The authors review the present status of self-consistent mean-field (SCMF) models for describing nuclear structure and low-energy dynamics. These models are presented as effective energy-density functionals. The three most widely used variants of SCMF's based on a Skyrme energy functional, a Gogny force, and a relativistic mean-field Lagrangian are considered side by side. The crucial role of the treatment of pairing correlations is pointed out in each case. The authors discuss other related nuclear structure models and present several extensions beyond the mean-field model which are currently used. Phenomenological adjustment of the model parameters is discussed in detail. The performance quality of the SCMF model is demonstrated for a broad range of typical applications.

Self-consistent mean-field models for nuclear structure

The fission barriers of the nuclei ${}^{254}\mathrm{Fm},$ ${}^{256}\mathrm{Fm},$ ${}^{258}\mathrm{Fm},$ ${}^{258}\mathrm{No}$, and ${}^{260}\mathrm{Rf}$ are investigated in a fully microscopic way up to the scission point. The analysis is based on the constrained Hartree-Fock-Bogoliubov theory and Gogny's D1S force. The quadrupole, octupole, and hexadecapole moments as well as the number of nucleons in the neck region are used as constraints. Two fission paths, corresponding to the bimodal fission, are found. The decrease with isotope mass of the half-life times of heavy Fm isotopes is also explained.

/pdf/self-consistent-calculations-of-fission-barriers-in-the-fm-3evp2gxn7g.pdf

Self-consistent calculations of fission barriers in the Fm region

We investigate the systematics of fission barriers in superheavy elements in the range $Z=108\char21{}120$ and $N=166\char21{}182$. Results from two self-consistent models for nuclear structure, the relativistic mean-field (RMF) model as well as the nonrelativistic Skyrme-Hartree-Fock approach are compared and discussed. We restrict ourselves to axially symmetric shapes, which provides an upper bound on static fission barriers. We benchmark the predictive power of the models examining the barriers and fission isomers of selected heavy actinide nuclei for which data are available. For both actinides and superheavy nuclei, the RMF model systematically predicts lower barriers than most Skyrme interactions. In particular, the fission isomers are predicted too low by the RMF, which casts some doubt on recent predictions about superdeformed ground states of some superheavy nuclei. For the superheavy nuclei under investigation, fission barriers drop to small values around $Z=110$, $N=180$, and increase again for heavier systems. For most of the forces, there is no fission isomer for superheavy nuclei, as superdeformed states are in most cases found to be unstable with respect to octupole distortions.

Systematics of fission barriers in superheavy elements

We present our current studies and future plans on microscopic potential based on effective nucleon-nucleon interaction and many-body theory. This framework treats in an unified way nuclear structure and reaction. It offers the opportunity to link the underlying effective interaction to nucleon scattering observables. The more consistently connected to a variety of reaction and structure experimental data the framework is, the more constrained the effective interaction will be. As a proof of concept, we present some recent results for both neutron and proton scattered from spherical target nucleus, namely 40Ca , using the Gogny D1S interaction. Possible fruitful cross-talks between microscopic potential, phenomenological potential and effective interaction are exposed. We then draw some prospective plans for the forthcoming years including scattering from spherical nuclei experiencing pairing correlations, scattering from axially deformed nuclei, and new effective interaction with reaction constraints.

Prospective study on microscopic potential with Gogny interaction

Hartree-Fock-Bogoliubov calculations of hot fission in $^{240}\mathrm{Pu}$ have been performed with a newly implemented code that uses the D1S finite-range effective interaction. The hot-scission line is identified in the quadrupole-octupole-moment coordinate space. Fission-fragment shapes are extracted from the calculations. A benchmark calculation for $^{226}\mathrm{Th}$ is obtained and compared with results in the literature. In addition, technical aspects of the use of HFB calculations for fission studies are examined in detail. In particular, the identification of scission configurations, the sensitivity of near-scission calculations to the choice of collective coordinates in the HFB iterations, and the formalism for the adjustment of collective-variable constraints are discussed. The power of the constraint-adjustment algorithm is illustrated with calculations near the critical scission configurations with up to seven simultaneous constraints.

Microscopic calculation of Pu 240 scission with a finite-range effective force

The random phase approximation (RPA) long-range correlations are known to play a significant role in understanding the depletion of single particle-hole states observed in ($e,{e}^{'}$) and ($e,{e}^{'}p$) measurements. Here the RPA theory, implemented using the D1S force is considered for the specific purpose of building correlated ground states and related one-body density matrix elements. These may be implemented and tested in a fully microscopic optical model for NA scattering off doubly closed-shell nuclei. A method is presented to correct for the correlations overcounting inherent to the RPA formalism. One-body density matrix elements in the uncorrelated (i.e., Hartree-Fock) and correlated (i.e., RPA) ground states are then challenged in proton scattering studies based on the Melbourne microscopic optical model to highlight the role played by the RPA correlations. Agreement between the parameter free scattering predictions and measurements is good for incident proton energies ranging from 200 MeV down to approximately 60 MeV and becomes gradually worse in the lower energy range. Those features point unambiguously to the relevance of the g-matrix method to build microscopic optical model potentials at medium energies, and emphasize the need to include nucleon-phonon coupling, that is, a second-order component of the Feshbach type in the potential at lower energies. Illustrations are given for proton scattering observables measured up to 201 MeV for the $^{16}\mathrm{O}$, $^{40}\mathrm{Ca}$, $^{48}\mathrm{Ca}$, and $^{208}\mathrm{Pb}$ target nuclei.

Correlations in microscopic optical model for nucleon elastic scattering off doubly closed-shell nuclei

A theoretical calculation of the partial lifetime of the fission isomer in $^{238}\mathrm{U}$ decaying via electric quadrupole radiation to the lowest ${2}^{+}$ state is presented and compared to experiment. This is a fully microscopic nuclear structure calculation using the constrained Hartree-Fock-Bogoliubov method, where the only input is the finite-ranged D1S interaction. The low-lying collective states are obtained via the Gaussian overlap approximation of the generator coordinate method. Rotational symmetry is restored within the rotor approximation. The computation of the transition rates provides an upper limit on the \ensuremath{\gamma}-decay lifetime of the isomer, since only axial quadrupole deformations are considered.

D. Gogny

Papers

Correlations in microscopic optical model for nucleon elastic scattering off doubly closed-shell nuclei

Limits on the lifetime of the shape isomer of 238U.