Exact solutions of mean-field plus various pairing interactions and shape phase transitions in nuclei
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-000014-5
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Review
Exact solutions of mean-field plus various pairing interactions and shape phase transitions in nuclei Feng Pan1,2,a , Xin Guan1 , Lian-Rong Dai1 , Yu Zhang1 , and Jerry P. Draayer2 1 2
Department of Physics, Liaoning Normal University, Dalian 116029, P.R. China Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA Received 1 February 2020 / Accepted 28 August 2020 Published online 23 October 2020 Abstract. The exact solutions of either spherical or deformed meanfield plus various types of pairing models are briefly reviewed. It is shown that, besides the standard pairing model, there are several special types of pairing interaction that can be solved exactly. In comparison to the standard pairing, the results of several pairing-driven quantities, such as pairing excitation energies, even–odd mass differences, the moment of inertia, etc., show that these types of pairing interaction can indeed be used to describe pairing correlations in nuclei either more accurately or efficiently. Moreover, the shape phase transitional behaviors of nuclei described by the consistent-Q formalism of the interacting boson model are summarized.
1 Introduction Nuclear pairing correlation, as an important part of the residual interactions necessary to augment any nuclear mean-field theory, represents one of the main and longstanding pillars of current understanding of nuclear structure [1]. For example, the pairing interaction of the nuclear shell model plays a key role to reproduce lowenergy spectroscopic properties of nuclei, such as binding energies, odd-even effects, single-particle occupancies, excitation spectra, and moments of inertia, etc. [2,3]. Bohr, Mottelson, Pines, and Belyaev were the first to introduce the Bardeen– Cooper–Schrieffer (BCS) theory for superconductivity in condensed matter [4] to descriptions of pairing phenomena in nuclei [2,5]. Though the BCS and the more refined Hartree–Fock–Bogolyubov (HFB) approximations provide simple and clear pictures in demonstrating pairing correlations in nuclei [2,6,7], tremendous efforts have been made in finding accurate solutions to the problem [8–14] to overcome serious drawbacks in the BCS and the HFB, such as spurious states, nonorthogonal solutions, etc. resulting from particle number-nonconservation effects in these approximations [9,10,14–16]. Driven by the importance of having exact solutions of either spherical or deformed mean-field plus the standard pairing Hamiltonian, much attention and progress, building on Richardson and Gaudin’s early work [17–21] and a
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extensions to it based on the Bethe ansatz, have been made [22–32]. For all these algebraic Bethe ansatz approaches, the solutions are provided by a set of highly non-linear Bethe–Gaudin–Richardson equations (BGREs). Though these applications demonstrate that the
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