Exotic shape symmetries around the fourfold octupole magic number N=136 : Formulation of experimental identification criteria

Abstract

We employ a realistic nuclear mean-field theory using the phenomenological, Woods-Saxon Hamiltonian with newly adjusted parameters containing no parametric correlations; originally present correlations are removed employing Monte-Carlo approach. We find very large neutron shell gaps at N = 136 for all the 4 octupole deformations α3µ=0,1,2,3. These shell-gaps generate well-pronounced double potential-energy minima in the standard multipole (α20, α22, α3µ, α40)-representation, often at α20 = 0, which in turn generate exotic symmetries: C2v, D 2d , T d and D 3h discussed in detail.The main goal of the article is to formulate spectroscopic criteria for experimental identification.Calculations employing macroscopic-microscopic method are performed for nuclei with Z ≥ 82 and N ≥ 126 in multi-dimensional deformation spaces to analyze the expected exotic symmetries and octupole shape-instabilities in the Mass Table 'north-east' of the doubly-magic 208 Pb nucleus. Whereas the proton-unperturbed properties of neutron-generated octupole shell effects are illustrated in detail for exotic Z=82PbN >126 nuclei, our discussion is extended into even-even Z > 82 nuclei approaching the less exotic Z/N ratios, to encourage experiments which could identify the predicted exotic symmetries. In addition to the tetrahedral point group symmetry, T d , of which experimental evidence has recently been published, we present D 2d -symmetry resulting from a superposition of axially-symmetric quadrupole and tetrahedral symmetries and two new point-group symmetries, D 3h and C2v, associated with the octupole α33 and α31 energy minima, respectively.The multi-dimensional n > 2 deformation spaces are treated as usual by projecting the total potential energies onto the n=2 sub-space. Using the representation theory of point groups we formulate quantum mechanical criteria for experimental identification of exotic symmetries through analysis of the specific properties of the collective rotational bands generated by the symmetries. The resulting band structures happen to be markedly distinct from the structure of the bands generated by ellipsoidal symmetry quantum rotors -those various rotational properties are discussed in detail.

Revisiting the island of hexadecapole-deformation nuclei in the A ≈ 150 mass region: focusing on the model application to nuclear shapes and masses

“…In figure 1(c), the direct quadrupole deformation parameters (α 20 , α 22 ) are adopted as the horizontal and vertical coordinates of the energy surface, which are widely used in the literature, cf e.g. [25,49]. It is further worth noting that, in figure 1, the three minima possess the same energies (for display purposes, the energies at each map are respectively normalized to their minima) and equilibrium shapes, though the adopted 'coordinates' are somewhat different (but equivalent).…”

“…So far, several popular parameterizations are available, such as spherical harmonics [15,19], the Cassinian ovals [20], the funny-Hills parameterization [21], the matched quadratic surfaces [22] and the generalized Lawrence shapes [12,23,24]. In this work, see equation (10) in section 2, we define the nuclear surface with the help of the spherical-harmonic functions and introduce some 'deformation parameters' to express nuclear shapes [25]. These deformation parameters play an important role in reliably describing many nuclear properties, such as nuclear mass, charge radii and moments of inertia.…”

“…Moreover, the MM models never stop developing, including the microscopic and macroscopic parts, e.g. cf [25,[32][33][34][35]. Part of the aims in this work are listed below: (1) to reproduce nuclear quadrupole deformations with two sets of Woods-Saxon (WS) parameters, and provide a comparison with experimental data and other theoretical results; (2) to improve the description of nuclear masses, by comparing with the results with and without consideration of the hexadecapole deformation; (3) to test the WS parameter sets and the present MM method to some extent.…”

Higher multipoles of highly symmetric charge distributions over Platonic solids

“…For the last two polyhedra, icosahedron and dodecahedron, all the moments through the 2 5 -pole or dotriacontapole are zero. The first non-vanishing is the 2 6 -pole (tetrahexacontapole) moment.…”

“…In the present work, we focus on the electrostatic potential generated by distribution of charges over Platonic solids. Symmetric charge distributions have not only a pedagogical and purely academic interest, cf [1], but such a problem is relevant in studies of properties of subatomic particles [2], highly symmetric nuclei [3][4][5], as well as larger complexes like molecules [6,7] and colloids [8].…”