Calculations using realistic mean-field methods suggest the existence of nuclear shapes with tetrahedral, , and/or octahedral, , symmetries, sometimes at only a few hundreds of keV above the ground-states in some Rare Earth nuclei around Gd and Yb. Underlying single-particle spectra manifest exotic four-fold rather than Kramers's two-fold degeneracies. The associated shell-gaps are very strong leading to a new form of shape coexistence in many Rare Earth nuclei. We present a possible experimental evidence of the new symmetries based on the published experimental results - although an unambigous confirmation will require dedicated experiments.
Symmetries of molecules, fullerenes, metal clusters, atomic nuclei and many other quantum objects can be conveniently described with the help of group theory that provides powerful means of classifying spectra in terms of the group representations. The symmetries of, e.g., fermion single-particle Hamiltonians are described with the help of double point groups whose irreducible representations determine the degeneracies of spectra and thus the underlying shell structure. Among all known double point groups, three only, i.e. tetrahedral, T ('pyramid'), octahedral, O ('diamond') and icosahedral (I) lead to 'exotic' four-fold degeneracies of single-fermion levels, all other symmetries leading to two-fold degeneracies only. This high-degeneracy aspect leads to high stability of implied nuclear shapes, as it turns out, Ref. . The symmetry plays a unifying role among distinct fields: in particular, experiments show that in the alkali metal clusters the observed magic numbers are 40, 70 and 112 (cf. e.g. Ref.  and references therein) while Ref.  predicts those (as well as some other) magic numbers for the atomic nuclei.
While an accidental discovery of the C fullerene, one of the 'most symmetric' objects in nature, took place over twenty years back, it remains to hope that an unambigous experimental discovery of the 'most symmetric' (tetrahedral and/or octahedral) nuclei will follow soon.
A possible existence of nuclei with exotic shapes that resemble round-edge pyramids (tetrahedral symmetry) has been a subject of a number of publications addressing so far mainly the theoretical aspects. In particular, Ref.  discussed for the first time the underlying four-fold degeneracies of single-particle levels in nuclei. In Ref.  group-theory aspects of tetrahedral symmetry in nuclei have been presented and existence of 'tetrahedral' magic numbers suggested. These numbers correspond to particularly large gaps in the nucleonic single-particle spectra related to the pyramid-like shapes. Calculations show that there exist magic tetrahedral gaps in many areas of the Periodic Table. Using the self-consistent Hartree-Fock approach solutions with the tetrahedral symmetry in light nuclei have been obtained in Ref. . In a more recent Ref.  it has been pointed out that some exotic nuclei around Zr may be tetrahedral-symmetric in their ground-states.
In this article we address the question of stable nuclear configurations with tetrahedral and octahedral symmetries corresponding to the groups of symmetry of tetrahedron, , and cube, , respectively. They belong to the groups of symmetry of regular polyhedra - the richest in terms of symmetry operations - among the point-groups currently used in physics. Any surface invariant with respect to all of those operations is called -invariant. The corresponding nucleonic mean-field Hamiltonian is invariant with respect to the double group, denoted , composed of 48 symmetry elements.
We performed systematic calculations for several hundreds of nuclei
using the standard Strutinsky method with the
Yukawa-folded macroscopic energy parametrisation of Ref.  and the
mean-field approach with the deformed Woods-Saxon (WS) potential
These calculations have been cross-checked with the help of the Hartree-Fock-Bogolyubov (HFB) method  using three types of the Skyrme interactions: SIII, SkM, and SLy4 and the contact pairing force. All results point clearly to strong shell effects associated with the tetrahedral symmetry in Rare Earth nuclei.
As it turns out, examining the octahedral symmetry will be important to learn
more about the tetrahedral symmetry in nuclei. Octahedral group
has 48 symmetry elements and the related double point group of
the mean-field Hamiltonian contains in total 96 symmetry elements. Both the
tetrahedral- and octahedral- invariant surfaces can be modelled with the help of
the standard spherical harmonic expansion
By requiring that expression (2) is invariant under one obtains the following conditions for odd : the lowest order parametrisation corresponds to ; there is no 5-order while the 7-order satisfies with . Similarly, invariance under implies that only even is allowed, and one may have and in the 4 order, and and in the 6 order.
The Skyrme-HFB results for three Gadolinum isotopes obtained by using the code HFODD (v2.20m)  are given in Table 1. Calculations were performed for the harmonic-oscillator basis of 16 spherical shells, and for the proton and neutron pairing strengths adjusted to reproduce the experimental pairing gaps in Gd. For comparison, the Table also shows the Strutinsky results obtained for the WS potential. In general, the shell effects obtained within the Skyrme-HFB method at tetrahedral (prolate) shapes are weaker (stronger) than those of the Strutinsky method. Moreover, at variance with the Strutinsky calculations, the HFB results show only a rather flat regions of the potential energy surfaces near the tetrahedral shapes that turn out to be unstable with respect to the quadrupole deformation. However, typical Skyrme parametrisations used in the present study do not have good enough spectroscopic-quality predictive power and better fine-tuned parametrisations should be found to be more realistic in the present context.
In our opinion, the puzzling feature of a negative-parity =2 band having no, or very weak, transitions can be explained by the tetrahedral shape of the rotating nucleus. In order to substantiate such a hypothesis, let us briefly discuss expected properties of the rotational bands associated with tetrahedral minima. The dipole and quadrupole moments of a nucleus possessing the exact -symmetry are equal to zero, and thus the corresponding electric radiation would be limited to octupole transitions. This is in contrast with the 'usual' pear-shape octupole deformation superposed with a sizeable quadrupole deformation leading to strong and transitions. However, let us emphasise that an ideal static-tetrahedral-symmetry picture needs to be modified when we wish to address, even semi-quantitatively, the problem of radiation. As it is known from the standard electric-radiation probability formulas, the transformation from the reduced transition probabilities to probabilities involves large factors such that the transitions very easily win the competition with the 's. Thus the presence of relatively small dipole polarisations, induced by rotation and/or zero-point motion, makes the corresponding transitions orders of magnitude stronger than the unmasked 's related to tetrahedral deformation.
The quadrupole zero-point vibration around the tetrahedral shape may indeed lead to a non-zero mass (and charge) dipole moment for =1. Within a simple geometrical picture of the nuclear surface described by Eq. (2), we have , where the tetrahedral () and dynamic quadrupole () deformations are assumed to be small. Of course, the final value of the electric dipole moment results from a balance between the mass and charge deformations and can only be estimated within a true microscopic calculation. However, the argument helps to understand qualitatively the presence of the dipole transitions and unobservably small quadrupole transitions in hypothetical tetrahedral bands of nuclei such as Gd.
Figure 5 illustrates the alignment curves for Gd at the tetrahedral and octahedral deformations, respectively and , as obtained by our multi-dimensional calculations, compared to quadrupole deformation typical for the ground-state minimum and . We conclude that at low-spin range the experimental moments of inertia should take roughly 30% to 40% lower values compared to the ground state moments of inertia.
In summary, the possible presence of the tetrahedral and octahedral symmetries in Rare Earth nuclei around Gd and Yb nuclei is predicted following the microscopic calculations using realistic nuclear mean-field methods. Predicted properties of the bands are discussed and compared with one of the band-candidates in Gd nucleus. In order to unambiguously identify the presence of the tetrahedral symmetry, the branching ratios of the related bands should be measured with sufficient precision. On the theoretical side, calculations of the induced dipole moments and transitions in the regime of small deformations should be undertaken and mean-field parametrisations should be fine-tuned to account for experimental positions of the hypothetical tetrahedral bands.
This work was supported in part by the Polish Committee for Scientific Research (KBN) under contract N0. 1 P03B 059 27 and by the Foundation for Polish Science (FNP). This work is a part of activities of the collaboration TETRANUC whose support through the INP, France, is acknowledged.