Density Functional Theory (DFT) for atomic nuclei: A simple introduction

Nuclear structure theory is a domain of physics faced at present with great challenges and opportunities. A larger and larger body of high-precision experimental data has been and continues to be accumulated. Experiments on very exotic short-lived isotopes are the backbone of activity at numerous large-scale facilities. Over the years, tremendous progress has been made in understanding the basic features of nuclei. However, the theoretical description of nuclear systems is still far from being complete and is often not very precise. Many questions, both basic and practical, remain unanswered.The goal of publishing this special focus issue of Journal of Physics G: Nuclear and Particle Physics on Open Problems in Nuclear Structure Theory (OPeNST) is to construct a fundamental inventory thereof, so that the tasks and available options become more clearly exposed and that this will help to stimulate a boost in theoretical activity, commensurate with the experimental progress. The requested format and scope of the articles on OPeNST was quite flexible. The journal simply offered the possibility to provide a forum for the material, which is very often discussed at conferences during the coffee breaks but does not normally have sufficient substance to form regular publications. Nonetheless, very often formulating a problem provides a major step towards its solution, and it may constitute a scientific achievement on its own.Prospective authors were therefore invited to find their own balance between the two extremes of very general problems on the one hand (for example, to solve exactly the many-body equations for a hundred particles) and very specific problems on the other hand (for example, those that one could put in one's own grant proposal). The authors were also asked not to cover results already obtained, nor to limit their presentations to giving a review of the subject, although some elements of those could be included to properly introduce the subject matter.The focus of these collected articles is therefore on the discussion of topics that are not yet understood, or that are poorly understood. We very much welcomed presentations on: (i) contradictory approaches, models, or theories that are, at present, difficult to reconcile,(ii) unsolved theoretical problems that hamper applications of existing methods, (iii) limitations of current approaches, (iv) difficulties in deriving and justifying models and theories, (v) generic problems in understanding or describing specific experimental data,and even (vi) all possible, wildest speculations and/or conjectures. The main idea behind the focus issue was to stimulate creative, unbounded thinking and provide young, but not only young, researchers with ideas that would promote further progress in this domain of science.The community of nuclear structure theorists enthusiastically responded to the idea of publishing the volume on OPeNST. It seemed that the idea struck the right chord and many colleagues were willing to share their observations on what research directions to follow and which problems to attack. The volume turned out to be a snapshot of the domain, revealing the burning questions that the community wants to address. All the articles also have a very interesting personal touch. They sometimes even present opposing or conflicting points of view, which is exactly what one would expect within a vibrant scientific discussion. All in all, the Editors of Journal of Physics G are very happy to offer you this unique collection, which will constitute very interesting reading for all those working in nuclear structure theory.

The Tenth International Spring Seminar on Nuclear Physics was held in Vietri sul Mare from May 21 to May 25, 2010. This Seminar was the tenth in a series of topical meetings held every two or three years in the Naples area. The series began with the Sorrento meeting in 1986 and continued with the Capri meeting in 1988, the Ischia meeting in 1990, the Amalfi meeting in 1992, the Ravello meeting in 1995, the S. Agata meeting in 1998, the Maiori meeting in 2001, the Paestum meeting in 2004, and the Vico Equense meeting in 2007.For this tenth meeting we returned to Salerno Bay and met in the small town of Vietri. While the location of the Conference has never been the same, what remained invariant is the aim of these meetings, which is to discuss recent advances and new perspectives in nuclear structure experiments and theory in a pleasant and friendly atmosphere.It is by now well established that we have entered a new era in Nuclear Physics research with the advent of radioactive ion beam facilities. While nuclear structure studies are currently being performed in several laboratories where RIBs are available, the development of new facilities, which will provide high-intensity beams, is in progress or under discussion in Europe, Asia and North America. At this meeting we had a comprehensive overview of this fascinating field and of future scenarios thanks to the participation of leaders of the most important projects.The results that are becoming available for nuclei far from stability are highlighting new themes of research, such as the evolution of the shell structure when moving towards the particle drip lines, and stimulating a proficuous interplay between experiment and theory. On the other hand, new ideas and the development of more powerful computational tools promise a deeper understanding of the structure of nuclei in terms of the basic interactions between their constituents.As usual, the program of the meeting consisted of general talks and of more specialized seminars, the latter including most of the contributions submitted by participants. The speakers covered five main topics: i) Nuclear Structure far from Stability: New Advances and Perspectives; ii) From Nuclear Forces to Nuclear Structure; iii) Exploring Nuclear Shell Structure: Experiment and Theory; iv) New Aspects of Collective Nuclear Motion; and v) Special Topics.We received 63 manuscripts out of the 77 invited papers and contributions presented at the Seminar. All of these have been peer reviewed and are collected in this volume. We would like to thank all the anonymous colleagues who have acted as referees to assess the suitability of the various articles for publication in the Journal of Physics: Conference Series. We are confident that the high quality of both invited and contributed papers contained in these Proceedings will be appreciated by the nuclear physics community.As was the case for most of the previous Seminars, the Vietri Seminar also ended with a Round Table Discussion on the theme ‘Trends and Perspectives in Nuclear Structure’. N Benczer-Koller, B A Brown, A Faessler, B Fornal, O Sorlin, and I Talmi kindly agreed to be on the panel and their remarks were essential in bringing about the active involvement of the audience.The Conference had about 100 participants from some 20 countries (please see PDF for list of participants). This is well in line with the tradition of these meetings, as is the fact that about 50% of the present participants attended one or more of the previous Seminars.We gratefully acknowledge the financial support of the Istituto Nazionale di Fisica Nucleare and the University of Naples Federico II who helped make the Seminar possible. We also acknowledge the support provided in various ways by the Dipartimento di Scienze Fisiche which acted as host to the Seminar.Aldo Covello Angela Gargano Editors LOCAL ORGANIZING COMMITTEEA Covello (Chair) A Gargano (Co-Chair) L Coraggio (Scientific Secretary) F Andreozzi N Itaco G La Rana N Lo Iudice A. Porrino INTERNATIONAL ADVISORY COMMITTEE J Äystö (Jyväskylä)D Morrissey (Michigan) A B Balantekin (Wisconsin)W Nazarewicz (Oak Ridge) B R Barrett (Tucson)P von Neumann-Cosel (Darmstadt) P G Bizzeti (Firenze)R Okamoto (Kyushu) Y Blumenfeld (CERN and IPN Orsay)A V Ramayya (Vanderbilt) J Dobaczewski (Warsaw)J Schiffer (Argonne) G Fiorentini (Ferrara)A C Shotter (Edinburgh) B Fornal (Kraków)Ch Stoyanov (Sofia) S Gales (GANIL)I Talmi (Rehovot) F Iachello (Yale)P van Duppen (Leuven) R Jolos (Dubna)A Vitturi (Padova) M Lattuada (Catania) SPONSORS OF THE SEMINARDipartimento di Scienze Fisiche, Università di Napoli "Federico II" Istituto Nazionale di Fisica Nucleare Università di Napoli Federico II

Background: Single-charge-exchange reactions are appropriate tools to study the nuclear response to one-particle/one-hole isospin probes, gaining additional interest from the connection to beta decay. This analogy has been recently extended to second order, connecting double-charge-exchange reactions and double beta decays. Nowadays, the availability of powerful experimental setups and advanced nuclear theories and models allows one to access precious information on key nuclear structure aspects embedded in the widely sought neutrinoless double beta decay.Purpose: We intend to bring to light the main nuclear structure and reaction features involved in the $^{18}\mathrm{O}+^{12}\mathrm{C}$ collision at 275 MeV incident energy. In this paper, the main focus is on the role of the initial- and final-state interactions in the overall reaction dynamics and on the single-particle nuclear structure properties accessed via the study of single-nucleon transfer reactions. Forthcoming articles will be devoted to go into the details of the response to one- and two-particle/hole isospin probes.Methods: Cross-section energy spectra and angular distributions were measured in a unique experimental setup for the $^{12}\mathrm{C}(^{18}\mathrm{O},^{18}\mathrm{O})^{12}\mathrm{C}$ elastic and inelastic scattering, the $^{12}\mathrm{C}(^{18}\mathrm{O},^{17}\mathrm{O})^{13}\mathrm{C}$ one-neutron stripping, the $^{12}\mathrm{C}(^{18}\mathrm{O},^{19}\mathrm{F})^{11}\mathrm{B}$ one-proton pickup, and the $^{12}\mathrm{C}(^{18}\mathrm{O},^{18}\mathrm{F})^{12}\mathrm{B}$ single-charge-exchange nuclear reactions. A unique comprehensive and coherent theoretical calculation, able to describe the whole network of direct reactions using state-of-the-art nuclear structure and reaction theories, was performed, and it is presented for the first time in this article. This holistic approach, applied both to the experimental and theoretical analysis, is the main feature and worth of the work here presented.Results: The energy and angular resolutions achieved in each reaction channel allowed us to isolate specific transitions and to map the diffraction patterns in the angular distributions. The cross-section calculations describe well the experimental data, both in terms of the absolute values and diffraction patterns. Although the distorted wave Born approximation (DWBA) calculations prove to be accurate in describing all the studied channels, better results are achieved when the couplings to inelastic transitions in both the incoming and outgoing partitions are introduced, as done in the coupled channels Born approximation (CCBA) calculations. Otherwise, no real improvement is found when the coupling effects among different partitions are explicitly taken into account in the coupled reaction channels (CRC) calculations.Conclusions: The multichannel approach proposed in this paper is a promising method for accurate investigations of direct reactions originating in heavy-ion collisions. This is quite appealing for the precise spectroscopy of heavy nuclei proposed in many areas of nuclear physics. An example is the NUMEN project with its challenging commitment to provide valuable information on neutrinoless double beta decay nuclear matrix elements from single- and double-charge-exchange cross-section measurements.

Three-nucleon forces: Implementation and applications to atomic nuclei and dense matter

Since the 1975 Nobel Prize in Physics, nuclear theory has evolved along two main directions. On the one hand, the energy–density functional (EDF) theory was established, which presently encompasses (by enlarging the EDF framework) all the mean-field and beyond-mean-field theories based on energy functionals produced by effective phenomenological interactions. Highly sophisticated structure and reaction models are currently available for the treatment of medium-mass and heavy nuclei. On the other hand, effective field theories (EFTs) have rendered possible the formulation of QCD as a low-energy hadronic theory. Ab initio methods have recently achieved remarkable success in the application of EFT or EFT-inspired potentials to structure analyses of light nuclei. Different but complementary competences have been developed during the past few decades in the EDF and EFT communities. Bridges and connections have in some cases been identified and constructed. We review here some of the developments that have been performed within the EDF theory and the EFT during recent years, with some emphasis on analogies and connections that may one day provide a unified picture of the two theories. Illustrations are given for infinite matter and finite nuclei.

Work in nuclear structure and reaction theory, specifically, the relation of reactions to the nuclear structure. Other work was in intermediate energy physics, few-body problems, and computational physics that heavy ions can be used to measure simultaneously both neutron and proton multipole matrix elements of the target nucleus has added new interest to this area of nuclear structure. Considerable attention to the is therefore paid to the to the methods for calculating multiple matrix elements.

By stipulating that like-fermions adjacent in space must be singlet paired, geometric considerations alone produce closed electron shells at the 6 noble gases and, similarly, closed nucleon shells and subshells isomorphic with those of the harmonic oscillator used in nuclear theory.

The central focus of this research was the interplay between three generic classes of degrees of freedom relevant to nuclear structure theory: single-particle degrees of freedom, collective degrees of freedom, and statistical degrees of freedom, which can be thought of as an incoherent mean field or a thermal bath.

The parity-violating asymmetry in polarized electron scattering from nuclei can be used to extract information on nuclear and nucleon structure, as well as to determine the values of Standard Model electroweak couplings. To achieve the latter, high precision is needed both in the measured asymmetry and in the underlying nuclear structure theory. For the former a few tenths of a percent may be attainable; for the latter the present discussions have the dual goal of ascertaining both the sizes of various nuclear structure related effects and of providing estimates of their uncertainties.

Background: There is renewed interest in electric dipole strength distributions for a variety of reasons including the extraction of the dipole polarizability related to properties of the symmetry energy and a measure for the neutron skin thickness, understanding the structure of low-energy E1 strength in nuclei with neutron excess, and establishing the systematics of the isovector giant dipole resonance (IVGDR). Inelastic proton scattering at energies of a few hundred MeV and very forward angles including 0∘ has been established as a tool for the study of electric and magnetic dipole strength distributions in nuclei. Purpose: The present work aims at a systematic investigation of the electric and magnetic dipole strength distributions in the chain of stable even-mass tin isotopes. Methods: Inelastic proton scattering experiments were performed at the Research Center for Nuclear Physics, Osaka, with a 295-MeV beam covering laboratory angles 0∘–6∘ and excitation energies 6–22 MeV. Cross sections due to E1 and M1 excitations were extracted with a multipole decomposition analysis (MDA) and then converted to reduced transition probabilities with the “virtual photon method” for E1 and the “unit cross section method” for M1 excitations, respectively. Including a theory-aided correction for the high-excitation-energy region not covered experimentally, the electric dipole polarizability was determined from the E1 strength distributions. Results: Total photoabsorption cross sections derived from the E1 and M1 strength distributions show significant differences compared to those from previous (γ,xn) experiments in the energy region of the IVGDR. The widths of the IVGDR deduced from the present data with a Lorentz parametrization show an approximately constant value of about 4.5 MeV in contrast to the large variations between isotopes observed in previous work. The IVGDR centroid energies are in good correspondence to expectations from empirical systematics of their mass dependence. Furthermore, a study of the dependence of the IVGDR energies on bulk matter properties is presented. The E1 strengths below neutron threshold show fair agreement with results from (γ,γ′) experiments on Sn112,116,120,124 in the energy region between 6 and 7 MeV, where also isoscalar E1 strength was found for Sn124. At higher excitation energies, large differences are observed, pointing to a different nature of the excited states with small ground-state branching ratios. The isovector spin-M1 strengths exhibit a broad distribution between 6 and 12 MeV in all studied nuclei. Conclusions: The present results contribute to the solution of a variety of nuclear structure problems including the systematics of the energy and width of the IVGDR, the structure of low-energy E1 strength in nuclei, new constraints to energy density functionals (EDFs) aiming at a systematic description of the dipole polarizability across the nuclear chart, from which properties of the symmetry energy can be derived, and the systematics of the isovector spin-M1 strength in heavy nuclei.16 MoreReceived 12 July 2020Accepted 2 September 2020DOI:https://doi.org/10.1103/PhysRevC.102.034327©2020 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCollective levelsElectromagnetic transitionsH & He induced nuclear reactionsNuclear density functional theoryNuclear structure & decaysPhotonuclear reactionsNuclear Physics

In addition to shape oscillations, low-energy excitation spectra of deformed nuclei are also influenced by pairing vibrations. The simultaneous description of these collective modes and their coupling has been a long-standing problem in nuclear structure theory. Here we address the problem in terms of self-consistent mean-field calculations of collective deformation energy surfaces, and the framework of the interacting boson approximation. In addition to quadrupole shape vibrations and rotations, the explicit coupling to pairing vibrations is taken into account by a boson-number non-conserving Hamiltonian, specified by a choice of a universal density functional and pairing interaction. An illustrative calculation for $^{128}$Xe and $^{130}$Xe shows the importance of dynamical pairing degrees of freedom, especially for structures built on low-energy $0^+$ excited states, in $\gamma$-soft and triaxial nuclei.

Nuclear structure theory has recently gone through a major renewal with the development of ab initio techniques that can be applied to a large number of atomic nuclei, well beyond the light sector that had been traditionally targeted in the past. Self-consistent Green's function theory is one among these techniques. The present work aims to give an overview of the self-consistent Green's function approach for atomic nuclei, including examples of recent applications and a discussion on the perspectives for extending the method to nuclear reactions, doubly open-shell systems and heavy nuclei.

Masses of unbound nuclear systems

We present examples and illustrate associated tests of statistical significance of the parameter fitting procedures in the nuclear mean-field context using a phenomenological toy-model based on the spherical Woods-Saxon Hamiltonian. We calculate the variance-covariance matrix and compare the standard estimates of the confidence intervals (and more generally – the uncertainty distributions) based on the standard χ2-fitting as opposed to a more general Monte-Carlo simulations. We give arguments for the superiority of the latter approach.

Helium-8 (8He) is the most neutron-rich helium isotope that can be synthesized on Earth: it consists of two protons and six neutrons, and remains stable for an average of 0.2 seconds. Unlike the abundant helium isotope, 4He, which has the two neutrons packed closely with two protons, the additional neutrons in 8He form a “halo” around a compact, 4He-like core. A similar neutron-halo structure also exists for 6He (half-life = 0.8 s). Because of their intriguing properties, 6He and 8He have the potential to reveal new aspects of the fundamental forces among the constituent nucleons. We have recently succeeded in laser trapping and cooling these exotic helium isotopes, and have performed precision laser spectroscopy on individual trapped atoms. Based on atomic frequency differences measured along the isotope chain 4He-6He-8He, the nuclear charge radius of 8He has been determined for the first time to be 1.96 fm and that of 6He to be 2.07 fm. The result shows that the charge radius of 8He is smaller than that of 6He, but notably larger than that of 4He (1.68 fm). This may be understood by noting that the four extra neutrons are more evenly distributed around the 4He core than the extra neutron pair in 6He as illustrated in Figure 1. The result is in good agreement with recent predictions of state-of-the-art nuclear structure theories, and provides a critical test of the present understanding of these loosely bound halo nuclei. The 6He measurement was first performed at the ATLAS accelerator facility of Argonne National Laboratory in 2004. An improved measurement on 6He and the first measurement on 8He were performed at GANIL in 2007.