Tetrahedral and Octahedral Degrees of Freedom Combined

Before proceeding, let us recall in passing an important mathematical relation between the tetrahedral and octahedral symmetries as represented in terms of spherical harmonics. This relation originates from the fact that the tetrahedral symmetry point-group is a sub-group of the octahedral one. Indeed, a surface with the octahedral symmetry is invariant under 48 symmetry elements, among others the inversion. It turns out that the ensemble of the 24 symmetry operations of the octahedral group that do not contain the inversion operation coincides with the 24 symmetry elements of the tetrahedral group. Consequently, all the surfaces invariant under the octahedral symmetry group are at the same time invariant under the tetrahedral symmetry group. One may show that the

Figure: Illustration of the interplay between the tetrahedral and octahedral geometrical symmetries. The figure shows a pure tetrahedral symmetry surface corresponding to the deformation $t_3=0.15$ (left), compared to the pure octahedral symmetry surface with $o_4=0.15$ (middle), compared to the surface obtained by superposition of the two (right). One can demonstrate that the latter surface is still tetrahedrally symmetric. This is also why combining the two symmetries simultaneously may strengthen the final tetrahedral symmetry effect.

mathematical expressions given in (12)-(17) are compatible in this sense, and thus an arbitrary combination of nuclear shapes defined by $t_3$, $t_7$, and $t_9$ on the one hand, and $o_4$, $o_6$, and $o_8$ on the other hand, preserves the tetrahedral symmetry while setting $t_3=0$, $t_7=0$ and $t_9=0$ we obtain surfaces of pure octahedral symmetry.

The above inter-relations are illustrated in Fig. 1 using typical sizes of the tetrahedral and octahedral deformations as predicted by microscopic calculations.