The Nuclear Shell Model 70 Years after Its Advent: Achievements and Prospects

A special issue of Physics (ISSN 2624-8174). This special issue belongs to the section "Atomic Physics".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 44982

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Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Interests: nuclear structure; nuclear shell model; effective interactions; beta decay

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Guest Editor
1. Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, viale Abramo Lincoln 5, I-81100 Caserta, Italy
2. Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Interests: nuclear structure; nuclear shell model; nuclear collective motions; beta decay

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Guest Editor
1. Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova, Via F. Marzolo 8, I-35131 Padova, Italy
2. Istituto Nazionale di Fisica Nucleare, Via F. Marzolo 8, I-35131 Padova, Italy
Interests: nuclear structure; nuclear shell model; nuclear collective motions; nuclear spectroscopy

Special Issue Information

Dear Colleagues,

The shell model (SM) entered into nuclear physics almost 70 years ago after Maria Goeppert Mayer and Hans Jensen - who shared the Nobel prize in 1963 for their work – were able to explain the mystery behind the magic numbers associated with the great stability in the ground state of some nuclides.

The SM is widely considered as the basic scheme of the microscopic description of the nucleus, and, starting from its introduction, has been successfully applied for investigating a variety of nuclear structure phenomena, which have also important implications in our understanding of both astrophysics and physics beyond the standard model.

Within the SM the complexity of the many-body system is simplified by considering the nucleus as a closed-shell core with additional valence nucleons that interact in a subspace of the Hilbert space through effective Hamiltonians, accounting for contributions from configurations outside the model space.

In the last two decades or so, thanks to the high-performance computing facilities available all around the world and the implementation of more and more efficient codes and methods, large-scale SM calculations have become a well-established approach to investigate medium- and heavy-mass nuclei whose description requires large model spaces with many valence nucleons. Today, it is possible to deal with huge matrices of dimension ~ 1011-12, unthinkable up to few years ago.

Over about the same period, the extraordinary improvements in sensitivity and efficiency of the experimental tools together with the development of a large variety of radioactive beams has allowed to explore new regions of the nuclide chart towards the drip lines. The richness of data emerging from these experimental studies has probed the reliability and robustness of the SM in describing the new phenomena observed far from stability such as the onset of collectivity at the historical magic numbers, the development of islands of inversion and the appearance of new magic numbers, the evidence of shape coexistence. These studies have revealed modifications in the shell structure as a function of proton and neutron number, leading to a paradigm shift away from the universality of the magic numbers. Large-scale SM calculations are required to investigate these phenomena and to understand the role of the different components of the nuclear force in determining the evolution of the shell structure towards the driplines.

While the SM has been used predominantly in the past with empirical effective Hamiltonians, substantial progress has been achieved in the last years in deriving the SM input from realistic bare interactions, including two and three-body forces based on chiral effective field theory. This has given a further impulse towards a fully microscopic description of atomic nuclei starting from the quantum chromodynamics degrees of freedom. Within this context, valence-space Hamiltonians can be derived by means of many-body perturbation theory, and only very recently nonperturbative approaches have been introduced.

In this Special Issue, we aim to collect contributions that, starting from the historical setting, may provide a clear overview of the current status and future developments of the nuclear shell model, including its applications in describing a variety of nuclear structure phenomena. For this purpose, we invite articles and reviews to contribute to both experimental and theoretical aspects for inclusion in this Special Issue.

Dr. Angela Gargano
Dr. Giovanni De Gregorio
Prof. Dr. Silvia Monica Lenzi
Guest Editors

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Keywords

  • nuclear shell model
  • effective interactions
  • nuclear forces
  • nuclear structure
  • many-body methods
  • high performance computing
  • exotic nuclei
  • nuclear spectroscopy

Published Papers (14 papers)

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Research

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8 pages, 232 KiB  
Article
The Very Long Lifetime of 14C in the Shell Model
by Igal Talmi
Physics 2022, 4(3), 940-947; https://doi.org/10.3390/physics4030062 - 24 Aug 2022
Viewed by 1531
Abstract
This is a fitting memory for our late friend and colleague Aldo Covello. For many years, he was our host in the series of Spring Seminars which he organized. In these conferences, the shell model was a central subject which was taken very [...] Read more.
This is a fitting memory for our late friend and colleague Aldo Covello. For many years, he was our host in the series of Spring Seminars which he organized. In these conferences, the shell model was a central subject which was taken very seriously. This paper is written after 70 years of successful shell model calculations of nuclear energies and also various transitions. The beta decay of 14C has been an enigma. The history and present situation are described. The importance check of any theory to yield the strength of the mirror transition of 14O is pointed out. Full article
17 pages, 504 KiB  
Article
Probing Different Characteristics of Shell Evolution Driven by Central, Spin-Orbit, and Tensor Forces
by Yutaka Utsuno
Physics 2022, 4(1), 185-201; https://doi.org/10.3390/physics4010014 - 9 Feb 2022
Cited by 3 | Viewed by 2315
Abstract
In this paper, the validity of the shell-evolution picture is investigated on the basis of shell-model calculations for the atomic mass number 25A55 neutron-rich nuclei. For this purpose, the so-called SDPF-MU interaction is used. Its central, two-body spin–orbit, and [...] Read more.
In this paper, the validity of the shell-evolution picture is investigated on the basis of shell-model calculations for the atomic mass number 25A55 neutron-rich nuclei. For this purpose, the so-called SDPF-MU interaction is used. Its central, two-body spin–orbit, and tensor forces are taken from a simple Gaussian force, the M3Y (Michigan 3-range Yukawa) interaction, and a π+ρ meson exchange force, respectively. Carrying out almost a complete survey of the predicted effective single-particle energies, it is confirmed here that the present scheme is quite effective for describing shell evolution in exotic nuclei. Full article
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17 pages, 1037 KiB  
Article
Low-Energy Coulomb Excitation for the Shell Model
by Marco Rocchini and Magda Zielińska
Physics 2021, 3(4), 1237-1253; https://doi.org/10.3390/physics3040078 - 15 Dec 2021
Cited by 2 | Viewed by 2885
Abstract
Low-energy Coulomb excitation is capable of providing unique information on static electromagnetic moments of short-lived excited nuclear states, including non-yrast states. The process selectively populates low-lying collective states and is, therefore, ideally suited to study phenomena such as shape coexistence and the development [...] Read more.
Low-energy Coulomb excitation is capable of providing unique information on static electromagnetic moments of short-lived excited nuclear states, including non-yrast states. The process selectively populates low-lying collective states and is, therefore, ideally suited to study phenomena such as shape coexistence and the development of exotic deformation (triaxial or octupole shapes). Historically, these experiments were restricted to stable isotopes. However, the advent of new facilities providing intense beams of short-lived radioactive species has opened the possibility to apply this powerful technique to a much wider range of nuclei. The paper discusses the observables that can be measured in a Coulomb-excitation experiment and their relation to the nuclear structure parameters with an emphasis on the nuclear shape. Recent examples of Coulomb-excitation studies that provided outcomes relevant for the Shell Model are also presented. Full article
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Review

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29 pages, 721 KiB  
Review
Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments
by Nadezda A. Smirnova
Physics 2023, 5(2), 352-380; https://doi.org/10.3390/physics5020026 - 31 Mar 2023
Cited by 4 | Viewed by 2920
Abstract
The paper reviews the recent progress in the description of isospin-symmetry breaking within the nuclear shell model and applications to actual problems related to the structure and decay of exotic neutron-deficient nuclei and nuclei along the N=Z line, where N is [...] Read more.
The paper reviews the recent progress in the description of isospin-symmetry breaking within the nuclear shell model and applications to actual problems related to the structure and decay of exotic neutron-deficient nuclei and nuclei along the N=Z line, where N is the neutron number and Z the atomic number. The review recalls the fundamentals of the isospin formalism for two-nucleon and many-nucleon systems, including quantum numbers, the spectrum’s structure and selection rules for weak and electromagnetic transitions; and at the end, summarizes experimental signatures of isospin-symmetry breaking effects, which motivated efforts towards the creation of a relevant theoretical framework to describe those phenomena. The main approaches to construct accurate isospin-nonconserving Hamiltonians within the shell model are briefly described and recent advances in the description of the structure and (isospin-forbidden) decay modes of neutron-deficient nuclei are highlighted. The paper reviews major implications of the developed theoretical tools to (i) the fundamental interaction studies on nuclear decays and (ii) the estimation of the rates of nuclear reactions that are important for nuclear astrophysics. The shell model is shown to be one of the most suitable approaches to describing isospin-symmetry breaking in nuclear states at low energies. Further efforts in extending and refining the description to larger model spaces, and in developing first-principle theories to deal with isospin-symmetry breaking in many-nucleon systems, seem to be indispensable steps towards our better understanding of nuclear properties in the precision era. Full article
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15 pages, 387 KiB  
Review
Double Beta Decay: A Shell Model Approach
by Mihai Horoi
Physics 2022, 4(4), 1135-1149; https://doi.org/10.3390/physics4040074 - 26 Sep 2022
Cited by 5 | Viewed by 2077
Abstract
Studies of weak interaction in nuclei are important tools for testing different aspects of the fundamental symmetries of the Standard Model. Neutrinoless double beta decay offers an unique venue of investigating the possibility that neutrinos are Majorana fermions and that the lepton number [...] Read more.
Studies of weak interaction in nuclei are important tools for testing different aspects of the fundamental symmetries of the Standard Model. Neutrinoless double beta decay offers an unique venue of investigating the possibility that neutrinos are Majorana fermions and that the lepton number conservation law is violated. Here, I use a shell model approach to calculate the nuclear matrix elements needed to extract the lepton-number-violating parameters of a few nuclei of experimental interest from the latest experimental lower limits of neutrinoless double beta decay half-lives. The analysis presented here could reveal valuable information regarding the dominant neutrinoless double beta decay mechanism if experimental half-life data become available for different isotopes. A complementary shell model analysis of the two-neutrino double beta decay nuclear matrix elements and half-lives is also presented. Full article
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13 pages, 451 KiB  
Review
Recent Progress of Shell-Model Calculations, Monte Carlo Shell Model, and Quasi-Particle Vacua Shell Model
by Noritaka Shimizu
Physics 2022, 4(3), 1081-1093; https://doi.org/10.3390/physics4030071 - 9 Sep 2022
Cited by 3 | Viewed by 2785
Abstract
Nuclear shell model is a powerful approach to investigate nuclear structure microscopically. However, the computational cost of shell-model calculations becomes huge in medium-heavy nuclei. I briefly review the theoretical framework and the code developments of the conventional Lanczos diagonalization method for shell-model calculations. [...] Read more.
Nuclear shell model is a powerful approach to investigate nuclear structure microscopically. However, the computational cost of shell-model calculations becomes huge in medium-heavy nuclei. I briefly review the theoretical framework and the code developments of the conventional Lanczos diagonalization method for shell-model calculations. In order to go beyond the conventional diagonalization method, the Monte Carlo shell model and the quasiparticle-vacua shell model were introduced. I present some benchmark examples of these models. Full article
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17 pages, 2594 KiB  
Review
Excited States in Isobaric Multiplets—Experimental Advances and the Shell-Model Approach
by Michael A Bentley
Physics 2022, 4(3), 995-1011; https://doi.org/10.3390/physics4030066 - 5 Sep 2022
Cited by 5 | Viewed by 1777
Abstract
A review of recent advances in the study of the energy splitting between excited isobaric analogue states is presented. Some of the experimental developments, and new approaches, associated with spectroscopy of the most proton-rich members of isobaric multiplets, are discussed. The review focuses [...] Read more.
A review of recent advances in the study of the energy splitting between excited isobaric analogue states is presented. Some of the experimental developments, and new approaches, associated with spectroscopy of the most proton-rich members of isobaric multiplets, are discussed. The review focuses on the immense impact of the shell-model in the analysis of energy differences and their interpretation in terms of nuclear structure phenomena. Full article
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13 pages, 380 KiB  
Review
Shell Model Applications in Nuclear Astrophysics
by Gabriel Martínez-Pinedo and Karlheinz Langanke
Physics 2022, 4(2), 677-689; https://doi.org/10.3390/physics4020046 - 17 Jun 2022
Cited by 6 | Viewed by 2969
Abstract
In recent years, shell model studies have significantly contributed in improving the nuclear input, required in simulations of the dynamics of astrophysical objects and their associated nucleosynthesis. This review highlights a few examples such as electron capture rates and neutrino-nucleus cross sections, important [...] Read more.
In recent years, shell model studies have significantly contributed in improving the nuclear input, required in simulations of the dynamics of astrophysical objects and their associated nucleosynthesis. This review highlights a few examples such as electron capture rates and neutrino-nucleus cross sections, important for the evolution and nucleosynthesis of supernovae. For simulations of rapid neutron-capture (r-process) nucleosynthesis, shell model studies have contributed to an improved understanding of half lives of neutron-rich nuclei with magic neutron numbers and of the nuclear level densities and γ-strength functions that are both relevant for neutron capture rates. Full article
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23 pages, 618 KiB  
Review
The Nuclear Shell Model towards the Drip Lines
by B. Alex Brown
Physics 2022, 4(2), 525-547; https://doi.org/10.3390/physics4020035 - 12 May 2022
Cited by 10 | Viewed by 4650
Abstract
Applications of configuration-mixing methods for nuclei near the proton and neutron drip lines are discussed. A short review of magic numbers is presented. Prospects for advances in the regions of four new “outposts” are highlighted: 28O, 42Si, 60Ca and 78 [...] Read more.
Applications of configuration-mixing methods for nuclei near the proton and neutron drip lines are discussed. A short review of magic numbers is presented. Prospects for advances in the regions of four new “outposts” are highlighted: 28O, 42Si, 60Ca and 78Ni. Topics include shell gaps, single-particle properties, islands of inversion, collectivity, neutron decay, neutron halos, two-proton decay, effective charge, and quenching in knockout reactions. Full article
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19 pages, 2333 KiB  
Review
Trends in the Structure of Nuclei near 100Sn
by Magdalena Górska
Physics 2022, 4(1), 364-382; https://doi.org/10.3390/physics4010024 - 21 Mar 2022
Cited by 7 | Viewed by 3647
Abstract
Inevitable progress has been achieved in recent years regarding the available data on the structure of 100Sn and neighboring nuclei. Updated nuclear structure data in the region is presented using selected examples. State-of-the-art experimental techniques involving stable and radioactive beam facilities have [...] Read more.
Inevitable progress has been achieved in recent years regarding the available data on the structure of 100Sn and neighboring nuclei. Updated nuclear structure data in the region is presented using selected examples. State-of-the-art experimental techniques involving stable and radioactive beam facilities have enabled access to those exotic nuclei. The analysis of experimental data has established the shell structure and its evolution towards N = Z = 50 of the number of neutrons, N, and the atomic number, Z, seniority conservation and proton–neutron interaction in the g9/2 orbit, the super-allowed Gamow–Teller decay of 100Sn, masses and half-lives along the rapid neutron-capture process (r-process) path and super-allowed α decay beyond 100Sn. The status of theoretical approaches in shell model and mean-field investigations are discussed and their predictive power assessed. The calculated systematics of high-spin states for N = 50 isotopes including the 5 state and N = Z nuclei in the g9/2 orbit is presented for the first time. Full article
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28 pages, 3498 KiB  
Review
Emerging Concepts in Nuclear Structure Based on the Shell Model
by Takaharu Otsuka
Physics 2022, 4(1), 258-285; https://doi.org/10.3390/physics4010018 - 22 Feb 2022
Cited by 8 | Viewed by 4049
Abstract
Some emerging concepts of nuclear structure are overviewed. (i) Background: the many-body quantum structure of atomic nucleus, a complex system comprising protons and neutrons (called nucleons collectively), has been studied largely based on the idea of the quantum liquid (à la Landau), where [...] Read more.
Some emerging concepts of nuclear structure are overviewed. (i) Background: the many-body quantum structure of atomic nucleus, a complex system comprising protons and neutrons (called nucleons collectively), has been studied largely based on the idea of the quantum liquid (à la Landau), where nucleons are quasiparticles moving in a (mean) potential well, with weak “residual” interactions between nucleons. The potential is rigid in general, although it can be anisotropic. While this view was a good starting point, it is time to look into kaleidoscopic aspects of the nuclear structure brought in by underlying dynamics and nuclear forces. (ii) Methods: exotic features as well as classical issues are investigated from fresh viewpoints based on the shell model and nucleon–nucleon interactions. The 70-year progress of the shell–model approach, including effective nucleon–nucleon interactions, enables us to do this. (iii) Results: we go beyond the picture of the solid potential well by activating the monopole interactions of the nuclear forces. This produces notable consequences in key features such as the shell/magic structure, the shape deformation, the dripline, etc. These consequences are understood with emerging concepts such as shell evolution (including type-II), T-plot, self-organization (for collective bands), triaxial-shape dominance, new dripline mechanism, etc. The resulting predictions and analyses agree with experiment. (iv) Conclusion: atomic nuclei are surprisingly richer objects than initially thought. Full article
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11 pages, 691 KiB  
Review
Reaching into the N = 40 Island of Inversion with Nucleon Removal Reactions
by Alexandra Gade
Physics 2021, 3(4), 1226-1236; https://doi.org/10.3390/physics3040077 - 8 Dec 2021
Cited by 5 | Viewed by 2585
Abstract
One ambitious goal of nuclear physics is a predictive model of all nuclei, including the ones at the fringes of the nuclear chart which may remain out of experimental reach. Certain regions of the chart are providing formidable testing grounds for nuclear models [...] Read more.
One ambitious goal of nuclear physics is a predictive model of all nuclei, including the ones at the fringes of the nuclear chart which may remain out of experimental reach. Certain regions of the chart are providing formidable testing grounds for nuclear models in this quest as they display rapid structural evolution from one nucleus to another or phenomena such as shape coexistence. Observables measured for such nuclei can confirm or refute our understanding of the driving forces of the evolution of nuclear structure away from stability where textbook nuclear physics has been proven to not apply anymore. This paper briefly reviews the emerging picture for the very neutron-rich Fe, Cr, and Ti isotopes within the so-called N=40 island of inversion as obtained with nucleon knockout reactions. These have provided some of the most detailed nuclear spectroscopy in very neutron-rich nuclei produced at rare-isotope facilities. The results indicate that our current understanding, as encoded in large-scale shell-model calculations, appears correct with exciting predictions for the N=40 island of inversion left to be proven in the experiment. A bright future emerges with predictions of continued shell evolution and shape coexistence out to neutron number N=50, below 78Ni on the chart of nuclei. Full article
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21 pages, 1291 KiB  
Review
Recent Progress in Gamow Shell Model Calculations of Drip Line Nuclei
by Jianguo Li, Yuanzhuo Ma, Nicolas Michel, Baishan Hu, Zhonghao Sun, Wei Zuo and Furong Xu
Physics 2021, 3(4), 977-997; https://doi.org/10.3390/physics3040062 - 8 Nov 2021
Cited by 10 | Viewed by 2995
Abstract
The Gamow shell model (GSM) is a powerful method for the description of the exotic properties of drip line nuclei. Internucleon correlations are included via a configuration interaction framework. Continuum coupling is directly included at basis level by using the Berggren basis, in [...] Read more.
The Gamow shell model (GSM) is a powerful method for the description of the exotic properties of drip line nuclei. Internucleon correlations are included via a configuration interaction framework. Continuum coupling is directly included at basis level by using the Berggren basis, in which, bound, resonance, and continuum single-particle states are treated on an equal footing in the complex momentum plane. Two different types of Gamow shell models have been developed: its first embodiment is that of the GSM defined with phenomenological nuclear interactions, whereas the GSM using realistic nuclear interactions, called the realistic Gamow shell model, was introduced later. The present review focuses on the recent applications of the GSM to drip line nuclei. Full article
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Other

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77 pages, 7464 KiB  
Opinion
To Shell Model, or Not to Shell Model, That Is the Question
by Andrew E. Stuchbery and John L. Wood
Physics 2022, 4(3), 697-773; https://doi.org/10.3390/physics4030048 - 29 Jun 2022
Cited by 11 | Viewed by 4431
Abstract
The present review takes steps from the domain of the shell model into open shell nuclei. The question posed in the title is to dramatize how far shell model approaches, i.e., many nucleons occupying independent-particle configurations and interacting through two-body forces (a configuration [...] Read more.
The present review takes steps from the domain of the shell model into open shell nuclei. The question posed in the title is to dramatize how far shell model approaches, i.e., many nucleons occupying independent-particle configurations and interacting through two-body forces (a configuration interaction problem) can provide a description of nuclei as one explores the structure observed where neither proton nor neutron numbers match closed shells. Features of doubly closed and singly closed shell nuclei and adjacent nuclei are sketched, together with the roles played by seniority, shape coexistence, triaxial shapes and particle–core coupling in organizing data. An illuminating step is taken here to provide a detailed study the reduced transition rates, B(E2;21+01+), in the singly closed shell nuclei with doubly closed shell plus or minus a pair of identical nucleons, and the confrontation between such data and state-of-the-art shell model calculations: this amounts to a review of the effective charge problem. The results raise many questions and point to the need for much further work. Some guidance on criteria for sharpening the division between the domain of the shell model and that of deformation-based descriptions of nuclei are provided. The paper is closed with a sketch of a promising direction in terms of the algebraic structure embodied in the symplectic shell model. Full article
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