Projected Generator Coordinate Method Research Articles

Ab initio description of monopole resonances in light- and medium-mass nuclei

The paper is the third of a series dedicated to the ab initio description of monopole giant resonances in mid-mass closed- and open-shell nuclei via the so-called projected generator coordinate method. The present focus is on the computation of the moments mk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_k$$\\end{document} of the monopole strength distribution, which are used to quantify its centroid energy and dispersion. First, the capacity to compute low-order moments via two different methods is developed and benchmarked for the m1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_1$$\\end{document} moment. Second, the impact of the angular momentum projection on the centroid energy and dispersion of the monopole strength is analysed before comparing the results to those obtained from consistent quasi-particle random phase approximation calculations. Next, the so-called energy weighted sum rule (EWSR) is investigated. First, the appropriate ESWR in the center-of-mass frame is derived analytically. Second, the intrinsic EWSR is tested in order to quantify the (unwanted) local-gauge symmetry breaking of the presently employed chiral effective field theory (χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}EFT) interactions. Finally, the infinite nuclear matter incompressibility associated with the employed χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}EFT interactions is extracted by extrapolating the finite-nucleus incompressibility computed from the monopole centroid energy.

Ab initio description of monopole resonances in light- and medium-mass nuclei

Giant resonances (GRs) are a striking manifestation of collective motions in atomic nuclei. The present paper is the second in a series of four dedicated to the use of the projected generator coordinate method (PGCM) for the ab initio determination of the isoscalar giant monopole resonance (GMR) in closed- and open-shell mid-mass nuclei. While the first paper was dedicated to quantifying various uncertainty sources, the present paper focuses on the first applications to three doubly-open shell nuclei, namely 46Ti\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{46}\\hbox {Ti}$$\\end{document}, 28Si\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{28}\\hbox {Si}$$\\end{document} and 24Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{24}\\hbox {Mg}$$\\end{document}. In particular, the goal is to investigate, starting from chiral effective field theory nuclear interactions, (i) the coupling of the GMR with the giant quadrupole resonance (GQR) in intrinsically-deformed nuclei, (ii) the possible impact of shape coexistence and shape mixing on the GMR, (iii) the GMR based on shape isomers and (iv) the impact of anharmonic effects on the monopole response. The latter is studied by comparing PGCM results to those obtained via the quasi-particle random phase approximation (QRPA), the traditional many-body approach to giant resonances, performed in a consistent setting. Eventually, PGCM results for sd-shell nuclei are in good agreement with experimental data, which is attributed to the capacity of the PGCM to capture the important fragmentation of the monopole response in light, intrinsically-deformed systems. Still, the comparison to data in 28Si\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{28}\\hbox {Si}$$\\end{document} and 24Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{24}\\hbox {Mg}$$\\end{document} illustrates the challenge (and the potential benefit) of extracting unambiguous experimental information.

Ab initio description of monopole resonances in light- and medium-mass nuclei

Giant resonances (GRs) are a striking manifestation of collective motions in mesoscopic systems such as atomic nuclei. Until recently, theoretical investigations have essentially relied on the (quasiparticle) random phase approximation ((Q)RPA), and extensions of it, based on phenomenological energy density functionals (EDFs). As part of a current effort to describe GRs within an ab initio theoretical scheme, the present work promotes the use of the projected generator coordinate method (PGCM). This method, which can handle anharmonic effects while satisfying symmetries of the nuclear Hamiltonian, displays a favorable (i.e. mean-field-like) scaling with system’s size. Presently focusing on the isoscalar giant monopole resonance (GMR) of light- and medium-mass nuclei, PGCM’s potential to deliver wide-range ab initio studies of GRs in closed- and open-shell nuclei encompassing pairing, deformation, and shape coexistence effects is demonstrated. The comparison with consistent QRPA calculations highlights PGCM’s unique attributes and sheds light on the intricate interplay of nuclear collective excitations. The present paper is the first in a series of four and focuses on technical aspects and uncertainty quantification of ab initio PGCM calculations of GMR using the doubly open-shell 46\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{46}$$\\end{document}Ti as an illustrative example. The second paper displays results for a set of nuclei of physical interest and proceeds to the comparison with consistent (deformed) ab initio QRPA calculations. While the third paper analyzes useful moments of the monopolar strength function and different ways to access them within PGCM calculations, the fourth paper focuses on the effect of the symmetry restoration on the monopole strength function.

Quantum-Number Projected Generator Coordinate Method for 21Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction

In this paper, we report a study of the low-lying states of deformed 21Ne within the framework of the quantum-number projected generator coordinate method (PGCM), starting from a chiral two-nucleon-plus-three-nucleon (NN+3N) interaction. The wave functions of states are constructed as a linear combination of a set of axially deformed Hartree–Fock–Bogliubov (HFB) wave functions with different quadrupole deformations. These HFB wave functions are projected onto different angular momenta and the correct neutron and proton numbers for 21Ne. The results of the calculations based on the effective Hamiltonians derived by normal-ordering the 3N interaction with respect to three different reference states, including the quantum-number projected HFB wave functions for 20Ne, 22Ne, and an ensemble of them with equal weights, are compared. This study serves as a key step towards ab initio calculations of odd-mass deformed nuclei with the in-medium GCM.

Electromagnetic properties of Λ hypernuclei with a beyond relativistic mean-field approach

We present a microscopic study of electromagnetic properties of the low-lying states of single-Λ hypernucleus Λ9Be using a quantum-number projected generator coordinate method (HyperGCM) based on covariant density functional theory. All the configurations are restricted to have axial symmetry, with the Λ hyperon in the lowest-energy state. The results are compared to the predictions of a simple particle-rotor model (PRM), for which analytic expressions are available. We show that the electromagnetic properties of Λ9Be from the HyperGCM calculation are close to those by the PRM. In particular, it is found that the magnetic moments and M1 transition strengths are less sensitive to the coupling strengths of the ΛN interaction than the electric quadrupole transition strengths. The latter may provide additional constraints on the ΛN interaction.

Towards microscopic optical potentials in deformed nuclei

A microscopic and consistent description of both nuclear structure and reactions is instrumental to extend the predictivity of models calculating scattering observables. In particular, this is crucial in the case of exotic nuclei not yet discovered.In this manuscript, we will present the plan of the Lund effort for a symmetry breaking description of bound and scattering observables using a generator coordinate method model based on an effective Hamiltonian constrained with a (Skyrme) functional. Following this, we will illustrate the steps to construct an optical potential for deformed nuclei from microscopic wavefunctions obtained with projected generator coordinate method from Hartree-Fock-Bogoliubov basis.

On the off-diagonal Wick’s theorem and Onishi formula

The projected generator coordinate method based on the configuration mixing of non-orthogonal Bogoliubov product states, along with more advanced methods based on it, require the computation of off-diagonal Hamiltonian and norm kernels. While the Hamiltonian kernel is efficiently computed via the off-diagonal Wick theorem of Balian and Brezin, the norm kernel relies on the Onishi formula (or equivalently the Pfaffian formula by Robledo or the integral formula by Bally and Duguet). Traditionally, the derivation of these two categories of formulae relies on different formal schemes. In the present work, the formulae for the operator and norm kernels are computed consistently from the same diagrammatic method. The approach further offers the possibility to address kernels involving more general states in the future.

Multi-reference many-body perturbation theory for nuclei

In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (\(e_{\mathrm {max}}=4\)) model space demonstrate that exact binding energies of closed- (\({}^{16}\mathrm {O}\)) and open-shell (\({}^{18}\mathrm {O}\), \({}^{20}\mathrm {Ne}\)) nuclei are reproduced within 0.5–\(1.5\%\) at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.

Multi-reference many-body perturbation theory for nuclei

The neon isotopic chain displays a rich phenomenology, ranging from clustering in the ground-state of the self-conjugate doubly open-shell stable $$^{20}$$ Ne isotope to the physics of the island of inversion around the neutron-rich $$^{30}$$ Ne isotope. This second (i.e. Paper II) of the present series proposes an extensive ab initio study of neon isotopes based on two complementary many-body methods, i.e. the quasi-exact in-medium no-core shell model (IM-NCSM) and the projected generator coordinate method (PGCM) that is ideally suited to capturing strong static correlations associated with shape deformation and fluctuations. Calculations employ a state-of-the-art generation of chiral effective field theory Hamiltonians and evaluate the associated systematic uncertainties. In spite of missing so-called dynamical correlations, which can be added via the multi-reference perturbation theory proposed in the first paper (i.e. Paper I) of the present series [1], the PGCM is shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing the physics of the island of inversion constitutes a challenge that seems to require the inclusion of dynamical correlations. This is addressed in the third paper (i.e. Paper III) of the present series [2].

Microscopic description of quadrupole collectivity in neutron-rich nuclei across the N = 126 shell closure

The quadrupole collectivity in Nd, Sm, Gd, Dy, Er, Yb, Hf and W nuclei with neutron numbers 122 ≤ N ≤ 156 is studied, both at the mean field level and beyond, using the Gogny energy density functional. Besides the robustness of the N = 126 neutron shell closure, it is shown that the onset of static deformations in those isotopic chains with increasing neutron number leads to an enhanced stability and further extends the corresponding two-neutron drip lines far beyond what could be expected from spherical calculations. Independence of the mean-field predictions with respect to the particular version of the Gogny energy density functional employed is demonstrated by comparing results based on the D1S and D1M parameter sets. Correlations beyond mean field are taken into account in the framework of the angular momentum projected generator coordinate method calculation. It is shown that N = 126 remains a robust neutron magic number when dynamical effects are included. The analysis of the collective wave functions, average deformations and excitation energies indicate that, with increasing neutron number, the zero-point quantum corrections lead to dominant prolate configurations in the 0 +1 , 0 +2 , 2 +1 and 2 +2 states of the studied nuclei. Moreover, those dynamical deformation effects provide an enhanced stability that further supports the mean-field predictions, corroborating a shift of the r-process path to higher neutron numbers. Beyond mean-field calculations provide a smaller shell gap at N = 126 than the mean-field one in good agreement with previous theoretical studies. However, the shell gap still remains strong enough in the two-neutron drip lines.

Global dynamical correlation energies in covariant density functional theory: Cranking approximation

The global dynamical correlation energies for 575 even-even nuclei with proton numbers ranging from Z=8 to Z=108 calculated with the covariant density functional theory using the PC-PK1 parametrization are presented. The dynamical correlation energies include the rotational correction energies obtained with the cranking approximation and the quadrupole vibrational correction energies. The systematic behavior of the present correlation energies is in good agreement with that obtained from the projected generator coordinate method using the SLy4 Skyrme force although our values are systematically smaller. After including the dynamical correlation energies, the root-mean-square deviation predicted by the PC-PK1 for the 575 even-even nuclei masses is reduced from 2.58 MeV to 1.24 MeV.

Application of the generator coordinate method to neutron-rich Se and Ge isotopes

The quantum-number projected generator coordinate method (GCM) is ap- plied to the neutron-rich Se and Ge isotopes, where the monopole and quadrupole pairing plus quadrupole-quadrupole interaction is employed as an effective interaction. The en- ergy spectra obtained by the GCM are compared to both the shell model results and the experimental data. The GCM reproduces well the energy levels of high-spin states as well as the low-lying states. The structure of the low-lying collective states is analyzed through the GCM wave functions.

Beyond-mean-field study of the possible “bubble” structure of34Si

Recent self-consistent mean-field calculations predict a substantial depletion of the proton density in the interior of 34Si. In the present study, we investigate how correlations beyond the mean field modify this finding. The framework of the calculation is a particle-number and angular-momentum projected Generator Coordinate Method based on Hartree-Fock-Bogoliubov+Lipkin-Nogami states with axial quadrupole deformation. The parametrization SLy4 of the Skyrme energy density functional is used together with a density-dependent pairing energy functional. For the first time, the generator coordinate method is applied to the calculation of charge and transition densities. The impact of pairing correlations, symmetry restorations and shape mixing on the density profile is analyzed step by step. All these effects significantly alter the radial density profile, and tend to bring it closer to a Fermi-type density distribution.

What can be learned from binding energy differences about nuclear structure: The example ofδVpn

We perform an analysis of a binding energy difference called $\ensuremath{\delta}{V}_{\mathit{pn}}(N,Z)\ensuremath{\equiv}\ensuremath{-}\frac{1}{4}[E(Z,N)\ensuremath{-}E(Z,N\ensuremath{-}2)\ensuremath{-}E(Z\ensuremath{-}2,N)+E(Z\ensuremath{-}2,N\ensuremath{-}2)]$ in the framework of a realistic nuclear model. It has been suggested that $\ensuremath{\delta}{V}_{\mathit{pn}}$ values provide a sensitive probe of nuclear structure, and it has been put forward as a primary motivation for the measurement of specific nuclear masses. Using the angular momentum and particle-number projected generator coordinate method and the Skyrme interaction SLy4, we analyze the contribution brought to $\ensuremath{\delta}{V}_{\mathit{pn}}$ by static deformation and dynamic fluctuations around the mean-field ground state. Our method gives a good overall description of $\ensuremath{\delta}{V}_{\mathit{pn}}$ throughout the chart of nuclei with the exception of the anomaly related to the Wigner energy along the $N=Z$ line. The main conclusions of our analysis of $\ensuremath{\delta}{V}_{\mathit{pn}}$, which are at variance with its standard interpretation, are that (i) the structures seen in the systematics of $\ensuremath{\delta}{V}_{\mathit{pn}}$ throughout the chart of nuclei can be easily explained combining a smooth background related to the symmetry energy and correlation energies due to deformation and collective fluctuations, (ii) the characteristic pattern of $\ensuremath{\delta}{V}_{\mathit{pn}}$ having a much larger size for nuclei that add only particles or only holes to a doubly magic nucleus than for nuclei that add particles for one nucleon species and holes for the other is a trivial consequence of the asymmetric definition of $\ensuremath{\delta}{V}_{\mathit{pn}}$ and not due to a the different structure of these nuclei, (iii) $\ensuremath{\delta}{V}_{\mathit{pn}}$ does not provide a very reliable indicator for structural changes, (iv)$\ensuremath{\delta}{V}_{\mathit{pn}}$ does not provide a reliable measure of the proton-neutron interaction in the nuclear energy density functional (EDF) or of that between the last filled orbits or of the one summed over all orbits, and (v) $\ensuremath{\delta}{V}_{\mathit{pn}}$ does not provide a conclusive benchmark for nuclear EDF methods that is superior or complementary to other mass filters such as two-nucleon separation energies or $Q$ values.

Shape ofAr44: Onset of deformation in neutron-rich nuclei nearCa48

The development of deformation and shape coexistence in the vicinity of doubly magic $^{48}\mathrm{Ca}$, related to the weakening of the $N=28$ shell closure, was addressed in a low-energy Coulomb excitation experiment using a radioactive $^{44}\mathrm{Ar}$ beam from the SPIRAL facility at GANIL. The ${2}_{1}^{+}$ and ${2}_{2}^{+}$ states in $^{44}\mathrm{Ar}$ were excited on $^{208}\mathrm{Pb}$ and $^{109}\mathrm{Ag}$ targets at two different beam energies. $B(E2)$ values between all observed states and the spectroscopic quadrupole moment of the ${2}_{1}^{+}$ state were extracted from the differential Coulomb excitation cross sections, indicating a prolate shape of the $^{44}\mathrm{Ar}$ nucleus and giving evidence of an onset of deformation already two protons and two neutrons away from doubly magic $^{48}\mathrm{Ca}$. New Hartree-Fock-Bogoliubov based configuration mixing calculations have been performed with the Gogny D1S interaction for $^{44}\mathrm{Ar}$ and neighboring nuclei using two different approaches: the angular momentum projected generator coordinate method considering axial quadrupole deformations and a five-dimensional approach including the triaxial degree of freedom. The experimental values and new calculations are furthermore compared to shell-model calculations and to relativistic mean-field calculations. The new results give insight into the weakening of the $N=28$ shell closure and the development of deformation in this neutron-rich region of the nuclear chart.

Collectivity-induced quenching of signatures for shell closures

Mass differences are an often used as signature and measure for shell closure. Using the angular-momentum projected generator coordinate method and the Skyrme interaction SLy4, we analyze the modification of mass differences due to static deformation and dynamic fluctuations around the mean-field ground state.

Large proton contribution to the2+excitation inMg20studied by intermediate energy inelastic scattering

Coulomb excitation of the proton-rich nucleus {sup 20}Mg was studied using a radioactive {sup 20}Mg beam at 58A MeV impinging on a lead target. The reduced transition probability B(E2;0{sub g.s.}{sup +}{yields}2{sub 1}{sup +}) was extracted to be 177(32) e{sup 2} fm{sup 4}, which agrees with the theoretical predictions by a cluster model assuming {sup 16}O + 2p + 2p structure, a mean-field approach based on the angular momentum projected generator coordinate method, and the USD shell model. The ratio of the neutron-to-proton multipole matrix elements M{sub n}/M{sub p} in the mirror nucleus {sup 20}O was deduced to be 2.51(25) with the M{sub n} value evaluated from the measured B(E2) value for {sup 20}Mg with the help of isospin symmetry. The results confirm the large M{sub n}/M{sub p} value previously reported in {sup 20}O, leading to the dominant role of the four valence nucleons in the 2{sub 1}{sup +} excitation and persistence of the {sup 16}O core in {sup 20}O and {sup 20}Mg.

Large-amplitudeQn-Qpcollectivity in the neutron-rich oxygen isotopeO20

By means of HFB calculations with independent constraints on axial neutron and proton quadrupole moments ${\stackrel{^}{Q}}_{n}$ and ${\stackrel{^}{Q}}_{p}$, we investigate the large amplitude isoscalar and isovector deformation properties of the neutron-rich isotope $^{20}\mathrm{O}$. Using the particle-number and angular-momentum projected generator coordinate method, we analyze the collective dynamics in the ${\ensuremath{\langle}{\stackrel{^}{Q}}_{n}\ensuremath{\rangle},\ensuremath{\langle}{\stackrel{^}{Q}}_{p}\ensuremath{\rangle}}$ plane. The parametrization SLy4 of the Skyrme interaction is used for all calculations in connection with a density-dependent zero-range pairing interaction. Our results show that already for this moderately neutron-rich nucleus the transition moments are modified when independent neutron and proton collective dynamics are allowed.

Global study of the spectroscopic properties of the first2+state in even-even nuclei

We discuss the systematics of the 2{sup +} excitation energy and the transition probability from this 2{sup +} to the ground state for most of the even-even nuclei, from {sup 16}O up to the actinides, for which data are available. To that aim we calculate their correlated J=0 ground state and J=2 first excited state by means of the angular-momentum and particle-number projected generator coordinate method, using the axial mass quadrupole moment as the generator coordinate and self-consistent mean-field states only restricted by axial, parity, and time-reversal symmetries. The calculation, which is an extension of a previous systematic calculation of correlations in the ground state, is performed within the framework of a nonrelativistic self-consistent mean-field model using the same Skyrme interaction SLy4 and a density-dependent pairing force to generate the mean-field configurations and mix them. To separate the effects due to angular-momentum projection and those related to configuration mixing, the comparison with the experimental data is performed for the full calculation and also by taking a single configuration for each angular momentum, chosen to minimize the projected energy. The theoretical energies span more than 2 orders of magnitude, ranging below 100 keV in deformed actinide nuclei to a few MeVmore » in doubly-magic nuclei. Both approaches systematically overpredict the experiment excitation energy, by an average factor of about 1.5. The dispersion around the average is significantly better in the configuration mixing approach compared to the single-configuration results, showing the improvement brought by the inclusion of a dispersion on the quadrupole moment in the collective wave function. Both methods do much better for the quadrupole properties; using the configuration mixing approach the mean error on the experimental B(E2) values is only 50%. We discuss possible improvements of the theory that could be made by introducing other constraining fields.« less

Pairing correlations in mesoscopic systems

The particle number projected generator coordinate method is applied to the study of generalized pairing Hamiltonians. The calculations reproduce the exact solution where available in the weak, crossover and strong pairing regimes. The behaviour of the wave functions is analysed showing large mixtures in the weak and smaller ones in the strong pairing regime. The physical insight of the ansatz and its numerical simplicity make this theory an excellent tool to study pairing correlations in complex situations and/or involved Hamiltonians.