Configuration-interaction Shell Model Research Articles

Recent progress in configuration–interaction shell model

Since Mayer and Jensen employed the single-particle shell model to interpret the magic numbers, various microscopic nuclear models have been developed to study the nuclear force and structure. The configuration–interaction shell model (CISM), performed in truncated model space with the inclusion of the residual interaction, is one widely-used nuclear structure model. In the last decade, CISM has progressed in investigating the cross-shell excitation in exotic light nuclei, the similarity and difference in mirror nuclei, and the isomerism and seniority conservation in medium and heavy nuclei. Additionally, researchers have attempted to construct effective Hamiltonians for nuclei near [Formula: see text]Sn and [Formula: see text]Pb through a unified way in the CISM framework. In parallel, related models, including the nucleon-pair approximation (NPA) approach, the Monte Carlo shell model (MCSM), the projected shell model (PSM), the Gamow shell model (GSM), etc., have also been extensively developed and validated in the last decade. This paper reviews the recent progress in CISM and some related models.

Nuclear chiral rotation within Relativistic Configuration-interaction Density functional theory

The Relativistic Configuration-interaction Density functional (ReCD) theory that combines the advantages of large-scale configuration-interaction shell model and relativistic density functional theory is extended to study nuclear chiral rotation. The energy spectra and transition probabilities of the chiral doublet bands are reproduced satisfactorily without any free parameters. By analyzing the probability amplitudes of the wavefunctions, the significant roles of configuration mixing and four quasiparticle states to the chiral doublets are revealed. The evolution from chiral vibration to static chirality is clearly illustrated by the K plot and azimuthal plot. The present investigation provides both microscopic and quantal descriptions for nuclear chirality for the first time and demonstrates the robustness of chiral geometry against the configuration mixing as well as the four quasiparticle states.

Nucleon-pair truncation of the shell model for medium-heavy nuclei

Background: Establishing computationally tractable models of atomic nuclei is a long-time goal of nuclear structure physics. A flexible framework which easily includes excited states and many-body correlations is the configuration-interaction shell model (SM), but the exponential growth of the basis means one needs an efficient truncation scheme, ideally one that includes both deformation and pairing correlations.Purpose: We propose an efficient truncation scheme of the SM: starting from a pair condensate variationally defined by Hartree-Fock single-particle states and the particle-number conserved Bardeen-Cooper-Schrieffer (NBCS) approximation, we carry out projection of states with good angular momentum.Methods: After generating Hartree-Fock single-particle states with Kramers degeneracy in a SM space, we optimize the pair amplitudes in the NBCS by minimizing the energy, and then use linear algebra projection (LAP) of states with good angular momentum. Both NBCS and LAP are computationally fast.Results: Our calculations yield good agreement with full configuration-interaction SM calculations for low-lying states of transitional and rotational nuclei with axially symmetric and triaxial deformation in medium- and heavy-mass regions: $^{44,46,48}\mathrm{Ti}$, $^{48,50}\mathrm{Cr}$, $^{52}\mathrm{Fe}$, $^{60,62,64}\mathrm{Zn}$, $^{66,68}\mathrm{Ge}$, $^{68}\mathrm{Se}$, and $^{108,110}\mathrm{Xe}$. We predict low-lying states of $^{112--114}\mathrm{Ba}$ and $^{116--120}\mathrm{Ce}$, nuclei difficult to reach by large-scale SM calculations.Conclusions: Both pair correlation and the configuration mixing between different intrinsic states play a key role in reproducing collectivity and shape coexistence, demonstrating the utility of this truncation scheme of the SM to study transitional and deformed nuclei.

How effective is the Brink–Axel hypothesis for astrophysical weak rates?

Abstract We explore the effectiveness of the Brink–Axel hypothesis (BAH) for the computation of stellar electron capture (EC) and β-decay (BD) rates, namely that the transition strength function depends only upon the transition energy and not upon the details of the initial state. For this purpose, we calculated Gamow–Teller (GT) strength distributions for a selection of sd-shell nuclides, using two different microscopic models, namely the proton–neutron quasiparticle random phase approximation and the full configuration-interaction shell model, taking into account the first 100 states of both the initial and final nuclides. The GT transition strengths among these levels evolve with initial state energy. These transition strength functions we folded into weak-interaction mediated rates in stellar matter, specifically EC and BD rates, for a range of densities 10 g cm−3 ⩽ ρ ⩽ 1011 g cm−3 and range of temperatures 1 GK ⩽ T ⩽ 30 GK. When transitions from excited states were approximated using the BAH, augmented by so-called ‘back-resonance’ transitions, the rates were affected by up to three orders of magnitude or more at high temperatures and densities. Thus the BAH is not a reliable approximation for the calculation of stellar rates, especially in high temperature–density environments.

Nuclear states projected from a pair condensate

Atomic nuclei exhibit deformation, pairing correlations, and rotational symmetries. To meet these competing demands in a computationally tractable formalism, we revisit the use of general pair condensates with good particle number as a trial wave function for even-even nuclei. After minimizing the energy of the condensate, we project out states with good angular momentum with a fast projection technique, allowing for general triaxial deformations. To show applicability, we present example calculations from pair condensates in several model spaces, and compare against projected Hartree-Fock and full configuration-interaction shell model calculations. This approach successfully generates spherical, vibrational and rotational spectra, demonstrating potential for modeling medium- to heavy-mass nuclei.

Initialization effects of nucleon profile on the π yields in heavy-ion collisions at medium energies

We study the problem of π production in heavy-ion collisions in the context of the isospin-dependent Boltzmann–Uehling–Uhlenbeck transport model. We generated nucleon densities using two different models, the Skyrme–Hartree–Fock (SHF) model and the configuration interaction shell model (SM). Indeed, inter-nucleon correlations are explicitly taken into account in SM, while they are averaged in the SHF model. As an application of our theoretical frameworks, we calculated the π − and π + yields in collisions of nuclei with A = 30–40 nucleons. We used different harmonic oscillator lengths b HO to generate the harmonic oscillator basis for SM in order to study both theoretical and experimental cases. It is found that the SM framework with b HO = 2.5 fm and SHF can be distinguished by the yield of π mesons, in this case the density distribution calculated by the SM produces more π in the collision. In comparison, SM with b HO = 2.0 fm is distinguished from SHF by the double π −/π + ratios with different large impact parameters, from which one can find that the double π −/π + ratios of SM change smoother and are less than those of SHF.

Finite-temperature mean-field approximations for shell model Hamiltonians: the code HF-SHELL

We present the code HF-SHELL for solving the self-consistent mean-field equations for configuration-interaction shell model Hamiltonians in the proton-neutron formalism. The code can calculate both ground-state and finite-temperature properties in the Hartree-Fock (HF), HF+Bardeen-Cooper-Schrieffer (HF+BCS), and the Hartree-Fock-Bogoliubov (HFB) mean-field approximations. Particle-number projection after variation is incorporated to reduce the grand-canonical ensemble to the canonical ensemble, making the code particularly suitable for the calculation of nuclear state densities. The code does not impose axial symmetry and allows for triaxial quadrupole deformations. The self-consistency cycle is particularly robust through the use of the heavy-ball optimization technique and the implementation of different options to constrain the quadrupole degrees of freedom.

Exact sum rules with approximate ground states

Electromagnetic and weak transitions tell us a great deal about the structure of atomic nuclei. Yet modeling transitions can be difficult: it is often easier to compute the ground state, if only as an approximation, than excited states. One alternative is through transition sum rules, in particular the non-energy-weighted and energy-weighted sum rules, which can be computed as expectation values of operators. We investigate by computing sum rules for a variety of nuclei, comparing the numerically exact full configuration-interaction shell model, as a reference, to Hartree–Fock, projected Hartree–Fock, and the nucleon pair approximation. These approximations yield reasonable agreement, which we explain by prior work on the systematics of transition moments.

Uncertainty quantification of an empirical shell-model interaction using principal component analysis

Recent investigations have emphasized the importance of uncertainty quantification (UQ) to describe errors in nuclear theory. We carry out UQ for configuration-interaction shell model calculations in the $1s$-$0d$ valence space, investigating the sensitivity of observables to perturbations in the 66 parameters (matrix elements) of a high-quality empirical interaction. The large parameter space makes computing the corresponding Hessian numerically costly, so we compare a cost-effective approximation, using the Feynman-Hellmann theorem, to the full Hessian and find it works well. Diagonalizing the Hessian yields the principal components of the interaction: linear combinations of parameters ordered by sensitivity. This approximately decoupled distribution of parameters facilitates theoretical error propagation onto structure observables: electromagnetic transitions, Gamow-Teller decays, and dark matter-nucleus scattering matrix elements.

Limits on assigning a shape to a nucleus

The interpretation of nuclear observables in the laboratory frame in terms of the intrinsic deformation parameters beta and gamma is a classical theme in nuclear structure. Here we use the quadrupole invariants (Kumar), calculated within the framework of the configuration-interaction shell model, to clarify the meaning and limitations of nuclear shapes. We introduce a novel method that enables us to calculate accurately higher-order invariants and, therefore, the fluctuations in both beta and gamma. We find that the shape parameter beta often has a non-negligible degree of softness, and that the angle gamma is usually characterized by large fluctuations, rendering its effective value not meaningful. Contrary to common belief, we conclude that doubly magic nuclei are not spherical because the notion of a well-defined shape does not apply to them.

Nuclear deformation in the configuration-interaction shell model

We review a method that we recently introduced to calculate the finite-temperature distribution of the axial quadrupole operator in the laboratory frame using the auxiliary-field Monte Carlo technique in the framework of the configuration-interaction shell model. We also discuss recent work to determine the probability distribution of the quadrupole shape tensor as a function of intrinsic deformation β,γ by expanding its logarithm in quadrupole invariants. We demonstrate our method for an isotope chain of samarium nuclei whose ground states describe a crossover from spherical to deformed shapes.

Nuclear deformation in the laboratory frame

We develop a formalism for calculating the distribution of the axial quadrupole operator in the laboratory frame within the rotationally invariant framework of the configuration-interaction shell model. The calculation is carried out using a finite-temperature auxiliary-field quantum Monte Carlo method. We apply this formalism to isotope chains of even-mass samarium and neodymium nuclei, and show that the quadrupole distribution provides a model-independent signature of nuclear deformation. Two technical advances are described that greatly facilitate the calculations. The first is to exploit the rotational invariance of the underlying Hamiltonian to reduce the statistical fluctuations in the Monte Carlo calculations. The second is to determine quadruple invariants from the distribution of the axial quadrupole operator. This allows us to extract effective values of the intrinsic quadrupole shape parameters without invoking an intrinsic frame or a mean-field approximation.

Shell model description of heavy nuclei and abnormal collective motions

In this contribution I present systematic calculations on the spectroscopy and electromagnetic transition properties of intermediate-mass and heavy nuclei around 100Sn and 208Pb. We employed the large-scale configuration interaction shell model approach with realistic interactions. Those nuclei are the longest isotopic chains that can be studied by the nuclear shell model. I will show that the yrast spectra of Te isotopes show a vibrational-like equally spaced pattern but the few known E2 transitions show rotational-like behaviour. These kinds of abnormal collective behaviors cannot be reproduced by standard collective models and provide excellent background to study the competition of single-particle and various collective degrees of freedom. Moreover, the calculated B(E2) values for neutron-deficient and heavier Te isotopes show contrasting different behaviours along the yrast line, which may be related to the enhanced neutron-proton correlation when approaching N=50. The deviations between theory and experiment concerning the energies and E2 transition properties of low-lying 0+ and 2+ excited states and isomeric states in those nuclei may provide a constraint on our understanding of nuclear deformation and intruder configuration in that region.

Pygmy and giant dipole resonances in the nitrogen isotopes

The configuration-interaction shell model with the WBP10 effective interaction has been used to investigate the pygmy and giant dipole resonances in the nitrogen isotopes. Large enhancement of low-lying dipole strength, i.e., pygmy dipole resonances (PDRs), is predicted in the neutron-rich $^{17,18,19,20}\mathrm{N}$. The nature of the PDRs is analyzed via the transition densities and transition matrix elements. It turns out these PDRs involve a larger amount of excitations between the $2s1d$ and loosely bound $1f2p$ shells. Combining with the transition densities, it is concluded that the PDRs in $^{17,18,19,20}\mathrm{N}$ are collective and due to the oscillation between the excess neutrons and the isospin saturated core. The isospin dependence of energy splitting and sum rule of isospin doublets is discussed. The theoretical energy splitting of isospin doublets can significantly deviate from the systematic values when nucleus is far away from the $\ensuremath{\beta}$-stability line. The ratios of ${T}_{l}$ and ${T}_{g}$ energy-weighted sum rule (EWSR) are consistently larger than the systematic values, and it is noticed that the calculated EWSR ratio over the systematic ratio increases with increasing isospin almost linearly. We also calculated the photoabsorption cross sections for the nitrogen isotopes. We proposed the normalization factors for $0--1\ensuremath{\hbar}\ensuremath{\omega}$ and $2--3\ensuremath{\hbar}\ensuremath{\omega}$ calculations. After the normalization, the shell model has well reproduced the experimental photoabsorption cross sections in $^{14,15}\mathrm{N}$, especially the detailed structure of resonances.

Level densities of heavy nuclei in the shell model Monte Carlo approach

Nuclear level densities are necessary input to the Hauser-Feshbach theory of compound nuclear reactions. However, the microscopic calculation of level densities in the presence of correlations is a challenging many-body problem. The configurationinteraction shell model provides a suitable framework for the inclusion of correlations and shell effects, but the large dimensionality of the many-particle model space has limited its application in heavy nuclei. The shell model Monte Carlo method enables calculations in spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods and has proven to be a powerful tool in the microscopic calculation of level densities. We discuss recent applications of the method in heavy nuclei.

Collective enhancement of nuclear state densities by the shell model Monte Carlo approach

The shell model Monte Carlo (SMMC) approach allows for the microscopic calculation of statistical and collective properties of heavy nuclei using the framework of the configuration-interaction shell model in very large model spaces. We present recent applications of the SMMC method to the calculation of state densities and their collective enhancement factors in rare-earth nuclei.

Calculating Level Densities of Heavy Nuclei by the Shell Model Monte Carlo Method

The microscopic calculation of nuclear level densities in the presence of correlations is a difficult many-body problem. The shell model Monte Carlo method provides a powerful technique to carry out such calculations using the framework of the configuration-interaction shell model in spaces that are many orders of magnitude larger than spaces that can be treated by conventional methods. We present recent applications of the method to the calculation of level densities and their collective enhancement factors in heavy nuclei. The calculated level densities are in close agreement with experimental data.

Collectivity in Heavy Nuclei in the Shell Model Monte Carlo Approach

The microscopic description of collectivity in heavy nuclei in the framework of the configuration-interaction shell model has been a major challenge. The size of the model space required for the description of heavy nuclei prohibits the use of conventional diagonalization methods. We have overcome this difficulty by using the shell model Monte Carlo (SMMC) method, which can treat model spaces that are many orders of magnitude larger than those that can be treated by conventional methods. We identify a thermal observable that can distinguish between vibrational and rotational collectivity and use it to describe the crossover from vibrational to rotational collectivity in families of even-even rare-earth isotopes. We calculate the state densities in these nuclei and find them to be in close agreement with experimental data. We also calculate the collective enhancement factors of the corresponding level densities and find that their decay with excitation energy is correlated with the pairing and shape phase transitions.

Configuration-interaction Monte Carlo method and its application to the trapped unitary Fermi gas

We develop a quantum Monte Carlo method to estimate the ground-state energy of a fermionic many-particle system in the configuration-interaction shell model approach. The fermionic sign problem is circumvented by using a guiding wave function in Fock space. The method provides an upper bound on the ground-state energy whose tightness depends on the choice of the guiding wave function. We argue that the antisymmetric geminal product class of wave functions is a good choice for guiding wave functions. We demonstrate our method for the trapped two-species fermionic cold atom system in the unitary regime of infinite scattering length using the particle-number projected Hartree-Fock-Bogoliubov wave function as the guiding wave function. We estimate the ground-state energy and energy-staggering pairing gap as a function of the number of particles. We compare our results with exact numerical diagonalization results and with previous fixed-node coordinate-space Monte Carlo calculations.

Recent developments in the shell model Monte Carlo approach to nuclei

The shell model Monte Carlo (SMMC) approach provides a powerful method for the microscopic calculation of statistical and collective nuclear properties in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods. We discuss recent applications of the method to describe the emergence of collectivity in the framework of the configuration-interaction shell model and the crossover from vibrational to rotational collectivity in families of rare-earth nuclei. We have calculated state densities of these rare-earth nuclei and find their collective enhancement factors to be correlated with the pairing and shape phase transitions. We also discuss an accurate method to calculate the ground-state energy of odd-even and odd-odd nuclei, circumventing the sign problem that originates in the projection on an odd number of particles. We have applied this method to calculate pairing gaps in families of isotopes in the iron region.