Adopted values for the reduced electric quadrupole transition probability, B(E2)↑, from the ground state to the first-excited 2+ state of even–even nuclides are given in Table I. Values of τ, the mean life of the 2+ state; E, the energy; and β, the quadrupole deformation parameter, are also listed there. The ratio of β to the value expected from the single-particle model is presented. The intrinsic quadrupole moment, Q0, is deduced from the B(E2)↑ value. The product E×B(E2)↑ is expressed as a percentage of the energy-weighted total and isoscalar E2 sum-rule strengths.

Table II presents the data on which Table I is based, namely the experimental results for B(E2)↑ values with quoted uncertainties. Information is also given on the quantity measured and the method used. The literature has been covered to November 2000.

The adopted B(E2)↑ values are compared in Table III with the values given by systematics and by various theoretical models. Predictions of unmeasured B(E2)↑ values are also given in Table III.

Transition probability from the ground to the first-excited 2^+ state of even-even nuclides

An overview is given on some of the main advances in the experimental methods, experimental results, theoretical models and ideas of the last few years in the field of nuclear fission. New approaches have considerably extended the availability of fissioning systems for the experimental study of nuclear fission, and have provided a full identification of all fission products in A and Z for the first time. In particular, the transition from symmetric to asymmetric fission around 226Th, some unexpected structures in the mass distributions in the fission of systems around Z  =  80-84, and an extended systematics of the odd-even effect in the fission fragment Z distributions have all been measured (Andreyev et al 2018 Rep. Prog. Phys. 81 016301). Three classes of model descriptions of fission presently appear to be the most promising or the most successful. Self-consistent quantum-mechanical models fully consider the quantum-mechanical features of the fission process. Intense efforts are presently being made to develop suitable theoretical tools (Schunck and Robledo 2016 Rep. Prog. Phys. 79 116301) for modeling the non-equilibrium, large-amplitude collective motion leading to fission. Stochastic models provide a fully developed technical framework. The main features of the fission-fragment mass distribution have been well reproduced from mercury to fermium and beyond (Moller and Randrup 2015 Phys. Rev. C 91 044316). However, limited computer resources still impose restrictions, for example, on the number of collective coordinates and on an elaborate description of the fission dynamics. In an alternative semi-empirical approach (Schmidt et al 2016 Nucl. Data Sheets 131 107), considerable progress in describing the fission observables has been achieved by combining several theoretical ideas, which are essentially well known. This approach exploits (i) the topological properties of a continuous function in multidimensional space, (ii) the separability of the influence of fragment shells and the macroscopic properties of the compound nucleus, (iii) the properties of a quantum oscillator coupled to a heat bath of other nuclear degrees of freedom, (iv) an early freeze-out of collective motion, and (v) the application of statistical mechanics for describing the thermalization of intrinsic excitations in the nascent fragments. This new approach reveals a high degree of regularity and allows the calculation of high-quality data that is relevant to nuclear technology without specifically adjusting the empirical data of individual systems.

https://iopscience.iop.org/article/10.1088/1361-6633/aacfa7/pdf

Review on the progress in nuclear fission-experimental methods and theoretical descriptions.

Nowadays, the Gogny force is a referent in the theoretical description of nuclear structure phenomena. Its phenomenological character manifests in a simple analytical form that allows for implementations of techniques both at the mean field and beyond all over the nuclide chart. Over the years, multiple applications of the standard many-body techniques in an assorted set of nuclear structure applications have produced results which are in a rather good agreement with experimental data. The agreement allows for a simple interpretation of those intriguing phenomena in simple terms and gives confidence on the predictability of the interaction. The present status on the implementation of different many body techniques with the Gogny force is reviewed with a special emphasis on symmetry restoration and large amplitude collective motion.

Mean field and beyond description of nuclear structure with the Gogny force: A review

Nowadays, the Gogny force is a referent in the theoretical description of nuclear structure phenomena. Its phenomenological character manifests in a simple analytical form that allows for implementations of techniques both at the mean field and beyond all over the nuclide chart. Over the years, multiple applications of the standard many-body techniques in an assorted set of nuclear structure applications have produced results which are in a rather good agreement with experimental data. The agreement allows for a simple interpretation of those intriguing phenomena in simple terms and gives confidence on the predictability of the interaction. The present status on the implementation of different many-body techniques with the Gogny force is reviewed with a special emphasis on symmetry restoration and large amplitude collective motion.

https://iopscience.iop.org/article/10.1088/1361-6471/aadebd/pdf

Mean field and beyond description of nuclear structure with the Gogny force: a review

The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Examples are strongly interacting atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. However, mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group theoretical techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. We present applications of the projection methods to simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the energy density functional theories are highlighted.

/pdf/symmetry-restoration-in-mean-field-approaches-5ae1s34kug.pdf

Symmetry restoration in mean-field approaches

We present a novel nuclear energy density functional method to calculate spectroscopic properties of atomic nuclei. Intrinsic nuclear quadrupole deformations and rotational frequencies are considered simultaneously as the degrees of freedom within a symmetry conserving configuration mixing framework. The present method allows the study of nuclear states with collective and single-particle character. We calculate the fascinating structure of the semimagic ^{44}S nucleus as a first application of the method, obtaining an excellent quantitative agreement both with the available experimental data and with state-of-the-art shell model calculations.

/pdf/collective-and-single-particle-motion-in-beyond-mean-field-7zfsn7ar78.pdf

Collective and Single-Particle Motion in Beyond Mean Field Approaches.

Symmetry conserving configuration mixing method with cranked states

Ground-state properties of even and odd Magnesium isotopes in a symmetry-conserving approach

We present a self-consistent theory for the description of the spectroscopic properties of odd nuclei which includes exact blocking, particle-number and angular-momentum projection and configuration mixing In our theory the pairing correlations are treated in a variation-after-projection approach and the triaxial deformation parameters are explicitly considered as generator coordinates The angular-momentum and particle-number symmetries are exactly recovered The use of the effective finite-range density-dependent Gogny force in the calculations provides an added value to the theoretical results 
We apply the theory to the textbook example of 25Mg and, although this nucleus has been thoroughly studied in the past, we still provide a novel view of nuclear phenomena taking place in this nucleus We obtain an overall good agreement with the known experimental energies and transition probabilities without any additional parameter such as effective charges In particular, we clearly identify six bands, two of which we interpret as collective gamma bands

/pdf/symmetry-conserving-configuration-mixing-description-of-odd-4ousundvzw.pdf

Symmetry conserving configuration mixing description of odd mass nuclei

The pairing correlations in odd-A nuclei are analyzed in the mean field approximation and beyond. In particular the role of symmetry conservation is investigated. We find that particle number projection after the variation (PN-PAV) has little effect on the pairing correlations specially in the weak pairing regime. This is in contrast to the variation after particle number projection (PN-VAP) approach where a strong effect is found. The situation is specially critical in odd nuclei because the pairing correlations vanish due to the blocking effect and the Hartree-Fock-Bogoliubov wave function collapses to the Hartree-Fock one. The PN-VAP, however, handles perfectly the exact blocking providing highly correlated wave functions. The role of the angular momentum projection is studied only in the PAV approach. We find small changes of the pairing correlation, at least at small angular momentum. In the calculations we use the Gogny interaction well suited to this kind of studies.

/pdf/pairing-correlations-and-symmetries-in-odd-a-nuclei-pqy3ov6k4y.pdf

Marta Borrajo

Papers

Collective and Single-Particle Motion in Beyond Mean Field Approaches.

Symmetry conserving configuration mixing method with cranked states

Ground-state properties of even and odd Magnesium isotopes in a symmetry-conserving approach

Symmetry conserving configuration mixing description of odd mass nuclei

Pairing correlations and symmetries in odd-A nuclei.