Focus on shape coexistence in nuclei

The present collection of articles focuses on new directions and developments under the title of shape coexistence in nuclei, following our 2011 Reviews of Modern Physics article (K Heyde and J L Wood).

Shape coexistence in neutron-rich nuclei in the N = 20 island of inversion, along the N = 28 isotone line, and in the region around neutron number N = 40 is reviewed. The present status, emerging experimental opportunities and challenges in the interpretation are discussed.

Low-energy Coulomb excitation provides a well-understood means of exciting atomic nuclei and allows measuring electromagnetic moments that can be directly related to the nuclear shape. The availability of radioactive ion beams (RIBs) at energies near the Coulomb barrier has made it possible to study shape coexistence in a variety of short-lived exotic nuclei. This review presents a short overview of the methods related to multi-step Coulomb excitation experiments, followed by a discussion of several examples. The focus is on two mass regions where recent Coulomb excitation experiments have contributed to the quantitative understanding of shape coexistence: nuclei with mass $A\approx 70$ near the N = Z line and nuclei with $A\;\approx \;100$ near neutron number N = 60. Experimental results are summarized and their significance for understanding shape coexistence is discussed. Experimental observables such as quadrupole moments and electromagnetic transition strengths represent furthermore important benchmarks for advancing theoretical nuclear structure models. With several new RIB facilities planned and under construction, Coulomb excitation will remain to be an important tool to extend the studies of nuclear shapes toward more exotic systems, and to obtain a more comprehensive and quantitative understanding of shape coexistence.

Shape coexistence, where different deformed minima compete within a small range of excitation energy, appears to be ubiquitous across the chart of nuclides. In many light alpha-conjugate nuclei, experimental data points to the coexistence of highly deformed nuclear configurations. It has long been suggested, with strong theoretical justification, that these deformed states are attributable to nuclear clustering based on building blocks of alpha particles. This short review will consider how well alpha clustering fits within the shape coexistence canon and point to future opportunities for experiments that can place the topic on a firmer footing.

In the present paper we focus on studies of shape coexistence in even-mass nuclei in the neutron-deficient Pb region. They are based on experiments carried out using tagging techniques in the Accelerator Laboratory of the University of Jyväskylä, Finland. Excited states in many of these nuclei can only be accessed via fusion-evaporation reactions employing high-intensity stable-ion beams. The key features in these experiments are high selectivity, clean spectra and instrumentation that enables high count rates. We review three spectroscopic highlights in this region.

A quantitative analysis of the evolution of nuclear shapes and shape phase transitions, including regions of short-lived nuclei that are becoming accessible in experiments at radioactive-beam facilities, necessitate accurate modeling of the underlying nucleonic dynamics. Important theoretical advances have recently been made in studies of complex shapes and the corresponding excitation spectra and electromagnetic decay patterns, especially in the 'beyond mean-field' framework based on nuclear density functionals. Interesting applications include studies of shape evolution and coexistence in N = 28 isotones, the structure of lowest 0⁺ excitations in deformed N ≈ 90 rare-earth nuclei, and quadrupole and octupole shape transitions in thorium isotopes.

Assuming that the time-evolution of the self-consistent mean field is determined by five pairs of collective coordinate and collective momentum, we microscopically derive the collective Hamiltonian for low-frequency quadrupole modes of excitation. We show that the five-dimensional collective Schrödinger equation is capable of describing large-amplitude quadrupole shape dynamics seen as shape coexistence/mixing phenomena. We focus on basic ideas and recent advances of the approaches based on the time-dependent mean-field theory, but relations to other time-independent approaches are also briefly discussed.

A forward-looking view is given on shape coexistence for N = 19, 21 isotones in the 12 ≤ Z ≤ 20 nuclei, based on the key magnetic moment measurements carried out for ^31,33Mg and in comparison to other moments measurements in the region. The need to characterize multi-particle-multi-hole configurations is underlined.

Shape coexistence has been a subject of great interest in nuclear physics for many decades. In the context of the nuclear shell model, intruder excitations may give rise to remarkably low-lying excited ${0}^{+}$ states associated with different intrinsic shapes. In heavy open-shell nuclei, the dimension of the shell-model configuration space that includes such intruder excitations becomes exceedingly large, thus requiring a drastic truncation scheme. Such a framework has been provided by the interacting boson model (IBM). In this article we address the phenomenon of shape coexistence and its relevant spectroscopy from the point of view of the IBM. A special focus is placed on the method developed recently which makes use of the link between the IBM and the self-consistent mean-field approach based on the nuclear energy density functional. The method is extended to deal with various intruder configurations associated with different equilibrium shapes. We assess the predictive power of the method and suggest possible improvements and extensions, by considering illustrative examples in the neutron-deficient Pb region, where shape coexistence has been experimentally studied.

We first review the shell evolution in exotic nuclei driven by nuclear forces. We then demonstrate that the underlying mechanism played by the balance of the tensor and central components in the effective nucleon–nucleon interaction is crucial when describing shape coexistence. This effect will be referred to as type II shell evolution, while the shell evolution passing through a series of isotopes or isotones is denoted as type I. We describe type II shell evolution in some detail for the case of the ⁶⁸Ni nucleus as an example. We present how the fission dynamics can be related to enhanced deformation triggered by type II shell evolution, at its initial stage. It is suggested that the island of stability may be related to the suppression of this mechanism.

We shall discuss the meaning of the 'nuclear shape' in the laboratory frame proper to the spherical shell model. A brief historical promenade will bring us from Elliott's SU3 breakthrough to today's large scale shell model calculations. A section is devoted to the algebraic model which extends drastically the field of applicability of Elliot's SU3, providing a precious heuristic guidance for the exploration of collectivity in the nuclear chart. Shape coexistence and shape mixing will be shown to occur as the result of the competition between the main actors in the nuclear dynamics; the spherical mean field, and the pairing and quadrupole–quadrupole interactions. These ideas will be illustrated with examples in magic nuclei (⁴⁰Ca and ⁶⁸Ni); neutron rich semi-magic (³²Mg, and ⁶⁴Cr); and in proton rich N = Z (⁷²Kr).

The many-nucleon quantum mechanics of a nucleus is infinite-dimensional and, although simply defined, it has the potential for unlimited complexity. Nevertheless, the low-energy states of heavy open-shell nuclei exhibit properties that are remarkably well described by simple collective models. This paper examines this emergent simplicity from a perspective that closely parallels the emergence of shell structure in the Mayer–Jensen model. The result is an expression of the many-nucleon Hilbert space of a nucleus as an energy-ordered sum of subspaces each of which carries a microscopic version of the Bohr–Mottelson unified model. Each of the subspaces is characterized by nuclear states with a common intrinsic shape defined by its quadrupole moments. An emergence of simplicity and shape-coexistence in nuclei is then explained if it can be demonstrated that there is a relatively small and coherent mixing of the states of different collective subspaces.

Neutron-deficient isotopes of Pt–Hg–Pb–Po–Rn are the classic region in the investigation of shape coexistence in atomic nuclei. A large programme of Coulomb-excitation experiments has been undertaken at the REX-ISOLDE facility in CERN with a number of even–even isotopes in this region. These experiments have been used to probe the electromagnetic properties of yrast and non-yrast states of even–even exotic nuclei, above and below Z = 82. Amongst a large amount of different complementary techniques used to study nuclear structure, Coulomb excitation brings substantial and unique information detailing shape coexistence. In this paper we review the Coulomb-excitation campaign at REX-ISOLDE in the light-lead region together with most recently obtained results. Furthermore, we present some new interpretations that arise from this data and show testing comparisons to state-of-the-art nuclear models.

We present, from a historical perspective, the evolution of instruments and techniques developed by our group, in conjunction with other collaborators, to establish a program in conversion electron spectroscopy that could be routinely implemented in radioactive decay studies. We focus here mainly on the investigations that bear upon the study of nuclear shape coexistence and its relation to electric monopole (E0) transitions. We show that many I^π → I^π (I ≠ 0)${\ }$transitions in both even and odd nuclei with mixed shape-coexisting configurations have large E0 components accompanying their M1 + E2 strength (in some cases nearly pure E0), and that this E0 enhancement is a clear 'signature' for nuclear shape coexistence.