No-core Shell Model Calculations Research Articles

Ab initio study of Z(N) = 6 magicity* *This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 12175280, 12250610193, 12375143, 11975282, 11705240, 11435014), the Natural Science Foundation of Gansu Province, China (Grant Nos. 20JR10RA067, 23JRRA675), the Chinese Academy of Sciences "Light of West China" Program, the Key Research Program of the Chinese Academy of

The existence of magic numbers of protons and neutrons in nuclei is essential for understanding the nuclear structure and fundamental nuclear forces. Over decades, researchers have conducted theoretical and experimental studies on a new magic number, Z(N)=6, focusing on observables such as radii, binding energy, electromagnetic transition, and nucleon separation energies. We performed ab initio no-core shell model calculations for the occupation numbers of the lowest single particle states in the ground states of Z(N)=6 and Z(N)=8 isotopes (isotones). The results of our calculations do not support Z(N)=6 as a magic number over a range of atomic numbers. However, 14C and 14O exhibit the characteristics of double-magic nuclei.

Machine Learning in the Problem of Extrapolating Variational Calculations in Nuclear Physics

A modified machine learning method is proposed, utilizing an ensemble of artificial neural networks for the extrapolation of energies obtained in variational calculations, specifically in the No-core Shell Model (NCSM), to the case of the infinite basis. A new neural network topology is employed, and criteria for selecting both the data used for training and the trained neural networks for statistical analysis of the results are formulated. The approach is tested by extrapolating the deutron ground state energy in calculations with the Nijmegen II NN interaction and provides statistically significant results. This technique is applied to obtain extrapolated ground state energies of 6Li and 6He nuclei based on the NCSM calculations with Daejeon16 NN interaction.

Improved Tensor Current Limit from ^{8}B β Decay Including New Recoil-Order Calculations.

A precision measurement of the β^{+} decay of ^{8}B was performed using the Beta-decay Paul Trap to determine the β-ν angular correlation coefficient a_{βν}. The experimental results were combined with new abinitio symmetry-adapted no-core shell-model calculations to yield the second-most precise measurement from Gamow-Teller decays, a_{βν}=-0.3345±0.0019_{stat}±0.0021_{syst}. This value agrees with the standard model value of -1/3 and improves uncertainties in ^{8}B by nearly a factor of 2. By combining results from ^{8}B and ^{8}Li, a tight limit on tensor current coupling to right-handed neutrinos was obtained. A recent global evaluation of all other precision β decay studies suggested a nonzero value for right-handed neutrino coupling in contradiction with the standard model at just above 3σ. The present results are of comparable sensitivity and do not support this finding.

Machine learning light hypernuclei

We employ a feed-forward artificial neural network to extrapolate at large model spaces the results of ab-initio hypernuclear No-Core Shell Model calculations for the Λ separation energy BΛ of the lightest hypernuclei, Λ3H, Λ4H and Λ4He, obtained in computationally accessible harmonic oscillator basis spaces using chiral nucleon-nucleon, nucleon-nucleon-nucleon and hyperon-nucleon interactions. The overfitting problem is avoided by enlarging the size of the input dataset and by introducing a Gaussian noise during the training process of the neural network. We find that a network with a single hidden layer of eight neurons is sufficient to extrapolate correctly the value of the Λ separation energy to model spaces of size Nmax=100. The results obtained are in agreement with the experimental data in the case of Λ3H and the 0+ state of Λ4He, although they are off of the experiment by about 0.3 MeV for both the 0+ and 1+states of Λ4H and the 1+ state of Λ4He. We find that our results are in excellent agreement with those obtained using other extrapolation schemes of the No-Core Shell Model calculations, showing this that an ANN is a reliable method to extrapolate the results of hypernuclear No-Core Shell Model calculations to large model spaces.

Machine learning for the prediction of converged energies from ab initio nuclear structure calculations

The prediction of nuclear observables beyond the finite model spaces that are accessible through modern ab initio methods, such as the no-core shell model, poses a challenging task in nuclear structure theory. It requires reliable tools for the extrapolation of observables to infinite many-body Hilbert spaces along with reliable uncertainty estimates. In this work we present a universal machine learning tool capable of capturing observable-specific convergence patterns independent of nucleus and interaction. We show that, once trained on few-body systems, artificial neural networks can produce accurate predictions for a broad range of light nuclei. In particular, we discuss neural-network predictions of ground-state energies from no-core shell model calculations for ▪, ▪ and ▪ based on training data for ▪, ▪ and ▪ and compare them to classical extrapolations.

SS-HORSE extension of the no-core shell model: Application to resonances in He7

Theoretical ab initio studies of resonances in the unbound $^{7}\mathrm{He}$ nucleus are presented. We perform no-core shell-model calculations with $NN$ interactions Daejeon16 and JISP16 and utilize the SS-HORSE method to calculate the $S$ matrix for two-body channels $n+^{6}\mathrm{He}$ and $n+^{6}\mathrm{He}^{*}$ with $^{6}\mathrm{He}$ respectively in the ground and excited ${2}^{+}$ states as well as for the four-body democratic decay channel $^{4}\mathrm{He}+n+n+n$. The resonant energies and widths are obtained by numerical location of the $S$-matrix poles. We describe all experimentally known $^{7}\mathrm{He}$ resonances and suggest an interpretation of an observed wide resonance of unknown spin-parity.

Two-dimensional extrapolation procedure for an ab initio study of nuclear size parameters and the properties of the halo nucleus He6

A new two-dimensional procedure for extrapolation of the values of matter, neutron, and proton radii obtained in no-core shell model (NCSM) calculations to infinite size of its basis is proposed. A relationship between the radii is used as an additional test. Together with the JISP16 potential, which is frequently used in NCSM calculations of the radii, the Daejeon16 potential is applied for these purposes for the first time. Halo nucleus 6He is the object of studies. The small spread of radii values, successful testing using relationship between the values of these three radii, and reasonable agreement between the obtained results and experimental data as well as the results of other advanced ab initio calculations demonstrate the high efficiency of the developed approach and, therefore, good prospects for its application. The performed investigations and analysis of the results of other studies indicate that the halo of 6He has a large size 0.7 - 0.8 fm.

Deformed in-medium similarity renormalization group

In the $m$-scheme Hartree-Fock (HF) basis, we have developed an ab initio deformed single-reference in-medium similarity renormalization group (IMSRG) approach for open-shell nuclei. A deformed wave function may be more efficient in describing the deformed nucleus. The broken rotational symmetry can be restored using the angular momentum projection. However, a full angular momentum projection at the IMSRG level is still a challenge in both theory itself and computation. The angular momentum restoration mainly recaptures the static correlations, and in the present work we estimate the angular momentum projection effect by projecting the HF state as a leading-order approximation. As a test ground, we have calculated the deformed $^{8,10}\mathrm{Be}$ isotopes with the optimized chiral interaction ${\mathrm{NNLO}}_{\mathrm{opt}}$. The results are benchmarked with the no-core shell model and valence-space IMSRG calculations. Then we systematically investigate the ground-state energies and charge radii of even-even nuclei from light beryllium to medium-mass magnesium isotopes. The calculated energies are extrapolated to infinite basis space by an exponential form, and compared with extrapolated valence-space IMSRG results and experimental data.

Rigorous constraints on three-nucleon forces in chiral effective field theory from fast and accurate calculations of few-body observables

We explore the constraints on the three-nucleon force (3NF) of chiral effective field theory $(\ensuremath{\chi}\mathrm{EFT})$ that are provided by bound-state observables in the $A=3$ and $A=4$ sectors. Our statistically rigorous analysis incorporates experimental error, computational method uncertainty, and the uncertainty due to truncation of the $\ensuremath{\chi}\mathrm{EFT}$ expansion at next-to-next-to-leading order. A consistent solution for the $^{3}\mathrm{H}$ binding energy, the $^{4}\mathrm{He}$ binding energy and radius, and the $^{3}\mathrm{H}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-decay rate can only be obtained if $\ensuremath{\chi}\mathrm{EFT}$ truncation errors are included in the analysis. The $\ensuremath{\beta}$-decay rate is the only one of these that yields a nondegenerate constraint on the 3NF low-energy constants, which makes it crucial for the parameter estimation. We use eigenvector continuation for fast and accurate emulation of no-core shell model calculations of the few-nucleon observables. This facilitates sampling of the posterior probability distribution, allowing us to also determine the distributions of the parameters that quantify the truncation error. We find a $\ensuremath{\chi}\mathrm{EFT}$ expansion parameter of $Q=0.33\ifmmode\pm\else\textpm\fi{}0.06$ for these observables.

Normal-ordering approximations and translational (non)invariance

Normal-ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this work we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for $^4\rm{He}$ and $^{16}\rm{O}$ by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for $^4\rm{He}$, it is also shown to be much smaller for $^{16}\rm{O}$ although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.

Ab initio benchmarks of neutrinoless double- β decay in light nuclei with a chiral Hamiltonian

We report ab initio benchmark calculations of nuclear matrix elements (NMEs) for neutrinoless double-beta ($0\nu\beta\beta$) decays in light nuclei with mass number ranging from $A=6$ to $A=22$. We use the transition operator derived from light-Majorana neutrino exchange and evaluate the NME with three different methods: two variants of in-medium similarity renormalization group (IMSRG) and importance-truncated no-core shell model (IT-NCSM). The same two-plus-three-nucleon interaction from chiral effective field theory is employed, and both isospin-conserving ($\Delta T=0$) and isospin-changing ($\Delta T=2$) transitions are studied. We compare our resulting ground-state energies and NMEs to those of recent ab initio no-core shell model and coupled-cluster calculations, also with the same inputs. We show that the NMEs of $\Delta T=0$ transitions are in good agreement among all calculations, at the level of 10%. For $\Delta T=2$, relative deviations are more significant in some nuclei. The comparison with the exact IT-NCSM result allows us to analyze these cases in detail, and indicates the next steps towards improving the IMSRG-based approaches. The present study clearly demonstrates the power of consistent cross-checks that are made possible by ab initio methodology. This capability is crucial for providing meaningful many-body uncertainties in the NMEs for the $0\nu\beta\beta$ decays in heavier candidate nuclei, where quasi-exact benchmarks are not available.

Ab initio no-core shell model study of B10–14 isotopes with realistic NN interactions

We report a comprehensive study of $^{10-14}$B isotopes within the \textit{ab-initio} no-core shell model (NCSM) using realistic nucleon-nucleon (\textit{NN}) interactions. In particular, we have applied the inside non-local outside Yukawa (INOY) interaction to study energy spectra, electromagnetic properties and point-proton radii of the boron isotopes. The NCSM results with the charge-dependent Bonn 2000 (CDB2K), the chiral next-to-next-to-next-to-leading order (N$^3$LO) and optimized next-to-next-to-leading order (N$^2$LO$_{opt}$) interactions are also reported. We have reached basis sizes up to $N_{\mbox{max}}$ = 10 for $^{10}$B, $N_{\mbox{max}}$ = 8 for $^{11,12,13}$B and $N_{\mbox{max}}$ = 6 for $^{14}$B with m-scheme dimensions up to 1.7 billion. We also compare the NCSM calculations with the phenomenological YSOX interaction using the shell model to test the predictive power of the \textit{ab-initio} nuclear theory. Overall, our NCSM results are consistent with the available experimental data. The experimental ground state spin $3^{+}$ of $^{10}$B has been reproduced using the INOY \textit{NN} interaction. Typically, the 3\textit{N} interaction is required to correctly reproduce the aforementioned state.

Experimental study of the low-lying negative-parity states in Be11 using the B12(d,He3)Be11 reaction

Low-lying negative-parity states in $^{11}\mathrm{Be}$ having dominant $p$-wave neutron configurations were studied using the $^{12}\mathrm{B}(d,^{3}\mathrm{He})\phantom{\rule{0.16em}{0ex}}^{11}\mathrm{Be}$ proton-removal reaction in inverse kinematics. The $1/{2}_{1}^{\ensuremath{-}}$ state at 0.32 MeV, the $3/{2}_{1}^{\ensuremath{-}}$ state at 2.56 MeV, and one or both of the states including the $5/{2}_{1}^{\ensuremath{-}}$ level at 3.89 MeV and the $3/{2}_{2}^{\ensuremath{-}}$ level at 3.96 MeV were populated in the present reaction. Spectroscopic factors were determined from the differential cross sections using a distorted wave Born approximation method. The $p$-wave proton removal strengths were well described by the shell model calculations while the Nilsson model calculation underestimates the spectroscopic factors for the higher excited states. Results from both variational Monte Carlo and no-core shell-model calculations were also compared with the experimental observations.

No-core shell model calculations of the photonuclear cross section of 10B

Results of ab initio no-core, shell model calculations for the photonuclear cross section of 10B are presented using realistic two-nucleon (NN) chiral forces up to next-to-next-to-next-order (N3LO) softened by the similarity renormalization group method (SRG) with \( \lambda= 2.02\) fm-1. The electric-dipole response function is calculated using the Lanczos method, with the effects of the continuum included via neutron escape widths derived from R-matrix theory and using the Lorentz integral transform method. The calculated cross section agrees well with experimental data in terms of structure as well as in absolute peak height, \( \sigma_{\max}=4.85\) mb at photon energy \( \omega= 23.61\) MeV, and integrated cross section 85.36 MeV. mb. We also test the Brink hypothesis by calculating the electric-dipole response for the first nine positive-parity states with \( J \ne 0\) in 10B and verify that dipole excitations built upon the ground and excited states have similar characteristics.

Extrapolation of nuclear structure observables with artificial neural networks

Calculations of nuclei are often carried out in finite model spaces. Thus, finite-size corrections enter, and it is necessary to extrapolate the computed observables to infinite model spaces. In this work, we employ extrapolation methods based on artificial neural networks for observables such as the ground-state energy and the point-proton radius. We extrapolate results from no-core shell model and coupled-cluster calculations to very large model spaces and estimate uncertainties. Training the network on different data typically yields extrapolation results that cluster around distinct values. We show that a preprocessing of input data, and the inclusion of correlations among the input data, reduces the problem of multiple solutions and yields more stable extrapolated results and consistent uncertainty estimates. We perform extrapolations for ground-state energies and radii in $^{4}$He, $^{6}$Li, and $^{16}$O, and compare the predictions from neural networks with results from infrared extrapolations.

Structure of Mg28 and influence of the neutron pf shell

Gamma-ray spectroscopy and lifetime measurements using the Doppler shift attenuation method (DSAM) were performed on the nucleus $^{28}\mathrm{Mg}$ near the $N=20$ ``island of inversion,'' which was populated using a $^{12}\mathrm{C}(^{18}\mathrm{O},2\mathrm{p})^{28}\mathrm{Mg}$ fusion-evaporation reaction to investigate the impact of shell evolution on its high-lying structure. Three new levels were identified at 7203(3), 7747(2), and 7929.3(12) keV along with several new gamma rays. A newly extracted $B(E2;{4}_{1}^{+}\ensuremath{\rightarrow}{2}_{1}^{+})$ of 42(7) ${e}^{2}{\mathrm{fm}}^{4}$ indicates reduced collectivity in the yrast band at high spin, consistent with ab initio symmetry adapted no-core shell model (SA-NCSM) calculations. At high excitation energy, evidence for the population of intruder orbitals was obtained through identification of negative parity levels $[{I}^{\ensuremath{\pi}}={(0,4)}^{\ensuremath{-}}$, ${(4,5)}^{\ensuremath{-}}]$. Calculations using the SDPF-MU interaction indicate that these levels arise from single neutron excitation to the $pf$ shell and provides evidence for the lowering of these intruder orbitals approaching the island of inversion.

Natural orbitals for ab initio no-core shell model calculations

We explore the impact of optimizations of the single-particle basis on the convergence behavior and robustness of ab initio no-core shell model calculations. Our focus is on novel basis sets defined by the natural orbitals of a correlated one-body density matrix that is obtained in second-order many-body perturbation theory. Using a perturbatively improved density matrix as starting point informs the single-particle basis about the dominant correlation effects on the global structure of the many-body state, while keeping the computational cost for the basis optimization at a minimum. Already the comparison of the radial single-particle wavefunctions reveals the superiority of the natural-orbital basis compared to a Hartree-Fock or harmonic oscillator basis, and it highlights pathologies of the Hartree-Fock basis. We compare the model-space convergence of energies, root-mean-square radii, and selected electromagnetic observables with all three basis sets for selected p-shell nuclei using chiral interactions with explicit three-nucleon terms. In all cases the natural-orbital basis provides the fastest and most robust convergence, making it the most efficient basis for no-core shell model calculations. As an application we present no-core shell model calculations for the ground-state energies of all oxygen isotopes and assess the accuracy of the normal-ordered two-body approximation of the three-nucleon interaction in the natural-orbital basis.

Open-shell nuclei from No-Core Shell Model with perturbative improvement

We introduce a hybrid many-body approach that combines the flexibility of the No-Core Shell Model (NCSM) with the efficiency of Multi-Configurational Perturbation Theory (MCPT) to compute ground- and excited-state energies in arbitrary open-shell nuclei in large model spaces. The NCSM in small model spaces is used to define a multi-determinantal reference state that contains the most important multi-particle multi-hole correlations and a subsequent second-order MCPT correction is used to capture additional correlation effects from a large model space. We apply this new ab initio approach for the calculation of ground-state and excitation energies of even and odd-mass carbon, oxygen, and fluorine isotopes and compare to large-scale NCSM calculations that are computationally much more expensive.

Large-scale exact diagonalizations reveal low-momentum scales of nuclei

Ab initio methods aim to solve the nuclear many-body problem with controlled approximations. Virtually exact numerical solutions for realistic interactions can only be obtained for certain special cases such as few-nucleon systems. Here we extend the reach of exact diagonalization methods to handle model spaces with dimension exceeding $10^{10}$ on a single compute node. This allows us to perform no-core shell model (NCSM) calculations for 6Li in model spaces up to $N_\mathrm{max} = 22$ and to reveal the 4He+d halo structure of this nucleus. Still, the use of a finite harmonic-oscillator basis implies truncations in both infrared (IR) and ultraviolet (UV) length scales. These truncations impose finite-size corrections on observables computed in this basis. We perform IR extrapolations of energies and radii computed in the NCSM and with the coupled-cluster method at several fixed UV cutoffs. It is shown that this strategy enables information gain also from data that is not fully UV converged. IR extrapolations improve the accuracy of relevant bound-state observables for a range of UV cutoffs, thus making them profitable tools. We relate the momentum scale that governs the exponential IR convergence to the threshold energy for the first open decay channel. Using large-scale NCSM calculations we numerically verify this small-momentum scale of finite nuclei.

Ab initio translationally invariant nonlocal one-body densities from no-core shell-model theory

[Background:] It is well known that effective nuclear interactions are in general nonlocal. Thus if nuclear densities obtained from {\it ab initio} no-core-shell-model (NCSM) calculations are to be used in reaction calculations, translationally invariant nonlocal densities must be available. [Purpose:] Though it is standard to extract translationally invariant one-body local densities from NCSM calculations to calculate local nuclear observables like radii and transition amplitudes, the corresponding nonlocal one-body densities have not been considered so far. A major reason for this is that the procedure for removing the center-of-mass component from NCSM wavefunctions up to now has only been developed for local densities. [Results:] A formulation for removing center-of-mass contributions from nonlocal one-body densities obtained from NCSM and symmetry-adapted NCSM (SA-NCSM) calculations is derived, and applied to the ground state densities of $^4$He, $^6$Li, $^{12}$C, and $^{16}$O. The nonlocality is studied as a function of angular momentum components in momentum as well as coordinate space [Conclusions:] We find that the nonlocality for the ground state densities of the nuclei under consideration increases as a function of the angular momentum. The relative magnitude of those contributions decreases with increasing angular momentum. In general, the nonlocal structure of the one-body density matrices we studied is given by the shell structure of the nucleus, and can not be described with simple functional forms.