A solvable model for octupole phonons
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-000026-x
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
A solvable model for octupole phonons P. Van Isackera Grand Acc´el´erateur National d’Ions Lourds, CEA/DRF-CNRS/IN2P3 Bvd Henri Becquerel, BP 55027, 14076 Caen, France Received 17 February 2020 / Accepted 31 August 2020 Published online 23 October 2020 Abstract. A solvable model is proposed for the description of octupole phonons in closed-shell nuclei, formulated in terms of shell-model particle–hole excitations. With some simple assumptions concerning single-particle energies and two-body interactions, closed expressions are derived for the energy and wave function of the octupole phonon. In particular, it is shown that the components of the octupole phonon are proportional to Wigner 3j coefficients. This analytic wave function is proven to be exactly valid in light nuclei, which have LS shell closures that coincide with those of the three-dimensional harmonic oscillator, and to be valid to a good approximation in heavier nuclei, which have jj shell closures due to the spin–orbit interaction. The properties of the solvable model are compared with the results of a realistic shell-model calculation for 208 Pb.
1 Introduction Nuclei with a closed-shell configuration for neutrons and/or protons frequently exhibit low-energy excitations with angular momentum J = 3 and negative parity. Such excitations are associated with nuclear shapes that break reflection symmetry and, in particular, with pear-like or octupole shapes [1]. Given the closed-shell configuration of at least one type of nucleon, the nucleus is thought to have a spherical equilibrium shape in its ground state and to exhibit reflection asymmetric oscillations of the octupole type around that shape. Nuclei with neutrons and protons in the valence shell may acquire a permanent ground-state deformation and an open question is whether they can assume a permanent pear-like deformation. Interest in this question was rekindled in 2013 by observed indications of such static octupole deformation in the ground-state configuration of 224 Ra [2]. By virtue of their supposed collective structure, octupole excitations are thought to exhibit phonon-like behavior [3], which renders them of particular interest, being at the cross-roads of microscopic and collective descriptions of nuclei. Consequently, many models of nuclear octupole excitations have been considered in the past (for a review, see Ref. [1]). From a shell-model point of view, an octupole vibration of a closed-shell nucleus corresponds to a coherent superposition of particle–hole excitations. This is the basis of the description of the 3− octupole state in 208 Pb proposed by Brown [4], leading to results in broad agreement with experimental findings. Such a
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The European Physical Journal Special Topics
calculations are, however, challenging when extended to more complex structures such as multiphonon states in terms of n-particle–n-hole exc
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