Single-particle Orbitals Research Articles

Backbending, seniority, and Pauli blocking of pairing correlations at high rotational frequencies in rapidly rotating nuclei

Garrett et al. systematically investigated band-crossing frequencies resulting from the rotational alignment of the first pair of i13/2 neutrons (AB) in rare-earth nuclei. In that study, evidence was found for an odd-even neutron number dependence attributed to changes in the strength of neutron pairing correlations. The present paper carries out a similar investigation at higher rotational frequencies for the second pair of aligning i13/2 neutrons (BC). Again, a systematic difference in band-crossing frequencies is observed between odd-N and even-N Er, Yb, Hf, and W nuclei, but in the BC case, it is opposite to the AB neutron-number dependence. These results are discussed in terms of a reduction of neutron pairing correlations at high rotational frequencies and of the effects of Pauli blocking on the pairing field by higher-seniority configurations. Also playing a significant role are the changes in deformation with proton and neutron numbers, the changes in location of single-particle orbitals as a function of quadrupole deformation, and the position of the Fermi surface with regard to the various Ω components of the neutron i13/2 shell.Received 22 March 2019DOI:https://doi.org/10.1103/PhysRevC.100.014302©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCollective levelsCollective modelsNuclear forcesNuclear structure & decaysProperties150 ≤ A ≤ 189Nuclear Physics

Backflow Transformations via Neural Networks for Quantum Many-Body Wave Functions.

Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this Letter, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman and Cohen [Phys. Rev. 102, 1189 (1956)10.1103/PhysRev.102.1189], adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to learn the optimal transformation via variational MonteCarlo calculations. NNB directly dresses a mean-field state, can be systematically improved, and directly alters the sign structure of the wave function. It generalizes the standard backflow [L. F. Tocchio etal., Phys. Rev. B 78, 041101(R) (2008)10.1103/PhysRevB.78.041101], which we show how to explicitly represent as a NNB. We benchmark the NNB on Hubbard models at intermediate doping, finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.

Pump-probe Auger-electron spectroscopy of Mott insulators

In high-resolution core-valence-valence (CVV) Auger electron spectroscopy from the surface of a solid at thermal equilibrium, the main correlation satellite, visible in the case of strong valence-electron correlations, corresponds to a bound state of the two holes in the final state of the CVV Auger process. We discuss the physical significance of this satellite in nonequilibrium pump-probe Auger spectroscopy by numerical analysis of a single-band Hubbard-type model system including core states and a continuum of high-energy scattering states. It turns out that the spectrum of the photo-doped system, due to the increased double occupancy, shares features with the equilibrium spectrum at higher fillings. The pumping of doublons can be watched when working with overlapping pulses at short $\Delta t$. For larger pump-probe delays $\Delta t$ and on the typical femtosecond time scale for electronic relaxation processes, spectra are hardly $\Delta t$-dependent, reflecting the high stability of bound two-hole states for strong Hubbard-$U$. We argue that taking into account the spatial expansion of single-particle orbitals when these are doubly occupied, as described by the dynamical Hubbard model, produces an oscillation of the barycenter of the satellite as a function of $\Delta t$. Pump-probe Auger-electron spectroscopy is thus highly sensitive to dynamical screening of the Coulomb interaction.

A triaxial critical point symmetry for odd-A nuclei

A new critical point symmetry (CPS) for odd-even systems is developed through coupling the T(4) CPS to a single particle within a spherical-j orbit. The resulting model is shown to be exactly solvable and serve as the triaxial CPS for transitional odd-A nuclei. A comparison between the model calculations and the available data on 135Ba indicates that the new CPS model at a large-γ deformation in the case of j=3/2 may yield the dynamical predictions being very similar to those from the E(5/4) CPS, thus providing an alternative explanation of the level structure built on the single-particle orbit 2d3/2 in 135Ba.

Nuclear decay studies of rare isotopes

This article overviews recent developments and achievements on decay spectroscopic studies at the Radioactive-Isotope Beam Factory (RIBF) in RIKEN Nishina Center, which has come online in 2007 as the first 3rd-generation in-flight separator facility. The research territory on the chart of nuclides has been dramatically expanded both towards the proton-rich and neutron-rich frontiers with the advent of RIBF. Taking advantage of the world’s currently strongest RI beams, in conjunction with new experimental techniques and state-of-the-art detector systems, more than 400 species of rare isotopes have been explored in experimental programs dedicated to radioactive-decay measurements performed in the last decade. Some selected topics are introduced in this review. Excited levels investigated by $\gamma$-ray measurements following $\beta$ decay have provided significant information on the evolution of shell structures and nuclear shapes, and thus serve as a stringent benchmark for testing microscopic and macroscopic nuclear model calculations. Systematic studies of $ \beta$-decay half-lives of very neutron-rich nuclei have an enormous impact on a better understanding of explosive nucleosynthesis in the rapid neutron-capture (r) process. On the proton-rich side of the $ \beta$-stability line, new proton emitters have been identified, allowing an exploration of atomic nuclei beyond the proton drip line. New results obtained in a series of decay experiments include various types of nuclear isomers, for example, seniority, spin-trap, shape, and K isomers. The identification and characterization of such exotic isomers have revealed the underlying single-particle orbits and nucleon-nucleon correlations, as well as the change of nuclear shape and spin orientation. These new results will offer further insight into exotic nuclei and contribute to the development of new theories of nuclear structure and nucleosynthesis. In this review article, some of the ongoing research projects and future prospects for decay studies at RIBF are also presented.

Probing the nuclear structure in the vicinity of 78Ni

Theoretical and experimental studies of neutron-rich nuclei have shown that the general concept of shell structure is not as robust and universal as earlier thought, but can exhibit significant changes as a function of neutron excess. New magic numbers appear and some other conventional ones disappear mainly because of a different ordering of the single-particle orbitals. In the present contribution, recent experimental studies of neutron-rich Cu isotopes, performed at RIKEN using β decay and one-proton knockout reactions, will be discussed. Neutron-rich nuclei near 78Ni were populated through in-flight fission of 238U on thick 9Be targets in both experiments. In the β-decay study, 75,77Ni nuclei were implanted into the WAS3ABi silicon array, while γ rays from excited states in 75,77Cu emitted after β decay of the implanted ions were detected with the EURICA Ge detector array that was surrounding the active stopper. In a second experiment within the SEASTAR campaign at RIKEN, the same 75,77Cu nuclei were produced in (p,2p) knockout reactions from 76,78Zn beam particles at around 250 MeV/nucleon impinging onto the MINOS liquid hydrogen target. In the latter experiment the DALI2 NaI array was used to detect de-excitation γ rays measured in coincidence with Cu nuclei identified in the Zero Degree Spectrometer. Both studies are complimentary and greatly contribute to our understanding on the nuclear structure in the 78Ni region.

New insights into triaxiality and shape coexistence from odd-mass Rh109

Rapid shape evolutions near A = 100 are now the focus of much attention in nuclear science. Much of the recent work has been centered on isotopes with Z <= 40, where the shapes are observed to transition between near-spherical to highly deformed with only a single pair of neutrons added. At higher Z, the shape transitions become more gradual as triaxiality sets in, yet the coexistence of varying shapes continues to play an important role in the low-energy nuclear structure, particularly in the odd-Z isotopes. This work aims to characterize competing shapes in the triaxial region between Zr and Sn isotopes using ultrafast timing techniques to measure lifetimes of excited states in the neutron-rich nucleus Rh-109. The measurements confirm the persistence at higher Z of similarly large deformations observed near Z = 40. Moreover, we show that new self-consistent mean-field calculations, with proper treatment of the odd nucleon, are able to reproduce the coexisting triaxial and highly deformed configurations revealing, for the first time, the important contribution of the unpaired nucleon to these different shapes based on the blocking of specific single-particle orbitals.

Non-orthogonal determinants in multi-Slater-Jastrow trial wave functions for fixed-node diffusion Monte Carlo.

The accuracy and efficiency of ab initio Quantum Monte Carlo (QMC) algorithms benefit greatly from compact variational trial wave functions that accurately reproduce ground state properties of a system. We investigate the possibility of using multi-Slater-Jastrow trial wave functions with non-orthogonal determinants by optimizing identical single particle orbitals independently in separate determinants. As a test case, we compute variational and fixed-node diffusion Monte Carlo (FN-DMC) energies of a C2 molecule. For a given multi-determinant expansion, we find that this non-orthogonal orbital optimization results in a consistent improvement in the variational energy and the FN-DMC energy on the order of a few tenths of an eV. In some cases, fewer non-orthogonal determinants are required compared to orthogonal ones in order to achieve similar accuracy in FN-DMC. Our calculations indicate that trial wave functions with non-orthogonal determinants can improve computed energies in a QMC calculation when compared to their orthogonal counterparts.

Empirical study of the shape evolution and shape coexistence in Zn, Ge and Se isotopes

The spectra of the isotopes of Zn, Ge and Se in the A=60–80 region of the nuclear landscape display interesting spectral variations with N and Z. At first we study these structures empirically. The state-wise deformation status of each isotope is deduced using the power index formula. The average value of the power index displays the variation with neutron number N. The linear plot of B(E2,01+→21+) versus [1/E(21+)] for a given Z gives further information on the spectral variation with N. The level energy spectra are used to study the role of the excited bands in the phase transitions and the possible shape co-existence. The common spectral features of the three isotopic chains and some unique features are pointed out from our study and the role of the Nilsson single particle orbits is discussed. A comparison with recent works for Ge and Se isotopes reveals the importance of our empirical studies vis-à-vis the high efficiency experiments and different theoretical interpretations of the shape changes with N and Z and spin.

Angle-resolved photoemission spectroscopy from first-principles quantum Monte Carlo.

Angle-resolved photoemission spectroscopy allows one to visualize in momentum space the probability weight maps of electrons subtracted from molecules deposited on a substrate. The interpretation of these maps usually relies on the plane wave approximation through the Fourier transform of single particle orbitals obtained from density functional theory. Here we propose a first-principle many-body approach based on quantum Monte Carlo (QMC) to directly calculate the quasi-particle wave functions (also known as Dyson orbitals) of molecules in momentum space. The comparison between these correlated QMC images and their single particle counterpart highlights features that arise from many-body effects. We test the QMC approach on the linear C2H2, CO2, and N2 molecules, for which only small amplitude remodulations are visible. Then, we consider the case of the pentacene molecule, focusing on the relationship between the momentum space features and the real space quasi-particle orbital. Eventually, we verify the correlation effects present in the metal planar complex.

Coupling of a j = 3/2 nucleon to an even–even core with Davidson potential

The properties of odd nuclei have been investigated within the collective model by assuming the system is composed of a single nucleon in the [Formula: see text] single particle orbit coupled to a [Formula: see text]-unstable even-core. The Davidson potential has been used in the corresponding Bohr Hamiltonian for the even core. The excitation energy spectrum and the electric quadrupole transition ratios have been obtained. The results have been used to predict the experimental data of the some selected odd isotopes.

Shells in a toroidal nucleus in the intermediate-mass region

Attention is fixed on shells in toroidal nuclei in the intermediate mass region using a toroidal single-particle potential. We find that there are toroidal shells in the intermediate mass region with large single-particle energy gaps at various nucleon numbers located at different toroidal deformations characterized by the aspect ratios of toroidal major to minor radius. These toroidal shells provide extra stability at various toroidal deformations. Relative to a toroidal core, Bohr-Mottelson spin-aligning particle-hole excitations may be constructed to occupy the lowest single-particle Routhian energies to lead to toroidal high-spin isomers with different spins. Furthermore, as a nucleon in a toroidal nucleus possesses a vorticity quantum number, toroidal vortex nuclei may be constructed by making particle-hole excitations in which nucleons of one type of vorticity are promoted to populate un-occupied single-particle orbitals of the opposite vorticity. Methods for producing toroidal high-spin isomers and toroidal vortex nuclei are discussed.

An efficient hybrid orbital representation for quantum Monte Carlo calculations.

The scale and complexity of the quantum system to which real-space quantum Monte Carlo (QMC) can be applied in part depends on the representation and memory usage of the trial wavefunction. B-splines, the computationally most efficient basis set, can have memory requirements exceeding the capacity of a single computational node. This situation has traditionally forced a difficult choice of either using slow internode communication or a potentially less accurate but smaller basis set such as Gaussians. Here, we introduce a hybrid representation of the single particle orbitals that combine a localized atomic basis set around atomic cores and B-splines in the interstitial regions to reduce the memory usage while retaining the high speed of evaluation and either retaining or increasing overall accuracy. We present a benchmark calculation for NiO demonstrating a superior accuracy while using only one eighth of the memory required for conventional B-splines. The hybrid orbital representation therefore expands the overall range of systems that can be practically studied with QMC.

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.

Phase diagram of a symmetric electron–hole bilayer system: a variational Monte Carlo study

We study the phase diagram of a symmetric electron–hole bilayer system at absolute zero temperature and in zero magnetic field within the quantum Monte Carlo approach. In particular, we conduct variational Monte Carlo simulations for various phases, i.e. the paramagnetic fluid phase, the ferromagnetic fluid phase, the anti-ferromagnetic Wigner crystal phase, the ferromagnetic Wigner crystal phase and the excitonic phase, to estimate the ground-state energy at different values of in-layer density and inter-layer spacing. Slater–Jastrow style trial wave functions, with single-particle orbitals appropriate for different phases, are used to construct the phase diagram in the (rs, d) plane by finding the relative stability of trial wave functions. At very small layer separations, we find that the fluid phases are stable, with the paramagnetic fluid phase being particularly stable at and the ferromagnetic fluid phase being particularly stable at . As the layer spacing increases, we first find that there is a phase transition from the ferromagnetic fluid phase to the ferromagnetic Wigner crystal phase when d reaches 0.4 a.u. at rs = 20, and before there is a return to the ferromagnetic fluid phase when d approaches 1 a.u. However, for rs < 20 and a.u., the excitonic phase is found to be stable. We do not find that the anti-ferromagnetic Wigner crystal is stable over the considered range of rs and d. We also find that as rs increases, the critical layer separations for Wigner crystallization increase.

Octupole deformation in neutron-rich actinides and superheavy nuclei and the role of nodal structure of single-particle wavefunctions in extremely deformed structures of light nuclei

Octupole deformed shapes in neutron-rich actinides and superheavy nuclei as well as extremely deformed shapes of the light nuclei have been investigated within the framework of covariant density functional theory. We confirmed the presence of new region of octupole deformation in neutron-rich actinides with the center around but our calculations do not predict octupole deformation in the ground states of superheavy nuclei. As exemplified by the study of 36Ar, the nodal structure of the wavefunction of occupied single-particle orbitals in extremely deformed structures allows to understand the formation of the α-clusters in very light nuclei, the suppression of the α-clusterization with the increase of mass number, the formation of ellipsoidal mean-field type structures and nuclear molecules.

Electron and Nucleon Localization Functions of Oganesson: Approaching the Thomas-Fermi Limit.

Fermion localization functions are used to discuss electronic and nucleonic shell structure effects in the superheavy element oganesson, the heaviest element discovered to date. Spin-orbit splitting in the 7p electronic shell becomes so large (∼10 eV) that Og is expected to show uniform-gas-like behavior in the valence region with a rather large dipole polarizability compared to the lighter rare gas elements. The nucleon localization in Og is also predicted to undergo a transition to the Thomas-Fermi gas behavior in the valence region. This effect, particularly strong for neutrons, is due to the high density of single-particle orbitals.

Kinetic energy classification and smoothing for compact B-spline basis sets in quantum Monte Carlo.

Quantum Monte Carlo calculations of defect properties of transition metal oxides have become feasible in recent years due to increases in computing power. As the system size has grown, availability of on-node memory has become a limiting factor. Saving memory while minimizing computational cost is now a priority. The main growth in memory demand stems from the B-spline representation of the single particle orbitals, especially for heavier elements such as transition metals where semi-core states are present. Despite the associated memory costs, splines are computationally efficient. In this work, we explore alternatives to reduce the memory usage of splined orbitals without significantly affecting numerical fidelity or computational efficiency. We make use of the kinetic energy operator to both classify and smooth the occupied set of orbitals prior to splining. By using a partitioning scheme based on the per-orbital kinetic energy distributions, we show that memory savings of about 50% is possible for select transition metal oxide systems. For production supercells of practical interest, our scheme incurs a performance penalty of less than 5%.

Non-invasive study of the three-dimensional structure of nanoporous triblock terpolymer membranes.

Nanoporous media are of great importance for drug delivery or filtration. Typically the pore structure of such media is characterized using high-resolution techniques such as electron microscopy or atomic force microscopy. However, these techniques are restricted to the surface of the material and/or are highly invasive. In a proof-of-concept experiment we have employed three-dimensional single-particle orbit tracking for testing the three-dimensional pore structure of a liquid filled nanoporous polystyrene-block-polyisoprene-block-poly(N-isopropylacrylamide) (PS-b-PI-b-PNiPAAm) triblock terpolymer membrane. Using fluorescent tracers with a diameter of about 10% of the relevant void structures, the tracking experiments yielded results that were comparable to those obtained from reference experiments using environmental scanning electron microscopy (eSEM). This testifies that single-particle orbit tracking can serve as a useful non-invasive alternative for characterising the structure of nanoporous materials.

Description of the 190–199Hg Nuclei Under the Frameworks of IBM-1 and IBFM-1

The energy levels and the electric quadrupole transition probabilities B(E2; Ji → Jf) for even–odd 191–199Hg have been investigated. The negative spin states of the even–odd 191–199Hg isotopes are studied within the framework of the interacting boson–fermion model (IBFM–1). The single fermion is assumed to be in one of the 2f5/2, 3p3/2, and 3p1/2 single-particle orbits. It is found that the calculated negative spin state energy spectra of the even–odd 191–199Hg isotopes agree quite well with the experimental data. The B(E2) values are also calculated and compared with the experimental data. In general, the results are in reasonably good agreement with the previous experimental values. Furthermore, the energy levels, electric quadrupole transition probabilities, and the wave function for even–even 1901–198Hg isotopes (as core for even–odd nuclei) have been calculated within the framework of the interacting boson model (IBM-1). The predicted energy levels and B(E2) transition probability results are reasonably consistent with the experimental data. These calculations have been compared with previous it and almost better than it. The study of the wave function’s structure shows that all interesting nuclei are deformed and have dynamical symmetry O(6) characters.