and proton, respectively. The rapid shape transition in Zr isotopes near N $$\approx $$ 60 is identified to be caused by the evolution of the shell structure associated with massive proton excitations to 1 $$\pi g_{9/2}$$ orbit. The present calculations also predict a deformed semi-bubble structure in the $$^{100}$$ Zr isotope.

Microscopic study of shape evolution and ground-state properties of Iodine isotopes

In this paper, we investigated the shape evolution and ground-state properties of 108−144I isotopes using Hartree–Fock-Bogoliubov Model while employing the axially deformed single-particle harmonic oscillator basis for the expansion of quasiparticle wave functions. We have used SKP and UNEDF2 models to perform the theoretical calculations. We have presented the shape evolution and final values of the quadrupole deformation parameter β 2 of iodine isotopes. We have used the final values of the β 2 for studying the ground-state properties of the iodine isotopic chain. The investigated ground-state properties are the nuclear electric quadrupole moment, single-particle energy levels, the binding energy per nucleon, pairing energy, one-neutron separation energy, two-neutron separation energy, nuclear charge radius, neutron rms radius, proton rms radius, and neutron skin thickness. After using the SKP parameterization on 109I, we have observed the shape coexistence of prolate and oblate shape. Similarly, after using the UNEDF2 parameterization on 110I, 112I, 114I, 115I, 116I, 117I, 118I, and 119I, we discover the shape coexistence of prolate and oblate shape.

Microscopic description of structural, surface, and decay properties of Z=124, 126 superheavy nuclei

A systematic investigation has been carried out for even-even isotopic series of two superheavy nuclei with Z=124 and Z=126 in the region 160≤N≤220, and 162≤N≤218, respectively, within the framework of covariant density functional theory (CDFT). Based on the potential energy curves calculated using the explicit density-dependent meson-exchange (DD-ME) and point-coupling (DD-PC) models, many sensitive observables have been analyzed to understand the structural evolution, shape coexistence, and to predict the possible candidates for neutron shell-closure within the isotopic series of Z=124 and Z=126. The shell correction energy and nuclear symmetry energy have been evaluated within the Strutinsky shell correction approach and the Coherent Density Fluctuation Model (CDFM), respectively. All the sensitive observables i.e. two-neutron separation energy, its differential, neutron-pairing energy, single-particle energy levels, and total shell-correction energy reproduce known spherical neutron magic numbers, and present evidence on new deformed neutron magic numbers at N=168, 174 and 178. The symmetry energy shows a direct correlation with the shape of the system. The α-decay chain of four isotopes 124298,302 and 126300,304 have been studied within three different semi-empirical approaches named UNIV2, SemFIS2 and ImSahu, and compared with the available experimental data, FRDM2012 and the WS4 mass model. All the models, DD-ME2, DD-PC1, FRDM2012, and WS4 have predicted around 8α to 10α decay before spontaneous fission (SF) can start.