From exotic mean-field symmetries to new classes of isomers in atomic nuclei

Abstract

In this article, we address the occurrence and properties of exotic point-group symmetries in nuclei. We focus on the relations between the specific gap openings in the single-nucleon spectra, which represent a measure of nuclear stability studied with the help of the nuclear mean-field theory and accompanying octupole shape properties manifesting the link between the particular stability configurations (magic octupole gaps) and resulting exotic geometrical forms. We employ a realistic phenomenological realisation of the nuclear mean-field theory with the so-called universal Woods–Saxon Hamiltonian and the group representation theory to formulate the experimental identification criteria of the addressed symmetries. We use the newest parameterisations of the Hamiltonian obtained employing the inverse problem theory. To stabilise the modelling predictions, we detect and eliminate parametric correlations. Following earlier articles introducing the octupole “fourfold magic numbers” and “universal magic numbers”, $N=136 {\text{ and }} 198$, examined in the heavy and super-heavy nuclei, we generalise these concepts for the whole mass table for the octupole magic chain $N=32, 40, 56, 64, 70, 90, 112, 136, 198$. They bring in the so-called high-rank tetrahedral and octahedral point groups strengthening the specific shell effects and gap openings and implying the unique hindrance factors: at the exact tetrahedral symmetry limit, the collective electric quadrupole and dipole reduced transition probabilities vanish provoking new isomerism. Under these circumstances, many rotational states which in other nuclei manifest strong decay probabilities, in the high-rank symmetry case become isomeric—forming a new class of nuclear high-rank symmetry isomers. The consequences for the future experimental studies of those isomers are discussed especially in the domain of exotic nuclei.

Appendix: Hints about T\(_{\textrm{d}}\)-symmetry seen from the outside of the mean-field considerations

We have presented so far a unified formulation of the point-group symmetries, in particular the high-rank symmetries in nuclei seen through a common strategy: Point-group symmetric configurations for given (Z, N)-nuclei were obtained from the realistic calculations of nuclear energies employing phenomenological mean-field Hamiltonian. Alternative microscopic mean-field calculations in this domain involve angular momentum and parity projected constrained Hartree–Fock self-consistent approaches. In this context we recalled Tagami and collaborators, Ref. [22], who, to our knowledge, were the first to show the variational solutions satisfying precisely the irreducible representation characteristics in Eq. (51), with approximately parabolic energy-vs.-spin relations, without any notion of group theory built in the Hartree–Fock computer codes. Indeed, the only information about the symmetry enters their spin-parity projected Hartree–Fock formalism merely implicitly via multipole constraints in the form of predefined $Q_{3 2}$-moments.

We do not wish to leave the reader with the impression the the mean field theory is the only approach leading to the tetrahedral symmetry mechanisms. We have no place to provide a review of alternatives but we believe that a short selection presented below might be useful for an interested reader.

The following examples address explicitly and from the start a molecular picture of molecular symmetries in nuclei beginning with the notions of tetrahedral symmetry of a very specific nucleus: $^{16}$O modelled by 4 $\alpha$-clusters. For example: The authors of Ref. [23] assume—following their own expression—the structure of $^{16}$O coinciding with that of the “naïve alpha particle model”, and construct their phenomenological tetrahedral symmetry Hamiltonian composed of various terms, which allow at the end to introduce the interpretation combining the rotation and vibration pictures. They assume the spin-parity band structures as in Eqs. (51–55) including A, E and F vibrational phonons. With their phenomenological parameter adjustment to the existing data the authors claim “matching rather well” 60 experimental energies, yet not entering into more detailed justification of the level by level theory-experiment correspondence.

An analogous approach has been adopted in Ref. [24] in which the spectrum of $^{40}$Ca in place of the of $^{16}$O of the previous references has been phenomenologically treated. The authors argue that manipulating with the concepts of A, E and F vibrational modes combined with rotation they interpret 100 experimental energies in $^{40}$Ca in terms of their rotation–vibration bands.

One of the earliest phenomenological alpha modelling of $^{16}$O goes back to Dennison, Ref. [25]. The author considers rotation/vibration structures with several adjustable parameters, predefined T$_{\textrm{d}}$ spin-parity sequences of the type of Eq. (51), but limited the empirical analysis to relatively low spins ($I < 5\,\hbar$). The author recognises the spherical harmonic $Y_{32}(\vartheta ,\varphi )$ as generating tetrahedral symmetry shapes and associates with certain $3^-$ excitations octupole-like oscillations. Considerations of this kind were further developed by Onishi and Sheline, Ref. [26], who moreover discussed in some detail the structures in terms of irreducible representations of the T$_{\textrm{d}}$ group. Slightly later, empirical analysis of the T$_{\textrm{d}}$ group symmetry concepts has been extended in Ref. [27] to higher spins to include $I \le 8\,\hbar$. A relatively complete discussion of the algebraic formulations of the tetrahedral symmetry problem seen through 4 point particle $\alpha$-clusters, which combines presentation of the explicit Hamiltonian forms and rotation–vibration interpretation of the discussed spectra can be found in more recent Refs. [28, 29]—treating all the states with A, E and F symmetries as well.

The issue of the presence and coexistence of the octupole deformations in nuclei has been approached in a yet alternative manner in Ref. [30], using as the starting point a formulation of the spherical modified oscillator (Nilsson) model, called by the authors shell model, and related symmetries and involving the algebraic methods. These concepts were combined with the collective model of Bohr and Mottelson. The authors formulated certain analytic approximations which allowed them to formulate related octupole-deformed solutions and suggest that certain deformation properties are reproduced naturally at the limits of the large oscillator shells.

A far going abstraction from what can be found in the literature in relation to nuclear surfaces and shapes has been formulated in Ref. [31], where the authors construct an abstract geometrical model of matter (atoms and/or nuclei) by introducing the notion of the complex algebraic surfaces and constructing related mathematical objects assuming that the constituents in question (nucleons or electrons) are not point-like particles. The reader willing to follow detailed argumentation would need to get accustomed with rather advanced abstract notions such as Chern, Betti or Hodge numbers and so on; the authors aim at no links with nuclear observables and the discussion remains focussed on mathematical and abstract concepts.

As stated at the beginning, our glimpse through the approaches to high-rank and in particular tetrahedral symmetry, which can be considered as an alternative with respect the mean-field theory formulation, does not pretend providing a review of the subject and can be considered as a starting point recalling a minimum of related vocabulary.