Dr Carlo Barbieri
University roles and responsibilities
- Chair of HPC Stakeholders Group
Courses I teach on
The newly developed Gorkov-Green’s function approach represents a promising path to the ab initio description of medium-mass open-shell nuclei. We discuss the implementation of the method at second order with a two-body interaction, with particular attention to the numerical solution of Gorkov’s equation. Different sources of theoretical error and degrees of self-consistency are investi- gated. We show that Krylov projection techniques with a multi-pivot Lanczos algorithm efficiently handle the growth of poles in the one-body Green’s function when Gorkov’s equation is solved self- consistently. The end result is a tractable, accurate and gently scaling ab initio scheme applicable to full isotopic chains in the medium-mass region.
We present results for charge form factors, the point-proton, charge, and single-nucleon momentum distributions of 4 He and 16 O obtained within the self-consistent Green's function approach. The removal of the center-of-mass contribution for both nuclei has been performed by using a metropolis Monte Carlo algorithm in which the center-of-mass coordinate can be exactly subtracted from the optimal reference state wave function generated during the self-consistent Green's function calculations. The spectral functions of the same two nuclei have been used to compute inclusive electron-nucleus cross sections. The formalism adopted is based on the factorization of the spectral function and the nuclear transition matrix elements. This allows us to provide an accurate description of nuclear dynamics and to account for relativistic effects in the interaction vertex. When final-state interactions for the struck particle are accounted for, we find nice agreement between the data and the theory for the inclusive electron- 16 O cross section. The results lay the foundations for future applications of the self-consistent Green's function method, in both closed and open shell nuclei, to neutrino data analysis.
Spectroscopic information has been extracted on the hole-states of 55Ni, the least known of the quartet of nuclei (55Ni, 57Ni, 55Co and 57Co), one neutron away from 56Ni, the N=Z=28 double magic nucleus. Using the 1H(56Ni,d)55Ni transfer reaction in inverse kinematics, neutron spectroscopic factors, spins and parities have been extracted for the f7/2, p3/2 and the s1/2 hole-states of 55Ni. This new data provides a benchmark for large basis calculations that include nucleonic orbits in both the sd and pf shells. State of the art calculations have been performed to reproduce the excitation energies and spectroscopic factors of the s1/2 hole state below Fermi energy.
Advances in the self-consistent Green's function approach to finite nuclei are discussed, including the implementation of three-nucleon forces and the extension to the Gorkov formalism. We report results on binding energies in the nitrogen and fluorine isotopic chains, as well as first ab-initio calculations for medium-mass open-shell chains ranging from argon to titanium. Results with chiral interactions put in evidence the important role played by 3NF in neutron rich isotopes.
We extend the formalism of self-consistent Green’s function theory to include three-body interactions and apply it to isotopic chains around oxygen for the first time. The third-order algebraic diagrammatic construction equations for two-body Hamiltonians can be exploited upon defining system-dependent one- and two-body interactions coming from the three-body force, and, correspondingly, dropping interaction-reducible diagrams. The Koltun sum rule for the total binding energy acquires a correction due to the added three-body interaction. This formalism is then applied to study chiral two- and three-nucleon forces evolved to low momentum cutoffs. The binding energies of nitrogen, oxygen, and fluorine isotopes are reproduced with good accuracy and demonstrate the predictive power of this approach. Leading order three-nucleon forces consistently bring results close to the experiment for all neutron rich isotopes considered and reproduce the correct driplines for oxygen and nitrogen. The formalism introduced also allows us to calculate form factors for nucleon transfer on doubly magic systems.
Preliminary ab-initio applications of many-body Green's function theory to the ground state of 4He suggest that high accuracy can be achieved in the so-called Faddeev-random-phase-approximation method. We stress the potentialities of this approach for microscopic studies of medium-large nuclei and report applications to 1s0d and 1p0f-shell nuclei. In particular, we discuss the role of long-range correlations on spectroscopic factors and their dependence on asymmetry.
The first nuclear structure application of the newly developed Gorkov self-consistent Green's function method is presented. The approach aims to describe many-nucleon systems from an ab-initio standpoint featuring an explicit treatment of pairing correlations. In the present work calculations of binding energies of calcium isotopes are reported and compared with experimental data and other theoretical references.
As ab-initio calculations of atomic nuclei enter the A=40-100 mass range, a great challenge is how to approach the vast majority of open-shell (degenerate) isotopes. We add realistic three-nucleon interactions to the state of the art many-body Green's function theory of closed-shells, and find that physics of neutron driplines is reproduced with very good quality. Further, we introduce the Gorkov formalism to extend ab-initio theory to semi-magic, fully open-shell, isotopes. Proof-of-principle calculations for Ca-44 and Ni-74 confirm that this approach is indeed feasible. Combining these two advances (open-shells and three-nucleon interactions) requires longer, technical, work but it is otherwise within reach.
Background: The possibility that an unconventional depletion (referred to as a “bubble”) occurs in the center of the charge density distribution of certain nuclei due to a purely quantum mechanical effect has attracted theoretical and experimental attention in recent years. Based on a mean-field rationale, a correlation between the occurrence of such a semibubble and an anomalously weak splitting between low angular-momentum spin-orbit partners has been further conjectured. Energy density functional and valence-space shell model calculations have been performed to identify and characterize the best candidates, among which 34 Si appears as a particularly interesting case. While the experimental determination of the charge density distribution of the unstable 34 Si is currently out of reach, ( d , p ) experiments on this nucleus have been performed recently to test the correlation between the presence of a bubble and an anomalously weak 1 / 2 − − 3 / 2 − splitting in the spectrum of 35 Si as compared to 37 S .Purpose: We study the potential bubble structure of 34 Si on the basis of the state-of-the-art ab initio self-consistent Green's function many-body method. Methods: We perform the first ab initio calculations of 34 Si and 36 S . In addition to binding energies, the first observables of interest are the charge density distribution and the charge root-mean-square radius for which experimental data exist in 36 S . The next observable of interest is the low-lying spectroscopy of 35 Si and 37 S obtained from ( d , p ) experiments along with the spectroscopy of 33 Al and 35 P obtained from knock-out experiments. The interpretation in terms of the evolution of the underlying shell structure is also provided. The study is repeated using several chiral effective field theory Hamiltonians as a way to test the robustness of the results with respect to input internucleon interactions. The convergence of the results with respect to the truncation of the many-body expansion, i.e., with respect to the many-body correlations included in the calculation, is studied in detail. We eventually compare our predictions to state-of-the-art multireference energy density functional and shell model calculations. Results: The prediction regarding the (non)existence of the bubble structure in 34 Si varies significantly with the nuclear Hamiltonian used. However, demanding that the experimental charge density distribution and the root-mean-square radius of 36 S be well reproduced, along with 34 Si and 36 S binding energies, only leaves the NNLO sat Hamiltonian as a serious candidate to perform this prediction. In this context, a bubble structure, whose fingerprint should be visible in an electron scattering experiment of 34 Si , is predicted. Furthermore, a clear correlation is established between the occurrence of the bubble structure and the weakening of the 1 / 2 − − 3 / 2 − splitting in the spectrum of 35 Si as compared to 37 S .Conclusions: The occurrence of a bubble structure in the charge distribution of 34 Si is convincingly established on the basis of state-of-the-art ab initio calculations. This prediction will have to be reexamined in the future when improved chiral nuclear Hamiltonians are constructed. On the experimental side, present results act as a strong motivation to measure the charge density distribution of 34 Si in future electron scattering experiments on unstable nuclei. In the meantime, it is of interest to perform one-neutron removal on 34 Si and 36 S in order to further test our theoretical spectral strength distributions over a wide energy range.
Quasifree one-proton knockout reactions have been employed in inverse kinematics for a systematic study of the structure of stable and exotic oxygen isotopes at the R3B=LAND setup with incident beam energies in the range of 300–450 MeV=u. The oxygen isotopic chain offers a large variation of separation energies that allows for a quantitative understanding of single-particle strength with changing isospin asymmetry. Quasifree knockout reactions provide a complementary approach to intermediate-energy onenucleon removal reactions. Inclusive cross sections for quasifree knockout reactions of the type AOðp; 2pÞA−1N have been determined and compared to calculations based on the eikonal reaction theory. The reduction factors for the single-particle strength with respect to the independent-particle model were obtained and compared to state-of-the-art ab initio predictions. The results do not show any significant dependence on proton-neutron asymmetry.
A precision mass investigation of the neutron-rich titanium isotopes 51 − 55 Ti was performed at TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN). The range of the measurements covers the N = 32 shell closure, and the overall uncertainties of the 52 − 55 Ti mass values were significantly reduced. Our results conclusively establish the existence of the weak shell effect at N = 32 , narrowing down the abrupt onset of this shell closure. Our data were compared with state-of-the-art ab initio shell model calculations which, despite very successfully describing where the N = 32 shell gap is strong, overpredict its strength and extent in titanium and heavier isotones. These measurements also represent the first scientific results of TITAN using the newly commissioned multiple-reflection time-of-flight mass spectrometer, substantiated by independent measurements from TITAN’s Penning trap mass spectrometer.
We perform ab initio self-consistent Green’s function calculations of the closed shell nuclei 4He, 16O and 40Ca, based on two-nucleon potentials derived from Lattice QCD simulations, in the flavor SU(3) limit and at the pseudo-scalar meson mass of 469 MeV/c 2 . The nucleon-nucleon interaction is obtained using the HAL QCD method and its short-distance repulsion is treated by means of ladder resummations outside the model space. Our results show that this approach diagonalises ultraviolet degrees of freedom correctly. Therefore, ground state energies can be obtained from infrared extrapolations even for the relatively hard potentials of HAL QCD. Comparing to previous Brueckner Hartree-Fock calculations, the total binding energies are sensibly improved by the full account of many-body correlations. The results suggest an interesting possible behaviour in which nuclei are unbound at very large pion masses and islands of stability appear at first around the traditional doublymagic numbers when the pion mass is lowered toward its physical value. The calculated one-nucleon spectral distributions are qualitatively close to those of real nuclei even for the pseudo-scalar meson mass considered here.
Advances in the self-consistent Green's function approach to finite nuclei are discussed, including the implementation of three-nucleon forces and the extension to the Gorkov formalism. We report results on binding energies in the nitrogen and fluorine isotopic chains, as well as spectral functions of 22O. The application to medium-mass open-shell systems is illustrated by separation energy spectra of two argon isotopes, which are compared to one-neutron removal experiments.
We present a systematic study of both nuclear radii and binding energies in (even) oxygen isotopes from the valley of stability to the neutron drip line. Both charge and matter radii are compared to state-of-the-art ab initio calculations along with binding energy systematics. Experimental matter radii are obtained through a complete evaluation of the available elastic proton scattering data of oxygen isotopes. We show that, in spite of a good reproduction of binding energies, ab initio calculations with conventional nuclear interactions derived within chiral effective field theory fail to provide a realistic description of charge and matter radii. A novel version of two- and three-nucleon forces leads to considerable improvement of the simultaneous description of the three observables for stable isotopes but shows deficiencies for the most neutron-rich systems. Thus, crucial challenges related to the development of nuclear interactions remain.
The differential cross section was measured for the 12C(e,e’pp)10Beg.s. reaction at energy and momentum transfers of 163MeV and 198MeV/c, respectively. The measurement was performed at the Mainz Microtron by using two high-resolution magnetic spectrometers of the A1 Collaboration and a newly developed silicon detector telescope. The overall resolution of the detector system was sufficient to distinguish the ground state from the first excited state in 10 Be. We chose a super-parallel geometry that minimizes the effect of two-body currents and emphasizes the effect of nucleon-nucleon correlations. The obtained differential cross section is compared to the theoretical results of the Pavia reaction code in which different processes leading to two-nucleon knockout are accounted for microscopically. The comparison shows a strong sensitivity to nuclear-structure input and the measured cross section is seen to be dominated by the interplay between long- and short-range nucleon-nucleon correlations. Microscopic calculations based on the ab initio self-consistent Green’s function method give a reasonable description of the experimental cross section.
We extend the self-consistent Green's functions formalism to take into account three-body interactions. We analyze the perturbative expansion in terms of Feynman diagrams and define effective one- and two-body interactions, which allows for a substantial reduction of the number of diagrams. The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fully correlated density matrices. We give examples up to third order in perturbation theory. To define nonperturbative approximations, we extend the equation of motion method in the presence of three-body interactions. We propose schemes that can provide nonperturbative resummation of three-body interactions. We also discuss two different extensions of the Koltun sum rule to compute the ground state of a many-body system.
We review some applications of self-consistent Green's function theory to studies of one- and two-nucleon structure in finite nuclei. Large-scale microscopic calculations that employ realistic nuclear forces are now possible. Effects of long-range correlations are seen to play a dominant role in determining the quenching of absolute spectroscopic factors. They also enhance considerably (e,e'pn) cross sections in superparallel kinematics, in agreement with observations
An ab initio calculation scheme for finite nuclei based on self-consistent Green's functions in the Gorkov formalism is developed. It aims at describing properties of doubly magic and semimagic nuclei employing state-of-the-art microscopic nuclear interactions and explicitly treating pairing correlations through the breaking of U(1) symmetry associated with particle number conservation. The present paper introduces the formalism necessary to undertake applications at (self-consistent) second order using two-nucleon interactions in a detailed and self-contained fashion. First applications of such a scheme will be reported soon in a forthcoming publication. Future works will extend the present scheme to include three-nucleon interactions and implement more advanced truncation schemes.
This contribution reviews a calculation of two-step rescattering events in (e,e'p) reactions. A semiclassical approach is employed for different kinematics, involving both medium and large missing energies. The effects of nuclear transparency and Pauli blocking are also included. The results are of interest for experiments aimed to study short-range correlations in the spectral distribution and suggest that the effects of rescattering can be strongly reduced in parallel kinematics. The comparison with the experimental data seem to confirm that sensible measurements could be achievable with a careful choice of the kinematics. However, contributions to final state interactions beyond the ones considered here become relevant for heavy nuclei. For transverse kinematics, rescattering induce large shifts of the spectral strength that can lead to a total experimental yield much larger than the direct signal. © Società Italiana di Fisica / Springer-Verlag 2005.
The nucleosynthesis of elements beyond iron is dominated by neutron captures in the s and r processes. However, 32 stable, proton-rich isotopes cannot be formed during those processes, because they are shielded from the s-process flow and r-process, β-decay chains. These nuclei are attributed to the p and rp process. For all those processes, current research in nuclear astrophysics addresses the need for more precise reaction data involving radioactive isotopes. Depending on the particular reaction, direct or inverse kinematics, forward or time-reversed direction are investigated to determine or at least to constrain the desired reaction cross sections. The Facility for Antiproton and Ion Research (FAIR) will offer unique, unprecedented opportunities to investigate many of the important reactions. The high yield of radioactive isotopes, even far away from the valley of stability, allows the investigation of isotopes involved in processes as exotic as the r or rp processes.
Background: Self-consistent Green’s function theory has recently been extended to the basic formalism needed to account for three-body interactions [A. Carbone, A. Cipollone, C. Barbieri, A. Rios, and A. Polls, Phys. Rev. C 88, 054326 (2013)]. The contribution of three-nucleon forces has then been included in ab initio calculations on nuclear matter and isotopic chains of finite nuclei. Purpose: ractical applications across post Hartree-Fock methods have mostly considered the contribution of three-nucleon interactions in an effective way, as averaged two-nucleon forces. We derive the working equations for all possible two- and three-nucleon terms that enter the expansion of the self-energy, including interactionirreducible (i.e. not averaged) three-nucleon diagrams. Methods: We employ the algebraic diagrammatic construction up to third order as the organization scheme for generating a non perturbative self-energy, in which ring (particle-hole) and ladder (particle-particle) diagrams are resummed to all orders. Results: We derive expressions of the static and dynamic self-energy up to third order, by taking into account also the set of diagrams required when the skeleton expansion of the single-particle propagator is not assumed. A hierarchy of importance among different diagrams is revealed, and a particular emphasis is given to a third-order diagram (see Fig. 2c) which is expected to play a significant role among those featuring an interaction-irreducible three-nucleon force. Conclusion: A consistent formalism to resum at infinite order correlations induced by three-nucleon forces in the self-consistent Green’s function theory is now available, and ready to be implemented in the many-body solvers. Work is in progress to include the first interaction-irreducible three-nucleon diagram in calculations of closed-shell medium-mass nuclei.
The self-consistent random phase approximation (RPA) based on a correlated realistic nucleon-nucleon interaction is used to evaluate correlation energies in closed-shell nuclei beyond the Hartree-Fock level. The relevance of contributions associated with charge exchange excitations as well as the necessity to correct for the double counting of the second order contribution to the RPA ring summation are emphasized. Once these effects are properly accounted for, the RPA ring summation provides an efficient tool to assess the impact of long-range correlations on binding energies throughout the whole nuclear chart, which is of particular importance when starting from realistic interactions.
Background: Microscopic calculations of the electromagnetic response of light and medium-mass nuclei are now feasible thanks to the availability of realistic nuclear interactions with accurate saturation and spectroscopic properties, and the development of large-scale computing methods for many-body physics. Purpose: To compute isovector dipole electromagnetic (E1) response and related quantities, i.e., integrated dipole cross section and polarizability, and compare with data from photoabsorption and Coulomb excitation experiments. To investigate the evolution pattern of the E1 response towards the neutron drip line with calculations of neutron-rich nuclei within a given isotopic chain. Methods: The single-particle propagator is obtained by solving the Dyson equation, where the self-energy includes correlations nonperturbatively through the algebraic diagrammatic construction (ADC) method. The particle-hole (ph) polarization propagator is treated in the dressed random phase approximation (DRPA), based on an effective correlated propagator that includes some 2p2h effects but keeps the same computation scaling as the standard Hartree-Fock propagator. Results: The E1 responses for 14,16,22,24O, 36,40,48,52,54,70Ca, and 68Ni have been computed: The presence of a soft dipole mode of excitation for neutron-rich nuclei is found, and there is a fair reproduction of the low-energy part of the experimental excitation spectrum. This is reflected in a good agreement with the empirical dipole polarizability values. The impact of different approximations to the correlated propagator used as input in the E1 response calculation is assessed. Conclusion: For a realistic interaction that accurately reproduces masses and radii, an effective propagator of the mean-field type computed by the self-consistent Green's function provides a good description of the empirical E1 response, especially in the low-energy part of the excitation spectrum and around the giant dipole resonance. The high-energy part of the spectrum improves and displays an enhancement of the strength when quasiparticle fragmentation is added to the reference propagator. However, this fragmentation (without a proper restoration of dynamical self-consistency) spoils the predictions of the energy centroid of the giant dipole resonance.
The factorization scheme, based on the impulse approximation and the spectral function formalism, has been recently generalized to allow the description of electromagnetic nuclear interactions driven by two-nucleon currents. We have extended this framework to the case of weakly charged and neutral currents, and carried out calculations of the double-differential neutrino-carbon and neutrino-oxygen cross sections using two different models of the target spectral functions. The results, showing a moderate dependence on the input spectral function, confirm that our approach provides a consistent treatment of all reaction mechanisms contributing to the signals detected by accelerator-based neutrino experiments.
A dispersive-optical-model analysis has been performed for both protons and neutrons on 40,42,44,48Ca isotopes. The fitted potentials describe accurately both scattering and bound quantities and extrapolate well to other stable nuclei. Further experimental infor- mation will be gathered to constrain extrapolations toward the driplines.
The accuracy of the Faddeev random phase approximation (FRPA) method is tested by evaluating total and ionization energies in the basis-set limit. A set of light atoms up to Ar is considered. Comparisons are made with the results of coupled-cluster singles and doubles (CCSD), with third-order algebraic diagrammatic construction [ADC(3)], and with the experiment. It is seen that even for two-electron systems, He and Be2+, the inclusion of RPA effects leads to satisfactory results, and therefore it does not overcorrelate the ground state. The FRPA becomes progressively better for larger atomic numbers, where it gives ≈5 mH more correlation energy, and it shifts ionization potentials by 2–10 mH with respect to the similar ADC(3) method. The ionization potentials from FRPA tend to reduce the discrepancies with the experiment.