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

Triaxiality near the 110Ru ground state from Coulomb excitation

A multi-step Coulomb excitation measurement with the GRETINA and CHICO2 detector arrays was carried out with a 430-MeV beam of the neutron-rich 110Ru (t1/2=12 s) isotope produced at the CARIBU facility. This represents the first successful measurement following the post-acceleration of an unstable isotope of a refractory element. The reduced transition probabilities obtained for levels near the ground state provide strong evidence for a triaxial shape; a conclusion confirmed by comparisons with the results of beyond-mean-field and triaxial rotor model calculations

Five-dimensional collective Hamiltonian with the Gogny force: An ongoing saga

International audienceWe provide a sample of analyses for nuclear spectroscopic properties based on the five-dimensional collective Hamiltonian (5DCH) implemented with the Gogny force. The very first illustration is dating back to the late 70's. It is next followed by others, focusing on shape coexistence, shape isomerism, superdeformation, and systematics over the periodic table. Finally, the inclusion of Thouless-Valatin dynamical contributions to vibrational mass parameters is briefly discussed as a mean of strengthening the basis of the 5DCH theory

Ab initio calculation of superdeformed bands in 192^{192}Hg

Three superdeformed (SD) bands including states with spins Iπ≤22+ are predicted for 192Hg. The SD and normally deformed levels are calculated as eigenstates of the Griffin-Hill-Wheeler equation treated in the Gaussian overlap approximation. Potential and tensor of inertia are deduced from constrained Hartree-Fock-Bogoliubov calculations based on Gogny's force. Rates for E2 in-band transitions are deduced and compared with recent measurements. The decay out of SD bands is discussed

Collective structure of Iπ=0+I^{\pi}=0^+ shape isomers in the 190,192,194^{190,192,194}Hg isotopes

Structure studies based on either the Strutinsky method or self-consistent Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) mean field theories suggest that superdeformed (SD) states might well develop at large quadrupole deformation (i.e. βgt-or-equal, slanted0.5) in mercury isotopes. In this work, we predict the existence of Iπ=0+ shape isomers which might take place at excitation energies of 4.4, 5.4 and 6.9 MeV in 190Hg, 192Hg and 194Hg, respectively. These SD levels are obtained by solving the Griffin-Hill-Wheeler equation in the gaussian overlap approximation. Inputs for that collective hamiltonian are tensors of inertia as well as potential energy surfaces which are deduced from constrained HFB calculations based on the finite-range, density-dependent effective force of Gogny

Microscopic description of superdeformed bands in 190,192,194^{190,192,194}Hg

Positive parity states at normal deformation and superdeformation (SD) are obtained for 190,192,194Hg by solving a microscopic collective Hamiltonian which is defined in terms of the five quadrupole degrees of freedom. Yrast and excited SD bands are predicted. Their properties are compared with experimental information. In particular, it is found that all the predicted dynamical moments of inertia J(2) are very similar, a feature consistent with that obtained from extrapolating the measured J(2)'s toward very low rotational frequencies. Excitations of two quasi-particle states are also considered