Summary and Conclusions

We have suggested that the new approach to the problem of the nuclear stability, in particular for the exotic nuclei projects, should take advantage the point group symmetries: especially those groups that possess large number of irreducible representations seem most promising. We have discussed the mechanism of high-rank geometrical symmetries, tetrahedral and octahedral ones, within the nuclear mean-field approach. We have presented in some detail how the conditions of invariance of the nuclear surfaces against the symmetry operations of a given symmetry group $\mathbf{G}$ can be implemented when using the spherical harmonics representation of the nuclear surfaces.

It turns out that the superposition of tetrahedral and octahedral nuclear surfaces still lead to a tetrahedrally-invariant deformed Woods-Saxon mean-field Hamiltonian. The same argument applies of course also for the self-consistent Hartree-Fock calculations as explicitly demonstrated through calculations for several nuclei in the considered range of the light Rare-Earth nuclei. Calculations with the Woods-Saxon Hamiltonian show that by combining the two symmetries an extra gain of up to about 1.5 MeV can be achieved. Illustrations related to the tetrahedral magic numbers and of the total potential energy surfaces have also been presented, fully confirming the general group-theoretical considerations as well as qualitative considerations related to the density of the nucleonic levels.

ACKNOWLEDGMENT

This work was partially supported through the collaboration program between the $IN_2P_3$, France, and the Polish partner Institutions; by the Polish Committee for Scientific Research (KBN) under Contract No. 1 P03B 059 27, and by the Foundation for Polish Science (FNP).