On the beauty of the f xyz orbital in T,T d,D2d and D2 symmetries

This paper focuses on the properties of the tesseral harmonic S³ ₂ which describes the f_xyz orbital. In a few symmetry groups: T, T_d, D_2d and D₂ f_xyz is a basis for the total symmetric representation Γ₁ as well as R which denotes a function going into itself under all proper and improper rotations. The f_xyz, tesseral harmonic is also the only function transforming into itself under a cyclic permutation × → y → z → x. This very unique property is discussed in the framework of the Judd-Ofelt theory. It is shown that the absorption or fluorescence intensity of the J = 2 ? J = 0 transitions in Eu³⁺ are concentrated in one, two or three lines with ratios 3; 2:1; 1:1:1 respectively for T, T_d; D_2d and D₂ because the only tesseral harmonic acting in the induced electric-dipole mechanism is S³ ₂ (without J mixing of course). Two notes are added with respect to the f_xyz, properties in applications as lasers and high-temperature superconductors (HTS).