Introduction

It is known that an increased nuclear binging is caused by the presence of large energy gaps in the single-particle (s.p.) nuclear spectra. The large gaps result in a decreased average density of s.p. levels and influence binding energies through the so-called shell effects. These effects can be further enhanced by high degeneracies of the s.p. levels above and/or below the energy gaps, which results in even larger fluctuations of the average level densities. Such degeneracies, in turn, are consequences of the conservation of certain symmetries in the s.p. Hamiltonian. Ordinary doubly-magic nuclei, for example, are spherically symmetric, i.e. characterized by degenarcies corresponding to the rotational group $O(3)$, and indeed the most strongly bound. Apart from the group of rotations, there exist only two other relevant symmetry groups whose conservation leads to s.p. degeneracies higher than the two-fold Kramers degeneracy. One of them is the point group, $T_d$, of the regular tetrahedron, which yields two-fold and four-fold degenerate s.p. levels. On this basis, Li and Dudek[1] suggested in 1994 that stable nuclear shapes characterized by the tetrahedral symmetry may exist in Nature. Expected experimental signatures thereof, like shape isomers or specific rotational bands, are discussed in Refs. [2,3,4,5].

The lowest-rank multipole deformation which does not violate the $T_d$ symmetry is $\beta _{32}$[3]. It represents a shape of a regular tetrahedron with "rounded edges and corners", and is usually called tetrahedral deformation. By using the deformed Woods-Saxon potential, several authors[1,2,3] examined the s.p. energies in function of $\beta _{32}$, and found that, indeed, large energy gaps, sometimes larger than the spherical ones, open up at neutron/proton numbers of N/Z=16, 20, 32, 40, 56-58, 70, 90-94, 100, 112, 126 or 136. They are sometimes referred to as tetrahedral magic numbers. In the vicinity of the tetrahedrally doubly-magic nuclei defined in this way, Strutinsky shell-correction calculations were performed[1,2,4], and energy minima corresponding to the tetrahedral deformation were found in even-even $^{80}$Zr, $^{106-112}$Zr, $^{160}$Yb, $^{222}$Rn, and $^{242}$Fm. Similarly, the Hartree-Fock+BCS (HF+BCS) calculations [6], found tetrahedral solutions in $^{80}$Zr, and Hartree-Fock-Bogolyubov (HFB) tetrahedral solutions in $^{80}$Zr and $^{106-112}$Zr were reported in Refs.[7] and [4], respectively.

The present paper reports on the first systematic study of the tetrahedral deformation in various regions of the nuclear chart, carried out by means of self-consistent methods. We focus our study on properties of the tetrahedral minima, mainly their energies and deformations, and analyze their dependence on the Skyrme force parameterizations.