Tetrahedral and Octahedral Instabilities within Self-Consistent HFB Method

It is worth emphasizing that self-consistent solutions also clearly manifest the presence of the tetrahedral symmetry. Table 1 shows the results of the Hartree-Fock-Bogolyubov (HFB) calculations performed for several Rare-Earth nuclei by using the SIII[15] parametrization of the Skyrme force and the zero-range density-dependent mixed pairing force[16,17]. We used the code HFODD (v2.20m)[18,19,20] to solve the self-consistent equations on the basis of $N_0=16$ spherical harmonic-oscillator shells.

Table 1: The HFB results for tetrahedral solutions in light Rare-Earth nuclei. The third column gives the energy difference between the spherical and the tetrahedral minima; for instance the energy difference $\Delta E=-1.387$ implies that the tetrahedral minimum lies 1.387MeV below the corresponding energy of the spherical configuration. Columns 4, 5, and 6 give the multipole moments indicated. For comments concerning column 7 - see text.

Z	N	$\Delta E$	$Q_{32}$	$Q_{40}$	$Q_{44}$	$Q_{40}\times\sqrt{\frac{5}{14}}$
		(MeV)	($b^{3/2}$)	($b^{2}$)	($b^{2}$)	($b^{2}$)
64	86	$-$1.387	0.941817	$-$0.227371	+0.135878	$-$0.135880
64	90	$-$3.413	1.394656	$-$0.428250	+0.255929	$-$0.255928
64	92	$-$3.972	0.000000	$-$0.447215	+0.267263	$-$0.267262
62	86	$-$0.125	0.487392	$-$0.086941	+0.051954	$-$0.051957
62	88	$-$0.524	0.812103	$-$0.218809	+0.130760	$-$0.130763
62	90	$-$1.168	1.206017	$-$0.380334	+0.227293	$-$0.227293

Contrary to non-selfconsistent methods, in the self-consistent calculations nuclear deformations are not parameters of the potential but result from dynamical effects related to the self-optimization of the nuclear shape. Therefore, the values of the multipole moments of the mass distribution, shown in Table 1, are the results of the calculation and not subject to any pre-assumption. The results shown in column no. 7 of the Table represent $Q_{40}\times\sqrt{\frac{5}{14}}$. The latter expression, according to Eq. (15), should be equal to $-Q_{44}$ if the solution possesses the octahedral symmetry. This relation is verified with a very high precision, as the comparison of columns 6 and 7 shows.

The solutions presented point to the presence of the tetrahedral instability around $Z=64$ and $N=90$ nuclei that may amount to about $-$4MeV as in the case of $^{156}$Gd. This latter case deserves particular attention since the corresponding solution is characteristic for its vanishing $Q_{32}$ moment and can be seen as an example of pure octahedral symmetry. However, in most cases we have at the same time the tetrahedral moment $Q_{32}\neq0$ and the hexadecapole moments $Q_{40}\neq0$ and $Q_{44}\neq0$, the latter appearing in the exact proportions characteristic for the octahedral symmetry.

Although the detailed properties of these solutions depend quite strongly on the parametrization of the Skyrme- and pairing interaction, the results presented here show all the characteristic features of the exotic symmetries as discussed earlier in the framework of the non self-consistent approaches. In particular, it illustrates very clearly that the combination of the tetrahedral and octahedral symmetries can lower the total energy of the system leading to the final tetrahedral symmetry as discussed earlier on the basis of the group-theoretical considerations in Sect. 5.