The nuclear energy density functional formalism

The present document focuses on the theoretical foundations of the nuclear energy density functional (EDF) method. As such, it does not aim at reviewing the status of the field, at covering all possible ramifications of the approach or at presenting recent achievements and applications. The objective is to provide a modern account of the nuclear EDF formalism that is at variance with traditional presentations that rely, at one point or another, on a {\it Hamiltonian-based} picture. The latter is not general enough to encompass what the nuclear EDF method represents as of today. Specifically, the traditional Hamiltonian-based picture does not allow one to grasp the difficulties associated with the fact that currently available parametrizations of the energy kernel $E[g',g]$ at play in the method do not derive from a genuine Hamilton operator, would the latter be effective. The method is formulated from the outset through the most general multi-reference, i.e. beyond mean-field, implementation such that the single-reference, i.e. "mean-field", derives as a particular case. As such, a key point of the presentation provided here is to demonstrate that the multi-reference EDF method can indeed be formulated in a {\it mathematically} meaningful fashion even if $E[g',g]$ does {\it not} derive from a genuine Hamilton operator. In particular, the restoration of symmetries can be entirely formulated without making {\it any} reference to a projected state, i.e. within a genuine EDF framework. However, and as is illustrated in the present document, a mathematically meaningful formulation does not guarantee that the formalism is sound from a {\it physical} standpoint. The price at which the latter can be enforced as well in the future is eventually alluded to.Comment: 64 pages, 8 figures, submitted to Euroschool Lecture Notes in Physics Vol.IV, Christoph Scheidenberger and Marek Pfutzner editor

Investigation of the ground-state spin inversion in the neutron-rich isotopes

16 pags., 9 figs., 5 tabs.A first -ray study of spectroscopy was performed at the Radioactive Isotope Beam Factory with projectiles at 217 MeV/nucleon, impinging on the liquid hydrogen target of the MINOS device. Prompt deexcitation rays were measured with the NaI(Tl) array DALI2. Through the one-proton knockout reaction , a spin assignment could be determined for the low-lying states of from the momentum distribution obtained with the SAMURAI spectrometer. A spin-parity is deduced for the ground state of , similar to the recently studied isotope . The evolution of the energy difference is compared to state-of-the-art theoretical predictions.We thank the RIKEN Nishina Center accelerator staff for their work in the primary beam delivery and the BigRIPS team for preparing the secondary beams. The development of MINOS has been supported by the European Research Council through the ERC Grant No. MINOS258567. B.D.L., L.X.C., and N.D.T. acknowledge support from the Vietnam Ministry of Science and Technology under Grant No. ĐTCB.01/21/VKHKTHN. M.G.R. and A.M.M. are supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (including FEDER funds) under project FIS2017-88410-P. F.B. was supported by the RIKEN Special Postdoctoral Researcher Program. Y.L.S. acknowledges the support of Marie Skłodowska-Curie Individual Fellowship (H2020-MSCAIF-2015-705023) from the European Union. I.G. has been supported by HIC for FAIR and Croatian Science Foundation. R.-B.G. is supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. BL 1513/1-1. K.I.H., D.K., and S.Y.P. acknowledge the support from the IBS grant funded by the Korea government (No. IBS-R031-D1). P.K. was supported in part by the BMBF Grant No. 05P19RDFN1 and HGS-HIRe. D.So. has been supported by the European Regional Development Fund Contract No. GINOP-2.3.3-15-2016-00034 and the National Research, Development and Innovation Fund of Hungary via Project No. K128947. This work was supported in part by JSPS KAKENHI Grants No. JP16H02179, No. JP18H05404, and No. JP20K03981. J.D.H. and R.S. acknowledge the support from NSERC and the National Research Council Canada. This work was supported by the Office of Nuclear Physics, U.S. Department of Energy, under Grants No. de-sc0018223 (NUCLEI SciDAC-4 collaboration) and the FieldWork Proposal ERKBP72 at Oak Ridge National Laboratory (ORNL). Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located at ORNL, which is supported by the Office of Science of the Department of Energy under Contract No. DE-AC05-00OR22725. GGF calculations were performed by using HPC resources from GENCI-TGCC (Contracts No. A007057392 and No. A009057392) and at the DiRAC Complexity system at the University of Leicester (BIS National E-infrastructure capital Grant No. ST/K000373/1 and STFC Grant No. ST/K0003259/1). This work was supported by the United Kingdom Science and Technology Facilities Council (STFC) under Grant No. ST/L005816/1 and in part by the NSERC Grants No. SAPIN-2016-00033, No. SAPIN-2018-00027, and No. RGPAS-2018-522453. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada. J.D.H. thanks S. R. Stroberg for the IMSRG++ code used to perform the VSIMSRG calculations [86]. N.T.T.P. was funded by Vingroup Joint Stock Company and supported by the Domestic Ph.D. Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA), code VINIF.2020.TS.52

“Southwestern” boundary of the <math><mrow><mi>N</mi><mo>=</mo><mn>40</mn></mrow></math> island of inversion: First study of low-lying bound excited states in <math><mmultiscripts><mi mathvariant="normal">V</mi><mprescripts/><none/><mn>59</mn></mmultiscripts></math> and <math><mmultiscripts><mi mathvariant="normal">V</mi><mprescripts/><none/><mn>61</mn></mmultiscripts></math>

International audienceThe low-lying level structure of V59 and V61 was investigated for the first time. The neutron knockout reaction and inelastic proton scattering were applied for V61 while the neutron knock-out reaction provided the data for V59. Four and five new transitions were determined for V59 and V61, respectively. Based on the comparison to our shell-model calculations using the Lenzi-Nowacki-Poves-Sieja (LNPS) interaction, three of the observed γ rays for each isotope could be placed in the level scheme and assigned to the decay of the first 11/2− and 9/2− levels. The (p,p′) excitation cross sections for V61 were analyzed by the coupled-channels formalism assuming quadrupole plus hexadecapole deformations. Due to the role of the hexadecapole deformation, V61 could not be unambiguously placed on the island of inversion

Extended p3/2{p}_{3/2} Neutron Orbital and the N=32N=32 Shell Closure in 52Ca^{52}\mathrm{Ca}

International audienceThe one-neutron knockout from Ca52 in inverse kinematics onto a proton target was performed at ∼230  MeV/nucleon combined with prompt γ spectroscopy. Exclusive quasifree scattering cross sections to bound states in Ca51 and the momentum distributions corresponding to the removal of 1f7/2 and 2p3/2 neutrons were measured. The cross sections, interpreted within the distorted-wave impulse approximation reaction framework, are consistent with a shell closure at the neutron number N=32, found as strong as at N=28 and N=34 in Ca isotopes from the same observables. The analysis of the momentum distributions leads to a difference of the root-mean-square radii of the neutron 1f7/2 and 2p3/2 orbitals of 0.61(23) fm, in agreement with the modified-shell-model prediction of 0.7 fm suggesting that the large root-mean-square radius of the 2p3/2 orbital in neutron-rich Ca isotopes is responsible for the unexpected linear increase of the charge radius with the neutron number

Extended p3/2 Neutron Orbital and the N=32 Shell Closure in Ca 52

The one-neutron knockout from Ca52 in inverse kinematics onto a proton target was performed at ∼230 MeV/nucleon combined with prompt γ spectroscopy. Exclusive quasifree scattering cross sections to bound states in Ca51 and the momentum distributions corresponding to the removal of 1f7/2 and 2p3/2 neutrons were measured. The cross sections, interpreted within the distorted-wave impulse approximation reaction framework, are consistent with a shell closure at the neutron number N=32, found as strong as at N=28 and N=34 in Ca isotopes from the same observables. The analysis of the momentum distributions leads to a difference of the root-mean-square radii of the neutron 1f7/2 and 2p3/2 orbitals of 0.61(23) fm, in agreement with the modified-shell-model prediction of 0.7 fm suggesting that the large root-mean-square radius of the 2p3/2 orbital in neutron-rich Ca isotopes is responsible for the unexpected linear increase of the charge radius with the neutron number. © 2022 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the https://creativecommons.org/licenses/by/4.0/Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article&apos;s title, journal citation, and DOI.11Nsciescopu

“Southwestern” boundary of the <math><mrow><mi>N</mi><mo>=</mo><mn>40</mn></mrow></math> island of inversion: First study of low-lying bound excited states in <math><mmultiscripts><mi mathvariant="normal">V</mi><mprescripts/><none/><mn>59</mn></mmultiscripts></math> and <math><mmultiscripts><mi mathvariant="normal">V</mi><mprescripts/><none/><mn>61</mn></mmultiscripts></math>

International audienceThe low-lying level structure of V59 and V61 was investigated for the first time. The neutron knockout reaction and inelastic proton scattering were applied for V61 while the neutron knock-out reaction provided the data for V59. Four and five new transitions were determined for V59 and V61, respectively. Based on the comparison to our shell-model calculations using the Lenzi-Nowacki-Poves-Sieja (LNPS) interaction, three of the observed γ rays for each isotope could be placed in the level scheme and assigned to the decay of the first 11/2− and 9/2− levels. The (p,p′) excitation cross sections for V61 were analyzed by the coupled-channels formalism assuming quadrupole plus hexadecapole deformations. Due to the role of the hexadecapole deformation, V61 could not be unambiguously placed on the island of inversion

Extended p3/2{p}_{3/2} Neutron Orbital and the N=32N=32 Shell Closure in 52Ca^{52}\mathrm{Ca}

International audienceThe one-neutron knockout from Ca52 in inverse kinematics onto a proton target was performed at ∼230  MeV/nucleon combined with prompt γ spectroscopy. Exclusive quasifree scattering cross sections to bound states in Ca51 and the momentum distributions corresponding to the removal of 1f7/2 and 2p3/2 neutrons were measured. The cross sections, interpreted within the distorted-wave impulse approximation reaction framework, are consistent with a shell closure at the neutron number N=32, found as strong as at N=28 and N=34 in Ca isotopes from the same observables. The analysis of the momentum distributions leads to a difference of the root-mean-square radii of the neutron 1f7/2 and 2p3/2 orbitals of 0.61(23) fm, in agreement with the modified-shell-model prediction of 0.7 fm suggesting that the large root-mean-square radius of the 2p3/2 orbital in neutron-rich Ca isotopes is responsible for the unexpected linear increase of the charge radius with the neutron number