Single-Nucleon Diagrams

An illustration of the single-particle proton-spectra for the nucleus $^{154}$Gd is shown in Fig. 2. The presence of the important ($\sim$2 MeV) tetrahedral-deformed gaps deserves noticing. These gaps correspond to tetrahedral 'magic' configurations or tetrahedral 'shell closures' at Z=56, 70, and 90/94. In the latter case a huge, about 3 MeV gap is crossed by a single level but the calculated effects of this structure are large; the related discussion will be presented elsewhere. Observe also the four-fold degenerate levels marked as continuous lines. These are the ones corresponding to the four-dimensional irreducible representations of the underlying double tetrahedral point group $T^D_d$. Referring to the same diagram let us emphasize that the spherical $Z=64$ gap is unstable: the corresponding gap increases with the increasing tetrahedral deformation.

The latter point brings us to a comparison between the spherical and tetrahedral 'magic' numbers. The discussion of tetrahedral magicity is a bit more complex as compared to the well known discussions of the spin-orbit splitting and the so-called spherical magic numbers 8, 20, 28, 50, 82 and 126. Indeed, the underlying physics arguments behind the spherical magicity are related to the strong main $N$-shell grouping of the single-particle levels together with the intervention of the 'un-natural' parity, highest [ $j=(N+1)+\frac{1}{2}$]-orbitals. The presence of these highest-$j$ (intruder) orbitals within the natural parity shells is caused by the strong spin-orbit interaction that is held responsible for such a strong intruder level repulsion and the corresponding spherical gaps.

Figure 3: Single-particle energy levels in function of the tetrahedral deformation for neutrons. Observe strong gaps at N=70, 90/94, and 112.

The calculated tetrahedral symmetry minima on total potential energy surfaces (discussed in the next Section) are a consequence of the spontaneous-symmetry breaking mechanism as the result of which the spherical mean field looses its stability in competition with the intrinsic-parity breaking deformations. The consequences thereof for the single-particle spectra is illustrated for the case of neutrons in Fig. 3, analogous to Fig. 2. It is worth emphasizing that magic tetrahedral gaps correspond to the same closures as in the case of protons; even the large gap structures at N=90/94 crossed by a single level look nearly the same as in the case of Z=90/94 configuration. We wish to note the presence of the $N=112$ gap whose size is comparable to the other tetrahedral gaps in the Figure.