Tetrahedral Symmetry vs. Hypothetical Neutron-Skin Effect

The lowest rank tetrahedral deformation corresponds to $t_3\equiv\alpha_{32}$, an example of octupole deformations. The influence of the octupole degrees of freedom on the single-nucleonic spectra comes mainly through a repulsion between the orbitals that differ in terms of the angular-momentum quantum number $\ell$ by $\Delta\ell=3$. This repulsion is caused by the presence of the $Y_{\lambda=3}$ components in the mean field and is related to the Clebsch-Gordan coupling in the matrix elements of the type $\langle\ell^{\,\prime}\vert Y_{\lambda=3}\vert\ell\rangle$. This effect increases with increasing $\ell$ quantum-number of the nucleonic orbitals and thus its influence on the intruder orbitals is the strongest.

Examples are the repulsion effects between $i_{\frac{13}{2}}$- $f_{\frac{7}{2}}$, $h_{\frac{11}{2}}$- $d_{\frac{5}{2}}$ or $g_{\frac{9}{2}}$- $p_{\frac{3}{2}}$, i.e., between the intruder orbitals and their $\Delta\ell=3$ partners, leading in all cases to an increase of the energy of the intruder level and at the same time an increase of the energy spacing between the doublets mentioned when the tetrahedral deformation increases.

Incidentally, there may exist a mechanism in exotic neutron-rich nuclei that influences the behaviour of the intruder orbitals in a very similar way. This mechanism, claimed to possibly occur in very neutron-deficient nuclei, is a hypothetical increasing of the neutron skin[14]. It is equivalent to an effective increase of the diffusivity parameter in the underlying Woods-Saxon potential. Indeed, an elementary estimate for the spherical Woods-Saxon potentials gives

$$V_{WS}\sim \frac{1}{1+e^{\frac{r-R_0}{a}}} \quad\to\quad ... ...frac{\vec{\ell}\cdot\vec{s}}{\cosh^{\,2}(\frac{r-R_0}{2a})}.$$ (17)

As can be seen from the above relation an increase of the diffusivity parameter $a$ by a certain fraction will immediately lead to a decrease of the spin-orbit matrix elements by a similar fraction; this decrease will be furthermore strengthened through the hyperbolic cosine function that is dependent even more strongly on the diffusivity factor $2a$. As a consequence, the skin mechanism, if indeed present in exotic nuclei, will diminish the importance of the spin-orbit potential thus bringing the intruder orbitals higher in energy - in qualitative agreement with the tendency predicted for the tetrahedral deformation effect.

In other words: should the neutron skin effect be discovered in exotic nuclei its presence may induce/strengthen the tetrahedral instability in those nuclei.