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 the Monte Carlo approach. We find very large neutron shell gaps at $N=136$ for all the four octupole deformations ${\alpha }_{3\mu =0,1,2,3}$. These shell gaps generate well-pronounced double potential-energy minima in the standard multipole $({\alpha }_{20},{\alpha }_{22},{\alpha }_{3\mu },{\alpha }_{40})$ representation, often at ${\alpha }_{20}=0$, which in turn generate exotic symmetries $C_{2v}$, $D_{2d}$, $T_d$, 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 $Z\ge 82$ and $N\ge 126$ in multidimensional deformation spaces to analyze the expected exotic symmetries and octupole shape instabilities in the mass table “northeast” of the doubly magic ${}^{208}\mathrm{Pb}$ nucleus. Whereas the proton-unperturbed properties of neutron-generated octupole shell effects are illustrated in detail for exotic ${}_{Z=82}{\mathrm{Pb}}_{N>126}$ 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 $C_{2v}$, associated with the octupole ${\alpha }_{33}$ and ${\alpha }_{31}$ energy minima, respectively. The multidimensional $n>2$ deformation spaces are treated as usual by projecting the total potential energies onto the $n=2$ subspace. 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.

Mean-field calculations in multidimensional deformation spaces are performed and the shape coexistence and isomers generated by exotic nuclear configurations and toroidal and superdeformed ones are addressed. We use a phenomenological mean-field Hamiltonian of Woods-Saxon type with its universal parametrization involving eight parameters fixed once for all for the full periodic table. Original parametric correlations existing in this type of Hamiltonians are removed using methods of inverse problem theory of applied mathematics. Stochastic analysis of uncertainties of the final nuclear energy predictions with the obtained correlation-free parametrization is performed and the viability tests are illustrated and discussed. Prediction capacities of resulting model related to the description of the nuclear shape properties are cross-checked using the experimental information available, revealing full coherence. Presented results encourage experimental verification of predicted exotic structures; suggestions related to identification possibilities are formulated and discussed.

We introduce the concept of the nuclear octupole fourfold (i.e., applying simultaneously to all the four octupole deformations ${\alpha }_{30}$, ${\alpha }_{31}$, ${\alpha }_{32}$, and ${\alpha }_{33}$) neutron “magic number” $N=196$ and discuss the physical consequences of its presence. Our theoretical predictions are obtained using the realistic phenomenological mean-field approach with the deformed Woods-Saxon potential, the latter employing the new parametrization optimized in our preceding articles. Correlations among 4 parameters in the set of 12 parameters of the Woods-Saxon potential are detected and removed employing Monte Carlo approach leading to stabilization of the predictive power of the modeling. Our main focus is examining the impact of the four-fold octupole magic number $N=196$ on the stability properties of superheavy nuclei with $114\le Z\le 130$ and $166\le N\le 206$. Calculations suggest that majority of the examined nuclei are either spherical or octupole deformed, octupole-tetrahedral geometry playing the dominating role lowering the ground-state energy by up to 8 MeV. The origin and manifestations of this domination are illustrated and discussed. It turns out that, in several cases, alternative point-group symmetries may lead to noticeable lowering of the nuclear energy; this concerns the $C_{2v}$ geometry associated with ${\alpha }_{31}$, the $D_{3h}$ geometry related to ${\alpha }_{33}$, and $D_{2d}$ corresponding to the combination of ${\alpha }_{32}$ and ${\alpha }_{20}$ quadrupole component.

The gamma-ray decay of the giant dipole resonance (GDR) in the compound nucleus ${}^{216}\mathrm{Rn}$ formed with the reaction ${}^{18}\mathrm{O}{+}^{198}\mathrm{Pt}$ at the bombarding energy of 96 MeV was investigated. High-energy gamma-ray spectra in coincidence with both prompt and delayed low-energy transitions were measured. The obtained GDR width at the average temperature $⟨T⟩\approx 1\mathrm{MeV}$ was found to be larger than that at $T=0\mathrm{MeV}$ and to be approximately constant as a function of spin. The measured width value of 7 MeV is found to be consistent with the predictions based on calculations of the nuclear shape distribution using the newest approach for the treatment of the fission barrier within the liquid drop model. The present study is the first investigation of the giant dipole resonance width from the fusion-evaporation decay channel in this nuclear mass range.

We present a theoretical analysis of the competition between the so-called nuclear Jacobi and Poincaré shape transitions as a function of spin at high temperatures. The latter condition implies the method of choice, a realistic version of the nuclear liquid drop model, here the Lublin-Strasbourg drop model. We address specifically the fact that the Jacobi and Poincaré shape transitions are accompanied by the flattening of the total nuclear energy landscape as a function of the relevant deformation parameters, which enforces large-amplitude oscillation modes that need to be taken into account. For that purpose we introduce an approximate form of the collective Schrödinger equation whose solutions are used to calculate the most probable deformations associated with the nuclear Jacobi and Poincaré transitions. We discuss selected aspects of the new description focusing on the critical-spin values for both types of these transitions.

In two recent articles we have formulated nuclear mean-field theory predictions of existence of a new form of magic numbers, referred to as fourfold magic numbers. These predictions stipulate the presence of strong shell closures at the neutron numbers $N=136$ (actinide region) and $N=196$ (superheavy region) simultaneously at nonvanishing all four octupole deformations ${\alpha }_{3\mu =0,1,2,3}\ne 0$. In contrast to the traditional notion of magic numbers, the new notion refers to simultaneous nonspherical configurations (${\alpha }_{3\mu }\ne 0, {\alpha }_{2\mu }=0$). In this article we study the nuclear equilibrium deformations with ${\alpha }_{33}\ne 0$ combined with nonvanishing quadrupole deformation, ${\alpha }_{20}\ne 0$. One easily shows that such geometrical shapes have a threefold symmetry axis and are invariant under the symmetry operations of the ${\mathrm{D}}_{3h}$ point group. We employ a realistic phenomenological mean-field approach with the so-called universal deformed Woods-Saxon potential and its recently optimized parametrization based on actualized experimental data with the help of the inverse problem theory methods. The presence of parametric correlations among 4 of 12 parameters in total was detected and removed employing Monte Carlo approach leading to stabilization of the modeling predictions. Our calculations predict the presence of three nonoverlapping groups of nuclei with ${\mathrm{D}}_{3h}$ symmetry, referred to as islands on the nuclear ($Z,N$) plane (mass table). These islands lie in the rectangle $110\le Z\le 138$ and $166\le N\le 206$. The “repetitive” structures with the ${\mathrm{D}}_{3h}$ symmetry minima are grouped in three zones of oblate quadrupole deformation, approximately, at ${\alpha }_{20}\in [-0.10,-0.20]$ (oblate normal deformed), around ${\alpha }_{20}\approx -0.5$ (oblate superdeformed) and ${\alpha }_{20}\approx -0.85$ (oblate hyperdeformed). Importantly, the energies of those latter exotic deformation minima are predicted to be very close to the ground-state energies. We illustrate, compare, and discuss the evolution of the underlying shell structures. Nuclear surfaces parametrized as usual with the help of real deformation parameters, $\{{\alpha }_{\lambda \mu }^{}={\alpha }_{\lambda \mu }^{*}\}$, are invariant under ${\mathcal{O}}_{xz}$-plane reflection, the symmetry also referred to as $y$ simplex (${\widehat{S}}_y$). For the shapes with odd-multipolarity ($\lambda \to {\lambda }_{\mathrm{odd}}=3,5,7,...$) it follows that $E(-{\alpha }_{{\lambda }_{\mathrm{odd}},\mu })=E(+{\alpha }_{{\lambda }_{\mathrm{odd}},\mu })$. It turns out that the predicted equilibrium deformations generate symmetric double (or “twin”) minima separated by potential barriers, whose heights vary with the nucleon numbers, possibly inducing the presence of parity-doublets in the spectra. To facilitate possible experimental identification of such structures, we examine the appearance of such doublets solving the collective Schrödinger equation. Implied suggestions are illustrated and discussed.