Shell-model Wave Functions Research Articles

Shell-model calculations for two-neutron transfer near the N=20 island of inversion

We describe connections between shell-model calculation results and predictions for two-nucleon transfer reactions. Measurements of the ^{30}Mg(t,p)^{32}Mg reaction were used to identify a low-lying shape-coexisting 0^+ state in ^{32}Mg. Interpretations of those early, limited data were based on simple empirical models. The cross sections for two-nucleon transfer are, however, extremely sensitive to the details of the nuclear-structure. Motivated by the possibility of new high-resolution two-nucleon transfer measurements around the N=20 Island of Inversion, we have performed shell-model calculations wave functions for states in the nuclei ^{30,32}Mg using the SDPF-M and SDPF-MU interactions, and from those calculations obtain the two-nucleon transfer amplitudes that we use to predict cross sections for the ^{28,30}Mg(t,p)^{30,32}Mg reactions. The combined results show how the nuclear structure properties influence experimental observables for the (t,p) reaction. We compare our results with the existing data.

Microscopic study of normal deformed bands in 167,169,171Lu

The normal deformed bands of [Formula: see text]Lu isotopes have been studied by using projected shell model approach. The band head spins, configurations and energies of all the normal deformed bands are reproduced well by the above said approach. The present calculations have predicted [Formula: see text] band head spin for [523]7/2− bands of [Formula: see text]Lu. The observed systematics of aligned angular momenta and the experimental differences in frequencies around the backbends for all the normal deformed bands are reproduced well by the theoretical results. Besides this, the observed backbends or upbends in these isotopes may be ascribed to the alignment of a pair of neutrons in the neutron [Formula: see text] orbital. Although the theoretical [Formula: see text] values obtained from the projected shell model wavefunctions overestimate the experimental [Formula: see text] values in [Formula: see text]Lu, yet they are close to the measured values, considering the precision of measurement. The reduction in the theoretical [Formula: see text] values around the band crossing region may be ascribed to change in nuclear structure of yrast bands due to neutron pair alignment in [Formula: see text] orbital.

Nuclear shell-model simulation in digital quantum computers

The nuclear shell model is one of the prime many-body methods to study the structure of atomic nuclei, but it is hampered by an exponential scaling on the basis size as the number of particles increases. We present a shell-model quantum circuit design strategy to find nuclear ground states by exploiting an adaptive variational quantum eigensolver algorithm. Our circuit implementation is in excellent agreement with classical shell-model simulations for a dozen of light and medium-mass nuclei, including neon and calcium isotopes. We quantify the circuit depth, width and number of gates to encode realistic shell-model wavefunctions. Our strategy also addresses explicitly energy measurements and the required number of circuits to perform them. Our simulated circuits approach the benchmark results exponentially with a polynomial scaling in quantum resources for each nucleus. This work paves the way for quantum computing shell-model studies across the nuclear chart and our quantum resource quantification may be used in configuration-interaction calculations of other fermionic systems.

Triaxiality and the nature of low-energy excitations in Ge76

The deformation properties of the low-lying states in Ge76 have been investigated following a safe-energy Coulomb excitation measurement with the GRETINA tracking array and CHICO2 heavy-ion counter at the ATLAS accelerator facility at Argonne National Laboratory. A comprehensive set of transition and static E2 matrix elements were extracted from the measured differential Coulomb cross sections and compared with results of configuration-interaction shell-model calculations and computations carried out within the framework of the generalized triaxial rotor model. The remarkable agreement between the calculated and experimental data supports a near-maximum triaxial deformation for the ground state of Ge76. In addition, the degree of softness of the asymmetry in Ge76 and Se76 was investigated using rotational invariants generated from configuration-interaction shell-model wave functions computed with the jj44b and JUN45 effective interactions. The resulting invariants are shown to be consistent with a stiff triaxial deformation in Ge76 and a predominantly soft triaxial potential for Se76, in agreement with the conclusions of recent works by this collaboration.5 MoreReceived 27 January 2023Accepted 3 April 2023DOI:https://doi.org/10.1103/PhysRevC.107.044314©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNuclear reactionsNuclear structure & decaysNucleon distributionProperties59 ≤ A ≤ 89TechniquesCollective modelsShell modelNuclear Physics

Nuclear structure properties of Si and P isotopes with the microscopic effective interactions

In the present work, we have reported comprehensive study of the sd-shell nuclei Si and P with neutron number varying from N=9 to N=20 using the microscopic effective valence shell interactions, namely, N3LO, JISP16 and DJ16A. These effective sd-shell interactions are developed using the ab initio no-core shell model wave functions and the Okubo-Lee-Suzuki transformation method. For comparison, we have also performed shell model calculations with the empirical USDB interaction. Energy spectra and electromagnetic properties of these isotopic chains have been studied. Theoretically calculated shell model results are compared with the available experimental data, to check the predictive strength of these microscopic interactions. It is found that the binding energies of the ground states are better reproduced with the DJ16A interaction as compared to other microscopic interactions and a proton subshell closure at Z=14 is obtained in Si. Spin-tensor decomposition of two-body interaction is presented to understand the contributions from central, vector and tensor components into these interactions. Spectroscopic strengths of 23Al(d,n)24Si are examined for the newly performed experiment at NSCL.

Study of structure and radii for 20−31Na isotopes using microscopic interactions

In this work, Na isotopes with mass varying from A=20 to 31 have been studied using DJ16A, JISP16 and N3LO microscopic effective interactions in the sd-shell. These effective interactions are derived for sd-shell using no-core shell model wavefunctions and a unitary transformation method. We have also performed calculation with IMSRG effective interactions targeted for a particular nucleus. The studies include a detailed analysis of ground state binding energy, low-lying spectra, and electromagnetic properties such as reduced electric quadrupole transition strengths, quadrupole and magnetic dipole moments of the Na chain. The results obtained from these microscopic effective interactions were compared with the experimental data as well as with the results of phenomenological interaction USDB. The charge radii of Na isotopes are evaluated using shell model harmonic oscillator wave functions. In addition to charge radii, matter radii and neutron skin thickness in the Na chain are discussed with a focus on neutron-deficient Na isotopes.

Constructing approximate shell-model wavefunctions by eigenvector continuation

Abstract Shell-model calculations play a key role in elucidating various properties of nuclei. In general, those studies require a huge number of calculations to be repeated for parameter calibration and quantifying uncertainties. To reduce the computational burden, we propose a new workflow of shell-model calculations using a method called eigenvector continuation (EC). It enables us to efficiently approximate the eigenpairs under a given Hamiltonian by previously sampled eigenvectors. We demonstrate the validity of EC as an emulator of the valence shell-model, including first application of EC to electromagnetic transition matrix elements. Furthermore, we propose a new usage of EC: preprocessing, in which we start the Lanczos iterations from the approximate eigenvectors, and demonstrate that this can accelerate subsequent research cycles. With the aid of the EC, the eigenvectors obtained during the parameter optimization are not necessarily to be discarded, even if their eigenvalues are far from the experimental data. Those eigenvectors can become accumulated knowledge.

Formulation of a shell–cluster overlap integral with the Gaussian expansion method

Abstract We formulate a computational method to evaluate the overlap integral of the shell-model and cluster-model wave functions. The framework is applied to the system of the core plus two neutrons, and the magnitude of the overlap of the shell-model configuration (core + $n$ + $n$) and the di-neutron cluster one (core + $2n$) is explored. We have found that the magnitude of the overlap integral is prominently enhanced when two neutrons occupy shell-model orbits with low orbital angular momenta, such as $s$- and $p$-wave orbits. The shell–cluster overlap is calculated in systems with $jj$-closed cores plus two neutrons, and the enhancement due to occupation of the $s$ or $p$ orbit also occurs in the systematic calculation. The effect of the configuration interaction on the shell–cluster overlap integrals is also discussed.

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.

Rotational bands beyond the Elliott model

Rotational bands are commonplace in the spectra of atomic nuclei. Inspired by early descriptions of these bands by quadrupole deformations of a liquid drop, Elliott constructed discrete nucleon representations of SU(3) from fermionic creation and annihilation operators. Ever since, Elliott’s model has been foundational to descriptions of rotation in nuclei. Later work, however, suggested the symplectic extension Sp(3, R) provides a more unified picture. We decompose no-core shell-model nuclear wave functions into symmetry-defined subspaces for several beryllium isotopes, as well as 20Ne, using the quadratic Casimirs of both Elliott’s SU(3) and Sp(3, R). The band structure, delineated by strong B(E2) values, has a more consistent description in Sp(3, R) rather than SU(3). In particular, we confirm previous work finding in some nuclides strongly connected upper and lower bands with the same underlying symplectic structure.

Consistent description for cluster dynamics and single-particle correlation

Cluster dynamics and single-particle correlation are simultaneously treated for the description of the ground state of $^{12}\mathrm{C}$. The recent development of the antisymmetrized quasicluster model (AQCM) makes it possible to generate $jj$-coupling shell-model wave functions from $\ensuremath{\alpha}$ cluster models. The cluster dynamics and the competition with the $jj$-coupling shell-model structure can be estimated rather easily. In the present study, we further include the effect of single-particle excitation; the mixing of the two-particle--two-hole excited states is considered. The single-particle excitation is not always taken into account in the standard cluster model analyses, and the two-particle--two-hole states are found to strongly contribute to the lowering of the ground state owing to the pairing-like correlations. By extending AQCM, all of the basis states are prepared on the same footing, and they are superposed based on the framework of the generator coordinate method (GCM).

Triaxial rigidity of Er166 and its Bohr-model realization

The triaxial nature of low-lying rotational bands of $^{166}\mathrm{Er}$ is presented from the viewpoint of the Bohr Hamiltonian and from that of many-fermion calculations by the Monte Carlo shell model and the constrained Hartree-Fock method with projections. A recently proposed novel picture of those bands suggests definite triaxial shapes of those bands, in contrast to the traditional view with the prolate ground-state band and the $\ensuremath{\gamma}$-vibrational excited band. Excitation level energies and $E2$ transitions can be described well by the Bohr Hamiltonian and by the many-fermion approaches, where rather rigid triaxiality plays vital roles, although certain fluctuations occur in shell-model wave functions. Based on the potential energy surfaces with the projections, we show how the triaxial rigidity appears and what the softness of the triaxiality implies. The excitation to the so-called double $\ensuremath{\gamma}$-phonon state is discussed briefly.

Structures and production cross sections of p-shell Lambda-hypernuclei calculated with multi-configuration shell model

The high-resolution data of the 10B (e, e′K +) Be experiment done at the Jefferson National Laboratory reported a new bump, which cannot be explained by the DWIA calculation by employing the conventional shell-model wave functions. To describe the new bump, we have extended the model space by introducing a new configuration with 1ħω excitation for both nuclear and hyperon states. We show that the DWIA calculation by using the extended shell-model wave functions successfully explains the new bump for the first time, and that the p-state Λ in the lower molecular orbit plays an important role in the appearance of the new bump.

Short-range correlations for 0νββ decay and low-momentum NN potentials

We approach the calculation of the nuclear matrix element of the neutrinoless double-β decay process, considering the light-neutrino-exchange channel, by way of the realistic shell-model. In particular the focus of our work is spotted on the role of the short-range correlations, which should be taken into account because of the short-range repulsion of the realistic potentials. Our shell-model wave functions are calculated using an effective Hamiltonian derived from the high-precision CD-Bonn nucleon-nucleon potential, the latter renormalized by way of the so-called Vlow-k approach. The renormalization procedure decouples the repulsive high-momentum component of the potential from the low-momentum ones by the introduction of a cutoff Λ, and is employed to renormalize consistently the two-body neutrino potentials to calculate the nuclear matrix elements of candidates to this decay process in mass interval ranging from A = 76 up to A = 136. We study the dependence of the decay operator on the choice of the cutoff, and compare our results with other approaches that can be found in present literature.

Second-forbidden nonunique β− decays of Na24 and Cl36 assessed by the nuclear shell model

We have performed a systematic study of the log$ft$ values, shape factors and electron spectra for the second-forbidden nonunique $\beta^-$ decays of $^{24}$Na$(4^+) \rightarrow ^{24}$Mg$(2^+)$ and $^{36}$Cl$(2^+) \rightarrow ^{36}$Ar$(0^+)$ transitions under the framework of the nuclear shell model. We have performed the shell model calculations in the $sd$ model space, using more recent microscopic effective interactions such as Daejeon16, chiral N3LO, and JISP16. These interactions are derived from the no-core shell model wave functions using Okubo-Lee-Suzuki transformation.For comparison, we have also shown the results obtain from the phenomenological USDB interaction. To test the predictive power of these interactions first we have computed low-lying energy spectra of parent and daughter nuclei involved in these transitions. The computed results for energy spectra, nuclear matrix elements, log$ft$ values, shape factors, electron spectra and decomposition of the integrated shape factor are reported and compare with the available experimental data.

Possibility of C14 cluster as a building block of medium-mass nuclei

The possibility of the $^{14}\mathrm{C}$ cluster being a basic building block of medium-mass nuclei is discussed. Although $\ensuremath{\alpha}$ cluster structures have been widely discussed in the light $N\ensuremath{\approx}Z$ mass region, the neutron-to-proton ratio deviates from unity in the nuclei near $\ensuremath{\beta}$-stability line and in neutron-rich nuclei. Thus, more neutron-rich objects with $N>Z$ could become the building blocks of cluster structures in such nuclei. The $^{14}\mathrm{C}$ nucleus is strongly bound and can be regarded as such a candidate. In addition, the path to the lowest shell-model configuration at short relative distances is closed for the $^{14}\mathrm{C}+^{14}\mathrm{C}$ structure, contrary to the case of the $^{12}\mathrm{C}+^{12}\mathrm{C}$ structure; this allows an appreciable separation distance between the $^{14}\mathrm{C}$ clusters. The recent development of the antisymmetrized quasicluster model (AQCM) allows us to utilize a $jj$-coupling shell-model wave function for each cluster in a simplified way. The AQCM results for the $^{14}\mathrm{C}+^{14}\mathrm{C}$ structure in $^{28}\mathrm{Mg}$ are compared with the ones of cranked relativistic mean field (CRMF) calculations. Although theoretical frameworks of these two models are quite different, they give similar results for the nucleonic densities and rotational properties of the structure under investigation. The existence of a linear chain three-$^{14}\mathrm{C}$ cluster structure in $^{42}\mathrm{Ar}$ has also been predicted in AQCM. These results confirm the role of the $^{14}\mathrm{C}$ cluster as a possible building block of cluster structures in medium-mass nuclei.

α decay to a doubly magic core in the quartetting wave function approach

We present a microscopic calculation of $\alpha$-cluster formation in heavy nuclei $^{104}$Te ($\alpha$+$^{100}$Sn), $^{212}$Po ($\alpha$+$^{208}$Pb) and their neighbors $^{102}$Sn, $^{102}$Te, $^{210}$Pb and $^{210}$Po by using the quartetting wave function approach. Improving the local density approximation, the shell structure of the core nucleus is considered, and the center-of-mass (c.o.m.) effective potential for the quartet is obtained self-consistently from the shell model wavefunctions. The $\alpha$-cluster formation and decay probabilities are obtained by solving the bound-state of the c.o.m. motion of the quartet and the scattering state of the formed $\alpha$-cluster in the Gurvitz approach. Striking shell effects on the $\alpha$-cluster formation probabilities are analyzed for magic numbers 50, 82 and 126. The computed $\alpha$-decay half-lives of these special nuclei are compared with the newest experimental data.

An analytical formula for proton momentum distribution in nuclei with A>4

A successful analytical formula for the proton momentum distribution in all nuclei with A>4 accounting for nucleon-nucleon correlation effects, is presented. In this formula the Isomorphic Shell Model wave functions are employed, which are readily available for all nuclei all the way up to 2 0 8Pb. However, other wave functions (e.g., shell model or Hartree-Fock) could be used with almost equivalent results. Available experimental data for 4He, 1 2C and 5 6Fe and predictions of other theories, e.g., for 4 0Ca, are used for comparison of the predictions of the present formula. A reservation is kept concerning the validity of this formula for the momentum distribution of exotic nuclei.

Effective interactions in the sd shell

We perform a quantitative study of the microscopic effective shell-model interactions in the valence sd shell, obtained from modern nucleon-nucleon potentials, chiral N3LO, JISP16 and Daejeon16, using No-Core Shell-Model wave functions and the Okubo-Lee-Suzuki transformation. We investigate the monopole properties of those interactions in comparison with the phenomenological universal sd-shell interaction, USDB. Theoretical binding energies and low-energy spectra of O isotopes and of selected sd-shell nuclei, are presented. We conclude that there is a noticeable improvement in the quality of the effective interaction when it is derived from the Daejeon16 potential. We show that its proton-neutron centroids are consistent with those from USDB. We then propose monopole modifications of the Daejeon16 centroids in order to provide an adjusted interaction yielding significantly improved agreement with the experiment. A spin-tensor decomposition of two-body effective interactions is applied in order to extract more information on the structure of the centroids and to understand the reason for deficiencies arising from our current theoretical approximations. The issue of the possible role of the three-nucleon forces is addressed.

Nonlocalized clustering and evolution of cluster structure in nuclei

We explain various facets of the THSR (Tohsaki-Horiuchi-Schuck-R\"opke) wave function. We first discuss the THSR wave function as a wave function of cluster-gas state, since the THSR wave function was originally introduced to elucidate the 3$\alpha$-condensate-like character of the Hoyle state ($0_2^+$ state) of $^{12}$C. We briefly review the cluster-model studies of the Hoyle state in 1970's in order to explain how there emerged the idea to assign the $\alpha$ condensate character to the Hoyle state. We then explain that the THSR wave function can describe very well also non-gaslike ordinary cluster states with spatial localization of clusters. This fact means that the dynamical motion of clusters is of nonlocalized nature just as in gas-like states of clusters and the localization of clusters is due to the inter-cluster Pauli principle which is against the close approach of two clusters. The nonlocalized cluster dynamics is formulated by the container model of cluster dynamics. The container model describes gas-like state and non-gaslike states as the solutions of the Hill-Wheeler equation with respect to the size parameter of THSR wave function which is just the size parameter of the container. When we notice that fact that the THSR wave function with the smallest value of size parameter is equivalent to the shell-model wave function, we see that the container model describes the evolution of cluster structure from the ground state with shell-model structure up to the gas-like cluster state via ordinary non-gaslike cluster states. For the description of various cluster structure, more generation of THSR wave function have been introduced and we review some typical examples with their actual applications.