Symmetry Arguments - Qualitatively

The main arguments related to exploiting the point-group symmetries and the implied group structure can be summarized as follows. Firstly, let us remark that the stability of an $N$-body quantum system depends on the excitation energy required to transform its ground-state into the first excited state. Within the mean-field approach, one of the most successful realistic methods in nuclear structure physics, such an excitation energy will be the larger the bigger the single-particle gap at the Fermi energy. Thus the problem of searching for the strongest nuclear stability can be approached, within the mean-field techniques, by the search for the possibly largest gaps in the single-particle spectra. As a consequence, the next step in our considerations must address the mechanisms that can possibly help in looking for the maximum gaps as functionals of nuclear shapes and associated symmetry. There are two such mechanisms that influence the solution: one is related to the saturation of the nuclear forces while another one, combined with the former, has to do with the point-group symmetries, the implied number of irreducible representations, and the dimensions of the irreducible representations in question.

The saturation property of the nuclear forces implies, among others, that the depth of the nuclear mean-field potential well, say $V_0$, remains, to an approximation, independent of the particle number, neither does it seem to depend on the nuclear deformation. Its depth of the order of $V_0\in [-60,-50]$ MeV can be considered as an average 'universal' estimate holding as the zero'th order approximation for all nuclei throughout the Periodic Table. Consequently, for a fixed particle number, say $N=N_p$ or $N_n$, the number of protons or neutrons, a rough estimate gives the density, $\rho_\lambda$, of single-particle proton and/or neutron levels in the vicinity of the Fermi energy as proportional to the ratio $N/V_0$, and the average level spacing $d\sim1/\rho_\lambda\sim V_0/N$. This zero-order estimate does not take into account symmetries. To clarify what is meant here, first consider two parities of the discussed states. If the numbers of states in both parities are $N^{\,+}$ and $N^{\,-}$, the average level spacings in both parities separately will be $d^{\,(+)}\sim V_0/N^{\,+}$ and similarly $d^{\,(-)}\sim V_0/N^{\,-}$. With the increasing nuclear mass $\vert N^{\,+}-N^{\,-}\vert\to0$ and therefore $N^{\,+} \sim N^{\,-} \sim N/2$: the average level spacings in the two parities will be twice as large as compared to the previous estimate without symmetries taken into account. Obviously what interests us in the present context is the average level spacing within each symmetry separately. This is because by combining relatively large gaps from both symmetries we may possibly arrive at a large common gap at the Fermi energy and therefore at the increased stability. Thus the necessary (although not sufficient) condition reads: Maximize the gaps in each symmetry separately - this may, although does not need to, lead to the common absolute large gaps in the single-particle spectra.

Suppose now that there exists a symmetry group $G$ of the mean field which has several, say $g$, irreducible representations $R_r;\,r=1,2,\;\ldots\;g$. An important property of the repartition of single-particle solutions over irreducible representations is that the energy levels belonging to distinct irreducible representations never repel each other (they cross freely in function of the symmetry-preserving deformations), while there are in general no crossings allowed within a given irreducible representation. Denote the numbers of levels corresponding to various irreducible representations by $n^{(r)}$; we find that there will be $r$ estimates of the average level spacings according to symmetry as given by $d^{\,(r)}\sim V_0/(N^{(r)})$. In general, the numbers $N^{(r)}$ corresponding to different irreducible representations will be different. Let us suppose, nevertheless, as a heuristic hypothesis, that the repartition of the original $N$ levels over the $g$ irreducible representations is approximately uniform in which case $N^{(r)}\approx N/g$. Let $g=6$ as an example. In such a case there will be a factor of 6 gain in terms of the average level spacings as compared to the originally considered situation. We thus conclude that the mean-field Hamiltonian invariant with respect to a point group symmetry with large number of irreducible representations has, on average, increased chances to lead to big gaps in the single-particle spectra, thus possibly leading also to the new class of nuclear stability and/or isomerism.

It is clear that the average level spacing can only serve as the first guide-line. The fluctuations around these averages are expected (and shown by numerical calculations with the realistic potentials) to be significant. This is precisely the effect sought: the bigger the deviation the better the chance to arrive at large gaps in the spectra. At this point, the argument about the saturation of the nuclear forces and the nearly constant potential depth can be helpful. To illustrate this aspect consider a fluctuation consisting in an increase of the local level density at a certain energy range by a certain factor. Because of the constant depth of the potential this automatically implies that the local density in some other energy range of the spectrum must decrease and the corresponding gaps become larger, i.e., exactly what we are looking for.

Another mechanism of importance related to the group-theoretical considerations corresponds to the dimensionality of the irreducible representations. More precisely, if in addition to the large number of irreducible representations, some of them have large dimensions, say $d_r$, the implied single-particle level energies will appear as $d_r$-fold degenerate. The presence of high degeneracies increases, on average, the single-particle level spacings thus acting in the desired direction. At this point it will be appropriate to recall that there are two types of point group symmetries sometimes referred to as 'simple' and 'double', the former applying to the 3D geometry of, e.g., macroscopic bodies, while the latter applying to systems and Hamiltonians composed of fermions. It is well known that the dimensions of the irreducible representations of the double point groups can be at most equal to 4. Consequently, the term 'large dimension $d_r$' may imply in the best case $d_r=4$. As it turns out, the octahedral double-group is characterized by $g=6$ irreducible representations, two of them with the dimension equal to 4, thus representing a good candidate case. The mathematically related (see below) tetrahedral double point group has $g=3$ and one four-dimensional irreducible representation.

On the basis of the above considerations, we would like to suggest that the point-group-symmetry guided research of nuclear stability appears as an interesting hypothesis, so far exploited to a very limited extent only. All the necessary details connected to the mathematical background of the point-group theory and related irreducible representations are well known, which facilitates the applications considerably. While a detailed study in this direction is in progress, in what follows we concentrate on two types of (related) symmetries, namely, on the tetrahedral and octahedral one. These two appear promising according to the calculations of the total potential energies, as illustrated further in this paper.